A Guide
Guide to
to
A
Fixed Income
Income
Fixed
Analysis
Analysis
Andrew R. Young
A Morgan Stanley
Guide to Fixed
Income Analysis
Andrew R. Young
This chapter is an excerpt from A Morgan Stanley Guide to Fixed Income Analysis by
Andrew R. Young, ©2003 Morgan Stanley & Co. Incorporated.
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
© Copyright 2003 Morgan Stanley & Co. Incorporated
Acknowledgments
This book represents a significant portion of what I have learned in this
industry. As such I should start by recognizing Jennifer Carpenter and
Stephen Lalli for helping me get started and fostering a sense of curiosity
and regard for the details.
Particular thanks are also due to Ben Wolkowitz for sparking the
development of this book and to Roy Campbell, David Chang, Yoon
Chang, Young-Sup Lee, Mike Mendelson, Kelly Thomas, Evan Tick, and
Joan Tse for their insight into the approach, additions of material, and
cheerful and detailed suggestions as to how to make this more useful.
In addition, I offer a warm appreciation for my editor, Sheila York, who
spent innumerable hours and applied all her skill and effort to bring this
to an (I hope) excellent level of quality and readability. It would have
truly been a different outcome without her guidance.
Thanks also to Steve Abrahams, Mark Childress, David Depew, Jeff
Jennings, Emily Kim, Joe Langsam, Krishna Memani, Louis Scott, Tim
Sears, John Scowcroft, Deb Shroyer, Eric Vandercar, and Andrew Waine
for their helpful advice, and to the dedicated team at Firm Graphics:
Ramona Boston, Bill Devine, Roger Adler, Vance Clarke, Paul Cohen,
Sharon Eng, Bryan Fernandez, Steve Feuerborn, Matt Foodim, Lucille
Harasti, Sasha Koren, Todd LeBlanc, Carol Murashige, Jane Seguin,
Neil Stillman, and Gail Vachon.
Finally, I would like to dedicate this book to my wife, Lisa, son, Zachary,
and daughter, Michelle, for all their love, support, and forbearance,
because I couldn’t have done it without them.
iii
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
159
Table of Contents
Chapter
Page
One
Zero-Coupon Bonds
1
Two
Coupon Bonds
Three
The Yield Curve, a Treasury Pack
and Fitted Curve Analysis
Sample Treasury Pack
71
101
Four
Forward Prices
123
Five
Yield Measurement and
Total Rate of Return
139
Six
Options
159
Seven
Futures
203
Eight
Corporate Bonds
235
Nine
Swaps
259
Ten
Mortgages
309
Eleven
Portfolio Theory and
Market Dynamics
355
27
Exercise Solutions
389
Glossary
489
Equation Reference
509
v
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Foreword
There is a
tremendous amount
of debt outstanding,
with a wide variety
of conventions for
pricing and
settlement
Fixed income is a business of details. The market is dominated by large
transactions and a number of well-capitalized competitors, so spreads in
some markets are thin. Accuracy and attention to detail are paramount
since the impact of using the wrong convention can easily exceed the
bid-ask spread.
Each sector of the market has its own conventions, which provide the
framework for internal and external communication. This material
illustrates some of the important ones. Keep in mind that conventions do
change over time, so it is important not to assume that last year’s still
hold. Always clarify your understanding of security pricing or mechanics
in markets in which you are not an active or regular participant with
sales, product management, research, or trading.
Total Debt Outstanding: $18 Trillion1
1 Source: Federal Reserve Flow of Funds—http://www.bog.frb.fed.us/releases/Z1
vi
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Foreword (Continued)
Morgan Stanley seeks to add value by helping customers formulate
investment strategy, seek return, and reduce risk, and by providing
efficient execution. Methods for doing this often require creative
problem-solving, both in identifying a new approach and working
through its ramifications. This material aims to provide tools and
techniques that are the foundation for that creativity and that will also
help you understand markets and market participants. It is fairly
mathematical, and requires some difficult algebra, but there are no partial
differential equations!
Your success depends on your ability to solve problems. At the end of
each chapter, you will find exercises that will help drive home concepts.
Use this material as an opportunity to enhance your skills. It is a cheap
way to gain experience. Better to make a mistake here than on an actual
trade. Do not ask for help too soon: struggle with each exercise on your
own first. Then, if you are still having trouble mastering a concept, do
not be afraid to consult with someone who is more experienced with
analytics.
Your feedback is important to us. Please let us know what you think of
the usefulness of this material and any suggestions you may have.
Andrew Young
Morgan Stanley
1585 Broadway
New York, NY 10036
Andrew.Young@morganstanley.com
vii
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 1
Zero-Coupon
Bonds
This chapter is an excerpt from A Morgan Stanley Guide to Fixed Income Analysis by
Andrew R. Young, ©2003 Morgan Stanley & Co. Incorporated.
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
In This Chapter, You Will Learn...
•
How to Calculate Present Value and Future Value of Single Cash
Flows (Zero-Coupon Bonds)
•
How to Compound Yields
•
How Prices Change When Yields Change
•
How to Estimate Price Changes Using
–
Duration
–
Convexity
2
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Future Value and Present Value
5% Per Annum (Simple Interest)
Future Value
Present value and
future value both
address the “time
value of money”
The basic concept of
future value is
“How much will I
receive in the future
for a fixed
investment today?”
The basic concept of
present value is
“How much do I
have to invest
today for a fixed
future cash
amount?”
Present Value
The present value is
also described as
the discounted
value of the future
cash flows (bond
payments)
3
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Compound Interest
10% Per Annum
Compounding
means that interest
earns interest
No Compounding
A 10% interest rate
compounded semiannually implies
two six-month 5%
interest periods per
year; $100 would
be worth $105
after six months
and $110.25 after
one year (the $105
earns 5%)
This is a higher
effective rate of
interest than the
same 10% rate
quotation with no
compounding
Semi-Annual Compounding
4
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Compounding $100 at 10% Interest
Compounding
Frequency
After
One Year
After
10 Years
Never
$110.00
$200.00
Annually
$110.00
$259.37
Semi-Annually
$110.25
$265.33
Quarterly
$110.38
$268.51
Monthly
$110.47
$270.70
Weekly
$110.51
$271.57
Daily
$110.52
$271.79
Continuously
$110.52
$271.83
Compounding
makes a larger
difference over a
longer period of
time
Given an
investment term,
successive divisions
of the compounding
frequency make
less and less
difference
The greater the time until maturity, the greater the difference
compounding makes. For example, over one year, the continuous
interpretation adds only $0.52 over the annual interpretation; over 10
years, it adds $12.46.
Although more frequent compounding increases return, it makes less
incremental difference as the compounding divisions get finer. For
example, over 10 years, the quarterly interpretation of a 10% rate would
produce $9.14 more than the annual interpretation; the continuous
interpretation only adds another $3.32.
5
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Mathematics of Compounding
For every
compounding
frequency f, there is
a different
annualized yield
quotation yf that
corresponds to a
given annual yield
The term annual
yield means that
the yield
compounds on an
annual basis (once
per year), as
opposed to
annualized, which
can apply to any
compounding
frequency
Most securities follow the example set by the most prevalent securities
in the market: Treasury notes and bonds. Their yields are quoted on a
semi-annual compounding basis.
The fundamental formula for converting an annualized yield yf —
compounding f times per year — to an annually compounded yield is
y ö
æ
1+ Annual Yield = ç 1+ f ÷
f ø
è
f
Note that more frequent yield compounding results in a higher annual
yield. Alternatively, given an annual yield, more frequent compounding
results in a lower yf .
Q: If a bond’s semi-annual yield is 7%, what is its quarterly yield? Its
monthly yield?
Hint: What is its annual yield?
6
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Future Value and Present Value
7% Per Annum, Semi-Annual Compounding
Future Value
$100 invested at
7%, compounded
semi-annually,
would return
$198.98 in 10
years; the amount
invested today is
50.26% of the final
value
Likewise, the price
of a 10-year zerocoupon investment
(paying $100 at
maturity) is $50.26
(50.26% of par) at
a 7% semi-annually
compounded yield
Present Value
where
v is par (100%)
y is yield (7%)
f is the
compounding
frequency (2)
n is the number of
compounding
periods (20)
7
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Pricing a Zero-Coupon Bond
v is the par amount
(usually 100%)
y is the yield
f is the
compounding
frequency
n is the number of
compounding
periods; n/f is the
number of years
until maturity
For zero-coupon bonds, the price is the present value—“How much do I
have to invest today to get the face value of the bond at maturity?” The
price can be calculated from the yield, and the yield can be calculated
from the price:
Price (Given Yield)
Price =
v
æ
yö
ç 1+ ÷
fø
è
n
Yield (Given Price)
1
é
ù
v
æ
ön
ê
Yield = f ´ ç
- 1ú
÷
êè Price ø
ú
úû
ëê
Example: A 10-year U.S. Treasury zero-coupon bond yielding 6%,
compounded semi-annually, has a price of:
100%
6% ö
æ
ç1 +
÷
è
2 ø
20
=
100%
0.06 ö
æ
ç1 +
÷
è
2 ø
20
=
100%
20
. )
(103
= 55.368%
Note: Price is generally quoted as a percent of par (face value). For
example, a zero-coupon bond with a price of 50 means that the security
costs 50% of par. Since the decimal representation of 50% is 0.50, the
cost of $1,000,000 face amount of the bond would be
$1,000,000 × 50% = $1,000,000 × 0.50 = $500,000
Frequently, the percent designation is not quoted. There are two ways to
reconcile this with economic reality: Take the percent designation as
implied, so that a price of 50 would really mean a price of 50% of par;
alternatively, the price could represent a cost of $50 for $100 face
amount of the bond.
8
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
A Quick Valuation of Future Cash Flows
Length of Time for Money to Double × Annual Yield @ 72
The Rule of 721
provides a quick
way to estimate the
value of future cash
flows
It states that over a
wide range of
interest rates, the
length of time it
takes money to
double is
approximately
72
71
_____
________
yAnnual » ySemi-Annual
For example, given
a 7% annual yield,
72
__
@ 10
7
Thus a dollar in
10 years is worth
approximately
50 cents today; this
is consistent with
the price of a 10year zero at 7%
(50.26%)
1 Do not confuse the Rule of 72 with the Rule of 78, which applies to proration of interest on
some consumer loan contracts.
9
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Determinants of
Market Yields and Prices
Market yields are
an imperfect, but
easily quoted,
measure of the
nominal rate of
return an investor
can earn by
purchasing a
security
When the market moves, prices and yields move simultaneously because
they are mathematically related. A market decline can thus be thought of
as either 1) investors demanding to pay less for fixed future cash flows
or 2) investors demanding to earn a higher rate on their investment.
The primary determinant of a change in yields is a change in the
expected rate of inflation. Investors ultimately care about the purchasing
power of their future cash flow receipts; in an inflationary environment,
their purchasing power decreases. Therefore, when inflation
expectations rise, fixed-income prices decline and yields rise. Inflation
The primary
components of yield expectations can be influenced by the current rate of inflation, the rate of
are “real” yield and growth of the economy, various industrial-capacity constraints, and
expected inflation
governmental policy, amidst a host of other potential factors.
Secondary
components include
the expected value
of credit losses, the
value of any
options embedded
in the security, tax
effects, and
compensation for
accepting additional
risk or illiquidity
Another cause of a change in yields is a change in “real” yields. Real
yields are what investors would demand to earn (risk-free) if there were
no prospect of inflation. Depending on changes in supply and demand,
investors may be able to command higher or lower real yields.
Investors also demand higher yields for taking additional risk. As the
market’s perception of risk changes, yields and prices for the affected
bonds will also change. These additional risks may arise through
extending credit to riskier borrowers, accepting payments that are not
fixed, participating in a less-liquid market, or granting rights to issuers
of bonds (embedded options).
10
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Calculating Prices
10-Year U.S. Treasury Zero-Coupon Bond
(Base Yield 7%, Compounded Semi-Annually)
Calculate the price of this zero-coupon bond at the given yields (answers As markets move,
prices and yields
on following page):
change
Yield (%)
Price (%)
6.99
?
7.00
50.257
7.01
?
For a bond, the
yield defines a
price, and the price
also defines a yield
Remember:
Price =
v
æ
yö
ç1+ ÷
fø
è
n
where
v is the par amount (usually 100%)
y is the yield
f is the compounding frequency (2 in this case)
n is the number of compounding periods (20 in this case)
11
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Estimating Price Changes
A basis point (bp) is
one-hundredth of a
percent (0.01%)
The price change for
a 10-basis-point
(0.10%) change in
yield is roughly 10
times as big as the
price change for a
one-basis-point
(0.01%) change in
yield
The change in value
for a one-basispoint change in
yield is also known
as the dollar value
of a basis point.
(DV01) or the
present value of a
basis point (PV01)
10-Year U.S. Treasury Zero-Coupon Bond
(Base Yield 7%, Compounded Semi-Annually)
We can estimate the price change of a bond when interest rates change
by extrapolating from the price change of the bond over small changes
in interest rates.
Yield (%)
Linearly Extrapolated
Price Estimate (%)
6.80
?
6.90
?
6.99
50.305
7.00
50.257
7.01
50.208
7.10
?
7.20
?
Note that prices decline when yields rise. This truism of fixed income
follows directly from the formula for converting a bond’s yield to a price.
Phrased differently, when yields rise, an investor would need to invest
less to produce a fixed future value.
For most fixedincome securities,
prices decline as
yields rise
12
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Comparing Estimated
and Actual Price Changes
10-Year U.S. Treasury Zero-Coupon Bond
(Base Yield 7%, Compounded Semi-Annually)
Linearly Extrapolated
Price Estimate (%)
Actual
Price (%)
Difference
in Price (%)
6.80
51.228
51.238
0.010
6.90
50.742
50.745
0.002
6.99
50.305
50.305
0.000
7.00
50.257
50.257
0.000
7.01
50.208
50.208
0.000
7.10
49.771
49.773
0.002
7.20
49.285
49.295
0.010
Yield (%)
For small changes
in interest rates, the
linear method of
estimating price
changes is very
accurate
Note that, for this
security, the actual
price is always
higher than the
estimated price
The price fell slightly
more when the
yield rose one basis
point (to 7.01%)
than it rose when
the yield fell
one basis point
(to 6.99%); this
table shows
extrapolation using
the average of the
change for a onebasis-point increase
and decrease in
yield
13
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Estimated Prices vs Actual Prices
10-Year U.S. Treasury Zero-Coupon Bond
(Base Yield 7%, Compounded Semi-Annually)
Our extrapolation is
simply a tangential
approximation of
the price/yield
curve at the given
(base) yield
The slope of the
price estimation
line, the change in
price for a change
in yield, is also
known as dollar
duration
The related
quantity, modified
duration, also
known as duration,
shows the same
price change as a
percent of the
current price
Price (%)
52
51
P = 0.486% ( of Par )
P
= 0.966% (of Price)
P
ar
50
49
6.80
y = −0.10%
6.90
7.00
7.10
7.20
Yield (%)
The price of the 10-year zero-coupon bond rose by 0.486% of par when
yields fell by 10 bp (0.10%). The dollar duration is then:
DurationDollar = -
dP
DP
0.486%
@== 486%
dy
Dy
- 0.10%
To estimate the price change if yields fall, for example 1%, multiply the
dollar duration by the yield change:
DP @ -DurationDollar ´ Dy = -486% ´ -1% = 4.86%
As a percent of initial price, the price of the zero rose by 0.966%. The
modified duration is then:
DurationModified
dP
DP
P
P = - 0.966% = 9.66
= Duration = @dy
- 0 .10%
Dy
14
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Duration
Interest Rate Sensitivity of a Security
Dollar duration estimates how much a security’s value changes for a
given change in interest rates. If the dollar duration is quoted as a percent
of face, then it can be used for an estimation of price in a different rate
environment. If it is quoted in dollars, then it illustrates how the dollar
value of a position or portfolio changes when rates change.
The modified
duration, or simply
duration, is defined
as:
Modified duration estimates how much the present value changes as a
percent of the current present value, and so it is more useful for for small changes in
comparing the interest rate sensitivity of the value of different securities y and estimates the
percentage change
or portfolios.
Mathematically, dollar duration is the slope of the line tangent to the
price/yield curve at the current yield (with the sign changed to produce
a positive number). Dollar duration, therefore, is given by the negative of
the first derivative of the price function with respect to yield.
in price for an
instantaneous,
parallel change in
yield
For a zero-coupon bond:
v
Price =
DurationDollar = -
æ
yö
ç1+ ÷
fø
è
vn
f
dP
DP
=
n +1 @ dy æ
Dy
yö
ç1 + ÷
fø
è
( for small Dy )
n
Duration = -
dP
n
f
DP
P=
P
@dy
Dy
æ
yö
ç1 + ÷
fø
è
( for small Dy)
Q: Calculate the dollar and modified duration of a 10-year zero-coupon
bond, using a semi-annual yield of 7%. How does your answer compare
to the duration on the graph on the preceding page? How does the
modified duration compare to the term of the zero?
15
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Estimating Price Changes (Revisited)
We can use dollar
duration to
estimate prices at
different yields
10-Year U.S. Treasury Zero-Coupon Bond
(Base Yield 7%, Compounded Semi-Annually)
We can quickly estimate the price change of a bond when interest rates
change by using the dollar duration of the bond.
The dollar duration is estimated by -
DP
for small changes in y.
Dy
Linearly Extrapolated
Price Estimate (%)
Yield (%)
5.00
?
6.00
?
6.99
50.305
7.00
50.257
7.01
50.208
8.00
?
9.00
?
16
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Comparing Estimated
and Actual Price Changes
10-Year U.S. Treasury Zero-Coupon Bond (Base Yield 7%)
Linearly
Extrapolated
Price Estimate (%)
Actual
Price (%)
Difference
in Price (%)
5.00
59.968
61.027
1.059
6.00
55.112
55.368
0.255
6.99
50.305
50.305
0.000
7.00
50.257
50.257
0.000
7.01
50.208
50.208
0.000
8.00
45.401
45.639
0.238
9.00
40.545
41.464
0.919
Yield (%)
Duration is the
linear estimate of
how price changes
when yield changes
The duration-based
estimates are
always lower than
the actual prices (for
bonds with no
embedded options)
The error in
estimation grows
larger with the
square of the
change in interest
rates: it quadruples
when the change in
interest rate
doubles
There is a secondorder correction
called convexity
that explains the
majority of the
difference between
the linear estimate
and the actual price
Q: What happens to
dollar duration as
yields change?
17
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Convexity
Second-Order Interest Rate Sensitivity of a Security
Convexity estimates
the difference
between the actual
price and the price
estimate obtained
using duration
Noncallable bonds
have positive
convexity; the
actual price is
always higher than
the duration-based
estimate
Trick question:
Why do zerocoupon bonds have
positive convexity if
their duration
always equals their
maturity?
Convexity measures the degree of curvature in a security’s price/yield
relationship, i.e., the rate of change in dollar duration. When the
price/yield relationship is curved, the linear estimate (using constant
dollar duration) will always have error.
Just as there were two related quantities for expressing a linear estimate
of price change, dollar duration and duration, there are two related
quantities for expressing the error: dollar convexity and convexity. Dollar
convexity estimates both the additional change in price and the change
in dollar duration for a given change in rates and is useful for refining
price estimates in a different interest rate environment. Convexity
estimates the same changes, but as a percent of price, and so it is more
useful for comparing which security or portfolio can expect a greater
percentage price boost above the duration estimate when rates change.
Mathematically, dollar convexity is the second derivative of the price
function with respect to yield:
ConvexityDollar =
d 2 P - d ( DurationDollar ) - DDurationDollar
=
@
(for small Dy)
dy 2
dy
Dy
For a zero-coupon bond:
Price =
ConvexityDollar
d 2P
= 2 =
dy
v
æ
yö
ç1 + ÷
fø
è
vn(n + 1)
æ
yö
f ç1+ ÷
fø
è
n
Convexity =
n+ 2
2
d2P
dy
P=
2
n( n+ 1)
2æ
yö
f ç1 + ÷
fø
è
2
Note that for zero-coupon bonds, convexity goes up approximately with
the square of maturity n/f.
18
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Using Duration and Convexity
The Taylor series expansion for price P1 given an initial price P0 and a The Taylor series
expansion for price
change in yield from y0 to y1 is
1 d 2P
dP
2
P1 = P0 +
´ (y1 - y0 )+ ´ 2 ´ (y1 - y0 ) + L
dy
2 dy
So,
P1 @ P0 - DurationDollar ´ (y1 - y0 )+
1
2
´ ConvexityDollar ´ (y1 - y0 )
2
shows how to use
duration and
convexity to
estimate the price
for a given change
in yield
Alternatively,
P1 @ P0 - P0 ´ Duration ´ (y1 - y0 )+
æ
ç
@ P0 ´ ç 1 - Duration ´ (y1 - y0 )+
ç
è
Since DurationDollar = ConvexityDollar =
1
2
´ P0 ´ Convexity ´ (y1 - y0 )
2
ö
1
2÷
´ Convexity ´ (y1 - y0 ) ÷
÷
2
ø
dP
= P ´ Duration and
dy
d 2P
= P ´ Convexity
dy 2
Some firms quote a gain from convexity, which is defined as:
Convexity
ConvexityGain=
2
Then,
P1 @ P0 - P0 ´ Duration ´ (y1 - y0 ) + P0 ´ ConvexityGain ´ (y1 - y0 )
2
19
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Using Duration and
Convexity (Continued)
Example: 10-Year U.S. Treasury Zero-Coupon Bond
Use the following formulas to answer the questions below:
Price =
v
æ
yö
ç1+ ÷
fø
è
n
Duration =
n
f
æ
yö
ç1 + ÷
fø
è
Convexity =
n(n + 1)
æ
yö
f ç1+ ÷
fø
è
2
2
Q1: At a yield of 7%, what is the price of the bond?
Q2: What is the duration?
Q3: What is the convexity?
Q4: Estimate the price if yields fall 200 bp using the following formula:
P1 @ P0 - P0 ´ Duration ´ (y1 - y0 ) +
1
2
´ P0 ´ Convexity ´ (y1 - y0 )
2
Q5: How does your estimate compare to the actual price of 61.027%?
20
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Duration and Convexity
(for a Longer-Duration Security)
30-Year U.S. Treasury Zero-Coupon Bond (Base Yield 7%)
The longer the
duration, the more
significant the gain
from convexity
Price (%)
30
Value from Convexity (Yields Down 250 bp)
Q1: What is the
slope of the dotted
estimator line for
the 30-year vs. the
10-year zerocoupon bond?
25
20
15
10
Actual Price
5
0
Estimated Price
4
5
6
7
8
9
Yield (%)
10-Year
30-Year
Price
50.26%
12.69%
Slope of Estimator Line
– 4.86
?
Dollar Duration
486%
?
Duration
9.66
?
Dollar Convexity
4926%
?
Convexity
98.02
?
10
Q2: What are the
dollar convexity
and convexity of
the 30-year U.S.
Treasury zerocoupon bond?
Q3: How do these
compare to the
dollar convexity
and convexity of
the 10-year zerocoupon bond?
21
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Contribution of
Convexity to Estimate of Price
30-Year U.S. Treasury Zero-Coupon Bond
(Base Yield 7%, Compounded Semi-Annually)
The contribution of
convexity increases
with the square of
the change in
interest rates
It underestimates
the excess price
appreciation in a
declining-interestrate environment
and overestimates
the appreciation in
a rising-interestrate environment
22
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Continuous Compounding
Advanced
The formula for one year’s worth of continuous compounding is
f
æ
yö
lim çç 1 + ÷÷ = e y
f ®4
fø
è
Therefore, the price of a 1-year zero is e –y and the price of a t-year zero
is e –yt, where y is a continuously compounded yield. Note that t is
measured in years because y is an annualized rate.
This formula has some interesting attributes:
d 2P
dP
Duration = -
Continuous
compounding
provides an
economy of
expression for
theoretical
analysis; no
security is quoted
using continuous
compounding
dy
=t
P
Convexity =
dy 2
= t2
P
Why are the duration and convexity formulas simpler when the yield is
continuously compounded than in the noncontinuous case? Define the
number of periods as tf where t is the term measured in years. Then
Duration = lim
f ®4
tf
f
æ
yö
çç 1 + ÷÷
fø
è
= lim
f ®4
t
æ
yö
çç 1 + ÷÷
fø
è
=t
Note that for a given change in quoted (semi-annual) yield, the
continuously compounded yield changes by less (because it compounds
more frequently). The estimated price change is the yield change
multiplied by the duration. Since the continuously compounded yield
changes by less, the security’s duration with respect to that yield must be
longer to estimate the same price change.
23
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 1 Exercises
1. Calculate the present value, modified duration, dollar duration, and
convexity of these two Treasury STRIPS (zero-coupon bonds).
Maturity
(Years)
Yield
(%)
5 Years
6.75
25 Years
7.50
Present
Value
(%)
Modified
Duration
Dollar
Duration (%) Convexity
2. What is the dollar duration of a 1-year STRIPS yielding 5%? What is
its modified duration? What is the dollar duration of a 30-year
STRIPS yielding 8%? What is its modified duration?
3. What are the price, modified duration, and convexity of a 30-year
STRIPS at a 7% and a 7½% yield? How do these numbers all fit
together?
4. A pension fund manager has a $23 million liability due in five years.
How much needs to be invested today if the manager can lock in an
annual interest rate of 6.75% for five years? How much if the rate
compounds semi-annually?
5. What is the semi-annually compounded yield of a Treasury STRIPS
that matures in 20 years and is priced at 23.111%?
6. If Manhattan was worth $24 in trade goods 360 years ago, what has
been the annual total rate of return on the investment if the island is
worth $100 billion today?
24
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 1 Exercises (Continued)
7. If a corporation expects to pay $100 million in the year 2020
(24 years from now) to its pension beneficiaries, what is the present
value of this liability at an annual discount rate of 7.25%? If rates
decline by 100 bp, what is the new value of the liability? What is the
error if we estimate the new liability value using duration?
8. A security that promises to pay $10,000 five years from now can be
purchased for $7,175.38 today. What is its semi-annually
compounded yield? If there is a secondary market for this security,
how will its market yield change if the credit quality of the issuer
deteriorates?
9. Should you pay $6 million today for a bond that promises to pay
$9 million in five years if you need to earn an 8.00% annual return?
10. A municipality has a $10 million liability payable July 15, 2020.
To satisfy the liability, the municipality must either set aside
$10 million cash today (June 26, 1996) or buy U.S. Treasury
securities disbursing $10 million to ensure that the debt will be paid.
If the following zero-coupon Treasury securities are available, what
must the municipality pay today to satisfy this liability, assuming
short rates rarely fall below 3%?
Maturity
Price (%)
Yield (%)
2/15/20
17.828
7.43
5/15/20
17.507
7.43
8/15/20
17.269
7.41
11/15/20
17.040
7.39
11. Derive a simple formula for convexity of a zero-coupon bond in
terms of its duration and yield.
25
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 2
Coupon Bonds
This chapter is an excerpt from A Morgan Stanley Guide to Fixed Income Analysis by
Andrew R. Young, ©2003 Morgan Stanley & Co. Incorporated.
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
In This Chapter, You Will Learn...
•
About Accrued Interest
•
Fixed-Income Calendar Conventions
•
How to Price a Coupon Bond
•
How to Value Annuities
•
How to Calculate a Yield
•
How to Amortize a Premium or Discount
•
Different Methods of Quoting Duration
•
The Durations of Coupon Bonds with Different Maturities
•
The Value of Convexity
28
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
U.S. Treasury Bond Structure
U.S. Treasury 8% Due November 15, 2021
The Treasury bond
is the most common
type of coupon
bond
A U.S. Treasury
bond pays interest
semi-annually (in
arrears)
Each coupon
payment is half the
nominal rate of
interest: 4% of face
value on this 8%
coupon bond
The present value of
the bond — how
much the buyer
must pay now to
get all the bond’s
future cash flows—
is the sum of the
present values of
the individual cash
flows
29
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Valuing a Coupon Bond
A Whole Number of Coupon Periods Remaining
Bearing in mind
that different cash
flows may have
different yields, we
can write a formula
for the present
value of a coupon
bond as the sum of
the values of the
individual cash
flows
Alternatively, we
can value all the
cash flows at the
same rate, called
the yield-tomaturity
In the example, the
yield-to-maturity is
6.28% and the
present value of the
bond equals
105.038%; if the
yield-to-maturity
had been equal to
the coupon (in this
case 9%), the
present value of the
bond would have
been 100% (par)
PV =
c
f
æ
y ö
ç1+ 1 ÷
fø
è
c
=
f
n
å
i=1
+
c
f
æ
y ö
ç1+ 2 ÷
fø
è
1
æ
yö
ç1+ i ÷
fø
è
i
+
2
+L+
v
æ
y ö
ç1+ n ÷
fø
è
c
f
æ
y ö
ç 1 + n-1 ÷
f ø
è
n-1
+
c
f
æ
y ö
ç1+ n ÷
fø
è
n
+
v
æ
y ö
ç1+ n ÷
fø
è
n
n
c is the annual coupon rate,
v is the redemption value,
yi is the yield for an i-period zero-coupon bond, quoted on a compound
basis,
f is the payment and compounding frequency,
n is the number of whole coupon periods between settlement and
maturity, and
n
is the number of years remaining until maturity.
f
If all the yi’s are the same, y is equal to the yield-to-maturity.
Example: U.S. Treasury 9% due May 15, 1998, with a yield-tomaturity of 6.28% and a settlement (funds-bond transfer) date of
May 15, 1996:
9%
9%
9%
9%
100% +
2
2
2
2
PV =
+
+
+
4
2
3
6.28% ö æ
æ
6
.
28
%
6
.
28
%
6
.
28
%
æ
ö
ö
æ
ö
1
+
ç
÷
1+
ç1 +
÷
÷
ç1 +
÷
2 ø çè
è
2 ø
2 ø
2 ø
è
è
= 105.038%
30
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Cash Flow Timeline
It is important to
understand exactly
when cash changes
hands (the
settlement date and
the future cash flow
payment dates)
Because of the time value of money, future cash flows are more valuable
the closer they are to settlement (the day when they are paid for). The
earlier the settlement date, the farther away the future cash flows, and the
lower the value of the bond (unless the earlier settlement entitles the
buyer to additional cash flows). The later the settlement date, the nearer
the future cash flows and the higher the value of the bond. Conversely,
the earlier the cash flows, the higher the value of the bond; and the later
the cash flows, the lower the value of the bond.
Every market
has a regular
settlement schedule;
currently, regular
settlement for
Treasuries is T
(trade) + 1 (next
business day), and
most other domestic
products settle T+3
For Treasuries,
skip-day means
T+2, and cash or
same-day means
T+0
31
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Quoting Bonds: Price and Present Value
For zero-coupon
bonds, price and
present value are
identical; however,
this is not true for
coupon bonds
(except on a coupon
payment date)
For a zero-coupon bond, price and present value are identical, and we
have used them interchangeably. For a coupon bond, price and present
value are the same only on a coupon payment date.
A bond’s present value (market value) identifies its total cost — or the
amount of money which must be given to the seller as compensation for
delivering the security. Present value is, therefore, the fundamental
measure of value for a bond.
However, present value is not the most convenient (or common) way to
quote bonds. As the next page shows, a bond’s present value fluctuates
dramatically over time, even when its yield remains constant.
So, for convenience, market participants quote a price that is more stable
than present value over time. The quoted price is slightly less than the
present value of a bond, but they are precisely related. All market
participants know how to transform a price into present value to
determine the cost of an acquisition or the proceeds of a sale.
32
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Present Value Over Time
(at Constant Yield)
U.S. Treasury 8% Due November 15, 2021, Priced to Yield 7%
Even when a
bond’s yield does
not change, its
present value
changes over time
because:
1) Value increases
as a coupon date
approaches,
2) Value decreases
after a coupon is
paid to the
bondholder, and
It would be nice if the value of a bond over time were smoother so we
could isolate the change in value due to a change in rates!
3) As the number
of coupons
remaining
decreases, the
value of the bond
drifts toward par
33
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Quoting Bonds: Price and Present Value
(Continued)
When securities are
traded, the money
exchanged is the
present value of
the securities
However,
transactions are
usually agreed
upon based on a
quoted price, which
is the present value
reduced for a
somewhat
arbitrary accrued
interest
Since present value
is the critical
quantity, the
precise
methodology for
calculating accrued
interest is
irrelevant, as long
as all market
participants
calculate it the
same way
Price
= Present Value – Accrued Interest
Convenient
for
quotations
Value of the bond
(exchanged at sale)
Just a definition
(to smooth price quotations)
Equivalent terminology used in the marketplace:
•
Flat Price
=
Full Price
–
Accrued Interest
• Clean Price
=
Dirty Price
–
Accrued Interest
•
=
Net
–
Accrued Interest
• Present Value
=
Price
+
Accrued Interest
•
Full Price
=
Flat Price
+
Accrued Interest
• Dirty Price
=
Clean Price
+
Accrued Interest
•
=
Principal
+
Accrued Interest
Principal
Alternatively,
Net
34
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Elements of Accrued Interest
Up until now, we have computed the present value of bonds with a whole
number of compounding periods until maturity. For the traditional bond,
whose compounding frequency equals its payment frequency, there are a
whole number of coupon periods remaining only when settlement lies on
a coupon payment date. In that situation, the bond has no accrued
interest, and the seller receives the coupon paid on the settlement date.
Accrued interest
represents the value
of interest earned
since the last
coupon payment
date
Between coupon payment dates, a bond will have accrued interest. The
mechanics of computing accrued interest depend on the calendar
conventions that hold for that particular type of security.
The following are the fundamental elements of accrued interest:
•
The size of the next coupon, usually c/f (if the coupon is irregular, the
size of the next coupon may be larger or smaller and could depend on
the calendar conventions);
•
The days on which coupons are paid. Most types of securities pay
every coupon at the end of the month, if the bond matures at the end
of the month. For example, a Treasury maturing on November 30,
1996 pays coupons on May 31 and November 30. An exception is
bonds issued by the Federal Home Loan Bank (FHLB);
•
The amount of time a (numerator) in the accrual period that has
elapsed since the last coupon date (or interest-accrual date, if the
settlement date falls prior to the first coupon payment of the bond).
The coupon date is determined without regard to business days, even
though a coupon scheduled to be paid on a weekend would be paid
on the following business day. The measurement of a depends on the
calendar; and
•
The amount of time b (denominator) in the full coupon accrual
period. This measurement also depends on the calendar.
The accrued interest would then be calculated as
c a
´ .
f b
35
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Actual/Actual Calendar
U.S. Treasury notes,
bonds, and STRIPS
use the
actual/actual
calendar convention
Because of the
predominance of
Treasury securities,
the actual/actual
calendar will
sometimes be
applied to other
securities with
different calendar
conventions so they
can be compared to
Treasuries on an
equal footing
•
The actual/actual calendar applies to U.S. Treasury notes, bonds and
STRIPS.
•
The “actual” number of days of interest in the accrual period a is the
number of calendar days that have elapsed since the last coupon date,
not including that date, up to and including the settlement date.
•
The “actual” number of days b is the number of days in the complete
coupon period containing the settlement date. A full six-month period
can have only 181, 182, 183, or 184 calendar days.
•
Example 1: With a settlement date of August 1, 1996, the
actual/actual accrued interest on the 8% due November 15, 2021
would be:
8%
Days Between May 15, 1996 and August 1, 1996
78
´
= 4% ´
= 1.696%
2
Days Between May 15, 1996 and November 15, 1996
184
•
Example 2: With a settlement date of September 30, 1996, the
actual/actual accrued interest on a 6% due January 31, 2007 would
be:
6% Days Between July 31, 1996 and September 30, 1996
61
´
= 3% ´
= 0.994%
2
Days Between July 31, 1996 and January 31, 1997
184
•
Leap day (February 29) counts as a calendar day.
36
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
30/360 Calendar
SIA Convention1
•
To determine the number of 30/360 days between two dates, Date1 For corporates,
(prior coupon date) and Date2 (settlement), where Date1 is earlier :
municipals, and
360 × (Year2 – Year1)
+ 30 × (Month2 – Month1)
+ DDays (from the following table)
= 30/360 days between Date1 and Date2
Day1
Not End of Month
End of Month
End of Month
Except:
End of Month (Excluding February)
Day2
DDays
Not End of Month
End of Month
Day2 – Day1
Day2 – 30
0
End of February
Day2 – 30
agencies, the
market uses a
30/360-day
calendar, where
every year is
composed of 12
30-day months
Using the 30/360
calendar, any bond
that matures at the
end of a month
accrues no interest
on the 31st day of
any month
•
The denominator always has 180 days for a semi-annual bond.
More generally, it has 360 f days.
•
Example 1: With a settlement date of August 1, 1996 and a 30/360- Q: What is the
day calendar, accrued interest on an 8% due November 15, 2021 impact of this
would be:
non-accrual on
8%
30 360 Days Between May 15, 1996 and August 1, 1996
76
´
= 4% ´
= 1.689%
2
30 360 Days Between May 15, 1996 and November 15, 1996
180
•
corporate bond
prices?
Example 2: With a settlement date of September 30, 1996 and a
30/360-day calendar, accrued interest on a 6% due January 31, 2007
would be:
6% 30 360 Days Between July 31, 1996 and September 30, 1996
60
´
= 3% ´
= 1.000%
2
30 360 Days Between July 31, 1996 and January 31, 1997
180
1
Jan Mayle, Standard Securities Calculation Methods, vol. 1, Third Edition. New York:
Securities Industry Association, 1993.
37
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Price Over Time
(at Constant Yield)
U.S. Treasury 8% Due November 15, 2021, Priced to Yield 7%
Now, we have a
convenient way of
quoting a price
that “behaves”
better than present
value
Price = PV – Accrued
To a first
approximation,
when yields
remain constant,
the quoted price of
a bond only
changes as it drifts
toward par
Q: Why isn’t this
line smooth?
38
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Actual vs Theoretical Accrued Interest
Accrued interest is
actually calculated
using the linear
rule on the prior
pages
The theoretical
accrued interest
grows according to
the rules of
compound interest,
the same way the
bond’s present
value grows
The actual accrued
interest is always
higher than the
theoretical accrued
interest
There is a “theoretical” accrued interest that would cause the price to The difference
drift smoothly towards par over time.
between actual
However, accrued interest is actually computed using the methodology
on the prior pages. Market participants always use the “actual”
calculation for accrued interest, so that is all you really need to know.
and theoretical has
been accentuated
here for
presentation
purposes
39
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Sample Confirm
Different securities
have different
conventions for
rounding
Treasury securities,
including STRIPS,
round price to
seven decimal
places, and round
accrued to eight
decimal places
Corporate bonds
round price to
three decimal
places
Note that U.S.
Treasury
transactions of
more than $50
million face
amount are broken
into lots no bigger
than $50 million;
this is the
maximum size for
the Fed-wire
system
You
Sold
Trade Date
Settle Date
Cusip
Symbol
07/05/1996
07/08/1996
U.S. Treasury Note,
5 3/8 11/30/1997
912827V90
Note
DTD 11/30/1995
Price 98 25/32
Acct Type
Qty
C.O.D.
50,000,000
Principal
Interest
Net Due (You)
49,390,625.00
279,030.05
49,669,655.05
You
Trade Date
Settle Date
Cusip
Symbol
Acct Type
Qty
Sold
07/05/1996
07/08/1996
912827V90
Note
C.O.D.
50,000,000
U.S. Treasury Note,
5 3/8 11/30/1997
DTD 11/30/1995
Price 98 25/32
Principal
Interest
Net Due (You)
49,390,625.00
279,030.05
49,669,655.05
You
Trade Date
Settle Date
Cusip
Symbol
Acct Type
Qty
Bought
07/05/1996
07/08/1996
912827V48
Note
C.O.D.
50,000,000
U.S. Treasury Note,
6 5/8 due 06/30/2001
DTD 07/01/1996
Price 99 12/32
Principal
Interest
Net Due (Us)
49,687,500.00
69,009.51
49,750,509.51
You
Trade Date
Settle Date
Cusip
Symbol
Acct Type
Qty
Bought
07/05/1996
07/08/1996
912827V48
Note
C.O.D.
50,000,000
U.S. Treasury Note,
6 5/8 due 06/30/2001
DTD 07/01/1996
Price 99 12/32
Principal
Interest
Net Due (Us)
49,687,500.00
63,009.51
49,750,509.51
40
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Coupon Bonds:
Pricing
41
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Basic Bond Pricing Timeline
The bond and its
cost (present value)
are transferred on
the settlement date
The new owner
receives all future
cash flows
The time-dependent
quantities n and x
are critical for
pricing the future
cash flows of a
bond
The calculations are
identical for zerocoupon bonds,
even though there
are no actual
“coupon” dates
where
n is the number of whole coupon periods between the next coupon date
and maturity.
x is the length of the accrual period (measured in units of whole coupon
periods).
42
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Pricing Coupon Bonds
U.S. Treasury 8% Due November 15, 2021
(7% Yield; May 15, 1996 Settlement)
The present value
of a bond is the
sum of the present
values of its
individual cash
flows
The present value
of a bond is also
the present value
of the principal
payment at
redemption plus
the present value
of an annuity
representing all the
coupon payments
104%
4%
4%
4%
4%
4%
+
+
+ LLLLLL +
+
+
2
3
49
50
51
7% ö æ
æ
7% ö
7% ö
7% ö
7% ö
æ
æ
æ
æ
ç1 +
÷ ç 1 + 7% ö÷
1
1
1
1
+
+
+
+
ç
÷
ç
÷
ç
÷
ç
÷
2 ø è
è
2 ø
2 ø
2 ø
2 ø
2 ø
è
è
è
è
43
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Valuing a Coupon Annuity
Valuing the Coupon Stream (S) on the Next Coupon Date
There is a
mathematical trick
for valuing a
coupon annuity
with a (relatively)
simple formula
c
c
f
S= +
+L+
f æ
yö
æ
1
+
ç
÷
ç1 +
fø
è
è
S
æ
yö
ç 1+ ÷
fø
è
=
c
f
æ
yö
ç 1+ ÷
fø
è
+L+
c
f
yö
÷
fø
n-1
c
f
æ
yö
ç 1+ ÷
fø
è
n
+
+
c
f
æ
yö
ç1 + ÷
fø
è
c
f
æ
yö
ç 1+ ÷
fø
è
c = annual coupon rate
y = yield, quoted on a compound basis
f = payment and compounding frequency
n = number of whole coupon periods
n
remaining until maturity
Dividing each side by 1 +
n+1
c
c
f
S= n+1
æ
yö f æ
yö
ç1+ ÷
1
+
ç
÷
fø
è
fø
è
S
é y
ê f
S
S=S´ê
æ
yö
ê1 + y
ç1 + ÷
êë
f
f
è
ø
ù
ú c
ú= ú f æ
ç1 +
úû
è
y
f
Taking the difference between the two
allows us to reduce the stream to a
simpler form.
c
f
yö
÷
fø
Simplifying the left-hand side.
n +1
c
cæ
yö
f
ç1+ ÷ n
fè
fø æ
yö
1
+
ç
÷
fø
è
S=
y
f
Solving for S.
æ
yö
c
cç 1 + ÷ n
fø æ
è
yö
ç1+ ÷
fø
è
=
y
Simplifying.
44
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Valuing a Coupon Bond
c is the annual coupon rate,
v is the redemption value,
y is the yield, quoted on a compound basis,
f is the payment and compounding frequency,
n is the number of whole coupon periods between the next coupon date
and maturity,
S is the value of the coupon stream on the next coupon date, and
x is the length of the accrual period (measured in units of whole
coupon periods), using the appropriate calendar, 0#x<1
The present value of
a bond is the
present value of the
coupons (S) plus the
present value of the
principal
The price of a bond
is the present value
less the accrued
interest
Securities with less
First, compute the value of the bond on the next coupon date by adding
than one coupon
the value of the principal to the value of the coupon annuity:
PVNext Coupon Date
æ
æ
yö
c
vy
yö
vy - c
cç 1 + ÷ cç 1 + ÷ +
n
n
n
fø æ
fø æ
è
è
æ
yö
yö
yö
ç1 + ÷
ç1 + ÷
ç1 + ÷
fø
fø
fø
è
è
è
v
=
=S+
+
n =
y
y
y
æ
yö
ç1 + ÷
fø
è
Then discount this value back to settlement using the fraction of a period
between settlement and the next coupon date (the complement of the
accrual period) according to the appropriate calendar:
PV =
æ
yö
vy - c
cç 1 + ÷ +
n
fø æ
è
yö
ç1+ ÷
fø
è
æ
yö
yç 1 + ÷
fø
è
1- x
Price =
æ
yö
vy - c
cç 1 + ÷ +
n
fø æ
è
yö
ç1 + ÷
fø
è
æ
yö
yç1 + ÷
fø
è
1- x
-x´
period until
maturity (i.e., n=0)
are valued using a
simple-interest
methodology,
described next
Q: Which inputs
are calendardependent?
c
f
45
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Compound- vs Simple-Interest Yield
The price difference
between a
compound- and a
simple-interest
interpretation of a
yield is less than a
32nd for periods less
than seven months
By convention,
securities in their
last coupon period
(n=0) are quoted
on a simple-interest
basis
This graph illustrates the difference between the price computed using a
compound- and simple-interest interpretation of yield. During the first
compounding period, the price using compound-interest yield is higher.
After one compounding period, the price using simple-interest yield is
higher, because the yield does not compound.
The following graph shows the difference between the price calculated
using a simple-interest yield interpretation and the price calculated using
a compound-interest yield interpretation.
Price Difference (%)
0.14
0.12
The difference is
small because of
the mathematical
approximation:
(1 + y ) t » 1 + t × y
for small
t
0.10
Price =
æ y
ö
ç1+ —————
÷
f
è
ø
Compound-Interest
0.08
0.06
0.04
c
v+—
f
Price =
n+1–x
c
– x´—
f
c
v+—
f
y Simple-Interest
1 + (n+1–x ) ´ ————
f
c
– x´—
f
0.02
0.00
(0.02)
0.0
0.5
1.0
Term (Years)
46
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Pricing a Bond When the
Yield and Coupon Are Equal
From the previous page,
Price =
æ
yö
vy - c
cç 1 + ÷ +
n
fø æ
è
yö
ç1+ ÷
fø
è
æ
yö
yç 1 + ÷
fø
è
1– x
-x´
c
f
Using the bond
price formula, it
can be proven that
the price of a bond
whose coupon
equals its yield is
par (100%) on a
coupon payment
date
If v = 100% and c = y, this formula reduces to:
x
æ
yö
y
Price = ç 1 + ÷ - x ´
fø
f
è
If x = 0, implying that the settlement date is a coupon date for the
security, then
0
æ
yö
y
Price = ç 1 + ÷ - 0 ´ = 100%
fø
f
è
Otherwise, the price will be near, but slightly below, par.
47
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Pricing a Bond Using the HP-17B II
U.S. Treasury 8% Due November 15, 2021
(7.252% Yield; June 26, 1996 Settlement)
Whenever you use
a tool to do
financial
computations, you
have the added
responsibility of
understanding all
the settings so you
can ensure they
are correct
Spreadsheets have
the advantage of
retaining inputs
and assumptions
for later
verification
•
Display the BOND menu: press “FIN” “BOND”
•
Press “CLEAR DATA”
•
Define the type of the bond. If the message in the display does not
match Treasury conventions, press “TYPE”
– Press “A/A” to set the calendar basis to actual/actual
– Press “SEMI” to set the coupon payment frequency to semiannual
– Press “EXIT” to restore the BOND menu
•
Enter the bond’s settlement date: “06.261996 SETT”
•
Enter the bond’s maturity date: “11.152021 MAT”
•
Enter the bond’s coupon (actually, the coupon × 100): “8 CPN%”
•
Move to the next screen: “MORE”
•
Enter the yield (actually, the yield × 100): “7.252 YLD%”
•
Request the price: “PRICE”; the calculator should respond
“108.611177”
•
Request the accrued interest: “ACCRU”; the calculator should
respond “0.913043”
•
For the present value, request “PRICE,” “+,” “ACCRU,” “=”; the
calculator should respond 109.524221. (Unless your calculator is
set for “Reverse Polish Notation”).
•
Did your calculator show enough significant digits?
•
How much would $100,000,000 bonds cost?
48
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Bond-Equivalent Yield (BEY)
Bond-equivalent yield is defined as the yield that equates the discounted
value of a bond’s actual future cash flows (determined using the
conventions and methodologies of that bond’s market) with the bond’s
present value in the market. The actual cash flows are discounted using
a semi-annual rate, and the lengths of the discounting periods are
determined using the actual/actual calendar to put the bond-equivalent
yield on the same footing as Treasury yields.
Because the formula for determining a bond’s price from its yield is not
“invertible,” there is no closed-form expression for determining a bond’s
yield from its price (except for bonds with only one future payment).
The yield is, therefore, found by trial and error. One algorithm begins
with an estimate of the yield (call it y0) and then computes the price and
dollar duration of the security. The algorithm calls for taking the
difference between two prices, which is equivalent to the difference
between two present values since they both have the same accrued
interest. Since dollar duration provides an estimate of the price’s
absolute sensitivity to yield changes, it can provide an estimate of the
yield change required to match the price of the bond, as follows:
y i +1 = y i +
Bond-equivalent
yield is a useful
measure for
comparing
securities from
different markets
with different
payment
frequencies and
conventions
Bond-equivalent
yield is calculated
using actual cash
flows, semi-annual
compounding, and
an actual/actual
calendar
Pricei - Price Actual
DurationDollar,i
The new yield is used as the starting point for the next iteration. When
the price is accurate enough, the algorithm stops. This is called the
Newton–Raphson method of equation solving.
49
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Newton–Raphson Equation Solving
The Newton–
Raphson method
seeks to solve an
equation by
iterating from an
initial guess y0 and
refining the guess
(y1, y2, y*) based
upon the slope of
the curve at each
successive point
For a bond, the
slope of the curve
is the dollar
duration
*Actual
50
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Accretion and Amortization
Under the effective-interest method, many investors record income that
is different from actual cash received. Bonds are frequently carried on a
company’s books at a carrying value derived from the acquisition price
and unrelated to the current market value of the security. The carrying
value starts at the acquisition price and drifts toward par over the life of
the bond. The income reported by the investor would be the bond’s
carrying value multiplied by the yield at acquisition (compounded for
the reporting period).
There are two equivalent methods for calculating income and updating
the carrying value according to the effective-interest method:
•
•
Accretion refers to
a growing carrying
value, while
amortization refers
to a declining
carrying value
Many investors
account for bonds
using the effectiveinterest method,
which amortizes or
accretes principal
toward par over
time
Under the effective-interest method, the carrying value on any date
can be determined by calculating the price of the bond at the
acquisition yield for settlement on that date. The income is then
defined as actual cash received plus the change in carrying value.
Neither accretion
Alternatively, the income can be calculated as the carrying value at
the beginning of the period multiplied by the acquisition yield
(compounded appropriately). The change in carrying value is then
income less actual cash received.
nor amortization
changes actual
cash flows
Use the second method to amortize the 8% due November 15, 2021, with
an acquisition yield of 7%.
Beginning Actual Cash
Period Start Carrying
Received
Date
Value (%)
(%)
5/15/96
111.814
Income
(%)
Ending
Amortization Carrying
(%)
Value (%)
4.000
11/15/96
4.000
5/15/97
4.000
51
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Accretion and Amortization (Continued)
The remaining
principal balance
on a bond that
makes level
principal and
interest payments
(like a mortgage)
can be calculated
in similar fashion
8% due November 15, 2021 with an acquisition yield of 7%:
Period Start
Date
Beginning Actual Cash
Ending
Carrying
Received
Amortization Carrying
Value (%)
(%)
Income (%)
(%)
Value (%)
5/15/96
111.814
4.000
3.913
(0.087)
111.727
11/15/96
111.727
4.000
3.910
(0.090)
111.638
5/15/97
..
.
111.638
..
.
4.000
..
.
3.907
..
.
(0.093)
..
.
111.545
..
.
11/15/10
107.584
4.000
3.765
(0.235)
107.349
5/15/11
107.349
4.000
3.757
(0.243)
107.106
11/15/11
..
.
107.106
..
.
4.000
..
.
3.749
..
.
(0.251)
..
.
106.855
..
.
5/15/20
101.401
4.000
3.549
(0.451)
100.950
11/15/20
100.950
4.000
3.533
(0.467)
100.483
5/15/21
100.483
4.000
3.517
(0.483)
100.000
Note that the bond amortized to par on its maturity date. As this example
illustrates, bonds always accrete or amortize toward par more quickly as
they approach maturity.
The same methodology can be used to allocate payments on a bond that
makes level payments of principal and interest. The interest is the
coupon on the security. Any difference between actual cash and interest
is a principal payment. The ending principal balance is the beginning
principal balance less the principal paid during the period.
52
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
More General Pricing Timeline
Every security has a dated date, from which interest begins accruing.
The dated date is not necessarily the issue or first settlement date. If the
dated date lies on the coupon payment cycle, then the first coupon will
usually be regular. If the dated date is not on the coupon cycle, the first
coupon may be larger than usual (“long” first coupon) or smaller than
usual (“short” first coupon). The size of the first coupon will be scaled
by the length of the first coupon period, measured as a number of regular
coupon periods plus a partial period using the appropriate calendar.
Some securities also have an irregular coupon at maturity.
The basic timeline
can be extended to
account for bonds
with an odd first
coupon period and
a coupon cycle that
does not coincide
with maturity
If the bond has
already paid a
coupon, then its
first coupon period
is regular (z=1)
z measures the length of the first coupon period and, therefore, the size
of the first coupon. x is the length of the accrual period, 0#x<z. n is the
number of full coupon periods between the next coupon and the final
regular coupon. w measures the length of the final coupon period and,
therefore, the size of the final coupon. z, x and w are all measured in units
of whole coupon periods.
2A
pseudo-coupon date is a date on which a generic coupon bond with the same maturity and
conventions would pay a coupon, but on which the specific bond does not pay a coupon.
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
General Coupon Bond Pricing Formula
Here is a more
general formula
that can price bonds
with odd first and
last coupons,
including mediumterm notes (MTNs)
c is the annual coupon rate,
v is the redemption value,
y is the yield, quoted on a compound basis,
f is the payment and compounding frequency,
n is the number of whole coupon periods between the next coupon date
(not including that coupon) and the final regular coupon,
x is the length of the accrual period, using the appropriate calendar,
0#x<z,
w is the length of the partial last coupon period, if any, using the
appropriate calendar, and
z is the length of the first coupon period (from the dated date), using the
appropriate calendar (if the first coupon is regular, z=1)
æ
wc ö
yç v +
÷
f ø
è
cy(z - 1) æ
yö
+ cç 1 + ÷ +
f
fø
è
PV =
æ
yö
yç1 + ÷
fø
è
æ
yö
ç1 + ÷
fø
è
z-x
-c
w
æ
yö
ç1 + ÷
fø
è
n
Price = PV - x ´
c
f
54
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Coupon Bonds:
Duration and
Convexity
55
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Modified Duration
When duration is
quoted, it is usually
quoted as modified
duration
The modified
duration of a bond
can be quoted as
either present-value
or price duration,
all relating to the
same dollar
duration; it is
important to be
clear about the
quoting convention
We generally use
modified presentvalue duration,
which estimates the
percentage change
in price for an
instantaneous,
parallel change in
yield
Modified duration is so named to differentiate it from Macaulay
duration (to be covered later). Modified duration is generally the only
type of duration that we use because it shows the sensitivity of a bond’s
value to changes in interest rates. There are three different methods of
quoting the same modified duration: dollar duration, present-value
duration, and price duration. The distinction is necessary because price
and present value are different for coupon bonds. For zero-coupon
bonds, there is no difference between present-value and price duration.
The difference between the three different methods is in how they
express the same price sensitivity.
Dollar duration estimates the price impact of a change in yield as a
percent of par and is the result of differentiating the formula for price
with respect to yield. Dollar duration is also sometimes quoted as an
absolute number: how much the dollar value of a security or portfolio
will change when yields change.
DurationDollar = -
dP
dPV
=dy
dy
The present-value duration of a bond estimates the price impact as a
percent of dollars invested (present value). Present-value duration can
be useful for evaluating the relative riskiness of a fixed-dollar
investment in different securities. This is the most common way of
quoting duration and is usually what is being described when “duration”
or “modified duration” is quoted without further description.
DurationPV =
dPV
dP
DurationDollar
PV = PV
=Present Value
dy
dy
56
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Modified Duration (Continued)
The price duration of a bond estimates the same price impact as a
percent of price. Price duration can be multiplied by quoted price to
compute dollar duration without ever calculating accrued interest or
present value. Therefore, it can be useful for estimating the price change
of a security when interest rates change. Price duration is also more
stable than present-value duration over time because price is more stable
than present value under constant interest rates. Because the price is
always less than the present value, a security’s price duration is always
greater than its present-value duration.
dP
dPV
DurationDollar
P
P
DurationPrice =
==Price
dy
dy
Modified and Price Durations of 8% Due
November 15, 2021, Priced to Yield 7% Over Time
57
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Weighted Averages
The duration of a
portfolio is the
average of the
durations of the
securities in the
portfolio, weighted
by market value
Likewise, the
duration of a
security is the
average of the
durations of the
security’s
individual cash
flows, weighted by
each cash flow’s
contribution to the
security’s market
value (its present
value at the bondequivalent yield)
There are many
other security and
portfolio
characteristics that
can be calculated
as weighted
averages
The average of an attribute xi weighted by wi is defined as:
1
´ å wi xi
w
å i i
i
An alternative definition would be to define normalized weights yi as:
yi =
wi
so that
å wi
i
åy
i
i
=1
Then the weighted average would be defined as:
åyx
i i
i
For example, as discussed on the next page, the duration of a security is
the average of the durations of the individual cash flows, weighted by
their respective present values:
DurationModified =
1
´ å PV ´ Durationi
å PVi i i
i
=
1
´ å PVi ´ Durationi
PV
i
This approach can provide worthwhile insight into how a security’s
individual cash flows affect the duration of the security. It works because
absolute dollar duration is additive: if we double our holdings in a
security, we will have twice the absolute market risk.
58
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Weighting Duration and Convexity
Because the duration of a security is the present-value-weighted duration We can apply the
of the individual cash flows, we can write an intuitive formula for the principles of
calculating
duration of a coupon bond (with no irregular coupons):
Par PV
DurationModified
×
Par Dur
+
Cpn PV × Cpn Dur
æ
n + 1- x
ç
n
1 ç
v
f
+
=
´ç
´
å
n + 1- x
PV ç æ
æ
yö
yö
1 + ÷ i = 0 æç 1 +
ç
1
+
ç
÷
çè
fø
è
fø
è
è
c
f
yö
÷
fø
i + 1- x
i + 1 - x ö÷
÷
f
´
÷
æ
yö ÷
ç1+ ÷ ÷
føø
è
ö
æ
÷
ç
n
1 çv
c
i + 1- x ÷
(n + 1 - x )
=
´ç ´
n + 2- x + 2 ´ å
i + 2- x ÷
PV ç f æ
f
i=0 æ
yö
yö
÷
ç1+ ÷
ç1+ ÷
÷
ç
f
f
è
ø
è
ø
ø
è
Where x is defined as the accrual period, 0#x<1, and n is the number of
whole coupon periods between the next coupon date and maturity.
weighted averages
to construct intuitive
formulas for
duration and
convexity
We will later
construct more
complicated, but
more
computationally
efficient, formulas
by differentiating
the closed-form
equation for price
as a function of
yield
Likewise, the convexity of a security is the present-value-weighted
convexity of the individual cash flows.
ö
æ
÷
ç
n
1 ç v (n + 1 - x ) ´ (n + 2 – x ) c
(i + 1 - x ) ´ (i + 2 – x )÷
Convexity =
´ç
´
+ 3 ´å
÷
n + 3– x
i + 3– x
PV ç f 2
f
i=0
æ
æ
yö
yö
÷
ç1+ ÷
ç1+ ÷
÷
ç
fø
fø
è
è
ø
è
These equations for duration and convexity can also be obtained as the
derivative of the summation expression for present value with respect to
yield.
59
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Modified Durations vs Maturity
Par Bonds Priced as of June 24, 1996:
30-Year Bond Yielding 7.09%
The modified
duration of a par
bond increases
with maturity, but
at a diminishing
rate
It is critical to build
an intuitive sense
for these durations
This curve shows
durations for
bonds priced at
par
The par-bond
construct is
necessary to avoid
comparing bonds
with similar
maturities but
different coupons
and, therefore,
different durations
60
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Modified Duration vs Yield
U.S. Treasury 8% Due November 15, 2021 for
Settlement May 15, 1996
Because of
convexity, duration
increases when
yield declines (for
this bond)
61
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
A Closed-Form
Expression for Dollar Duration
Dollar duration can
be calculated simply
by taking the first
derivative of the
price formula
DurationDollar = -
DurationDollar =
DurationPV × PV
Alternatively,
=
DurationDollar =
DurationPrice × Price
It is usually easier
and more reliable
to estimate dollar
duration as the
change in price for
a small change in
yield:
DurationDollar @ –
1
æ
yö
y çç 1 + ÷÷
fø
è
2
z- x
dPV
dP
DP
=@dy
dy
Dy
é 2æ
ê y çç v +
ê è
ê
ê
ê
ê
´ê
ê
ê
ê
ê
ê
ê
êë
ù
ú
æ
ö ú
y
+
+
+
c
1
n
z
x
1
(
)
ç
÷
w
ç
÷ ú
f
è
ø
æ
yö
f çç 1 + ÷÷
ú
fø
è
+ú
n+ 1
ú
æ
yö
ú
çç 1 + ÷÷
fø
ú
è
ú
ú
ö
ú
cy 2 (z - 1)(z - x ) æ
y
+ cçç 1 + (z - x )÷÷
ú
f
yö
2æ
è
ø
ú
f çç 1 + ÷÷
fø
úû
è
wc ö
÷(n + w + z - x )
f ÷ø
Where the variables are defined as follows:
c is the annual coupon rate,
v is the redemption value,
DP y is the yield, quoted on a compound basis,
Dy f is the payment and compounding frequency,
n is the number of whole coupon periods between the next coupon date
(not including that coupon) and the final regular coupon,
x is the length of the accrual period, using the appropriate calendar,
0#x<z,
w is the length of the last coupon period, if any, using the appropriate
calendar, and
z is the length of the first coupon period (from the dated date), using the
appropriate calendar (if the first coupon is regular, z = 1).
62
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Macaulay Duration
U.S. Treasury 8% Due November 15, 2021
(7% Yield, May 15, 1996 Settlement)
The Macaulay
duration is defined
as the presentvalue-weighted
time to payment of
a bond’s cash
flows
It happens to be
related to modified
duration by a
simple formula
The Macaulay
duration of a zerocoupon bond is its
term
DurationMacaulay =
n
1
´ å PVi ´ Ti
PV i =1
where Ti is the time (in years) until the ith cash flow
æ
yö
DurationMacaulay = DurationModified ´ ç 1 + ÷
fø
è
DurationModified =
DurationMacaulay
y
1+
f
63
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Convexity
Convexity is the
second-order
correction to
estimated price
change given a
change in yield
Convexity is on the
order of the square
of duration (for
bonds without
embedded options)
For bonds (without
embedded options)
having the same
duration, the bond
with the wider
dispersion of cash
flows will have the
higher convexity; a
zero-coupon bond,
with the lowest
possible dispersion,
has the lowest
possible convexity
for a given
duration
We have already noted that convexity goes up with the square of
maturity for a zero-coupon bond. More generally, convexity is on the
order of the square of duration.
For bonds with the same duration, the bond with the wider dispersion of
cash flows will have the higher convexity.
Example: A 10-year STRIPS has approximately the same duration as a
portfolio of 50% cash and 50% 20-year STRIPS. Because convexity
increases with the square of duration, the 20-year STRIPS has four times
the convexity of the 10-year STRIPS. The cash and STRIPS portfolio,
therefore, has twice the convexity of the 10-year STRIPS portfolio.
Likewise, a portfolio of two-thirds cash and one-third STRIPS also has
a duration of 10, but has a convexity three times as great as for the
10-year STRIPS.
The “value of convexity” lies in the fact that the higher the convexity, the
more the expected rate of return exceeds the yield. This is because the
average of the price of a portfolio in both a “down” and “up” interest rate
scenario will be higher than the current price. The higher the convexity,
the more the average price will exceed the current price.
Since convexity has value, we should expect the more convex portfolio
of cash and STRIPS to have a lower yield. In fact, it does: a 10-year
STRIPS (May 15, 2006) has a yield of 7.13%, and a portfolio of cash
(yielding 5.25%) and 20-year STRIPS (May 15, 2016, yielding 7.459%)
has a market-value-weighted-average yield of 6.355%, significantly
lower.
64
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Value of Convexity
30-Year U.S. Treasury Zero-Coupon Bond
When there is
volatility in yields,
positive convexity
implies that a
portfolio’s
expected return is
greater than its
yield
When there is anticipated yield volatility, a convex portfolio has a shortterm expected return that is greater than its yield. The greater the
expected volatility and the greater the convexity, the greater this effect.
Q: Why would
anyone choose to
buy a bullet (10year STRIPS) rather
than a barbell
(50% cash and
50% 20-year
STRIPS) portfolio
with the same
duration and
higher convexity?
65
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
A Closed-Form
Expression for Dollar Convexity
Dollar convexity
can be calculated
simply by taking
the second
derivative of the
price formula
Convexity would
then be calculated
as:
d 2PV 2
dy
Convexity = PV
Convexity is
divided by two in
the Taylor
expansion for
price; some firms
quote it already
divided by two
Convexity Dollar =
=
1
æ
yö
y 3 ç1 + ÷
fø
è
DDurationDollar
d2P
@Dy
dy 2
z - x +1
é
ê
ê
ê
ê
ê y 3 æ v + wc ö (n + w + z - x )(n + w + z - x + 1)
÷
ê çè
æ
f ø
æ
öö
y
y
- cç 2 + (n + z - x + 2)ç 2 + (n + z - x + 1)÷ ÷
ê
w
f
f
è
øø
è
æ
yö
ê
f 2 ç1 + ÷
ê
fø
è
´ê
n +1
ê
æ
yö
ê
ç1 + ÷
fø
è
ê
ê
ê
ê
ê
cy 3 (z - 1)(z - x )(z - x + 1) æ
æ
öö
y
y
ê
+ cç 2 + (z - x + 1)ç 2 + (z - x )÷ ÷
f
f
æ
ö
è
øø
y
è
ê
f 3 ç1 + ÷
ê
f
è
ø
ë
ù
ú
ú
ú
ú
ú
ú
ú
ú
ú
+ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
û
where the variables are determined as follows:
c is the annual coupon rate,
v is the redemption value,
y is the yield, quoted on a compound basis,
f is the payment and compounding frequency,
n is the number of whole coupon periods between the next coupon date
(not including that coupon) and the final regular coupon,
x is the length of the accrual period using the appropriate calendar,
0#x<z,
w is the length of the last coupon period, if any, using the appropriate
calendar, and
z is the length of the first coupon period (from the dated date), using the
appropriate calendar (if the first coupon is regular, z=1).
66
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 2 Exercises
1. Calculate the present value, modified duration, dollar duration, and
convexity of these Treasury STRIPS for settlement on June 26,
1996.
Maturity
BondEquivalent
Present
Yield (%) Value (%)
11/15/99
6.63
11/15/22
7.38
02/15/23
7.36
Modified
Duration
Dollar
Duration
Convexity
2. Using the bond price formula, what is the price of a 10-year 7%
coupon bond at an 8% bond-equivalent yield?
3. What is the price of an 8% semi-annual-pay coupon bond that
matures in exactly 15 years if the required bond-equivalent yield-tomaturity is 6%?
4. Many bonds pay interest twice per year, but their coupons are quoted
on an annual basis. That is, an 8% 2-year U.S. Treasury note pays a
4% coupon twice per year. What is the bond’s annual yield-tomaturity if it is priced at par on a coupon date?
5. If a 10-year Treasury bond with a 7% coupon is issued today at a
price of 99-24 (99.750%), what is its bond-equivalent yield-tomaturity? Its annual yield-to-maturity?
6. For settlement on June 26, 1996, the price of the February 15, 1997
STRIPS was 96.444%. The yield is quoted as the yield to the stated
maturity date, but that day is a Saturday and the cash is not delivered
until the following Monday. What is the difference between the
quoted yield and the yield actually earned by the investor?
67
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 2 Exercises (Continued)
7. Is the price of a bond above or below par if its yield is less than its
coupon?
8. Which has a longer duration, a 7-year zero-coupon bond yielding
7.20% (BEY) or a 10-year 7.25% coupon bond yielding 7.20%
(BEY)?
9. As long as you can safely stuff cash under your mattress (nonnegative interest rates), what is the most you would ever pay for a
bond that matures in eight years and has a 7% coupon paid annually?
What if the bond paid a semi-annual coupon? Could interest rates
ever become negative?
10. A bond issued by company A has a 6% coupon and matures
February 15, 2026. The U.S. Treasury bond that matures the same
date also has a coupon of 6% and is priced at 86-18+ (86.578125%).
Is the price of company A’s bond greater or less than 86-18+?
11. If three bonds promise the following cash flows, which is worth the
most? Estimate the duration of each at a 7% semi-annual yield.
Years from Now
1
Cash Flow A
($)
Cash Flow B
($)
1,000
400
2
500
1,000
1,000
3
1,000
600
4
1,000
700
5
Cash Flow C
($)
800
1,000
12. A perpetual bond pays coupons forever, but never matures. If a
perpetual bond pays a 7% coupon annually and is priced at 95%,
what is its yield? What is its duration? What is its convexity? How
does its convexity compare to a zero-coupon bond with the same
duration?
68
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 2 Exercises (Continued)
13. One year ago, a bank loaned you enough to purchase a home with a
30-year fixed-rate mortgage requiring a payment of $1,000 per
month. Mortgage payments are level across the life of the note, so
each payment comprises both interest and principal. The monthly
interest rate on the mortgage is 8%. What was its original face
value? What is the balance today? What is the BEY? Who is the
issuer?
69
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 3
The Yield Curve,
a Treasury Pack
and Fitted Curve
Analysis
This chapter is an excerpt from A Morgan Stanley Guide to Fixed Income Analysis by
Andrew R. Young, ©2003 Morgan Stanley & Co. Incorporated.
71
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
In This Chapter, You Will Learn...
•
The Relationship Between STRIPS and Coupon Bonds
•
How to Use a Treasury Pack
•
How to Hedge Using Duration
•
Butterfly Hedging
•
How to Construct and Use a Fitted Curve
72
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Yield vs Maturity
U.S. Treasury Coupon Bond Yields from
Tuesday, June 25, 1996 Pack
The most liquid,
fundamental type
of security is a U.S.
Treasury coupon
bond
These securities are
regularly issued by
the Treasury to
finance the U.S.
government
The yields of these
securities can be
plotted against
maturity to create a
yield curve; this
particular yield
curve is upward
sloping
Note: Callable Treasuries at a premium are plotted to the call date.
Yield can also be
plotted against
duration to adjust
better for different
coupon rates
73
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
The STRIPS Curve and the
Coupon Bond Curve
Benchmark Treasury Yields from Tuesday, June 25, 1996 Pack
There is a yield
curve for both
coupon bonds and
STRIPS (zero-coupon
bonds)
The coupon bond
curve shown here is
for fairly-priced
hypothetical par
bonds (priced at
100%)
Two bonds with the
same maturity but
different coupons
(and, therefore,
prices and
durations) will
usually have
different yields
The coupon curve is usually upward sloping (positive), although there
have been times when it has been flat or inverted. Theories for why the
yield curve should be positive include:
•
“rational expectations,” where investors generally believe inflation
will rise in the future,
•
“term premiums,” where investors need to earn a higher expected
rate of return to compensate them for the risks of owning longer-term
securities, and
•
“investor segmentation,” where different sets of investors are
restricted to or have preference for different parts of the yield curve,
so there are actually different supply and demand equilibria. For
example, money-market funds have a greater demand for short-term
securities than for long-term securities.
74
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Stripped Treasury Securities
$100 Million of an 8% 30-year Treasury Bond
Suppose the above security is purchased to create stripped zero-coupon
Treasury securities. The cash flow from this bond is 60 semi-annual
payments of $4 million each plus the repayment of $100 million
principal at maturity:
The process of
selling individual
coupon and
principal payments
separately is called
coupon stripping
This process creates
STRIPS, which are
separately traded
zero-coupon
securities
Most “strippable” securities (10-years and 30-years original-issue only)
mature on February, May, August or November 15, so only the STRIPS
that mature on these dates have significant liquidity. In 1996, the
Treasury began issuing strippable securities maturing in July and
October, so there is potential for STRIPS maturing on these cycles as
well. Coupon STRIPS are separated coupon payments, and principal
STRIPS are separated principal payments. While it is impossible to
determine which issue was the source of a coupon STRIPS, principal
STRIPS correspond directly to the specific bond from which they were
created. A bond can be reconstituted from the correct amount of each of
its component STRIPS. Principal STRIPS are always priced so that the
value of all of a bond’s STRIPS added together is very close to the value
of the bond itself. This is sometimes called STRIPS-bonds parity. If all
bonds were priced consistently, a coupon and a principal STRIPS with
the same maturity would have identical prices, because they are both
U.S. Treasury obligations.
STRIPS are bought,
sold, and held the
same way as the
underlying
Treasuries, except
they are most often
quoted on a yield
basis
75
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Understanding the STRIPS Curve
The Bootstrap Method
By doing some
simple calculations,
we can understand
how the shape of
the coupon curve
affects the shape of
the STRIPS curve
This is called the
bootstrap method
because we start
with what we
know and then
“pull up” our
knowledge one step
at a time
The present value of
a UST coupon bond
is very close to the
total value of its
cash flows, each
discounted at the
relevant STRIPS
yield
Therefore, the
coupon bond’s yield
is some kind of
average of the
STRIPS yields
Example 1
Term
Coupon Rate
Coupon Bond Yield
STRIPS Yield
6-Month
6.000%
6.000%
?
1-Year
6.500%
6.500%
?
Hints: What is the present value of each bond?
What is the value of the first coupon payment of the 1-year
security (assuming UST cash flows are priced consistently,
regardless of their source)?
Example 2
Term
Coupon Rate
Coupon Bond Yield
STRIPS Yield
20-Year
8.000%
8.000%
8.500%
8.000%
7.950%
?
20½ -Year
Hints: What is the present value of each bond?
What is the value of the first 40 coupon payments on the
20-year?
What is the value of the first 40 coupon payments on the
20½-year (assuming UST cash flows are priced consistently,
regardless of their source)?
76
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Understanding the STRIPS Curve
(Continued)
The Bootstrap Method
Given:
Term
Coupon
Rate
Coupon
Bond Yield
Coupon
Bond Price
STRIPS Yield
6-Month
6.000%
6.000%
100%
?
1-Year
6.500%
6.500%
100%
?
Our task is to calculate the 1-year STRIPS’ yield.
The 1-year bond comprises two cash flows; the value of the bond is the
sum of the values of the individual flows:
PV1-Year Bond = PV6 -Month Cash Flow + PV1-Year Cash Flow = 100%
=
6.500% ö
6.500%
æ
´ PV6 -Month STRIPS + ç 100% +
÷ ´ PV1-Year STRIPS
2 ø
2
è
Determining the
yield of the 1-year
STRIPS using the
bootstrap method
depends critically
on the assumption
that the 6-month
coupon from the
1-year bond is
priced the same as
the coupon and
principal from the
6-month bond
Assuming consistent pricing, the yield of the 6-month STRIPS is the
same as the yield of the 6-month bond since both securities have a single
cash flow on the same date. The present value of the 6-month STRIPS
is then:
PV6 -Month STRIPS =
100%
= 97.087379%
6.000% ö
æ
ç1 +
÷
2 ø
è
The yield of the 1-year STRIPS is then derived from the following two
equations:
PV1-Year STRIPS =
6.500%
´ PV6 - Month STRIPS
2
= 93.796281%
6.500% ö
æ
ç 100% +
÷
2 ø
è
100% -
æ
ö
100%
- 1÷ = 6.508%
y1-Year STRIPS = 2 ´ ç
ç PV
÷
1-Year STRIPS
è
ø
77
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Understanding the STRIPS Curve
(Continued)
The Bootstrap Method
The effect of a
change in coupon
bond yields on
STRIPS yields can be
understood by
estimating how
much STRIPS yields
would have to
change to produce
a given change in
the value of the
whole bond
The principal
payment contributes
less and less to the
present value of a
bond as maturity
increases; the yield
of that principal
must, therefore,
change by more to
affect the yield of
the security
Example 1
Term
Coupon Rate
Coupon Bond Yield
STRIPS Yield
6-Month
6.000%
6.000%
6.000%
1-Year
6.500%
6.500%
6.508%
Because the majority of the value of the 1-year bond is in the final
payment, the steepness of the coupon bond curve does not imply a
significantly steeper STRIPS yield curve.
Example 2
Term
Coupon Rate
Coupon Bond Yield
STRIPS Yield
20-Year
8.000%
8.000%
8.500%
20½ -Year
8.000%
7.950%
8.355%
Because the majority of the value of the 20-year bond is in its coupons,
the yield of the final cash flow has to fall dramatically to affect the
overall yield of the bond.
Using the bootstrap method, if an n-period bond with coupon c pays f
times per year and has present value PVBond , and given prices of the
STRIPS with shorter maturities PVi-Period STRIPS , where i<n, then the
present value of the n-period STRIPS is
c n-1
PVBond – ´ å PVi-Period STRIPS
f i =1
PVn-Period STRIPS =
c
100% +
f
and the yield follows from the present value.
78
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
STRIPS Quote Sheets in a Treasury Pack
U.S. Treasury Prices from Tuesday, June 25, 1996 Pack
At the end of this chapter, there is a representative Treasury pack for A Treasury pack
trade on June 25, 1996 (settling on June 26, 1996). The prices are as of provides a large
quantity of
the close on June 24, 1996.
information that is
useful on a daily
basis
The first page of the pack contains a STRIPS quote sheet for both
coupon and principal STRIPS. STRIPS have fewer differentiating
features and so are often presented in a compressed form.
A pack for a given
•
The top section of the sheet shows contemporaneous price and yield
closes for the Treasury benchmark issues. This shows the context of
the STRIPS yields.
– There is a listing of the benchmarks. Between each successive
pair of benchmarks is the yield spread between those benchmarks.
trade date usually
contains closing
prices for the prior
business day for
settlement on the
business day after
the trade date
– Under each benchmark is its closing bond-equivalent yield and its At the end of this
chapter, there is a
closing price. For the 1-year bill, the “price” is the discount
representative pack
(quoted) yield (covered later).
•
for June 25, 1996
(trade date)
The main section of the report lists coupon STRIPS in maturity order,
The first page in the
followed by principal STRIPS in maturity order.
– The first column of this section shows the STRIPS maturity.
These STRIPS mature on either February 15, May 15, August 15,
or November 15.
Treasury pack
contains yields for
STRIPS
– The second column lists the bid yield for each STRIPS, i.e., the
yield at which the provider of the quote sheet is willing to buy the
STRIPS. The firm would usually offer to sell the STRIPS at a
lower yield (translating to a higher price) in order to create the
potential for a profit.
79
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Coupon Quote Sheet in a Treasury Pack
U.S. Treasury Prices from Tuesday, June 25, 1996 Pack
The rest of the pack
contains prices,
yields, and other
information for
Treasury coupon
notes and bonds
and Treasury bills
There is substantially more useful detail about coupon securities than
that contained on the compressed page. The next set of pages shows
information about Treasury coupon notes and bonds. Some of the
information, notably duration and CUSIP, would also be useful for the
STRIPS; similar pages could be constructed for the STRIPS, but are not
shown here to save (a little) space. Each security takes at least two rows.
The second page in the pack provides information regarding the
benchmark Treasury coupon notes and bonds and bills. The most
recently auctioned Treasury issue for each maturity is referred to as the
current or on-the-run issue and is so indicated in the first column. The
Treasury currently issues 3- and 6-month bills, 1-year bills, 2-, 3-, 5- and
10-year notes, and 30-year bonds.
Term
When Issued
Issuance Cycle
3-Month Bill
1929–Current
Issued every Thursday; mature 13 weeks later
6-Month Bill
1958–Current
Issued every Thursday; mature 26 weeks later
1-Year Bill
1967–Current
Issued every fourth Thursday; mature 52 weeks later
2-Year Note
1974–Current
Usually issued on last day of every month or on the next
following business day
3-Year Note
1978–Current
Usually issued on 15th of February, May, August, and
November
5-Year Note
1991–Current
Usually issued on last day of every month or on the next
following business day; prior to 1991, there were
similar maturities
10-Year Note
1978–Current
Traditionally issued on 15th of February, May, August,
and November; in 1996, extended to include July and
October 15 issues
30-Year Bond
1978–Current
Usually issued February, May, August, and November
15; May and November issuance was cut in 1993, but
November was reinstated in 1996
80
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Coupon Quote Sheet in a Treasury Pack
(Continued)
U.S. Treasury Prices from Tuesday, June 25, 1996 Pack
The same information about a complete list of Treasury notes and bonds
follows on the next pages, broken down into the following sectors: 0- to
3-, 3- to 5-, 5- to 15-, and 15- to 30-year securities. Each sector lists all
the benchmark bonds, even those not in the sector.
•
The first column on each Treasury sheet flags various characteristics
of the Treasury security.
– The term of the issue, if it is a current benchmark.
– “B” indicates a “bad” end date. The security matures on a
weekend or holiday, and the money will not be available until the
next business day. The bond pays the same amount regardless of
the actual payment date. For example, the 6¼% of August 31,
1996 yields 5.545%, 39 basis points more than a bond just 15
days shorter. However, August 31, 1996 is a Saturday. The yield
of the bond to a September 3, 1996 (remember Labor Day!)
receipt date is much more fair: 5.304%.
– “O” signifies an odd first coupon. An odd first coupon means that
the bond began accruing interest from a date that does not lie on
the coupon cycle, so the first coupon’s size is irregular. The first
coupon may be either long (greater than normal) or short (less
than normal). The size of the first coupon has no consequence
after it has been paid.
– “F” represents a “phantom,” or old (but recent) on-the-run, issue.
These securities tend to have better-than-average liquidity.
81
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Coupon Quote Sheet in a Treasury Pack
(Continued)
U.S. Treasury Prices from Tuesday, June 25, 1996 Pack
– “WI” indicates that the issue is trading on a when-issued basis and
has an original settlement date that is later than the next business
day. No investor actually owns the issue. If the security has not
yet been auctioned, the coupon is not known, and trades are done
on a yield basis; actual prices are computed after the auction sets
the coupon. In the auction for 2- and 5-year notes, every winning
bidder buys at the same price; this is called a Dutch auction. In all
the other auctions, the highest bidders win and buy at the
(different) prices bid in the auction. During the auction process,
the Treasury sets the coupon to be the average fill level (average
of winning yields), rounded down to the nearest eighth; most
winners will buy at a discount.
•
The second column identifies the coupon (or “BILL” if the security
is a Treasury bill benchmark), and the third column identifies the
maturity. If the bond is callable, there is a third row with the call year.
Note that all existing callable Treasuries are callable at par five years
prior to maturity.
•
The fourth column shows the previous day’s closing bid price, as
well as the bid yield-to-maturity. If the Treasury is callable, there is a
third row that contains the bid yield-to-call. The firm providing the
quote sheet would generally stand ready to sell at a higher price
(lower yield).
– Treasury coupon securities usually trade on price in units of 1/8 of
a 32nd of a percent of par. Usually, par is assumed to be $100, so
that a price of 99% and a price of 99 mean the same thing. For
example, the quote of 99-03 for the current five-year note refers
to a price of 99 and 3/32 percent of par.
82
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Coupon Quote Sheet in a Treasury Pack
(Continued)
U.S. Treasury Prices from Tuesday, June 25, 1996 Pack
– A plus sign (“+”) following the number of 32nds means that a 64th,
or ½ of a 32nd, is added to the price. For example, the price of the
current 30-year bond is 86-18+, which refers to a price of 86 and
37
64 percent of par.
– A number (1–7) following the number of 32nds indicates how
many 8ths of a 32nd are added to the price. For example, the price
of the current 3-year note is 99-222, which indicates a price of
99 +
22 2
32
8
or 99 and
178
256
Treasury securities
are quoted on both
a yield and a price
basis; since they are
mathematically
related, one can
always be
converted to the
other
percent of par.
This is read “99-22 and a quarter” ( 2 8 is simplified to ¼). Another
example is the 6% of August 31, 1997 with a price of 99-303,
which indicates a price of 99 and 243 256 percent of par. This would
be read “99-30 and three-eighths.”
– Treasury coupon securities are also quoted on a yield-to-maturity
basis. The yield-to-maturity is the discount rate that equates the
present value of the cash flows (interest and principal) to the
market price plus accrued interest. If the yield-to-maturity is
given, then the present value of the bond is the present value of
all the cash flows using that yield, and the price is the present
value less accrued interest. Treasury securities with less than six
months until maturity follow a slightly different simple-interest
convention.
83
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Coupon Quote Sheet in a Treasury Pack
(Continued)
U.S. Treasury Prices from Tuesday, June 25, 1996 Pack
– Some Treasury securities are callable and are also quoted using a
yield-to-call. The yield-to-call is the interest rate that will make
the present value of the cash flows of the bond, if called at the first
possible opportunity, equal to the actual present value of the bond.
The price of the bond is determined in the market and does not
change regardless of how investors analyze or quote yield. If the
bond is currently trading at a premium, we usually quote the yield
assuming that the bond will be called, and so it “trades to call.” If
the bond is currently trading at a discount, then we assume that it
will not be called, so it “trades to maturity.” The lower of yieldto-maturity and yield-to-call is called the yield-to-worst and is
another measure of potential return. The yield-to-worst is
identified for each price in the quote sheet by an “r” preceding it;
since most callable bonds have high coupons, the yields-to-call
are identified with the “r.” For example, the yield-to-maturity of
the 11¾% due November 15, 2014 is 7.71%, but the yield-to-call
is 7.13%. Therefore, the bond trades to call; the bond’s yield-tocall is applied to the call date to calculate the market price.
Any callable
security, given a
market price, can
be quoted on a YTM
or YTC basis; the
yields are different,
but both will
produce the actual
price of the bond in
the market
•
In the fifth column, the first row contains the yield value of a 32nd
increase in dollar price (YV32). The value of a 32nd, also known as
the value of a tick, measures the change in yield of a security if its
dollar price increases by one 32nd of a percent. For example, if the
2-year note’s price rises by one 32nd to 99-15+, its yield will fall to
6.302% – 0.0175% = 6.284%. The second row contains the CUSIP, a
unique nine-digit security identifier in wide use for identifying and
settling domestic securities.
84
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Coupon Quote Sheet in a Treasury Pack
(Continued)
U.S. Treasury Prices from Tuesday, June 25, 1996 Pack
•
The sixth column contains the amount outstanding, in millions, and
the Macaulay duration. If the bond is callable, the third row contains
the Macaulay duration-to-call. In order to use the quote-sheet
duration correctly, it must be converted using the formula
DurationModified =
DurationMacaulay
yö
æ
ç1 + ÷
fø
è
For example, the listed duration of the long bond is 12.82. The
modified duration is 12.38.
•
Columns seven through 11 show the yield for various incremental
changes in price. This information is useful for quickly determining
the yield given a price quote.
For example, the closing price of the 5-year was 99-03. In the
“Tic–1” column, its price is “31.” This means 98-31, a different
handle (the most significant part of price, i.e., “99” or “98”). Always
know your handle! If the bond is callable, the third row shows the
yield-to-call for that scenario.
85
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Coupon Quote Sheet in a Treasury Pack
(Continued)
Other Useful Concepts
•
Current yield, which is defined as coupon divided by price
•
Modified duration (present value or price)
•
Convexity (or gain from convexity)
•
Dollar value of a basis point (DV01 or PV01). DV01 is the change in
price for a one-basis-point change in yield, i.e., 0.01% ×
DurationDollar. It is directly related to YV32. Based on the definition
of dollar duration:
DurationDollar
Thus,
DV 01 =
1
32
@
-
DP
Dy
=
DV 01
0.01%
=
1
32
YV 32
´ 0.01%
YV 32
86
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Treasury Bills in a Treasury Pack
U.S. Treasury Prices from Tuesday, June 25, 1996 Pack
The final two pages in the representative Treasury pack are the bill A Treasury pack
pages. Treasury bills are discount securities and pay no coupon.
also contains
•
The first column indicates which bills are the most recently issued
benchmarks and which bills are trading on a when-issued basis. It
also has the name of the security, “BILL.”
•
The second column displays the maturity of the bill.
•
The third column shows the closing bid discount yield and bondequivalent yield. Treasury bills are usually quoted on a discount-yield
basis; that is how the Treasury auctions them.
information about
Treasury bills
– For the purpose of discussing these yields, define d to be the
actual number of days between settlement and maturity.
– Discount yield and price are related by the following formulas:
Yield Discount = æç
è
Par - Price ö æ 360 ö
÷ ´ç
÷
ø è d ø
Par
Price = Par - Yield Discount ´ Par ´
d
360
87
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Treasury Bills in a Treasury Pack
(Continued)
U.S. Treasury Prices from Tuesday, June 25, 1996 Pack
– For three reasons, the discount yield is not a meaningful measure
of the return from holding a Treasury bill. First, the measure is
based on the face value, rather than the market value, of the
investment. Second, the yield is annualized according to a
360-day year instead of a 365-day (or 366-day) year, making it
difficult to compare Treasury bill yields with those of Treasury
notes and bonds, which pay interest on an actual/actual basis. And
third, discount yield does not compound.
Bond-equivalent
yield usually allows
us to compare
yields of securities
with different
quoting conventions
on an equal footing
However, the
“bond-equivalent
yield” for a
Treasury bill is
calculated according
to unique
conventions that
only approximate
putting it on an
equal footing
– For Treasury bills, bond-equivalent yield is only an estimate of
the actual/actual semi-annual yield that discounts the future cash
flows to their market value. Nevertheless, the only bondequivalent yield that is ever used for a Treasury bill is this
estimate:
d £ 182 Þ yBEY º
d > 182 Þ y BEY
Par – Price 365
´
Price
d
182 .5
é
ù
æ
ö
Par – Price d
ú
ê
÷
º 2 ´ êç 1 +
– 1ú
Price ø
êè
ú
ë
û
Unfortunately, this convention produces a different bond-equivalent
yield for T-bills than STRIPS with the same maturity and price, so the
BEY for bills is not on “equal footing.”
•
The fourth column shows the CUSIP of the Treasury bill.
•
The fifth through ninth columns contain the bond-equivalent yields
corresponding to discount yield up and down two basis points, in
one-basis-point increments.
88
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Hedging
U.S. Treasury Prices from Tuesday, June 25, 1996 Pack
Dollar duration quantifies a bond’s or portfolio’s sensitivity to a parallel
change in interest rates. A first-order hedge would be to offset a risk
position with a hedge security with the same dollar duration. Recall that
dollar duration and present value of a basis point (PV01) measure the
same thing: the change in value for a small change in yield. For this
reason, the hedge can also be described as matching the PV01.
Use the following table to hedge a long position (owning) the 5.125% of
November 30, 1998 by shorting (selling) the 6% of May 31, 1998:
Par
Coupon
(%)
Maturity
Price Accrued PV
(%)
(%)
(%)
Modified
Dollar
PV
Duration
Duration
(%)
A bond is a hedge if
its dollar-duration
exposure offsets
that of another
bond or portfolio
Deviation of the
value of the hedge
from the value of
the hedged asset is
called basis risk; the
quality of a hedge
is often measured
by its basis risk
Long
100.000
5.125
11/30/98 97-037
6.000
5/31/98 99-14+
Short
89
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Hedging (Continued)
U.S. Treasury Prices from Tuesday, June 25, 1996 Pack
A single-bond
hedge is exposed to
the yield
relationship of the
two securities
If the yield curve
has a non-parallel
shift, any difference
in the durations of
the securities may
cause a change in
the yield
relationship
Hedges often age
differently than the
underlying position
and require
rebalancing (for
example, the
durations of two
bonds decline at
different rates as
time passes)
Par
Coupon
(%)
Maturity
Price Accrued
(%)
(%)
PV
(%)
Modified Dollar
PV
Duration
Duration
(%)
Long
100.000
5.125
11/30/98
97-037
0.364
97.485
2.233
217.710
6.000
5/31/98
99-14+
0.426
99.879
1.787
217.710
Short
121.998
The hedge security has a duration almost half a year shorter than the
underlying position. Therefore, hedging a long position in the 51/8% bond
would require selling a greater par amount of the 6% bond. If the yield
curve steepens, the hedge security should “rally” more. Since this
strategy is short the hedge security, the hedge would lose money against
the underlying position.
The hedge is also affected by the passage of time. If yields remain
unchanged, the dollar duration of the underlying position would be
211.058 in one month. The dollar duration of the hedge would be
209.088. The hedge’s “efficiency” with respect to a parallel shift in
interest rates would decline from 100% to 99%.
90
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
More Complex Hedges
U.S. Treasury Prices from Tuesday, June 25, 1996 Pack
Suppose you own the 5.125% of November 30, 1998. You want to hedge
using the current 2-year and 3-year, such that the trade is proceedsneutral. The duration (and dollar duration) of the long and short
positions will cancel out. The long—single-security—position is called
a bullet (because its cash flows are more concentrated), while the short—
combination—position is called a barbell (because its cash flows are
more dispersed).
Weight the trade:
Par
Coupon (%)
Maturity
Price (%)
PV (%)
Modified PV
Duration
Long
100.000
5.125
11/30/98
97-037
6.000
5/31/98
99-14+
6.375
5/15/99
99-222
Short
A butterfly is a
proceeds- and
duration-neutral
three-bond trade
where a bond is
hedged with both a
longer-duration and
a shorter-duration
bond
A butterfly has
much less basis risk
than a hedge with
only one bond
Butterfly analysis
can also provide a
good methodology
for intra-day
Treasury repricing
Q1: Which side of the trade has higher convexity?
Q2: A common measure (discussed in Chapter 5) for estimating the
internal rate of return for a portfolio is dollar-duration-weighted yield.
In an upward-sloping yield curve, which side of this trade has a higher
dollar-duration-weighted yield?
91
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
More Complex Hedges (Continued)
U.S. Treasury Prices from Tuesday, June 25, 1996 Pack
A barbell always
has higher
convexity than a
bullet because it has
more dispersion of
cash flows
(Example: a 5-year
zero vs. cash and a
10-year zero)
In an upwardsloping yield curve,
a barbell usually
has higher dollarduration-weighted
yield because the
higher yield of the
long leg gets more
weight
The barbell portfolio is constructed so that it has the same proceeds and
dollar duration as the bullet security:
Proceeds
Dollar Duration
ParBarbell-Short ´ PVBarbell-Short
ParBarbell-Short ´ PVBarbell-Short ´ DBarbell-Short
+ ParBarbell-Long ´ PVBarbell-Long
+ ParBarbell-Long ´ PVBarbell-Long ´ DBarbell-Long
= ParBullet
= ParBullet
´ PVBullet
´ PVBullet
´ DBullet
Solving for the par amounts:
ParBarbell-Short =
ParBarbell-Long =
Par
(
ParBullet ´ PVBullet ´ DBarbell-Long - DBullet
(
PVBarbell-Short ´ DBarbell-Long - DBarbell-Short
)
)
ParBullet ´ PVBullet ´ (DBullet - DBarbell-Short )
(
PVBarbell-Long ´ DBarbell-Long - DBarbell-Short
)
Coupon (%)
Maturity
Price (%)
PV (%)
Modified PV
Duration
5.125
11/30/98
97-037
97.485
2.233
6.000
6.375
5/31/98
5/15/99
99-14+
99-222
99.879
100.423
1.787
2.579
Long
100.000
Short
42.621
54.685
This trade will roughly break even if the yield change on the long
position equals the dollar-duration-weighted yield change on the short
position. Because the barbell is approximately replicating the bullet, the
change in value of the barbell should approximate the change in value of
the bullet. This suggests a useful algorithm for estimating intra-day offthe-run Treasury prices when the market has not shifted too dramatically.
92
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
The Fitted Curve
U.S. Treasury Prices from Tuesday, June 25, 1996 Pack
The fitted yield curve is a smoothed discount (zero-coupon) curve for the The fitted yield
entire Treasury market; the curve can be used to price coupon bond cash curve is an
flows, thus producing a fitted yield and a fitted price for coupon bonds. internally consistent
Some relative-value trading strategies compare actual market yields to
the fitted curve.
curve that, in
aggregate, does the
“best” job of pricing
individual Treasury
securities
Actual bond prices
can be compared to
their fitted prices to
determine if the
bonds are richer or
cheaper than
“average”
93
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Valuing Treasuries on the Fitted Curve
U.S. Treasury Prices from Tuesday, June 25, 1996 Pack
A fitted-curve
analysis can
identify bonds with
cheaper or richer
cash flows for
investment
purposes
However, a fittedcurve analysis
provides limited
insight into shortterm trading
phenomena: cheap
bonds can cheapen,
and rich bonds can
richen
Another phase of
the analysis would
be a comparison of
richness or
cheapness to
historical levels
This graph depicts what happens when we subtract fitted yield from
actual yield.
If actual yield is higher than fitted, we have a “cheap” security. If actual
yield is lower than fitted, we have a “rich” one.
Before deciding whether to buy or sell, we must also look at the
historical trading ranges.
94
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Constructing a Fitted Zero Curve
Advanced
The objective of the curve-fitting is to find a function for zero-coupon
yields (or, equivalently, prices) for pricing individual cash flows. Pricing
each cash flow of a bond independently and summing them provides the
bond’s fitted present value. A reasonable measure to gauge the success
of the fit is the total of the squared pricing errors of the bonds (the
difference between the bond’s actual and fitted price, multiplied by the
amount outstanding).
A fitted zero curve
contains all the
information
necessary to build a
fitted par curve, but
is more efficient to
compute
One method for
Define the function f(t) to be the fitted present value of a zero-coupon constructing a fitted
bond of term t. Then the problem would be to choose f(t) to minimize zero curve is to
the total squared error E:
estimate a curve
E=
(Number of Cash Flows )i
æ
öù
êOutstandingi ´ ç PVi Cash Flowsij ´ f Termij ÷ ú
å
ç
÷ú
ê
j =1
è
øû
ë
Number of Bonds é
å
i =1
(
2
)
The following two pages define a methodology for fitting f(t) as the
exponential of a cubic spline. The Treasury fitted curve in the prior
pages was built using this technique on a five-segment spline with knot
points at ¾, three, six, and 12 years. The spline has seven independent
parameters, which must be chosen using a finicky general optimizer,
such as the variable metric method.1 Experts agree that when you need
a general optimizer, call an expert.
that minimizes the
total squared
difference between
each Treasury
bond’s value in the
market and its
value according to
the curve
The same technique can be used to fit corporate, mortgage, and other
types of curves.
1 William Press et al., Numerical Recipes in C, 2nd ed. New York: Cambridge University
Press, 1995.
95
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Constructing a Fitted Zero Curve
(Continued)
Advanced
By taking the log of
the fitted price
function, we get a
nearly linear
relationship that is
easier to fit
Most curve-fitting methodologies do better the closer the actual function
is to a straight line. For zero-coupon bonds, price is much more linear
than yield, and log of price more linear still. Consequently, this approach
constructs a concise model for the log of price. Note that if a function
cannot be linearized, it helps to look for an estimating function shaped
like the data.
A cubic spline is a common fitting function. It has different segments,
and each segment has its own cubic (third degree) polynomial. The
polynomials are constrained so that, at each intersection of two segments
(a knot point), both functions have the same value, slope, and curvature
(the resulting function is continuous and twice differentiable
everywhere). The cubic spline has reasonable flexibility and appears
smooth.
96
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Constructing a Fitted Zero Curve
(Continued)
Advanced
Mathematically, define the function for segment k for time tk#t#tk+1 to be
gk(t)=ak+bkt+ckt2+dkt3. As an end condition, t0=0. Then if the
unconstrained variables are defined to be b0 , c0 and dk (with a0/0 so that
f(0)=eg(0)=e0=100%), then ak , bk , and ck are defined as follows for k > 0:
g k¢¢(t k ) = g k¢¢-1 (t k )
gk¢ (t k ) = gk¢- 1 (t k )
gk (tk ) = gk -1 (tk )
ß
ß
ß
ck + 3 d k t k
bk + 2ck t k + 3d k t k2
ak + bk tk + ck tk2 + dk tk3
= ck - 1 + 3 d k - 1 t k
= bk - 1 + 2ck - 1t k + 3d k - 1t k2
= ak -1 + bk -1tk + ck -1tk2 + dk -1tk3
ß
ß
ß
ck = ck -1 + 3t k (d k -1 - d k )
ak = ak -1 + tk3 (dk -1 - dk )
bk = bk - 1 - 3t k2 (d k - 1 - d k )
A common function
for fitting a curve is
the cubic spline, a
“smooth” set of
cubic polynomials
Choosing the spline
parameters to
minimize error
requires a general
optimizer (more
complicated than
ordinary least
squares)
The function for gk (t) and the fitted-price curve fk (t) are, therefore,
defined piece-wise in terms of the unconstrained variables for
t k £ t £ t k + 1 as
k
é
ù
gk (t ) = å t ´ (dm - 1 - dm ) + êb0 - 3å tm2 ´ (dm - 1 - dm )ú ´ t +
m=1
m=1
û
ë
k
3
m
k
ù 2
é
3
êc0 + 3å t m ´ (dm-1 - dm )ú ´ t + d k ´ t
m=1
û
ë
or, alternatively,
k
(
)
g k (t ) = b0 ´ t + c0 ´ t 2 + å t m3 - 3t m2 t + 3t m t 2 ´ (dm-1 - dm ) + d k ´ t 3
m=1
Now that we have gk(t), we can substitute it in fk(t)=egk(t) to obtain our
fitted discount function.
97
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Constructing a Fitted Par Curve
U.S. Treasury Prices from Tuesday, June 25, 1996 Pack
A fitted par curve is
equivalent to a
fitted zero curve
The fitted par curve
tells what coupon
would equal the
yield-to-maturity
(and thus price the
bond at par) at
each maturity along
the curve
Recall that the present value of a bond with regular coupons can be
expressed as:
c
f
n
Price+Accrued = å
i =0
æ
ö
y
ç 1+ i +1- x ÷
f ø
è
i +1- x
+
100%
æ
ö
y
ç 1+ n+1- x ÷
f ø
è
n+1- x
where x is the length of the accrual period (0#x<1) and yt is the yield
for a t-period zero-coupon bond.
c
After substituting Accrued = x ×
and Price = 100%, this can be
f
solved for c:
æ
100%
ç 100% –
n+1- x
æ
ç
y n + 1- x ö
ç 1+
÷
ç
f
è
ø
c= f ´ç n
ç
1
-x
ç å
i +1- x
i =0 æ
ö
y
ç
ç 1+ i +1- x ÷
ç
f ø
è
è
ö
÷
÷
÷
÷
÷
÷
÷
÷
ø
The U.S. Treasury par-coupon curve provides a good benchmark for
relative-value analysis of corporate or mortgage new issues. For
example, there have been no 7-year bonds issued recently, and all bonds
in the 12- to 25-year range are very old. A par-coupon curve provides a
consistent benchmark for pricing issues in these regions.
98
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 3 Exercises
U.S. Treasury Prices from Tuesday, June 25, 1996 Pack
1. Under what conditions will the STRIPS curve lie above the coupon
curve on a plot of maturity vs. yield (maturity on the x-axis)? When
will it lie below the coupon curve?
2. Estimate the closing price and accrued interest for the UST 6.875%
of July 31, 1999 if its yield-to-maturity falls 10 bp (from 6.548%).
3. Two separate Treasury issues mature on August 15, 1997. Why do
their durations differ?
4. Given the quote sheet price for the UST 6.875% of July 31, 1999
(100-285), calculate the bond’s yield, modified duration, price
duration, Macaulay duration, accrued interest, and the value of an 01
and a 32nd.
5. Is the price of a 2-year fixed-rate bond more or less sensitive to
movements in interest rates than the price of a 2-year floating-rate
bond? Why?
6. If the 11¾% of November 15, 2014 falls in price to 135-00, what is
its yield-to-call for settlement on June 26, 1996?
7. A trader has given you the 5¾% of August 15, 2003 as a benchmark
for a corporate bond. On your Telerate screen, the 5-year is now
trading at 100, the 10-year is now trading at 101, and the trader looks
very busy. How would you estimate the current price of your
benchmark?
8. You sell the 5¾% of August 15, 2003 (at the closing price) and hedge
with the 5-year and the 10-year. The Fed tightens, and the curve
flattens. Do you hang your head in shame or do a victory lap?
99
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
U.S. Treasury Benchmarks
Settlement Date: 6/26/96
1yr +47
5.827
5.505
2yr
+19
6.302
99-14+
Coupons
8/96 5.112
11/96 5.522
2/97
5/97
8/97
11/97
5.762
5.872
5.987
6.157
2/98
5/98
8/98
11/98
6.272
6.327
6.397
6.467
2/99
5/99
8/99
11/99
6.524
6.564
6.594
6.634
2/00 6.662
5/00 6.682
11/00 6.717
2/01
5/01
8/01
11/01
6.722
6.742
6.762
6.782
2/02
5/02
8/02
11/02
6.805
6.825
6.845
6.865
2/03
5/03
8/03
11/03
6.885
6.905
6.920
6.940
Trade Date: 6/25/96
3 yr +19
6.489
99-222
5yr +22
6.717
99-03
10yr
6.935
99-18
+15
7.150
96-20+
Principals
2/04
5/04
8/04
11/04
6.975
7.005
7.025
7.045
2/12
5/12
8/12
11/12
7.364
7.369
7.374
7.379
2/20
5/20
8/20
11/20
7.469
7.469
7.449
7.434
2/05
5/05
8/05
11/05
7.060
7.080
7.085
7.085
2/13
5/13
8/13
11/13
7.394
7.399
7.404
7.409
2/21
5/21
8/21
11/21
7.439
7.434
7.429
7.414
2/06
5/06
8/06
11/06
7.110
7.130
7.140
7.150
2/14
5/14
8/14
11/14
7.414
7.419
7.424
7.429
2/22
5/22
8/22
11/22
7.409
7.404
7.399
7.384
2/99
5/99
8/99
11/99
6.514
6.579
6.629
6.659
2/07
5/07
8/07
11/07
7.160
7.170
7.180
7.190
2/15
5/15
8/15
11/15
7.434
7.439
7.444
7.449
2/23
5/23
8/23
11/23
7.364
7.349
7.334
7.314
2/00
5/00
8/00
11/00
6.687
6.697
6.732
6.742
2/08
5/08
8/08
11/08
7.205
7.215
7.225
7.235
2/16
5/16
8/16
11/16
7.454
7.459
7.459
7.464
2/24
5/24
8/24
11/24
7.294
7.284
7.274
7.264
2/01
5/01
8/01
11/01
6.757
6.772
6.797
6.817
2/09
5/09
8/09
11/09
7.245
7.255
7.265
7.275
2/17
5/17
8/17
11/17
7.469
7.474
7.474
7.474
2/25 7.224
8/25 7.104
5/02
8/02
2/03
8/03
6.836
6.851
6.891
6.931
2/10
5/10
8/10
11/10
7.289
7.299
7.309
7.319
2/18 7.474
5/18 7.474
11/18 7.474
2/04
5/04
8/04
11/04
6.961
7.001
7.011
7.060
2/11
5/11
8/11
11/11
7.329
7.339
7.344
7.354
2/05
5/05
8/05
11/05
7.026
7.096
7.106
7.015
2/19
5/19
8/19
11/19
2/26 6.964
7.474
7.474
7.469
7.469
11/96 5.532
5/97 5.882
8/97 6.002
11/97 6.162
2/98 6.297
5/98 6.327
11/98 6.472
-6 30yr
7.089
86-18+
2/06 7.016
5/06 6.990
11/09
2/15
8/15
11/15
7.360
7.404
7.419
7.429
2/16 7.444
5/16 7.414
11/16 7.434
8/17 7.464
5/18 7.464
11/18 7.464
2/19 7.464
8/19 7.454
2/20 7.459
5/20 7.459
8/20 7.459
2/21
5/21
8/21
11/21
7.429
7.429
7.424
7.414
8/22
11/22
2/23
8/23
7.384
7.374
7.344
7.299
11/24 7.249
2/25 7.184
8/25 7.064
2/26 6.899
101
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
U.S. Treasury Benchmarks
Settlement Date: 6/26/96
Trade Date: 6/25/96
Maturity Ds/Yld
*90DY* BILL
WI
*180DY* BILL
*WI*
BILL
Cusip
Tic–2
Tic–1 Ds/Yld Tic+1 Tic+2
5.095
09/26/96 5.234 9127943H5
5.075
5.213
5.085
5.224
5.095
5.234
5.105
5.244
5.115
5.255
5.225
12/26/96 5.441
9127943T9
5.205
5.420
5.215
5.431
5.225
5.441
5.235
5.452
5.245
5.463
5.505
06/26/97 5.827 9127942R4
5.485
5.804
5.495
5.815
5.505
5.827
5.515
5.838
5.525
5.849
Coupon Maturity Pr/Yld
32nd/Cusip Size/Dur Tic–2 Tic–1
Pr/Yld Tic+1 Tic+2
FB
*2YR*
6.000
05/31/98
99-14+
6.302
(0.0175)
912827X98
1.85
10+
6.372
12+
6.337
99-14+
6.302
16+
18+
6.267 6.232
FB
*3YR*
6.375
05/15/99
99-222
6.489
(0.0121)
912827X72
19011
2.67
14+
6.583
18+
6.534
99-22+
6.486
26+
30+
6.438 6.390
5.500
04/15/00
96-082
6.626
(0.0096)
912827K43
9761
3.44
0+
6.701
4+
6.662
96-08+
6.624
12+
16+
6.585 6.547
6.500
05/31/01
99-03
6.717
(0.0076)
912827Y22
4.28
27
6.778
31
6.748
99-03
6.717
7
11
6.687 6.657
F
*10YR* 6.875
05/15/06
99-18
6.935
(0.0044)
912827X80
7.27
10
6.971
14
6.953
99-18
6.935
22
26
6.918 6.900
FB
*30YR* 6.000
02/15/26
86-18+
(0.0028)
7.089 912810EW4
12.82
10+
7.112
14+
7.100
86-18+
7.089
22+
26+
7.077 7.066
B
F
*5YR*
FED FUNDS 5.250
102
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
U.S. Treasury Notes and Bonds
(0–3 Years)
Settlement Date: 6/26/96
Coupon Maturity Pr/Yld
B
Trade Date: 6/25/96
32nd/Cusip Size/Dur Tic–2 Tic–1
Pr/Yld Tic+1 Tic+2
100-01
4.843
7.875
06/30/96
100-01
4.843
(2.7381)
912827B43
9250
0.01
100
7.583
0+
6.213
7.875
07/15/96
100-045
(0.5788)
4.925 912827XT4
7250
0.05
3+
5.577
4
100-04+
5
5+
5.287 4.998 4.708 4.419
7.875
07/31/96
100-082
(0.3156)
5.021 912827B76
9250
0.10
7+
5.258
8
100-08+
9
9+
5.100 4.942 4.784 4.627
6.125
07/31/96
100-031
(0.3183)
4.981 912827Q54
17304
0.10
2
5.340
2+
5.180
100-03
5.021
3+
4
4.862 4.703
4.375
08/15/96
99-282
5.152
(0.2257)
912827L75
15782
0.14
27+
5.322
28
5.209
99-28+
5.096
29
29+
4.983 4.870
6.250
08/31/96
100-033
(0.1723)
5.545 912827Q96
17257
0.18
2+
5.696
3
100-03+
4
4+
5.610 5.523 5.437 5.351
7.250
08/31/96
100-09
5.538
(0.1715)
912827C34
9250
0.18
8
5.709
8+
5.623
100-09
5.538
9+
10
5.452 5.366
7.000
09/30/96
100-136
(0.1183)
5.252 912827C59
9250
0.27
13
5.341
13+
5.281
100-14
5.222
14+
15
5.163 5.104
6.500
09/30/96
100-092
(0.1186)
5.301 912827R38
17267
0.27
8+
5.390
9
100-09+ 10
10+
5.330 5.271 5.212 5.153
8.000
10/15/96
100-243
(0.1023)
5.363 912827YB2
7500
0.31
23+
5.453
24 100-24+ 25
25+
5.402 5.350 5.299 5.248
6.875
10/31/96
100-15
5.433
(0.0908)
912827C83
17271
0.35
14
5.524
14+
5.479
7.250
11/15/96
100-20+
(0.0815)
5.509 912827UF7
20258
0.39
19+
5.590
20 100-20+ 21
21+
5.550 5.509 5.468 5.427
4.375
11/15/96
99-176
5.526
(0.0826)
912827M74
17008
0.39
17
5.588
17+
5.547
11/30/96
100-112
(0.0740)
5.635 912827D41
9000
0.43
10+
5.690
11
100-11+
12
12+
5.653 5.616 5.579 5.542
7.250
11/30/96
100-21+
(0.0737)
5.617 912827R95
17316
0.43
20+
5.691
21 100-21+ 22
22+
5.654 5.617 5.580 5.543
7.500
12/31/96
100-307
(0.0622)
5.557
912827S37
17300
0.50
29
5.673
30
5.611
100-31
5.549
101
1
5.487 5.424
6.125
12/31/96
100-091
(0.0627)
5.549 912827D66
9000
0.50
7
5.683
8
5.620
100-09
5.557
10
11
5.495 5.432
B
B
B
6.500
B
100-15
5.433
99-18
5.505
1+
2
3.474 2.105
15+
16
5.388 5.343
18+
19
5.464 5.423
103
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
U.S. Treasury Notes and Bonds
(0–3 Years) (Continued)
Settlement Date: 6/26/96
Coupon Maturity Pr/Yld
Trade Date: 6/25/96
32nd/Cusip Size/Dur Tic–2 Tic–1
Ds/Yld Tic+1 Tic+2
8.000
01/15/97
101-085
(0.0575)
5.623 912827YK2
7500
0.54
6+
5.745
7+ 101-08+ 9+
10+
5.688 5.630 5.573 5.516
6.250
01/31/97
100-116
(0.0537)
5.604 912827D90
9250
0.58
10
5.698
11
5.644
7.500
01/31/97
101-022
(0.0534)
5.635
912827S52
17257
0.58
0+
5.729
1+ 101-02+ 3+
4+
5.675 5.622 5.568 5.515
4.750
02/15/97
99-122
5.739
(0.0508)
912827N73
17008
0.63
10+
5.828
11+
5.777
6.750
02/28/97
100-213
(0.0471)
5.717 912827E57
9750
0.67
19+
5.805
20+ 100-21+ 22+
23+
5.758 5.711 5.664 5.617
6.875
02/28/97
100-241
(0.0471)
5.711
912827S94
17251
0.67
22
5.811
23
5.764
100-24
5.717
25
26
5.670 5.623
6.625
03/31/97
100-196
(0.0421)
5.770 912827T36
17251
0.75
18
5.844
19
5.801
100-20
5.759
21
22
5.717 5.675
6.875
03/31/97
100-257
(0.0421)
5.759 912827E73
10250
0.75
24
5.838
25
5.796
100-26
5.754
27
28
5.712 5.670
8.500
04/15/97
102-027
(0.0396)
5.781 912827YT3
7500
0.79
1
5.856
2
5.816
102-03
5.776
4
5
5.737 5.697
6.875
04/30/97
100-275
(0.0381)
5.801
912827F23
10250
0.83
25+
5.882
26+ 100-27+ 28+
29+
5.843 5.805 5.767 5.729
6.500
04/30/97
100-177
(0.0382)
5.801 912827T51
17751
0.83
16
5.873
17
5.834
8.500
05/15/97
102-083
(0.0360)
5.831 912827UW0
9921
0.87
6+
5.898
7+ 102-08+ 9+
10+
5.862 5.826 5.790 5.754
6.500
05/15/97
100-177
(0.0365)
5.834
912827P71
17000
0.87
16
5.903
17
5.866
100-18
5.830
19
20
5.793 5.757
6.750
05/31/97
100-247
(0.0348)
5.871
912827F64
10300
0.92
23
5.937
24
5.902
100-25
5.867
26
27
5.832 5.797
6.125
05/31/97
100-067
(0.0350)
5.878 912827T93
17750
0.92
5
5.944
6
5.909
100-07
5.874
8
9
5.839 5.804
5.625
06/30/97
99-235
5.895
(0.0323)
912827U34
17753
0.97
19+
6.028
21+
5.963
99-23+
5.899
25+
27+
5.834 5.769
6.375
06/30/97
100-151
(0.0322)
5.886
912827F80
10517
0.97
11
6.018
13
5.954
100-15
5.890
17
19
5.825 5.761
B
B
B
100-12
5.590
99-12+
5.726
100-18
5.796
13
14
5.536 5.483
13+
14+
5.676 5.625
19
20
5.758 5.720
104
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
U.S. Treasury Notes and Bonds
(0–3 Years) (Continued)
Settlement Date: 6/26/96
Coupon Maturity Pr/Yld
Trade Date: 6/25/96
32nd/Cusip Size/Dur Tic–2 Tic–1
Pr/Yld Tic+1 Tic+2
8.500
07/15/97
102-19
5.915
(0.0304)
912827ZB1
8000
1.00
15
6.037
17
5.976
102-19
5.915
21
23
5.854 5.794
5.875
07/31/97
99-295
5.940
(0.0298)
912827U59
17754
1.06
25+
6.063
27+
6.003
99-29+
5.943
31+
1+
5.884 5.824
5.500
07/31/97
99-171
5.938
(0.0299)
912827G30
10506
1.06
13
6.062
15
6.002
99-17
5.942
19
21
5.882 5.822
8.625
08/15/97
102-276
(0.0282)
5.968 912827VE9
9358
1.08
24
6.074
26
6.018
102-28
5.961
30
103
5.905 5.849
6.500
08/15/97
100-171
(0.0287)
5.997 912827Q70
17010
1.09
13
6.116
15
6.058
100-17
6.001
19
21
5.944 5.886
5.625
08/31/97
99-17+
6.020
(0.0279)
912827G71
10588
1.14
13+
6.132
15+
6.076
99-17+
6.020
19+
21+
5.964 5.909
6.000
08/31/97
99-303
6.036
(0.0278)
912827U91
17794
1.14
26+
6.144
28+
6.088
99-30+
6.033
0+
2+
5.977 5.921
5.500
09/30/97
99-111
6.036
(0.0262)
912827G97
10514
1.23
7
6.144
9
6.092
99-11
6.039
13
15
5.987 5.935
5.750
09/30/97
99-20
6.054
(0.0261)
912827V33
17752
1.22
16
6.159
18
6.106
99-20
6.054
22
24
6.002 5.950
8.750
10/15/97
103-103
(0.0247)
6.047 912827ZK1
8503
1.25
6+
6.142
8+ 103-10+ 12+
14+
6.093 6.044 5.994 5.945
5.750
10/31/97
99-19+
6.050
(0.0246)
912827H47
10753
1.31
15+
6.148
17+
6.099
99-19+
6.050
21+
23+
6.000 5.951
5.625
10/31/97
99-133
6.075
(0.0246)
912827V58
1.31
9+
6.171
11+
6.121
99-13+
6.072
15+
17+
6.023 5.974
7.375
11/15/97
101-19+
(0.0236)
6.138 912827R79
17158
1.34
15+
6.233
17+ 101-19+ 21+
23+
6.186 6.138 6.091 6.044
11/15/97
103-196
(0.0232)
6.104 912827VN9
9800
1.33
16
6.192
18
6.145
103-20
6.099
22
24
6.052 6.006
6.000
11/30/97
99-26+
6.123
(0.0232)
912827H88
10750
1.39
22+
6.216
24+
6.170
99-26+
6.123
28+
30+
6.077 6.031
5.375
11/30/97
98-306
6.142
(0.0233)
912827V90
18250
1.39
27
6.230
29
6.183
98-31
6.136
1
3
6.089 6.043
5.250
12/31/97
98-23
6.150
(0.0222)
912827W32
18254
1.44
19
6.239
21
6.195
98-23
6.150
25
27
6.106 6.062
B
B
B
B
8.875
B
B
FO
105
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
U.S. Treasury Notes and Bonds
(0–3 Years) (Continued)
Settlement Date: 6/26/96
Coupon Maturity Pr/Yld
32nd/Cusip Size/Dur Tic–2 Tic–1
Pr/Yld Tic+1 Tic+2
6.000
12/31/97
99-255
6.139
(0.0220)
912827J29
10540
1.43
21+
6.230
23+
6.186
7.875
01/15/98
102-13+
(0.0211)
6.210 912827ZT2
9126
1.45
9+
6.295
11+ 102-13+ 15+
17+
6.253 6.210 6.168 6.126
5.000
01/31/98
98-04+
(0.0211)
6.238 912827W57
19086
1.53
0+
6.322
2+
6.280
98-04+
6.238
6+
8+
6.196 6.154
5.625
01/31/98
99-033
6.218
(0.0210)
912827J45
11507
1.52
31+
6.299
1+
6.257
99-03+
6.215
5+
7+
6.173 6.131
7.250
02/15/98
101-17
6.244
(0.0201)
912827S78
17123
1.54
13
6.325
15
6.285
101-17
6.244
19
21
6.204 6.164
8.125
02/15/98
102-28
(0.0200)
6.242 912827VW9
9151
1.53
24
6.322
26
6.282
102-28
6.242
30
103
6.203 6.163
5.125
02/28/98
98-065
6.261
(0.0201)
912827J94
11686
1.61
2+
6.344
4+
6.303
98-06+
6.263
8+
10+
6.223 6.183
5.125
03/31/98
98-03+
6.269
(0.0192)
912827K35
11008
1.69
31+
6.346
1+
6.307
98-03+
6.269
5+
7+
6.230 6.192
6.125
03/31/98
99-24+
6.262
(0.0190)
912827X31
18250
1.68
20+
6.338
22+
6.300
99-24+
6.262
26+
28+
6.224 6.186
7.875
04/15/98
102-212
(0.0183)
6.281 912827A44
8530
1.70
17+
6.349
19+ 102-21+ 23+
25+
6.313 6.276 6.240 6.203
5.125
04/30/98
98-006
6.271
(0.0184)
912827K68
11024
1.77
29
6.340
31
6.303
98-01
6.267
3
5
6.230 6.193
5.875
04/30/98
99-092
6.283
(0.0183)
912827X56
18777
1.76
5+
6.352
7+
6.315
99-09+
6.279
11+
13+
6.242 6.206
6.125
05/15/98
99-22+
6.290
(0.0178)
912827T77
1.80
18+
6.361
20+
6.325
99-22+
6.290
24+
26+
6.254 6.218
05/15/98
104-241
(0.0173)
6.281 912827WE8
8750
1.77
20
6.352
22
6.317
104-24
6.283
26
28
6.248 6.214
6.000
05/31/98
99-14+
6.302
(0.0175)
912827X98
1.85
10+
6.372
12+
6.337
99-14+
6.302
16+
18+
6.267 6.232
5.375
05/31/98
98-11
6.297
(0.0176)
912827L26
11034
1.85
7
6.368
9
6.333
98-11
6.297
13
15
6.262 6.227
5.125
06/30/98
97-251
6.315
(0.0170)
912827L42
11007
1.89
21
6.385
23
6.351
97-25
6.317
27
29
6.283 6.249
FB
B
B
B
B
FO
F
9.000
FB
*2YR*
Trade Date: 6/25/96
B
99-25+
6.142
27+
29+
6.098 6.054
106
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
U.S. Treasury Notes and Bonds
(0–3 Years) (Continued)
Settlement Date: 6/26/96
Coupon Maturity Pr/Yld
Trade Date: 6/25/96
32nd/Cusip Size/Dur Tic–2 Tic–1
Pr/Yld Tic+1 Tic+2
99-28+
6.309
WI F O
*2YRWI* 6.250
06/30/98
99-28+
6.310
(0.0169)
912827Y30
1.91
24+
6.377
26+
6.343
8.250
07/15/98
103-195
(0.0161)
6.341 912827B50
9000
1.87
15+
6.408
17+ 103-19+ 21+
23+
6.376 6.343 6.311 6.279
5.250
07/31/98
97-286
6.333
(0.0164)
912827L67
11023
1.97
25
6.394
27
6.362
97-29
6.329
31
1
6.296 6.264
5.875
08/15/98
99-01
6.362
(0.0160)
912827U75
18003
2.00
29
6.426
31
6.394
99-01
6.362
3
5
6.330 6.298
9.250
08/15/98
105-217
(0.0153)
6.357 912827WN8
11326
1.94
18
6.416
20
6.385
105-22
6.355
24
26
6.324 6.294
4.750
08/31/98
96-236
6.370
(0.0159)
912827M25
11000
2.07
20
6.430
22
6.398
96-24
6.366
26
28
6.335 6.303
4.750
09/30/98
96-196
6.375
(0.0154)
912827M41
11015
2.15
16
6.433
18
6.402
96-20
6.371
22
24
6.340 6.310
7.125
10/15/98
101-192
(0.0146)
6.360 912827C67
9280
2.14
15+
6.415
17+ 101-19+ 21+
23+
6.386 6.357 6.327 6.298
4.750
10/31/98
96-14
6.407
(0.0149)
912827M66
11013
2.23
10
6.466
12
6.436
96-14
6.407
16
18
6.377 6.347
5.500
11/15/98
97-312
6.425
(0.0145)
912827V74
2.26
27+
6.479
29+
6.450
97-31+
6.421
1+
3+
6.392 6.363
8.875
11/15/98
105-113
(0.0139)
B 9893
6.413 912827WW8 2.19
7+
6.467
9+ 105-11+ 13+
15+
6.439 6.411 6.383 6.356
5.125
11/30/98
97-037
6.422
(0.0144)
912827N24
11023
2.31
97
6.477
2
6.449
97-04
6.420
6
8
6.391 6.363
5.125
12/31/98
97-007
6.425
(0.0139)
912827N40
11042
2.33
29
6.479
31
6.452
97-01
6.424
3
5
6.396 6.368
01/15/99
99-273
6.435
(0.0135)
912827D74
9507
2.33
23+
6.488
25+
6.461
99-27+
6.434
29+
31+
6.407 6.380
5.000
01/31/99
96-182
6.453
(0.0135)
912827N65
12029
2.42
14+
6.504
16+
6.477
96-18+
6.450
20+
22+
6.423 6.396
5.000
02/15/99
96-16
6.461
(0.0134)
912827W73
2.46
8
6.568
12
6.515
96-16
6.461
20
24
6.408 6.355
8.875
02/15/99
105-242
(0.0126)
6.460 912827XE7
9702
2.35
20+
6.507
22+ 105-24+ 26+
28+
6.482 6.457 6.431 6.406
B
B
B
B
B
6.375
B
F
30+
0+
6.275 6.242
107
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
U.S. Treasury Notes and Bonds
(0–3 Years) (Continued)
Settlement Date: 6/26/96
Coupon Maturity Pr/Yld
B
32nd/Cusip Size/Dur Tic–2 Tic–1
Pr/Yld Tic+1 Tic+2
5.500
02/28/99
97-206
6.466
(0.0131)
912827P22
11021
2.49
17
6.515
19
6.489
97-21
6.463
23
25
6.436 6.410
5.875
03/31/99
98-156
6.475
(0.0126)
912827P48
11003
2.56
12
6.523
14
6.497
98-16
6.472
18
20
6.447 6.422
7.000
04/15/99
101-093
(0.0123)
6.483 912827E81
9750
2.57
1+
6.580
5+ 101-09+ 13+
17+
6.530 6.481 6.432 6.383
6.500
04/30/99
100-00+
(0.0122)
6.490
912827P63
11004
2.62
24+
6.587
28+ 100-00+ 4+
8+
6.538 6.490 6.441 6.392
9.125
05/15/99
106-24
6.516
(0.0116)
912827XN7
10030
2.59
16
6.609
20
6.563
106-24
6.516
28
107
6.470 6.424
6.375
05/15/99
99-222
6.489
(0.0121)
912827X72
19011
2.67
14+
6.583
18+
6.534
99-22+
6.486
26+
30+
6.438 6.390
6.750
05/31/99
100-192
(0.0118)
6.519 912827Q21
11000
2.70
11+
6.610
15+ 100-19+ 23+
27+
6.563 6.516 6.468 6.421
6.750
06/30/99
100-196
(0.0116)
6.520 912827Q47
11000
2.69
12
6.610
16
6.564
100-20
6.518
24
28
6.471 6.425
6.375
07/15/99
(0.0115)
912827F98
9750
2.75
12
6.603
16
6.557
99-20
6.511
24
28
6.465 6.419
6.875
07/31/99
100-285
(0.0112)
6.548 912827Q62
11014
2.77
20+
6.640
24+ 100-28+ 0+
4+
6.594 6.549 6.504 6.460
B
FB
*3YR*
Trade Date: 6/25/96
B
99-20
6.511
108
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
U.S. Treasury Notes and Bonds
(3–5 Years)
Settlement Date: 6/26/96
Coupon Maturity Pr/Yld
Trade Date: 6/25/96
32nd/Cusip Size/Dur Tic–2 Tic–1
Pr/Yld Tic+1 Tic+2
F
*2YR*
5.875
04/30/98
99-092
6.283
(0.0183)
912827X56
18777
1.76
5+
6.352
7+
6.315
99-09+
6.279
11+
13+
6.242 6.206
FB
*3YR*
6.375
05/15/99
99-222
6.489
(0.0121)
912827X72
19011
2.67
14+
6.583
18+
6.534
99-22+
6.486
26+
30+
6.438 6.390
5.000
02/15/99
96-16
6.461
(0.0134)
912827W73
2.46
8
6.568
12
6.515
96-16
6.461
20
24
6.408 6.355
7.000
04/15/99
101-093
(0.0123)
6.483 912827E81
9750
2.57
1+
6.580
5+ 101-09+ 13+
17+
6.530 6.481 6.432 6.383
6.500
04/30/99
100-00+
(0.0122)
6.490
912827P63
11004
2.62
24+
6.587
28+ 100-00+ 4+
8+
6.538 6.490 6.441 6.392
9.125
05/15/99
106-24
6.516
(0.0116)
912827XN7
10030
2.59
16
6.609
20
6.563
6.750
05/31/99
100-192
(0.0118)
6.519 912827Q21
11000
2.70
11+
6.610
15+ 100-19+ 23+
27+
6.563 6.516 6.468 6.421
6.750
06/30/99
100-196
(0.0116)
6.520 912827Q47
11000
2.69
12
6.610
16
6.564
100-20
6.518
24
28
6.471 6.425
6.375
07/15/99
(0.0115)
912827F98
9750
2.75
12
6.603
16
6.557
99-20
6.511
24
28
6.465 6.419
6.875
07/31/99
100-285
(0.0112)
6.548 912827Q62
11014
2.77
20+
6.640
24+ 100-28+ 0+
4+
6.594 6.549 6.504 6.460
8.000
08/15/99
104-00+
(0.0109)
6.558 912827XW7
10163
2.77
24+
6.646
28+ 104-00+ 4+
8+
6.602 6.558 6.515 6.471
6.875
08/31/99
100-27
6.572
(0.0110)
912827R20
11012
2.85
19
6.660
23
6.616
100-27
6.572
31
3
6.528 6.484
7.125
09/30/99
101-196
(0.0107)
6.561 912827R46
11009
2.93
12
6.644
16
6.601
101-20
6.559
24
28
6.516 6.473
(0.0108)
912827H21
9754
3.01
6
6.616
10
6.573
98-14
6.530
18
22
6.487 6.444
11019
3.00
13
6.681
17
6.639
102-21
6.598
25
29
6.556 6.515
F
B
B
B
6.000
10/15/99
B
99-20
6.511
98-14
6.530
106-24
6.516
28
107
6.470 6.424
7.500
10/31/99
102-206
(0.0104)
6.600 912827R61
7.875
11/15/99
103-256
(0.0102)
6.599 912827YE6
10771
3.02
18
6.678
22
6.637
103-26
6.596
30
2
6.556 6.515
7.750
11/30/99
103-146
(0.0101)
6.603
912827S29
11000
3.07
7
6.682
11
6.641
103-15
6.601
19
23
6.560 6.520
109
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
U.S. Treasury Notes and Bonds
(3–5 Years) (Continued)
Settlement Date: 6/26/96
Coupon Maturity Pr/Yld
Trade Date: 6/25/96
32nd/Cusip Size/Dur Tic–2 Tic–1
Pr/Yld Tic+1 Tic+2
7.750
12/31/99
103-17
6.606
(0.0099)
912827S45
11000
3.04
9
6.686
13
6.646
103-17
6.606
21
25
6.567 6.527
6.375
01/15/00
99-086
6.606
(0.0101)
912827J37
9752
3.15
1
6.684
5
6.644
99-09
6.604
13
17
6.564 6.523
7.750
01/31/00
103-17+
(0.0097)
6.622
912827S60
11000
3.12
9+
6.700
13+ 103-17+ 21+
25+
6.661 6.622 6.584 6.545
8.500
02/15/00
105-306
(0.0095)
6.625 912827YN6
10012
3.13
23
6.698
27
6.660
105-31
6.622
3
7
6.584 6.547
7.125
02/29/00
101-176
(0.0096)
6.638 912827T28
11001
3.24
10
6.712
14
6.674
101-18
6.635
22
26
6.597 6.558
6.875
03/31/00
100-242
(0.0095)
6.640 912827T44
11000
3.33
16+
6.713
20+ 100-24+ 28+
0+
6.675 6.637 6.600 6.562
5.500
04/15/00
96-082
6.626
(0.0096)
912827K43
9761
3.44
0+
6.701
4+
6.662
96-08+
6.624
12+
16+
6.585 6.547
6.750
04/30/00
100-096
(0.0093)
6.655 912827T69
11500
3.42
2
6.728
6
6.690
100-10
6.653
14
18
6.616 6.579
8.875
05/15/00
107-176
(0.0088)
6.634 912827YW6
10503
3.37
10
6.703
14
6.667
107-18
6.632
22
26
6.596 6.561
6.250
05/31/00
98-196
6.654
(0.0092)
912827U26
11502
3.53
12
6.726
16
6.689
98-20
6.652
24
28
6.615 6.578
5.875
06/30/00
97-086
6.661
(0.0092)
912827U42
11505
3.52
1
6.732
5
6.695
97-09
6.658
13
17
6.622 6.585
6.125
07/31/00
98-016
6.673
(0.0089)
912827U67
11501
3.59
26
6.743
30
6.707
98-02
6.671
6
10
6.635 6.600
8.750
08/15/00
107-14+
(0.0084)
6.656 912827ZE5
10503
3.49
6+
6.723
10+ 107-14+ 18+
22+
6.689 6.656 6.622 6.589
08/31/00
98-142
6.679
(0.0088)
912827V25
11922
3.67
6+
6.747
10+
6.712
98-14+
6.677
18+
22+
6.642 6.607
6.125
09/30/00
97-31
6.677
(0.0086)
912827V41
11500
3.76
23
6.746
27
6.712
97-31
6.677
3
7
6.642 6.608
5.750
10/31/00
96-162
6.687
(0.0086)
912827V66
12081
3.87
8+
6.753
12+
6.719
96-16+
6.685
20+
24+
6.650 6.616
8.500
11/15/00
106-256
(0.0080)
6.681 912827ZN5
11000
3.75
18
6.743
22
6.711
106-26
6.679
30
2
6.647 6.615
B
B
B
6.250
B
110
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
U.S. Treasury Notes and Bonds
(3–5 Years) (Continued)
Settlement Date: 6/26/96
Coupon Maturity Pr/Yld
Trade Date: 6/25/96
32nd/Cusip Size/Dur Tic–2 Tic–1
Pr/Yld Tic+1 Tic+2
5.625
11/30/00
95-30+
(0.0085)
6.694 912827W24
12000
3.96
22+
6.762
26+
6.728
95-30+
6.694
2+
6+
6.661 6.627
5.500
12/31/00
95-14
6.688
12821
3.94
6
6.755
10
6.722
95-14
6.688
18
22
6.655 6.621
5.250
01/31/01
94-16+
(0.0083)
6.653 912827W65
4.04
8+
6.719
12+
6.686
94-16+
6.653
20
24
6.620 6.587
7.750
02/15/01
104-04+
(0.0078)
6.693 912827ZX3
11000
3.90
28+
6.756
0+ 104-04+ 8+
12+
6.724 6.693 6.662 6.631
5.625
02/28/01
95-23
6.703
(0.0081)
912827X23
4.09
15
6.768
19
6.736
95-23
6.703
27
31
6.671 6.639
6.375
03/31/01
98-196
6.717
(0.0078)
912827X49
12006
4.13
12
6.777
16
6.746
98-20
6.715
24
28
6.683 6.652
6.250
04/30/01
98-03
6.715
(0.0077)
912827X64
15
4.21
27
6.777
31
6.746
7
6.715
11
15
6.684 6.653
F
*5YR*
6.500
05/31/01
99-03
6.717
(0.0076)
912827Y22
15
4.28
27
6.778
31
6.748
7
6.717
11
15
6.687 6.657
WI FBO
*WI*
6.750
06/30/01
100-06
6.705
(0.0075)
912827Y48
18
4.32
30
6.765
2
6.735
10
6.705
14
18
6.675 6.646
5.875
11/15/05
92-233
6.941
(0.0048)
912827V82
13500
7.21
15+
6.979
19+
6.959
92-23+
6.940
27+
31+
6.921 6.902
02/15/26
86-18+
(0.0028)
7.089 912810EW4
12.82
10+
7.112
14+
7.100
86-18+
7.089
22+
26+
7.077 7.066
FBO
F
F
FBO
F
FB
*30YR*
6.000
(0.0084)
912827W40
111
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
U.S. Treasury Notes and Bonds
(5–15 Years)
Settlement Date: 6/26/96
Coupon Maturity Pr/Yld
Trade Date: 6/25/96
32nd/Cusip Size/Dur Tic–2 Tic–1
Pr/Yld Tic+1 Tic+2
FB
*2YR*
6.000
05/31/98
99-14+
6.302
(0.0175)
912827X98
1.85
10+
6.372
12+
6.337
99-14+
6.302
16+
18+
6.267 6.232
FB
*3YR*
6.375
05/15/99
99-222
6.489
(0.0121)
912827X72
19011
2.67
14+
6.583
18+
6.534
99-22+
6.486
26+
30+
6.438 6.390
6.875
08/31/99
100-27
6.572
(0.0110)
912827R20
11012
2.85
19
6.660
23
6.616
100-27
6.572
31
3
6.528 6.484
7.125
09/30/99
101-196
(0.0107)
6.561 912827R46
11009
2.93
12
6.644
16
6.601
101-20
6.559
24
28
6.516 6.473
7.500
10/31/99
102-206
(0.0104)
6.600 912827R61
11019
3.00
13
6.681
17
6.639
102-21
6.598
25
29
6.556 6.515
7.750
11/30/99
103-146
(0.0101)
6.603
912827S29
11000
3.07
7
6.682
11
6.641
103-15
6.601
19
23
6.560 6.520
6.375
01/15/00
99-086
6.606
(0.0101)
912827J37
9752
3.15
1
6.684
5
6.644
99-09
6.604
13
17
6.564 6.523
7.750
01/31/00
103-17+
(0.0097)
6.622
912827S60
11000
3.12
9+
6.700
13+ 103-17+ 21+
25+
6.661 6.622 6.584 6.545
8.500
02/15/00
105-306
(0.0095)
6.625 912827YN6
10012
3.13
23
6.698
27
6.660
105-31
6.622
3
7
6.584 6.547
7.125
02/29/00
101-176
(0.0096)
6.638 912827T28
11001
3.24
10
6.712
14
6.674
101-18
6.635
22
26
6.597 6.558
6.875
03/31/00
100-242
(0.0095)
6.640 912827T44
11000
3.33
16+
6.713
20+ 100-24+ 28+
0+
6.675 6.637 6.600 6.562
5.500
04/15/00
96-082
6.626
(0.0096)
912827K43
9761
3.44
0+
6.701
4+
6.662
96-08+
6.624
12+
16+
6.585 6.547
6.750
04/30/00
100-096
(0.0093)
6.655 912827T69
11500
3.42
2
6.728
6
6.690
100-10
6.653
14
18
6.616 6.579
05/15/00
107-176
(0.0088)
6.634 912827YW6
10503
3.37
10
6.703
14
6.667
107-18
6.632
22
26
6.596 6.561
6.250
05/31/00
98-196
6.654
(0.0092)
912827U26
11502
3.53
12
6.726
16
6.689
98-20
6.652
24
28
6.615 6.578
5.875
06/30/00
97-086
6.661
(0.0092)
912827U42
11505
3.52
1
6.732
5
6.695
97-09
6.658
13
17
6.622 6.585
6.125
07/31/00
98-016
6.673
(0.0089)
912827U67
11501
3.59
26
6.743
30
6.707
98-02
6.671
6
10
6.635 6.600
B
B
B
B
8.875
112
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
U.S. Treasury Notes and Bonds
(5–15 Years) (Continued)
Settlement Date: 6/26/96
Coupon Maturity Pr/Yld
B
Pr/Yld Tic+1 Tic+2
11500
3.76
23
6.746
27
6.712
97-31
6.677
3
7
6.642 6.608
11/15/00
106-256
(0.0080)
6.681 912827ZN5
11000
3.75
18
6.743
22
6.711
106-26
6.679
30
2
6.647 6.615
5.625
11/30/00
95-30+
(0.0085)
6.694 912827W24
12000
3.96
22+
6.762
26+
6.728
95-30+
6.694
2+
6+
6.661 6.627
5.250
01/31/01
94-16+
(0.0083)
6.653 912827W65
4.04
8+
6.719
12+
6.686
94-16+
6.653
20+
24+
6.620 6.587
11.750
02/15/01
119-29
6.682
(0.0071)
912810CT3
1500
3.67
21
6.739
25
6.711
119-29
6.682
1
5
6.654 6.626
7.750
02/15/01
104-04+
(0.0078)
6.693 912827ZX3
11000
3.90
28+
6.756
0+ 104-04+ 8+
12+
6.724 6.693 6.662 6.631
19
6.736
95-23
6.703
27
31
6.671 6.639
09/30/00
8.500
F
F
5.625
02/28/01
95-23
6.703
6.375
03/31/01
98-196
6.717
(0.0078)
912827X49
12006
4.13
12
6.777
16
6.746
98-20
6.715
24
28
6.683 6.652
6.250
04/30/01
98-03
6.715
(0.0077)
912827X64
4.21
27
6.777
31
6.746
98-03
6.715
7
11
6.684 6.653
13.125
05/15/01
126-145
(0.0066)
6.684 912810CU0
1800
3.86
6+
6.737
10+ 126-14+ 18+
22+
6.711 6.685 6.659 6.632
8.000
05/15/01
105-096
(0.0074)
6.706 912827A85
11750
4.13
2
6.763
6
6.734
105-10
6.704
14
18
6.675 6.645
6.500
05/31/01
(0.0076)
912827Y22
4.28
27
6.778
31
6.748
99-03
6.717
7
11
6.687 6.657
13.375
08/15/01
128-185
(0.0062)
6.702 912810CW6
1800
3.90
10+
6.752
14+ 128-18+ 22+
26+
6.728 6.703 6.678 6.653
08/15/01
104-293
(0.0071)
6.724 912827B92
12000
4.24
21+
6.780
25+ 104-29+ 1+
5+
6.751 6.723 6.695 6.667
7.500
11/15/01
103-105
(0.0069)
6.749 912827D25
23000
4.51
2+
6.805
6+
6.778
103-10
6.750
14+
18+
6.723 6.695
15.750
11/15/01
140-12
6.693
(0.0056)
912810CX4
1800
4.04
4
6.738
8
6.715
140-12
6.693
16
20
6.670 6.648
14.250
02/15/02
134-21
6.751
(0.0056)
912810CZ9
1800
4.14
13
6.796
17
6.774
134-21
6.751
25
29
6.729 6.707
FBO
F
F
*5YR*
32nd/Cusip Size/Dur Tic–2 Tic–1
(0.0086)
912827V41
6.125
97-31
6.677
Trade Date: 6/25/96
7.875
99-03
6.717
(0.0081)
912827X23
4.09
15
6.768
113
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
U.S. Treasury Notes and Bonds
(5–15 Years) (Continued)
Settlement Date: 6/26/96
Coupon Maturity Pr/Yld
Trade Date: 6/25/96
32nd/Cusip Size/Dur Tic–2 Tic–1
Pr/Yld Tic+1 Tic+2
7.500
05/15/02
103-17+
(0.0064)
6.757
912827F49
11500
4.85
9+
6.808
13+ 103-17+ 21+
25+
6.783 6.757 6.732 6.706
6.375
08/15/02
97-31+
6.780
(0.0064)
912827G55
22337
5.05
23+
6.831
27+
6.805
97-31+
6.780
3+
7+
6.754 6.729
11.625
11/15/02
124-196
(0.0053)
6.804 912810DA3
2800
4.83
12
6.846
16
6.824
124-20
6.803
24
28
6.782 6.761
10.750
02/15/03
120-202
(0.0053)
6.829 912810DC9
3000
4.93
12+
6.869
16+ 120-20+ 24+
28+
6.848 6.827 6.806 6.785
6.250
02/15/03
97-00+
6.814
(0.0060)
912827J78
21519
5.39
24+
6.863
28+
6.839
97-00+
6.814
4+
8+
6.790 6.766
10.750
05/15/03
121-037
(0.0051)
6.849 912810DD7
3250
5.18
28
6.889
121
6.868
121-04
6.848
8
12
6.828 6.807
11.125
08/15/03
123-24
6.858
(0.0049)
912810DE5
3500
5.17
16
6.897
20
6.877
123-24
6.858
28
124
6.838 6.819
5.750
08/15/03
93-282
6.846
(0.0058)
912827L83
23099
5.78
20+
6.891
24+
6.868
93-28+
6.845
0+
4+
6.821 6.798
11.875
11/15/03
128-166
(0.0046)
6.881 912810DG0
3500
5.36
9
6.917
13
6.899
128-17
6.880
21
25
6.861 6.843
5.875
02/15/04
(0.0055)
912827N81
12001
6.08
28
6.919
94
6.897
94-04
6.875
8
12
6.853 6.831
12009
6.12
28+
6.933
0+ 102-04+ 8+
12+
6.912 6.892 6.871 6.851
B
B
B
B
B
94-04
6.875
7.250
05/15/04
102-042
(0.0051)
6.893
912827P89
12.375
05/15/04
132-233
(0.0043)
6.917 912810DH8
3750
5.57
15+
6.951
19+ 132-23+ 27+
31+
6.934 6.917 6.899 6.882
13.750
08/15/04
141-301
(0.0040)
6.919 912810DK1
4000
5.46
22
6.952
26
6.936
141-30
6.920
2
6
6.904 6.887
08/15/04
102-031
(0.0050)
6.907 912827Q88
12073
6.16
27
6.948
31
6.927
102-03
6.907
7
11
6.887 6.867
7.875
11/15/04
105-317
(0.0048)
6.919 912827R87
12051
6.31
24
6.957
28
6.937
106-00
6.918
4
8
6.899 6.880
11.625
11/15/04
129-165
(0.0042)
6.923 912810DM7
8301
5.89
8+
6.957
12+ 129-16+ 20+
24+
6.940 6.924 6.907 6.890
7.500
02/15/05
103-213
(0.0048)
6.927
912827S86
12045
6.39
13+
6.964
17+ 103-21+ 25+
29+
6.945 6.926 6.907 6.888
B
B
B
7.250
114
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
U.S. Treasury Notes and Bonds
(5–15 Years) (Continued)
Settlement Date: 6/26/96
Coupon Maturity Pr/Yld
B
Trade Date: 6/25/96
32nd/Cusip Size/Dur Tic–2 Tic–1
Pr/Yld Tic+1 Tic+2
6.500
05/15/05
97-06+
6.925
(0.0049)
912827T85
6.81
30+
6.964
2+
6.945
97-06+
6.925
10+
14+
6.906 6.886
8.250
05/15/05
104-067
(0.0048)
7.587 912810BU1
4200
6.47
31
7.625
3
7.606
104-07
7.587
11
15
7.568 7.549
3.39
r7.061 r7.024
r6.988
r6.952 r6.915
4260
6.09
25+
6.982
29+ 133-01+ 5+
9+
6.966 6.950 6.934 6.919
B
YTC in 00
B
r6.99
12.000
05/15/05
133-015
(0.0040)
6.950 912810DQ8
10.750
08/15/05
125-075
(0.0040)
6.967 912810DR6
9269
6.21
31+
7.000
3+ 125-07+ 11+
15+
6.984 6.968 6.951 6.935
6.500
08/15/05
97-021
6.937
(0.0048)
912827U83
13010
6.83
26
6.976
30
6.957
97-02
6.938
6
10
6.919 6.900
5.875
11/15/05
92-233
6.941
(0.0048)
912827V82
13500
7.21
15+
6.979
19+
6.959
92-23+
6.940
27+
31+
6.921 6.902
5.625
02/15/06
91-09+
(0.0047)
6.873 912827W81
7.30
1+
6.912
5+
6.892
91-09+
6.873
13+
17+
6.854 6.836
F
*10YR* 6.875
05/15/06
99-18
6.935
(0.0044)
912827X80
7.27
10
6.971
14
6.953
99-18
6.935
22
26
6.918 6.900
*20YR* 9.375
02/15/06
117-00+
(0.0041)
6.924 912810DU9
4755
6.62
24+
6.956
28+ 117-00+ 4+
8+
6.940 6.924 6.908 6.891
FB
*30YR* 6.000
02/15/26
86-18+
(0.0028)
7.089 912810EW4
12.82
10+
7.112
14+
7.100
F
86-18+
7.089
22+
26+
7.077 7.066
115
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
U.S. Treasury Notes and Bonds
(Long)
Settlement Date: 6/26/96
Coupon Maturity Pr/Yld
Trade Date: 6/25/96
32nd/Cusip Size/Dur Tic–2 Tic–1
Pr/Yld Tic+1 Tic+2
FB
*2YR*
6.000
05/31/98
99-14+
6.302
(0.0175)
912827X98
1.85
10+
6.372
12+
6.337
99-14+
6.302
16+
18+
6.267 6.232
F
*5YR*
6.500
05/31/01
99-03
6.717
(0.0076)
912827Y22
4.28
27
6.778
31
6.748
99-03
6.717
7
11
6.687 6.657
12073
6.16
27
6.948
31
6.927
102-03
6.907
7
11
6.887 6.867
106-00
6.918
4
8
6.899 6.880
B
7.250
08/15/04
102-031
(0.0050)
6.907 912827Q88
7.875
11/15/04
105-317
(0.0048)
6.919 912827R87
12051
6.31
24
6.957
28
6.937
11.625
11/15/04
129-165
(0.0042)
6.923 912810DM7
8301
5.89
8+
6.957
12+ 129-16+ 20+
24+
6.940 6.924 6.907 6.890
7.500
02/15/05
103-213
(0.0048)
6.927
912827S86
12045
6.39
13+
6.964
17+ 103-21+ 25+
29+
6.945 6.926 6.907 6.888
05/15/05
104-067
(0.0048)
7.587 912810BU1
4200
6.47
31
7.625
3
7.606
104-07
7.587
11
15
7.568 7.549
3.39
r7.061 r7.024
r6.988
r6.952 r6.915
B
8.250
YTC in 00
B
r6.99
12.000
05/15/05
133-015
(0.0040)
6.950 912810DQ8
4260
6.09
25+
6.982
29+ 133-01+ 5+
9+
6.966 6.950 6.934 6.919
10.750
08/15/05
125-075
(0.0040)
6.967 912810DR6
9269
6.21
31+
7.000
3+ 125-07+ 11+
15+
6.984 6.968 6.951 6.935
6.500
05/15/05
97-06+
6.925
(0.0049)
912827T85
6.81
30+
6.964
2+
6.945
97-06+
6.925
10+
14+
6.906 6.886
6.500
08/15/05
97-021
6.937
(0.0048)
912827U83
13010
6.83
26
6.976
30
6.957
97-02
6.938
6
10
6.919 6.900
5.875
11/15/05
92-233
6.941
(0.0048)
912827V82
13500
7.21
15+
6.979
19+
6.959
92-23+
6.940
27+
31+
6.921 6.902
F
*10YR* 6.875
05/15/06
99-18
6.935
(0.0044)
912827X80
7.27
10
6.971
14
6.953
99-18
6.935
22
26
6.918 6.900
7.625
02/15/07
102-21
7.260
(0.0042)
912810BX5
4200
7.33
13
7.294
17
7.277
102-21
7.260
25
29
7.244 7.227
4.58
r7.097 r7.070
r7.043
r7.016 r6.990
1500
7.76
25+
7.247
5.14
r6.930 r6.907
B
*YTC in 02
7.875
11/15/07
*YTC in 02
r7.04
105-01+
(0.0039)
7.216 912810BZ0
r6.88
29+ 105-01+ 5+
9+
7.232 7.216 7.200 7.184
r6.883
r6.812 r6.835
116
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
U.S. Treasury Notes and Bonds
(Long) (Continued)
Settlement Date: 6/26/96
Coupon Maturity Pr/Yld
8.375
08/15/08
*YTC in 03
B
8.750
11/15/08
*YTC in 03
9.125
05/15/09
*YTC in 04
B
10.375
11/15/09
*YTC in 04
11.750
02/15/10
*YTC in 05
B
10.000
05/15/10
*YTC in 05
12.750
11/15/10
YTC in 05
B
13.875
05/15/11
*YTC in 06
14.000
11/15/11
*YTC in 06
10.375
11/15/12
*YTC in 07
12.000
08/15/13
*YTC in 08
108-01
7.361
Trade Date: 6/25/96
32nd/Cusip Size/Dur Tic–2 Tic–1
Pr/Yld Tic+1 Tic+2
(0.0037)
912810CC0
r6.93
109-20
7.539
(0.0037)
912810CE6
r7.05
112-10+
(0.0035)
7.606 912810CG1
r7.06
120-24+
(0.0033)
7.840 912810CK2
r7.05
129-06
(0.0032)
8.160 912810CM8
r7.16
119-04
7.728
(0.0033)
912810CP1
r7.06
138-18+
(0.0029)
8.146 912810CS5
r7.06
147-26+
(0.0028)
8.238 912810CV8
r7.07
150-13+
(0.0027)
8.178 912810CY2
r7.06
124-28+
(0.0029)
7.678 912810DB1
r7.14
138-31
7.829
2100
7.81
25
7.391
29
7.376
108-01
7.361
5
9
7.346 7.331
5.44
r6.970 r6.949
r6.927
r6.906 r6.884
5200
7.95
12
7.569
16
7.554
109-20
7.539
24
28
7.524 7.510
5.64
r7.095 r7.074
r7.053
r7.033 r6.912
4600
8.06
2+
7.634
5.87
r7.096 r7.077
4200
7.98
16+
7.867
6.00
r7.086 r7.068
r7.050
r7.033 r6.915
5900
7.63
30
8.186
2
8.173
129-06
8.160
10
14
8.148 8.135
5.85
r7.191 r7.174
r7.158
r7.141 r7.125
3000
8.23
28
7.754
119
7.741
119-04
7.728
8
12
7.715 7.702
6.29
r7.087 r7.080
r7.063
r7.046 r7.029
4700
7.92
10+
8.169
6.24
r7.087 r7.072
4600
7.92
18+
8.260
6.36
r7.094 r7.081
4500
8.06
5+
8.199
6.56
r7.090 r7.077
11200
8.95
7.36
(0.0026)
912810DF2
r7.14
15300
8.68
7.29
20+
7.701
6+ 112-10+ 14+
18+
7.620 7.606 7.591 7.577
r7.057
r7.038 r7.019
20+ 120-24+ 28+
0+
7.853 7.840 7.827 7.813
14+ 138-18+ 22+
26+
8.158 8.146 8.134 8.122
r7.067
r7.042 r7.027
22+ 147-26+ 30+
2+
8.249 8.238 8.227 8.216
r7.067
r7.053 r7.040
9+ 150-13+ 17+
21+
8.189 8.178 8.168 8.157
r7.064
r7.051 r7.038
24+ 124-28+ 0+
4+
7.689 7.678 7.666 7.655
r7.170 r7.156
r7.142
r7.128 r7.114
23
7.850
27
7.839
138-31
7.829
3
7
7.818 7.808
r7.168 r7.156
r7.143
r7.130 r7.118
117
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
U.S. Treasury Notes and Bonds
(Long) (Continued)
Settlement Date: 6/26/96
Coupon Maturity Pr/Yld
13.250
05/15/14
*YTC in 09
12.500
08/15/14
*YTC in 09
B
11.750
11/15/14
*YTC in 09
B
150-23
7.903
Trade Date: 6/25/96
32nd/Cusip Size/Dur Tic–2 Tic–1
Pr/Yld Tic+1 Tic+2
(0.0024)
912810DJ4
r7.16
144-30+
(0.0024)
7.819 912810DL9
r7.16
139-12+
(0.0025)
7.706 912810DN5
r7.13
4750
8.88
15
7.922
19
7.913
150-23
7.903
27
31
7.894 7.884
7.58
r7.179 r7.168
r7.156
r7.145 r7.134
4750
8.87
22+
7.838
7.59
r7.184 r7.172
6006
9.26
4+
7.726
7.94
r7.155 r7.144
26+ 144-30+ 2+
6+
7.828 7.819 7.809 7.799
r7.161
r7.149 r7.138
8+ 139-12+ 16+
20+
7.716 7.706 7.696 7.686
r7.132
r7.120 r7.109
11.250
02/15/15
141-09+
(0.0024)
7.191 912810DP0
10.625
08/15/15
135-05+
7.207
(0.0024)
912810DS
7149
9.63
29+
7.226
1+ 135-05+ 9+
13+
7.217 7.207 7.197 7.188
9.875
11/15/15
127-16
7.217
(0.0025)
912810DS
6899
10.01
8
7.237
12
7.227
9.250
02/15/16
121-00+
(0.0026)
7.228 912810DV7
7266
10.00
24+
7.249
28+ 121-00+ 4+
8+
7.239 7.228 7.218 7.208
7.250
05/15/16
100-04+
(0.0030)
7.235 912810DW5
18823
10.75
28+
7.259
0+ 100-04+ 8+
12+
7.247 7.235 7.224 7.212
7.500
11/15/16
102-22+
(0.0029)
7.243 912810DX3
18864
10.79
14+
7.266
18+ 102-22+ 26+
30+
7.255 7.243 7.232 7.220
8.750
05/15/17
116-02
7.245
(0.0026)
912810DY1
18117
10.58
26
7.266
30
7.255
8.875
08/15/17
117-15+
(0.0026)
7.245 912810DZ8
14000
10.42
7+
7.266
11+ 117-15+ 19+
23+
7.255 7.245 7.235 7.224
9.125
05/15/18
120-13+
(0.0025)
7.249 912810EA2
8500
10.72
5+
7.269
9+ 120-13+ 17+
21+
7.259 7.249 7.239 7.229
9.000
11/15/18
119-06+
(0.0025)
7.252 912810EB0
9000
10.85
30+
7.271
2+ 119-06+ 10+
14+
7.261 7.252 7.242 7.232
8.875
02/15/19
117-28
7.254
(0.0025)
912810EC8
19000
10.73
20
7.274
24
7.264
109-20+
(0.0026)
7.258 912810ED6
19750
11.00
12+
7.279
16+ 109-20+ 24+
28+
7.268 7.258 7.247 7.237
B
B
B
8.125
08/15/19
12667
9.41
1+
7.210
5+ 141-09+ 13+
17+
7.200 7.191 7.182 7.172
127-16
7.217
116-02
7.245
117-28
7.254
20
24
7.207 7.197
6
10
7.234 7.224
118
4
7.244 7.234
118
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
U.S. Treasury Notes and Bonds
(Long) (Continued)
Settlement Date: 6/26/96
Coupon Maturity Pr/Yld
B
Trade Date: 6/25/96
32nd/Cusip Size/Dur Tic–2 Tic–1
Pr/Yld Tic+1 Tic+2
8.500
02/15/20
113-29+
(0.0025)
7.258 912810EE4
10000
11.00
21+
7.278
25+ 113-29+ 1+
5+
7.268 7.258 7.248 7.238
8.750
05/15/20
116-26
7.257
(0.0025)
912810EF1
10000
11.20
18
7.277
22
7.267
116-26
7.257
30
2
7.247 7.237
8.750
08/15/20
116-28
7.257
(0.0024)
912810EG9
21000
11.04
20
7.277
24
7.267
116-28
7.257
117
4
7.247 7.238
7.875
02/15/21
107-01+
(0.0026)
7.256 912810EH7
11000
11.34
25+
7.277
29+ 107-01+ 5+
9+
7.266 7.256 7.245 7.235
8.125
05/15/21
109-29+
(0.0025)
7.257
912810EJ3
11750
11.53
21+
7.277
25+ 109-29+ 1+
5+
7.267 7.257 7.247 7.237
8.125
08/15/21
109-31+
(0.0025)
7.254 912810EK0
12000
11.36
23+
7.275
27+ 109-31+ 3+
7+
7.265 7.254 7.244 7.234
8.000
11/15/21
108-19+
(0.0025)
7.252 912810EL8
32000
11.65
11+
7.272
15+ 108-19+ 23+
27+
7.262 7.252 7.242 7.232
7.250
08/15/22
100-00+
(0.0027)
7.248 912810EM6
10000
11.77
24+
7.269
28+ 100-00+ 4+
8+
7.258 7.248 7.237 7.226
7.625
11/15/22
104-14+
(0.0026)
7.243 912810EN4
10298
11.91
6+
7.264
10+ 104-14+ 18+
22+
7.254 7.243 7.233 7.223
7.125
02/15/23
98-21+
7.237
(0.0027)
912810EP9
17590
11.89
13+
7.259
17+
7.248
98-21+
7.237
25+
29+
7.226 7.216
6.250
08/15/23
88-14+
7.226
(0.0029)
912810EQ7
22053
12.27
6+
7.249
10+
7.237
88-14+
7.226
18+
22+
7.214 7.202
7.500
11/15/24
103-16
7.208
(0.0025)
912810ES3
11000
12.27
8
7.228
12
7.218
103-16
7.208
20
24
7.198 7.188
7.625
02/15/25
105-07+
(0.0025)
7.190 912810ET1
11017
12.07
31+
7.210
3+ 105-07+ 11+
15+
7.200 7.190 7.180 7.170
08/15/25
96-19+
7.152
11500
12.38
11+
7.174
15+
7.163
96-19+
7.152
23+
27+
7.142 7.131
02/15/26
86-18+
(0.0028)
7.089 912810EW4
12.82
10+
7.112
14+
7.100
86-18+
7.089
22+
26+
7.077 7.066
B
B
B
B
6.875
FB
*30YR* 6.000
(0.0026)
912810EV6
119
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
U.S. Treasury Bills
Settlement Date: 6/26/96
Trade Date: 6/25/96
Maturity
Ds/Yld
Cusip
Tic–2
Tic–1
Ds/Yld
Tic+1
Tic+2
BILL
06/27/96
4.870
4.952
912794Z56
4.850
4.931
4.860
4.942
4.870
4.952
4.880
4.962
4.890
4.972
BILL
07/05/96
4.690
4.774
9127942Y9
4.670
4.753
4.680
4.764
4.690
4.774
4.700
4.784
4.710
4.794
BILL
07/11/96
4.710
4.798
9127942Z6
4.690
4.778
4.700
4.788
4.710
4.798
4.720
4.808
4.730
4.818
BILL
07/18/96
4.550
4.639
9127943A0
4.530
4.618
4.540
4.629
4.550
4.639
4.560
4.649
4.570
4.659
BILL
07/25/96
4.810
4.909
912794Z64
4.790
4.889
4.800
4.899
4.810
4.909
4.820
4.919
4.830
4.930
BILL
08/01/96
4.870
4.975
9127943B8
4.850
4.955
4.860
4.965
4.870
4.975
4.880
4.986
4.890
4.996
BILL
08/08/96
4.980
5.093
9127943C6
4.960
5.073
4.970
5.083
4.980
5.093
4.990
5.104
5.000
5.114
BILL
08/15/96
4.985
5.103
9127943D4
4.965
5.083
4.975
5.093
4.985
5.103
4.995
5.114
5.005
5.124
BILL
08/22/96
5.030
5.155
912794Z72
5.010
5.134
5.020
5.145
5.030
5.155
5.040
5.165
5.050
5.176
BILL
08/29/96
5.040
5.156
9127943E2
5.020
5.136
5.030
5.146
5.040
5.156
5.050
5.167
5.060
5.177
BILL
09/05/96
5.105
5.229
9127943F9
5.085
5.208
5.095
5.218
5.105
5.229
5.115
5.239
5.125
5.249
BILL
09/12/96
5.115
5.244
9127943G7
5.095
5.223
5.105
5.234
5.115
5.244
5.125
5.255
5.135
5.265
BILL
09/19/96
5.130
5.265
912794Z80
5.110
5.244
5.120
5.255
5.130
5.265
5.140
5.275
5.150
5.286
09/26/96
5.095
5.234
9127943H5
5.075
5.213
5.085
5.224
5.095
5.234
5.105
5.244
5.115
5.255
BILL
10/03/96
5.135
5.281
9127943J1
5.115
5.260
5.125
5.270
5.135
5.281
5.145
5.291
5.155
5.302
BILL
10/10/96
5.150
5.302
9127943K8
5.130
5.281
5.140
5.291
5.150
5.302
5.160
5.312
5.170
5.323
BILL
10/17/96
5.160
5.318
912794Z98
5.140
5.297
5.150
5.307
5.160
5.318
5.170
5.328
5.180
5.339
10/24/96
5.150
5.313
9127943L6
5.130
5.292
5.140
5.302
5.150
5.313
5.160
5.323
5.170
5.334
*90DY* BILL
BILL
120
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
U.S. Treasury Bills (Continued)
Settlement Date: 6/26/96
BILL
BILL
BILL
BILL
BILL
BILL
BILL
BILL
WI
*180DY* BILL
BILL
BILL
BILL
BILL
BILL
*360DY* BILL
WI
BILL
Maturity DS/Yld
5.160
10/31/96 5.329
5.175
11/07/96 5.350
5.215
11/14/96 5.398
5.205
11/21/96 5.393
5.215
11/29/96 5.410
5.250
12/05/96 5.452
5.260
12/12/96 5.468
5.260
12/19/96 5.474
5.225
12/26/96 5.441
5.280
1/09/97
5.501
5.315
2/06/97
5.545
5.365
3/06/97
5.609
5.410
4/03/97
5.671
5.445
5/01/97
5.724
5.475
5/29/97
5.774
5.505
6/26/97
5.827
Trade Date: 6/25/96
Cusip
9127943M4
9127943N2
9127942A1
9127943P7
9127943Q5
9127943R3
9127942B9
9127943S1
9127943T9
9127942K9
9127942L7
9127942N5
9127942N3
9127942P8
9127942Q6
9127942R4
Tic-2
5.140
5.308
5.156
5.329
5.195
5.377
5.185
5.372
5.195
5.388
5.230
5.430
5.240
5.447
5.240
5.452
5.205
5.420
5.260
5.480
5.295
5.523
5.345
5.587
5.390
5.649
5.425
5.702
5.455
5.753
5.485
5.804
Tic-1
5.150
5.318
5.165
5.339
5.205
5.387
5.195
5.382
5.205
5.399
5.240
5.441
5.250
5.457
5.250
5.463
5.215
5.431
5.270
5.491
5.305
5.534
5.355
5.598
5.400
5.660
5.435
5.713
5.465
5.764
5.495
5.815
Ds/Yld
5.160
5.329
5.175
5.350
5.215
5.398
5.205
5.393
5.215
5.410
5.250
5.452
5.260
5.468
5.260
5.474
5.225
5.441
5.280
5.501
5.315
5.545
5.365
5.609
5.410
5.671
5.445
5.724
5.475
5.774
5.505
5.827
Tic+1
5.170
5.339
5.185
5.360
5.225
5.408
5.215
5.403
5.225
5.420
5.260
5.462
5.270
5.479
5.270
5.484
5.235
5.452
5.290
5.512
5.325
5.556
5.375
5.620
5.420
5.681
5.455
5.735
5.485
5.785
5.515
5.838
Tic+2
5.170
5.350
5.195
5.371
5.235
5.419
5.225
5.414
5.235
5.431
5.270
5.473
5.280
5.489
5.280
5.495
5.245
5.463
5.300
5.523
5.335
5.566
5.385
5.630
5.430
5.692
5.465
5.746
5.495
5.796
5.525
5.849
121
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 4
Forward Prices
This chapter is an excerpt from A Morgan Stanley Guide to Fixed Income Analysis by
Andrew R. Young, ©2003 Morgan Stanley & Co. Incorporated.
123
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
In This Chapter, You Will Learn...
•
Repurchase Agreements (Repo)
•
Arbitrage-Free Methodologies
•
How to Price a Bond for Forward Settlement
•
Forward Yields
124
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
What Is a Forward Price?
Prices generally quoted on Treasury securities are for regular settlement. Forward rates and
In regular settlement, the securities are actually transferred and paid for prices implied by
the current interest
on the business day following the trade.
Traders can also quote a price for forward settlement. In a forwardsettled transaction, the buyer maintains use of funds for longer than in a
spot transaction, implying different prices depending on the date of
settlement. Forward prices are an important input into many different
types of valuation models, including options and derivatives.
An incorrect forward price can provide arbitrage, or guaranteed profits,
to one of the participants. Obviously, it is important to recognize
arbitrage opportunities when they appear and to avoid giving those
opportunities to others.
rate environment
are used for
transactions settling
later than regular
settlement
Forward rates are
key inputs into the
valuation of
1) bonds with
embedded options,
2) swaps, and
3) other derivatives
Parties engage in forward settlement when they believe prices are more
favorable than for spot settlement, when they seek leverage, or when the
The cost of
way the transaction is reported would be better for their purposes.
financing or
borrowing bonds
depends on the
short-term rate, or
repo rate
If a trader sells a security for forward settlement, the trader can hedge
(cover) by buying the same security in the regular market, which is more
liquid. The trader will then finance the position for the period between
regular settlement and forward settlement in the repo market.
The forward price the trader will quote incorporates both the coupon We can determine
income the trader will receive and the financing cost the trader will pay the forward price of
during this holding period.
securities from their
Alternatively, if a trader buys a security forward, the trader can hedge by
selling the same security in the regular market, either out of inventory or
short. The trader will have to borrow the bonds in the repo market. The
net cost of borrowing the bonds is a consideration in quoting the forward
price.
price for regular
settlement, the term
of the forward
transaction, and the
repo rate
125
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
A Repo Is an Example of a
Simple-Interest Security
Repo agreements
trade on a simpleinterest basis
The assets traded on a simple-interest basis include repurchase
agreements, Eurodollar deposits, and securities in their last coupon
period.
Simple-interest
securities are priced
without
compounding
Repurchase Agreements
(Repo):
One entity sells securities on a temporary
basis, with an agreement to repurchase the
securities at a later date at a specified price.
The forward price is determined such that
interest is earned at the market-determined
repo rate.
Eurodollar Deposits:
Eurodollar deposits are dollar-denominated
deposits accepted by banks outside the U.S.
These deposits range in maturity from
overnight to as long as five years. The
minimum denomination is $1,000,000, and
rates are usually quoted as a spread to
LIBOR (London Inter-Bank Offered Rate),
the deposit rate offered among leading
international banks.
Securities in Their
Last Coupon Period:
By convention, securities in their last
coupon period are quoted using a simpleinterest yield. The price approximates the
price obtained using the compounding
yield. The formula is (recall that x is the
length of the accrual period):
Other types of
securities also use a
simple-interest yield
v+
PV =
c
f
æ
yö
ç 1 + (1 - x ) ´ ÷
fø
è
v+
@
c
f
æ
yö
ç1+ ÷
fø
è
1- x
126
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Simple-Interest Repo
Assuming No Coupons During the Repo Term
When there are no coupons paid on the security collateralizing the repo,
the repo is simply a loan, repaid with interest. The exact collateral that
was posted initially is returned at term, the maturity of the agreement.
The repayment amount is
rd ö
FV = PV ´ æç 1 +
÷
è
360 ø
where r = repo rate and d = actual days to maturity of the repo.
Basic Repo Agreement Mechanics
(Assumes no marks-to-market or intervening coupons)
Financial institutions
use repo to finance
inventory and
create leverage;
investors use repo
to invest cash in a
Treasurycollateralized
investment at a
positive spread to
Treasuries
Repo agreements
are characterized
by the side taken
by the system;
when the system
borrows money, the
transaction is a
repo, and when the
system lends
money, the
transaction is a
reverse repo
127
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
The Repo Market
The generic or general collateral (GC) market provides investors with
an opportunity to invest at a yield higher than the short-term Treasury
rate and still have a highly creditworthy obligation. General collateral
trades range in term from overnight to several years, with the largest
volumes traded in the overnight to three-month sector. Generally, the
investor would receive 102% of the value of the investment in U.S.
Treasury collateral. There are higher spreads, and sometimes other
It provides a way to
regulatory issues, for other types of collateral. There would usually be
borrow at a low
daily additions and reductions of securities to maintain the market value
rate by using
of the investor’s collateral and, potentially, substitutions of collateral at
inventory as
the whim of the borrower.
collateral, while
The repo market
exists to facilitate
financing Treasury
and other collateral
held by leveraged
institutions
offering investors a
positive spread to
short-term
Treasuries
There is also a specific collateral market, which provides an investor
with a specific security as collateral for the loan. The rate the investor
will receive is always lower than the general collateral rate, and
sometimes it is significantly lower (as low as 0%) for securities that are
in short supply relative to demand. Investors will participate in this
market when they need that specific collateral, often to make delivery on
a short. When the rate for a specific security falls, that security is called
on special. This type of repo is more often for an unspecified (or open)
term and would not be substitutable. Margin calls on special trades are
typically satisfied by exchanges of cash, which effectively change the
loan amount.
Repo has a bid/ask spread. When a repo trader bids, the trader is offering
to take securities as collateral for a loan. The trader will want to earn as
much on the loan as possible. When a repo trader offers, the trader is
offering to borrow money and send out securities as collateral. The
trader will want to pay as low an interest rate as possible on the loan.
The bid-side repo rate is, therefore, higher than the offered-side rate, just
like in the Treasury market.
128
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Term Repo Timeline
The flow of funds for
a repo agreement:
Broker/Dealer
(System)
(Borrower)
Money
Market
Investor
Bonds (Collateral)
1
Treasury
Bond
Now
PriceSpot AccruedSpot
c
2
f
Intervening
Coupons
Additional Collateral
c
Additional Collateral
Bonds
(Collateral Returned)
3
1) The investor loans
money to the
borrower by
investing price plus
accrued today in a
repo agreement; the
investor receives the
bonds as collateral
Issuer
f
Treasury
Bond
Term
FVCoupons
PriceSpot AccruedSpot
At term, the borrower
has the bonds plus the
forward value of the coupons
1
rd 360 Time
At term, the investor
has earned the repo rate
on the investment
At inception, the borrower could purchase the bonds without incurring
any cost by borrowing the proceeds in the repo market.
At term, the borrower will hold the original bonds and the forward value
of the coupons, and will have to repay the loan with interest.
2) the investor
receives the coupons
from the issuer and
forwards them to the
borrower, who can
reinvest them; since,
when a coupon is
paid, the present
value of the collateral
falls and the repo
loan becomes
undersecured, a
typical
“reinvestment”
would be to pay
down the repo and
preserve the
borrower’s security
3) at term, the
borrower pays back
the loan and the
bonds are returned
129
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Arbitrage
Financial theory
holds that if an
arbitrage is
available in the
market, investors
will execute it, and
by so doing,
squeeze out the
arbitrage
An arbitrage is a riskless investment strategy under which one may make
money and is certain not to lose it. More technically, an arbitrage
strategy, including all expenses:
•
is costless to put on initially and has non-negative, and possibly
positive, cash flows (including closing out the transaction) between
now and some definite time in the future, or
•
generates cash when it is put on initially and has non-negative cash
flows (including closing out the transaction) between now and some
definite time in the future.
Example: Buying Treasury bonds and creating and selling STRIPS can
be an arbitrage if the proceeds from the STRIPS are greater than the cost
of the bond because the strategy creates cash and entails no future cash
flows. Another example is 100% non-recourse financing of an asset,
which is arbitrage because it is costless today, has no future negative
cash flows (unless the investor has management responsibility for the
project), and has positive probability of future cash flows.
An arbitrage is a “money pump,” also known as a “free lunch.” But
remember, there is no such thing as a free lunch. At least, not if you
spend too long studying it.
There are many constructs for financial equilibrium that are predicated
on no arbitrage being available in the market. Theoretically, if there
were an arbitrage, investors would buy the cheap asset and sell the rich
asset. The buying of the cheap asset would tend to increase its price, and
the selling of the rich asset would tend to reduce its price. Eventually,
the gap would close, and the arbitrage would vanish. Note that
somebody made some money removing the arbitrage.
130
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Repo and the
Forward Price of U.S. Treasuries
Under arbitrage-free pricing, one should be unable to guarantee profit
(or loss, which would be the counterparty’s guaranteed profit) by buying
a security, financing it in the repo market, and simultaneously agreeing
to sell it at the term of the repo agreement. This is equivalent to
indifference between lending $100 in the repo market at the repo rate and
investing $100 in the collateral security and simultaneously selling it at
the forward price. The following diagram illustrates this relationship:
Investing $100 in
the repo market
must give the same
forward value as
investing $100 in
the collateral
security, holding it,
and then selling it
on the forward date
at the agreed-upon
forward price
If not, an arbitrage
opportunity would
exist, which would
be driven away by
the actions of
market participants
131
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Calculating the Forward Price for STRIPS
U.S. Treasury STRIPS Due November 15, 2021
Using arbitrage-free
pricing, we can
compute the
forward price and
yield of STRIPS
We can solve for the forward price of STRIPS using a special case of the
principle of arbitrage-free pricing: that investing the PriceSpot in the repo
market should provide the same future value as investing PriceSpot in the
securities.
rd ö
æ
PriceSpot ´ ç 1 +
÷ = PriceForward
360 ø
è
Example: Let the spot settlement date be June 26, 1996, YieldSpot for
the November 15, 2021 STRIPS be 7.410%, forward settlement be
March 17, 1997, and the term repo rate be 5%. On spot settlement, the
number of full coupon periods until maturity n is 50, and the partial
period (1 – x) = 142/184
PriceSpot =
100%
æ 7.41% ö
ç1 +
÷
2 ø
è
50 +
142
184
= 15.770%
5% ´ 264 ö
æ
15.770% ´ ç 1 +
÷ = PriceForward = 16 .348%
è
360 ø
The forward yield can be determined according to:
100%
16 .348% =
yf ö
æ
ç1 +
÷
2ø
è
49 + 59
181
Solving for the forward yield gives us: yf = 7.480%
132
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Calculating the
Forward Price for a Treasury
U.S. Treasury 8% Due November 15, 2021
We can solve for the forward price of a coupon security using the Using arbitrage-free
principle of arbitrage-free pricing: investors should be indifferent pricing, we can
compute the
between investing in the repo and the security.
(Price
Spot
rd ö
æ
+ AccruedSpot ´ ç 1 +
÷ = PriceForward + Accrued Forward + FVCoupons
è
360 ø
)
Example: What is the forward price for the UST 8% due November 15,
2021, given: a settlement date of June 26, 1996, the spot price of
108-20 (yielding 7.251%), a forward settlement date of March 17, 1997,
and a term repo rate (r) of 5%.
AccruedSpot =
Days from Last Coupon to Spot Settlement
8% 42
´
=
´ 4% = 0.913%
Days Between Last Coupon and Next Coupon
2
184
Accrued Forward =
Days Accrued on Forward Settlement Date 8% 122
´
=
´ 4% = 2.696%
Days in That Coupon Period
2
181
forward price of a
Treasury bond
Q1: How does the
change in yield for
the coupon bond
compare to the
change in yield for
the STRIPS?
Q2: What would
happen to the
forward yield for
very short-term
securities?
FVCoupons is the November 15, 1996 coupon plus its reinvestment to
the forward settlement date (d´ days):
FVCoupons
=
8% æ
r ´ d ¢ ö 8% æ
5% ´ 122 ö
´ ç1 +
´ ç1 +
÷=
÷ = 4.068%
2 è
360 ø
2 è
360 ø
Note that this is a different investment period and, therefore, a
different money rate, or forward repo rate, could be used in this
formula; however, most market participants just use the repo rate
itself.
Substituting into the formula gives:
(108.625% + 0.913%)´ æçè 1 +
5% ´ 264 ö
÷ = PriceForward + 2.696% + 4.068%
360 ø
PriceForward = 106.791%; yForward = 7.396%
133
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
An Alternative Definition
of the Forward Price
There are two
conventions for
calculating the
forward price
Their results are
almost identical
The remainder of
this material uses
the “traditional”
convention
Traditional convention (same as before):
(Price
Spot
rd ö
c
+ AccruedSpot ´ æç 1 +
÷ = PriceForward + AccruedForward +
è
360 ø
f
)
ri di
å æçè 1 + 360 ö÷ø
i
where di is the number of days between the i th coupon and the forward
date, and ri is the forward repo rate (money rate) for that period. Assume
that ri= r for all i. This formula is easily solved for the forward price.
This formulation sets the repo repayment amount equal to the forward
value of the bond plus the reinvested value of its coupons.
Alternative convention:
PriceSpot + AccruedSpot =
PriceForward + AccruedForward c
+
f
æç 1 + rd ö÷
è
360 ø
1
r (d - di )ö
ç1+
÷
360 ø
è
åæ
i
This is almost the same thing, except that it sets the present value of the
bond equal to the discounted forward value plus the present value of the
coupons. This statement is equivalent to:
(Price
Spot
+ AccruedSpot
rd ö
c
´ æç 1 +
÷ = PriceForward + AccruedForward +
è
ø
360
f
)
æç 1 + rd ö÷
è
ø
åi æ r(d360
- di )ö
ç1+
÷
360 ø
è
The traditional convention and the alternative convention are not
identical. They would be the same if the coupon-reinvestment rates were
compounded; however, since they are simple interest, there is a
difference of up to 15 bp on the effective rate used to forward the
coupons. This would usually affect the forward price by less than
0.01%, or 1/4 of 1/32. The rest of this material uses the traditional
convention.
134
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Estimating the
Forward Yield of a Treasury
Instead of precisely calculating the forward yield, we can use duration to It is often
make a quick estimate of it.
convenient to
If an investor buys a long-term security that yields y and holds it for a
relatively short period of time t when the repo rate is r, the investor could
believe that approximately (y – r) ´ t will be earned above the short-term
investment rate. To try to earn this excess return, the investor bears the
risk of holding the longer security. The potential excess return is a return
on the present value of the initial investment.
If, however, the investor sells the security at inception for forward
settlement at time t, the investor has a guaranteed return. If there is no
arbitrage, however, that guaranteed return would be the repo rate.
Selling at the arbitrage-free (lower) forward price would surrender this
excess return to the market, and the forward yield would be higher than
the spot yield. The price differential, - DPV PV , should approximate the
“excess” return, (y – r) ´ t.
estimate the
forward yield using
the duration of the
security and the
spread between the
bond’s yield and its
repo rate (this
works in any yieldcurve environment)
This estimate is not
a substitute for
actually computing
the forward price
for a transaction
Q: Why can’t you
We can estimate the yield change by using the definition of modified estimate the
duration (all quantities as of the forward settlement date):
forward price by
Dy = -
DPV
PV
DurationModified PV
(y - r )´ t
@
DurationModified PV
subtracting the price
differential from the
current price?
For example, the UST 8% due November 15, 2021 had a yield of 7.251%
and a modified present-value duration of 11.24 for settlement on June
26, 1996. If the repo rate until March 17, 1997 is 5%,
Dy »
(7.251% - 5%)´ 264 365 = 0.145% = 14.5 bp
11.24
which is precisely the same as the actual yield differential. This method
works well even with rough estimates of the inputs.
135
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Market-Expectations
Forward Yields and Prices
The forward yields
and prices we have
just calculated are
arbitrage-free
The forward yields and prices we have just calculated are arbitrage-free.
This means that it is impossible to buy the securities forward at a higher
yield without incurring risk, because this would give the short-term
investor a higher return than the repo rate.
There is an
alternative
definition of
forward prices and
yields that follows
the logic of market
expectations
There is another theory, market expectations, that states that in order for
there to be balance between the spot market and the forward market,
investors as a whole must be indifferent between 1) buying a shorter
security and rolling any cash received until a given term and 2) investing
to that term directly. Of course, individual investors’ views can vary
widely.
Marketexpectations
forward yields are
usually higher than
arbitrage-free
forward yields
(Why?)
Marketexpectations
forward yields are
usually a better
input for models
The theories need not be inconsistent. The bid-ask spread in a farforward price can be significant, leaving room for other theories to try to
pin down the “true” forward price.
For zero-coupon bonds, the rule that must be kept is
y0,m ö
æ
ç1 +
÷
f ø
è
f ´m
y
æ
ö
´ ç 1 + m,m+ n ÷
f ø
è
f ´n
y
æ
ö
= ç 1 + 0,m + n ÷
f ø
è
f ´ (m+ n )
where ya,b is the (forward) yield from time a to b.
The arbitrage-free methodology can distort forward yields due to the
negative economics of investing in Treasuries and borrowing at a spread
to Treasuries. The market-expectations methodology does not distort
forward yields.
Since the Treasury market is highly efficient, we can determine forward
prices and yields for coupon bonds by valuing the individual cash flows
from those bonds at the appropriate forward zero-coupon rates.
136
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
The Forward Curve
U.S. Treasury Coupon Bonds as of June 25, 1996
The shape of
today’s yield curve
determines forward
yields
The forward rates
on this graph are
for the same
maturity on
June 25, 1997
Why does the forward curve lie above the current curve in this case?
When would it lie below the current curve?
Why do shorter yields rise more than longer yields, resulting in a
“flatter” curve?
137
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participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 5
Yield Measurement
and Total Rate
of Return
This chapter is an excerpt from A Morgan Stanley Guide to Fixed Income Analysis by
Andrew R. Young, ©2003 Morgan Stanley & Co. Incorporated.
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
In This Chapter, You Will Learn...
•
Various Methods of Measuring the Return of a Portfolio, Including:
– Yield-to-Maturity
– Internal Rate of Return
– Yield-to-Call
– Yield-to-Worst
– Dollar-Duration-Weighted Yield
– Market-Value-Weighted Yield
– Total Rate of Return
– Current Yield
140
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Measuring Performance
There are many different models for measuring performance. Each has
as its objective the ability to compare different securities or portfolios.
Some of these measures are specifically for fixed-income investments,
while others have more general applicability.
The most basic measure of potential return is current yield, also called
cash-on-cash return. Current yield measures the annual cash flow as a
percent of the amount invested and is the most flawed measure of
performance, except for under some very specific conditions mentioned
later in this chapter.
The most common measure of potential return is yield-to-maturity
(YTM). It is also called internal rate of return (IRR) and only makes
sense for an asset with known cash flows. For callable bonds, there are
several extensions to the concept of YTM that better measure yield.
IRR, but not YTM, is also defined as a measure of potential return for a
portfolio. Weighting the yield of each security in a portfolio by its dollar
duration approximates portfolio IRR. Another common weighting of
yields, market-value weighting, approximates the next year’s income
from the portfolio.
There are many
different
measurements of
asset performance
Some of these
measures are
specific to fixedincome assets
Some measures
apply directly to a
portfolio; others can
be applied indirectly
by weighting the
measures for
individual securities
The broadest measure of performance is total rate of return (ROR). It
uses all the information available to project the cash flows from the
security over a fixed time period, including cash-flow reinvestment.
Total rate of return is applicable to any type of asset (unlike IRR) and
generalizes to provide a measure of performance for a portfolio.
It is important to recognize that a return on a larger base has a more
significant impact than a return on a smaller base. For example, an
investment manager with outstanding performance one year on
$1 million and mediocre performance the next year on $100 million had
mediocre overall impact on client wealth.
141
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Yield-to-Maturity (YTM) and Other Yield
Measures for a Single Security
YTM is a basic, but
flawed, estimate of
total rate of return
for a single security
The only way to
calculate YTM is by
trial and error
There are
extensions of YTM
to better measure
return for a callable
bond
We can calculate the yield-to-maturity of STRIPS by inverting the zerocoupon bond price formula.
Unfortunately, however, there is no way to invert the coupon bond price
formula. Instead, we start by making an initial guess for the yield-tomaturity. Based on the difference between the value of the bond’s cash
flows using that yield and the market value of the bond, we refine the
estimate of the yield. When the estimated price is “close enough” to the
actual price, we stop. This iterative process is the Newton–Raphson
approach discussed in Chapter 2:
yi + 1 = yi +
Pricei - PriceActual
DurationDollar , i
The concept of yield-to-maturity is often extended to better measure return
for callable securities by applying the same techniques to the bond’s cash
flows, assuming the bond is called. Yield-to-call (YTC) thus refers to the
yield that properly prices the cash flows of the bond, assuming that it is
called on its next call date. Often, there is a call premium (a call price
greater than 100%), and this modified redemption value needs to be taken
into account in the calculation of YTC. It is possible to define a yield to
each of a security’s call dates, at the appropriate call price. The lowest of
these yields (including yield-to-maturity) is called the yield-to-worst
(YTW) and is the most realistic of the “yield-to” measures because it
assumes that the issuer will minimize the yield to the investor.
The final measure of yield is called option-adjusted yield (OAY).
It measures the yield that discounts future cash flows to the value of
1) the bond, plus 2) the value of the embedded option (assuming it is a
call option). This produces the cash flows and value of an option-free
bond. OAY is the best measure of yield because it adjusts for the
economics of random interest rates; however, computing it requires an
option model. OAY always lies below YTW (and YTM).
142
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Weighting Yields in a Portfolio
Like a bond, a portfolio also has an internal rate of return (although it
does not have a yield). The IRR is the rate that discounts all the future
cash flows to the actual value of the portfolio. Like yield, there are
different measures for internal rate of return. The usual definition
discounts the cash flows of the bonds if they remain outstanding until
maturity. However, it is possible to define an IRR-to-worst by using
each security’s cash flows to worst (the cash flows that, when discounted
by the yield-to-worst, produce the actual market value of the bond). It is
also possible to define an option-adjusted IRR.
A common approximation of IRR is to take the dollar-duration-weighted
yield of the individual securities. Because the yield is dollar-weighted,
securities with larger market values have a larger effect on the estimate
of the IRR. Because it is duration-weighted, securities that will be held
in the portfolio for a longer time also have a larger effect on the estimate.
The duration should match up with the yield: yield-to-maturity with
duration, yield-to-worst with duration-to-worst, and option-adjusted
yield with option-adjusted duration. The total dollar duration of a
position is Par ´ PV ´ Duration .
å Par ´ PV ´ Duration ´ y
Dollar-Duration-Weighted Yield =
å Par ´ PV ´ Duration
i
i
i
i
i
i
i
i
There are two
primary methods
for weighting
individual yields to
come up with a
weighted-average
portfolio yield:
dollar duration and
market value
Dollar-durationweighted yield
approximates the
internal rate of
return for the entire
portfolio
Market-valueweighted yield
approximates the
next year’s income
for the entire
portfolio
i
Another measure of portfolio return is market-value-weighted yield.
This can deviate substantially from the IRR, but approximates short-term
income from the portfolio.
å Par ´ PV ´ y
Market-Value-Weighted Yield =
å Par ´ PV
i
i
i
i
i
i
i
143
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Total Rate of Return (ROR)
For shorter holding
periods, ROR can be
a better estimate of
expected return
than YTM or IRR
YTM can be more
attractive for longer
holding periods
because ROR
requires more
speculative
reinvestment
assumptions
Yield analysis seeks to understand what investors can earn on their
money by buying securities. The most basic estimate of potential
earning power is YTM/IRR. However, the only way to actually earn the
internal rate of return on a security or a portfolio is if 1) all cash flows
are reinvested at that same rate, and 2) all assets are priced at that same
rate at the end of the holding period, which is highly unlikely.
Furthermore, for shorter holding periods, we have market-based
estimates of both the reinvestment rate and the forward prices.
Therefore, YTM/IRR is a questionable measure of expected return for
shorter time horizons.
Total rate of return (ROR) allows a more flexible analysis of expected
return than YTM. Many investors use expected ROR as a framework for
making investment decisions. ROR is the rate an investor would earn on
a security or a portfolio by buying and holding it, reinvesting cash flows,
and valuing it at the end of a holding period.
Often, investors will try to use different reinvestment and terminal value
assumptions to better understand the sensitivity of ROR to varying
market conditions. Since the ROR calculation uses specific assumptions
about intermediate cash-flow reinvestment rates and the future prices of
assets, the appropriateness of these assumptions is critical to the
usefulness of the ROR analysis.
Thus, ROR can be a better estimate of expected return than YTM for
shorter holding periods, where there is a reasonable ability to make
accurate assumptions about reinvestment rates and future prices.
However, YTM is more attractive for longer holding periods because it
is simpler and because ROR would require problematic reinvestment
and horizon-pricing assumptions.
144
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Total-Rate-of-Return Inputs
Holding Period
(Horizon)
The length of time (frequently one to three years)
over which return is going to be measured. The
longer the holding period, the more significant the
reinvestment-rate assumption becomes; however,
there is often less uncertainty in the horizon
prices for fixed-income investments because they
have a shorter term at the horizon.
Reinvestment
Rate(s)
The rate at which cash flows paid during the holding
period are reinvested, usually related to today’s
market investment rate to the horizon. For longer
holding periods, the lengths of the different
reinvestment periods and the forward-yield curves on
the future payment dates are both important in
determining future reinvestment rates.
It is important to be
specific about the
assumptions
underlying rate-ofreturn analysis
Horizon Price(s) The price(s) of the security at the end of the holding
or Yield(s)
period. Frequently, the forward price of the security
is used, although the horizon price can also be
specified according to a given scenario, for example,
rates up 100 bp. The horizon prices can also account
for some expected rate of default.
Scenario
Weights
Often, we will calculate the rate of return for several
scenarios. We then need to decide how to weight the
different scenarios to estimate the expected rate of
return. The scenario weights assign relative
importance to the various reinvestment and horizon
scenarios. The expected return uses a weighting of
the horizon value (including reinvestment) in each
scenario.
145
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Horizon Yields and Prices
The horizon yield or
price is often the
most significant
contributor to total
rate of return
There are many
factors to consider
when modeling
horizon yields and
prices, including the
forward-yield
curve, yield
volatility, maturity
roll-down, option
exercise, and credit
loss
There are many factors to consider when modeling horizon yields:
ForwardYield Curve
The spot-yield curve implies a market-expectationsneutral forward-yield curve. Any horizon
assumptions that vary from the forward curve (at
least in expected value) imply a market view.
Volatility
Frequently, there is a distribution of horizon yields to
reflect uncertainty about future prices. The higher
the volatility, the more diverse the horizon yields.
Volatility also has an impact on horizon prices if the
security has embedded options, because, as we will
see, option values change when volatility changes.
Maturity
Securities age during the holding period. For
example, a 5-year security will have a 4-year
maturity at the end of a one-year holding period. The
horizon yields should be chosen based on the
remaining term of the security on the horizon date.
Under a static (upward-sloping) yield-curve
assumption, this roll-down results in each security’s
having a horizon yield lower than its spot yield,
which can provide a substantial boost to return.
Recall that IRR assumes that the horizon yields are
the same as the spot yields.
Option Exercise Any options exercised during the holding period can
and Credit Loss change the amount of the security outstanding on the
horizon date. If a bond is called, the bond would
have no value on the horizon date, but the reinvested
proceeds would have value. Any credit losses or
restructurings would also change the horizon values.
146
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Total-Rate-of-Return Timeline
Rate of return
allows complete
flexibility as to
reinvestment
assumptions,
borrowing
assumptions (if
any), and horizon
prices
147
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Components of Horizon Value
30-Year 8% Coupon Bond, 8% Horizon Yield, 8% Reinvestment Rate
Coupon cash flow
and its reinvestment
provide an
overwhelming
portion of horizon
value over a long
holding period;
however, for a
fixed-rate bond,
there is no risk in
coupon cash flows
The underlying
bond’s contribution
to horizon value
stays constant but
declines as a
percentage of total
value, as does its
contribution to
horizon risk
Reinvestment
income contributes
a growing
proportion of return
and horizon risk as
the holding period
extends
148
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Comparison of
Measures of Portfolio Yield
Total Rate
of Return
Measures the return over a fixed holding period,
including the effects of horizon value, intervening
cash flows, and reinvestment. Readily extends to an
uncertain world by allowing a probability
distribution for each of its components. Can be used
on any asset class. Probably used by the broadest
range of investors.
Internal Rate
of Return
The rate which discounts a portfolio’s future cash
flows to the portfolio’s current market value. Only
effective as a measure of return if reinvestment rates
over the holding period equal the IRR and horizon
price risk is minimal.
Total rate of return
is the most
sophisticated
analysis of potential
future earnings
from an investment
Dollar-Duration- One way to weight individual security yields
Weighted Yield to arrive at a measurement of portfolio return. The
dollar-duration weights should match the yields:
“to-maturity,” “to-worst,” or “option-adjusted.”
The dollar-duration-weighted yield-to-maturity
approximates the internal rate of return.
Market-ValueWeighted Yield
Yields can also be weighted by market values. The
market-value-weighted yield shows the rate of
income that certain classes of investors, including
insurance companies, will show over the short term.
Current Yield
(Cash-on-Cash)
Measures the annual cash flow of any asset as a
percent of investment. Does not account for
reinvestment or price risk. An important measure of
yield for leveraged accounts or other high-cost
(especially relative to return on assets) borrowers
who need to pay down debt rapidly.
149
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Calculating Expected Total Rate of Return
The rate of return of
a bond can be
calculated from
basic inputs
While laborious, it
is important to be
thoroughly familiar
with the calculations
Expected rate of
return is especially
important in
analyzing securities
with significant
positive or negative
convexity like long
STRIPS, callable
corporate bonds,
and mortgage
products
U.S. Treasury 8% Maturing November 15, 2021
One-Year Holding Period
Today is June 25, 1996. An investor is considering purchasing the U.S.
Treasury 8% due November 15, 2021 at 108-19+. The investor has a
one-year holding period and thinks that there is an equal chance of the
following three horizon yields: the forward yield, the forward yield plus
100 basis points, and the forward yield minus 100 basis points. The
investor foresees reinvesting any cash flows at 5% simple interest, the
one-year repo rate. What is the expected total rate of return?
Scenario
Reinvested
Horizon Horizon
Spot
Cash
Horizon Yield
Value
ROR
PV (%) Flows (%) Price (%) (%)
(%)
(BEY) (%)
Forward – 100 bp
Forward
Forward + 100 bp
Average
Hints: The first two columns are the same for each row (start here).
The column titled “Reinvested Cash Flows” is the value of the
coupons on the horizon date.
The horizon price (“forward” scenario) is the arbitrage-free
forward price.
Think about what you can determine from what you now know.
What are the components of horizon value?
Which of the last two columns would be the correct one to
average?
150
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Calculating Expected Total
Rate of Return (Continued)
U.S. Treasury 8% Maturing November 15, 2021
One-Year Holding Period
Settlement: June 26, 1996
Q1: Can you
Horizon: June 26, 1997
explain the rate of
Scenario
Reinvested
Horizon Horizon
Spot
Cash
Horizon Yield
Value
ROR
PV (%) Flows (%) Price (%) (%)
(%)
(BEY) (%)
Forward – 100 bp
109.522
8.147
118.772
6.460
127.833 16.072
Forward
109.522
8.147
106.014
7.460
115.075
5.007
Forward + 100 bp 109.522
8.147
95.271
8.460
104.331
–4.798
115.746
5.604
Average
109.522
return in the
forward scenario?
Q2: Can you
explain why the
average rate of
return is higher
than the rate of
return in the
forward scenario?
The final rate of return is calculated from an average horizon value, not
an average of the scenario returns (the average of the scenario returns is
5.427%). The difference is due to compounding. Averaging annual
returns matches the average scenario return of 5.604%.
We just calculated the expected rate of return given today’s price. In
Chapter 6, we will answer the reverse question: If we hypothesize a
distribution of rates over time and a return requirement, what does that
imply about the fair price for a security today?
151
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Factors Affecting Returns
Calculating
expected return for
even a simple
security depends
heavily on the
assumptions and
may be complicated
There are many factors that affect security returns. For example,
calculating the expected return of a Treasury security requires
assumptions about the distribution of horizon prices (which in turn depend
on yields and the convexity of the instrument) and assumptions about
reinvestment. At the horizon date, the length of time until the security
matures will have shortened by the length of the holding period. This
means that, even if the yield curve remains constant, the horizon yield for
the security may be different than the spot yield because the horizon yield
would be the spot yield of a shorter-time-until-maturity bond.
General Factors
•
•
The length of the horizon
Any rebalancing strategy (potentially involving targeting duration
over time, etc.)
Factors Affecting Horizon Price
• The overall level of rates
• The steepness of the yield curve
• The curvature of the yield curve
• The spread to Treasuries
• The level of volatility — for both the overall market and spreads
• The duration, convexity, and embedded options of the security
Factors Affecting Intervening Cash Flows
• Option exercise, including prepayments (Chapters 6 and 10)
• Indices for floating-rate payments (Chapter 8)
• Defaults
Factors Affecting Reinvestment
• The reinvestment instrument
• The pricing of that instrument over time (which depends on many of
the factors affecting horizon price)
152
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Averaging Returns
1 n
The arithmetic average is defined as å xi
n i =1
It is the most common definition of average. If an investor has a oneyear horizon, the arithmetic average of horizon values leads to the
expected return over the one-year holding period.
The geometric average is defined as
n
n
Õx
i =1
i
Geometric averaging is usually used when the portfolio will be held for
a long time and returns will compound; as such, given a return of 8%, a
dollar would be worth $1.08 after one year, so xi=1.08. (This definition
of xi would also be acceptable for the arithmetic average formula.) If an
investor will be rolling the portfolio over (continuing the strategy) at the
end of the holding period, the geometric average of potential returns tells
the expected long-run average return. The geometric average is always
less than the arithmetic average. For example, a 10% gain and a 10%
loss leaves an investor a 1% net loss. In this case, the geometric average
return is negative and the arithmetic average return is zero. The larger
the volatility of returns, the greater the difference between arithmetic and
geometric averaging.
If the various outcomes have different probabilities associated with
them, the formulas need to include these weights pi :
We illustrate two
ways to average
returns:
arithmetically and
geometrically
Arithmetic
averaging is
appropriate when
only one of the
scenarios will come
true
Geometric
averaging is
appropriate when
the investor has a
long horizon but
measures results
frequently, because
many of the
scenarios may come
true over time
n
Arithmetic average:
åpx
i i
i=1
n
åp
i
i=1
n
Geometric average: å p
i
i=1
n
Õ (x )
i =1
i
pi
153
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Calculating Expected Return — Case Study
U.S. Treasury Prices from Tuesday, June 25, 1996 Pack
This page illustrates
some of the
decisions required
to investigate a
relatively simple
question: Which has
a higher expected
return, a 5-yearduration portfolio or
a 17-year-duration
portfolio?
There are many
different factors to
consider in
calculating expected
return
Next step: model
real rates instead of
nominal rates?
Does a zero-coupon bond portfolio with a duration of 17 have a higher
expected return than a zero-coupon bond portfolio with a duration of 5?
The strategy we followed to investigate this question was to average
simulated returns over 1,000 random interest rate scenarios. We
originally anticipated the 17-year-duration portfolio would be the
winner; however, the results of this experiment were far from clear. The
major assumptions were
•
Holding period (1-, 10-, and 30-year horizon)
•
Portfolio composition (achieve duration target by weighting the two
nearest STRIPS)
•
Pricing strategy (log-normal rate evolution, with and without mean
reversion (Chapter 11) of 1% per month, with no roll-down. Roll-down
would improve the attractiveness of the 5-year relative to the 17-year)
•
Rebalancing strategy (monthly adjustment to attain duration target, with
buys and sells executed at the same yield)
•
Volatility (actual 5-year (17.47%) and 17-year (11.25%) yield
volatilities for the last nine years)
Arithmetic Mean
Holding Period
No Mean Reversion
1-Year
10-Year
30-Year
Mean Reversion
1-Year
10-Year
30-Year
5-Year
17-Year
Duration
Duration
Return (%) Return (%)
Geometric Return
5-Year
17-Year
Duration
Duration
Return (%) Return (%)
6.45
6.82
14.76
7.76
7.23
7.85
6.31
6.74
7.80
6.95
6.77
7.41
6.42
6.88
7.71
7.41
7.38
7.67
6.31
6.81
7.30
6.76
7.17
7.48
154
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Indexation
A broad government and corporate index comprises thousands of fixedincome securities, each of which is priced on at least a monthly basis to
provide index returns. An investor whose performance is measured against
this index could replicate it by buying each of its components in exactly the
correct proportion. However, most investors do not have enough
information to do this or enough assets to get good execution in each issue.
Many investors
either index
outright or measure
their performance
against an index
The monthly index release does contain a sector breakdown, with a
duration and other attributes for each sector. An investor could seek to
replicate index returns by investing in each sector according to its weight.
The investor would then nearly replicate the index, while still owning a
manageable and efficient portfolio. Any deviation of actual portfolio
returns from index returns is called tracking error, which can be historically
quantified.
Some managers specialize in index replication and usually charge a
relatively low fee. Others seek to enhance index returns through a variety
of strategies and often charge higher fees. Usually, these managers are
evaluated based on their ability to beat the index by a significant amount
(after fees) to compensate the investor for the extra risk. Some evidence
suggests that it is rare for managers to consistently beat the index — rare
enough to be attributable to chance alone.
Designing an arrangement to align the investor’s and the manager’s
incentives can be difficult, but it is critical. For example, a one-year
performance-based fee is common, but it can create an incentive for
managers to take unreasonable risk because of the option-like component
of their fee (participating in success, but not in failure). Investment banks
have recently begun to re-examine how they compensate and provide
incentives for their asset managers (the traders).
155
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Beating the Index
There are many
strategies that
investors pursue to
beat the index
against which they
are measured
There are a variety of strategies that investors may follow to beat an
index:
•
Buying asset classes not included in the index. Many investors used
this strategy in the 1980’s by replacing corporates with higheryielding asset-backed securities that were not in the index.
•
Market-directional bets. The index has an interest rate sensitivity and
a duration determined by its component bonds. Investors, on the
other hand, have duration flexibility (although sometimes constraints
as well). Investors can, therefore, position themselves to outperform
the index in a rally or decline by owning more or less duration than
the index. Additionally, investors can distribute securities along the
yield curve in a different way than the index. Investors can also
hedge or speculate with options to provide a different return profile.
•
Overweighting sectors relative to the index — either as a relative
value play or as a permanent decision. Some investors continually
rotate between sectors to try to buy cheap assets at the expense of rich
assets. Some believe that they are well-compensated for risk in
higher-spread products and make a permanent asset-allocation
decision to own more of those products than the index.
•
Individual asset selection. Many investors believe that by security
analysis and careful research, they can select securities that will have
less risk and better performance than the index.
•
Using structured notes and derivatives. These securities often
provide exposures that cannot be determined, given only the basic
security description of issuer, coupon, and maturity. Investors may
use these to enhance returns while technically fitting within their
investment guidelines.
156
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 5 Exercises
1. What is the yield-to-maturity or internal rate of return of a portfolio of
$2,000 face amount of 2-year STRIPS priced at 88.60% and $1,000 face
amount of 3-year STRIPS priced at 82.30%? What is the dollar-durationweighted yield? What is the market-value-weighted yield?
2a. What is the total bond-equivalent rate of return of a 3-year 7% annual
coupon bond selling at par and held by the investor until maturity? Do
three cases: 1) reinvest all cash flows at 5%, 2) reinvest all cash flows at
7%, and 3) reinvest all cash flows at 9%.
b. What is the expected rate of return if all three scenarios are equally likely?
Assume June 26, 1996 settlement:
3a. Which has a higher yield-to-maturity (bond-equivalent internal rate of
return):
•
The 5-year Treasury (6.500% due May 31, 2001, priced at 99-03), or
•
The same-duration portfolio comprising the
– 2-year Treasury (6.000% due May 31, 1998, priced at
99-14+), and
– 10-year Treasury (6.875% due May 15, 2006, priced
at 99-18)?
b. Which portfolio has a higher one-year rate of return if cash flow is
reinvested at 5½% and horizon yields equal spot yields?
c. Which portfolio has a higher one-year rate of return if cash flow is
reinvested at 5½% and horizon yields equal forward yields (assuming a
5½% repo rate)?
d. Which portfolio has a higher one-year rate of return if each of the two
preceding scenarios are equally likely? What is the expected rate of return
for each portfolio?
157
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 6
Options
This chapter is an excerpt from A Morgan Stanley Guide to Fixed Income Analysis by
Andrew R. Young, ©2003 Morgan Stanley & Co. Incorporated.
159
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
In This Chapter, You Will Learn...
•
What an Option Is
•
Call and Put Options
•
Intrinsic Value and Time Value
•
Put/Call Parity
•
Option Strategies
•
Option-Valuation Models
– Black–Scholes
– Black–Derman–Toy
– Heath–Jarrow–Morton
•
Measuring Volatility
•
Hedging Options
160
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Options Overview
An option gives the holder a right, not an obligation, to buy or sell a
security. More generally, a derivative is any contract whose payoff is
defined in terms of prices of other securities, rates, or contracts that are
observable in the market (for example, an option, a futures contract, or a
swap).
Options provide
market participants
with alternative
ways to gain
market exposure
and manage risks
Options provide an opportunity to create an asymmetrical payoff pattern:
investors, hedgers, and speculators can gain exposure to security price
movements in some scenarios, while removing exposure in others.
Hedgers use options to reduce their downside risk while maintaining
exposure to favorable price movements. For example, banks and other
financial institutions that could be harmed by sharp rate increases can
buy options that will pay off in those, and only those, situations. The
options can be tailored to trade off the economic value of the protection
against the cost of the option.
Speculators use options to provide leverage. A speculator who believes
that a security will increase in value may choose to buy a call option (the
right, but not the obligation, to buy the security for a fixed price) instead
of the security itself. The price of a call is always less than the price of
the security, so the speculator can get more upside participation for the
same investment. If the price of the security declines, however, the
speculator will lose money, and the option can expire worthless.
Investors may purchase options because they have a view on volatility.
Increasing volatility directly increases the value of options.
161
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Options Terminology
Q: Which are more
valuable —American
or European
options?
Call Option
Gives the holder the right, but not the obligation, to
buy the underlying asset by a certain date for a
certain price.
Put Option
Gives the holder the right, but not the obligation, to
sell the underlying asset by a certain date for a
certain price.
Strike Price or
Exercise Price
The price at which the underlying asset can be
bought or sold.
Expiration Date
or Exercise Date
The date by which the option must be exercised,
before it expires worthless.
American Options Options that can be exercised at any time up to and
including the expiration date.
European Options Options that can only be exercised on the
expiration date.
Premium
The price of the option.
There are two sides to every option. On one side is the buyer who has
taken the long position (i.e., has bought the option). On the other side is
the seller who has taken the short position (i.e., has sold or written the
option). The buyer of the option pays the premium up front, but has
substantial upside. The writer (seller) of the option receives the
premium up front, but has potential liability if the buyer exercises the
option. The option writer’s profit or loss is the opposite of that for the
buyer of the option.
162
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Call and Put Option Payoffs
Owning Options at Expiration
Strike Price = 100%
Call Option
The owner of an
option has almost
unlimited upside
potential and no
downside other
than lost premium
An option can never
have negative value
or payoff — at
worst, it can expire
worthless
The owner of a call option has no downside other than lost premium and These graphs do not
has upside potential that is limited only if the underlying security has a include premium
maximum value (e.g., a bond with a yield of 0%).
expense
Put Option
The owner of a put option has no downside other than lost premium and
has upside potential that is limited only by the fact that the underlying
asset has a minimum price of zero.
163
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Call and Put Option Payoffs (Continued)
Writing European Options
Strike Price = 100%
The writer of an
option has almost
unlimited liability
but receives the
premium up front
Call Option
Note that these
charts measure
strategy profitability
(payoff plus
premium) rather
than payoff at
expiration
Q: Why is the put
premium higher
than the call
premium (both
struck at the current
market price)?
The writer of a call option has liability that is limited only if the
underlying security has a maximum value (e.g., a bond with a yield of
0%), but the writer receives the premium up front.
Put Option
The writer of a put option has liability that is limited only by the fact that
the underlying asset has a minimum price of zero, but the writer receives
the premium up front.
164
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Recognizing Options
Corporate bonds can have a multitude of embedded options. Callable
bonds can be redeemed by the issuer at a fixed price under certain
conditions. There are also putable bonds, where the investor can return
the bonds to the issuer at a fixed price, and extendible bonds, where the
option holder can increase the term of the bonds. Some corporate bonds
have sinking funds (mandatory prepayments of principal prior to
maturity), and issuers may have the option to double up or triple up,
which is a partial call (see Chapter 8). There can also be a provision
known as a make-whole call: the issuer can call the bond at the present
value of its future cash flows discounted at a fixed spread to Treasuries.
This is not an interest rate option; rather, it is an option on the spread of
the bond to Treasuries.
Many different
fixed-income
products and
situations have an
option component
Mortgages are generally subject to prepayment of principal (option
exercise) at any time in whole or in part (see Chapter 10). Mortgage
holders are, therefore, short call options. The behavior of mortgage
prepayments depends on the history of interest rates and prior
prepayments; the option is path-dependent. This increases the
complexity of analyzing mortgage securities. Generic mortgage passthroughs (pools) traded on a to be announced (TBA) basis are subject to
variation, where the seller can deliver the promised amount ±1%. The
seller, therefore, retains a put and a call option on a percentage of the
assets sold.
Sellers always have the right to fail to deliver on time. The seller would
then have to deliver the security (plus any intervening cash flows) at a
later date for the original cost, which amounts to an interest-free loan to
the buyer. This option would be exercised if the asset were so scarce it
had a negative repo rate.
165
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Recognizing Options (Continued)
U.S. Treasury note and bond futures include an option to deliver any one
of a number of qualifying securities (see Chapter 7). The amount of
securities that the short seller of the futures must deliver is normalized
for variations in coupon (relative to 8%), but at different interest rates,
there will be a clear preference to deliver a specific security even with
the normalization factors. Futures have another option-like component
because the margin that futures owners post when yields rise is
“expensive” cash, while the margin that they receive when yields fall is
reinvested at a low rate.
Owners of highly leveraged companies have an option on the assets of
the company. If the company does well, the owners will earn a
tremendous return on their investment. However, the owners’ downside
is limited by the size of their investment, which is small compared to the
value of the assets of the company. Higher volatility increases the value
of options in general, so owners of leveraged companies have an
incentive to maximize the value of their “option” by pursuing a volatile,
risky strategy. The bondholders of a highly leveraged firm are short the
option and try to constrain the behavior of the owners through covenants.
GICs (Guaranteed Investment Contracts), a fixed-rate investment
alternative in many 401k plans, are reallocated among plan participants
as their investment elections change. If more investors reduce their
exposure to the GICs, which would tend to occur when higher-yielding
investments become available, then the GICs can be redeemed at par.
166
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Option Valuation Prior to Expiration
A primary determinant of option value is how the forward price of the
underlying security relates to the strike price of the option. For example,
a call option is more valuable the lower the strike price and the higher
the forward price. The forward price of the security is itself a function
of the current (spot) market price, the short-term interest rate, and the
term of the option. Therefore, a European call option is less valuable the
steeper the yield-curve environment because the forward yield is higher
than the spot yield.
Option values are a
function of the
underlying asset, its
market price, the
strike price, the
term, short-term
interest rates, and
volatility
Because of the way that put and call options and underlying securities
are related, a long European call position (providing upside exposure
above the strike price) combined with a short European put position with
the same strike (providing downside exposure below the strike price) has
the same value as an agreement to purchase the security at the strike
price on the expiration date. Because of this relationship (put/call
parity), calculating the value of a European call option immediately
gives the value of the corresponding put option.
A portion of the value of an option comes from the value of what
“might” happen. This is called time value. The rest of the option’s value
is called intrinsic value. As the remaining life of an option declines,
there is less time for volatility to move the price of the underlying asset
in a way that would be favorable for the option owner. In an upwardsloping-yield-curve environment, call values may decline or increase as
the term of the option increases because the total volatility rises, but the
forward price of the security declines.
Similarly, the more volatility there is in the underlying asset’s value, the
higher the probability a favorable event will occur, hence the more
valuable the option. Buying options, therefore, is often said to be “buying
volatility,” and writing options is often said to be “selling volatility.”
167
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Option Valuation Prior to Expiration
(Continued)
Put/Call Parity (European Options)
A long call option
and a short put
option with the
same strike and
expiration is
equivalent to a long
security position for
settlement on
expiration at the
strike price
A short call option
and a long put
option with the
same strike and
expiration is
equivalent to a
short security
position for
settlement on
expiration at the
strike price
This implies that
knowing the
forward price of the
bond and the price
of either the call or
the put determines
the price of the
other option (for
European options)
Underlying Long Asset Position
The owner of an asset has both the upside and downside of that asset.
Long Call/Short Put Option
The payoff pattern of the long call/short put strategy, taking premiums
into account, is identical to the long asset position.
Put/call parity (European options with identical strike and expiration; no
dividends or counterparty risk):
PriceCall – PricePut = PriceUnderlier – PVStrike
168
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Option Valuation Prior to Expiration
(Continued)
Premium comprises intrinsic value and time value. Intrinsic value is the
amount of gain, if any, that would result if the option were exercised
immediately (even if not currently exercisable). We also refer to an
option with intrinsic value as being in-the-money (spot). Time value is
the remainder of the premium and is almost always positive since an
option owner always has more upside than downside (although that
upside may not be accessible for a long time). Increased volatility does
not affect intrinsic value, but it has a large impact on time value. Time
value almost always decays as the remaining life of the option declines.
An option with a strike price equal to the current price of the underlying
asset is called at-the-money (spot), and an option which would result in
a loss if it were exercised immediately is called out-of-the-money (spot).
The value of an
option is the sum of
its intrinsic value
and its time value
An option can also be described relative to the forward price of the
underlying asset. This, in fact, can make more sense, because the only
way to capture the amount that an option is in-the-money (spot) is to
exercise the option; the in-the-money (forward) amount can be hedged
(locked in), and the option holder can continue to have the benefit of
future volatility. The holder of an option that is in-the-money (forward)
will have a gain if the forward-curve prediction comes true. Similarly, an
option with a strike price equal to the forward price of the underlying
asset is called at-the-money (forward), and an option that would result in
a loss on exercise at expiration if the forward-curve prediction comes
true is called out-of-money (forward).
For example, an American put option struck at 100%, given a price of
the underlying asset of 90% and a premium of 15%, is in-the-money
(spot), and has intrinsic value of 10% and time value of 5%. However, if
the asset’s forward price on the expiration date is 85%, then the in-themoney (forward) amount is 15%, which the owner could lock in, so the
option does not place any value on volatility and must be cheap.
169
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Option Valuation Prior to Expiration
(Continued)
The longer the time
until expiration of
an option, the
greater the time
value and the
higher the option
price (almost
always)
Option Value (%)
35
The value of a call option decays as
the time to expiration decreases. As
expiration approaches, the value
approaches the payoff curve.
30
25
20
Value Three
Years Before
Expiration
Value One Year
Before Expiration
Value at
Expiration
15
10
Intrinsic Value
5
Time Value
0
70
80
90
100
110
120
130
Underlying Asset Value (%)
170
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Option Strategies: Covered Call Writing
Long Asset and Short Covered Call
A short call position
is called covered if
the writer also
owns the
underlying asset
The owner of an asset sometimes writes a covered call to increase
income, although the strategy limits participation in the upside of an
asset.
Covered Call Compared to Underlying Asset
Gain/Loss (%)
30
15
Writing covered
calls allows
investors to earn
extra income
(premium) in return
for foregoing future
price appreciation
above the strike
price
The investor can
retain some of the
upside by selling
the call out-of-themoney
The covered-callwriting strategy is
“selling volatility”
Asset Combined
with Short
Covered Call
0
Underlying Asset
-15
-30
70
80
90
100
110
120
130
Asset Price (%)
Although writing covered calls is held by regulators to be the least risky
option strategy, its payoff pattern is as risky as writing a put.
Q: Where is the
break-even price
for selling a covered
call struck at 100%
for a 5% premium?
171
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Option Strategies:
Option Combinations
Options can be
combined in many
different ways to
provide different
return profiles
Straddle
Strike Price = 100%
One example of a combination is a straddle. Buying a straddle involves
buying a call and a put with the same strike price and expiration date.
The payoff diagram is shown below:
Long Straddle
A long or short
straddle involves
taking the same
side of the market
on both calls and
puts on the same
security with the
same strike and is
only one of many
types of options
combinations
Other strategies
include collars,
strangles, and
butterflies
Short Straddle
172
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Other Types of Options
Cap
Pays out, over time, any excess of a given short-term There are many
rate over the cap rate. Because it pays out when rates other types of
options that can be
rise, it is similar to a put option.
Floor
Pays out, over time, any shortfall of a given short-term return profiles or
rate under the floor rate. Because it pays when rates hedge specific risks
fall, it is similar to a call option.
Spread
Pays when the relationship between two different
assets changes beyond the strike spread. The strike
spread and the method for determining the payout
must be carefully specified. These options allow
hedgers to efficiently buy specific price protection.
Binary
Pays a large fixed sum if the option is exercised in-themoney. They can be used to provide lump-sum
insurance against an unwanted risk.
Look-Back
Pays based on the maximum or minimum price during
the life of the option.
As a general rule,
options on interest
rates (as opposed
to options on
securities) that
benefit in a
declining-rate
environment are
known as calls, and
options that benefit
in a rising-rate
environment are
known as puts
Knock-Out
Expires worthless if the knock-out event occurs. For
example, an investor who does not need protection if
short rates rise 50 bp may buy an option that knocks
out in that situation to reduce premium over the
option’s life.
Q: Which product
would a bank
consider in order to
protect itself against
rising deposit rates?
Knock-In
Can only be exercised if the knock-in event occurs.
Asian
Exercisable based on the average price of the asset.
Bermudan
Exercisable periodically and so is a blend of an
American and European option.
used to give specific
173
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Option Sensitivities
Options provide
leveraged exposure
to assets and,
therefore, are
sensitive to the
levels of various
pricing parameters
Factor
Call Value
Put Value
­ Volatility
­ A call option is a long ­ A put option is a long
­ Strike Price
¯ The higher the strike price, ­ The higher the strike price,
­ Underlying
­ The higher underlying price ¯ The higher underlying price
­ Time Until
­ The longer the time period, ­ The longer the time period,
Price
Expiration
­ Short-Term
Rate
volatility position. Thus, the
higher the volatility, the
higher the option value.
the lower the profit on the
call option given any price on
the underlying asset.
raises the value of the call
option.
the higher the absolute level
The price of an
American option always
increases with time until
expiration. However, a deepin-the-money (high intrinsic
value) European call option
on a high-yielding security
could decline in value as time
until expiration increases.
? of volatility.
volatility position. Thus, the
higher the volatility, the
higher the option value.
the higher the profit on the
put option given any price on
the underlying asset.
lowers the value of the put
option.
the higher the absolute level
? of volatility. The price of an
American option always
increases with time until
expiration. However, since
securities have a minimum
price of zero, the maximum
value of a put option is the
present value of the strike
price; thus, a European put
could decline in value as time
until expiration increases.
­ The higher the short-term ¯ The higher the short-term
rate, the higher the forward
price, which raises the value
of a call option.
rate, the higher the forward
price, which lowers the value
of a put option.
174
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Option Values Depend on Future
Interest-Rate-Distribution Assumptions
Mean of 8%, Standard Deviation of 80 bp (10% volatility)
Most optionvaluation models,
including
Black–Scholes, rely
on either a normal
or a log-normal
distribution of prices
or yields
The log-normal
distribution implies
that percentage
changes, rather
than absolute
changes, are
normally distributed
For the log-normal
The tails on the normal distribution have small, but positive, distribution,
probabilities of events such as negative prices and yields. The log- volatility is
normal distribution has no probability of these events.
proportional to the
A log-normally distributed variable is the exponentiation of a normally
distributed variable: Log-Normal » e Normal ; thus the log of a lognormally distributed variable is a normally distributed variable. If a
normal distribution has mean m and standard deviation s , the
associated log-normal distribution has mean m ¢ and standard
deviation s ¢ :
m¢ = e
s¢ = e
m+
s2
2
m+
s2
2
level of interest
rates
The log-normal
distribution is
skewed: large
increases outweigh
large declines
2
es - 1
175
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Comparing and
Contrasting Option-Valuation Models
Different optionvaluation models
are appropriate for
different securities
and situations
A large number of different option-valuation models have been proposed
at one time or another. They all make different assumptions and have
different levels of complexity and are, therefore, appropriate for different
situations and securities. A good option-valuation model:
• uses market observables as parameters,
• accurately prices a range of liquid options,
• is based on assumptions that are realistic, to the extent possible, and
• is simple and intuitive.
The following material presents three different option-valuation models:
• The simplest and most elegant, Black–Scholes1, provides a closedform solution for the option price. The assumptions are relatively
reasonable for European equity options, but inappropriate for
American options or fixed-income options.
• The intermediate model, Black–Derman–Toy2, is more complicated
to implement because it requires building a binary interest rate tree.
It is conceptually simple and works especially well for American
options and fixed-income securities without path-dependency.
• The final model, Heath–Jarrow–Morton3, is the most complicated
to implement because it can only be analyzed using Monte Carlo
simulation techniques. This facet makes it significantly less efficient
for evaluating American options. However, it is an excellent model
for pricing path-dependent options or options that are based on the
various points on the yield curve and their correlation.
1 Fischer Black and Myron Scholes, “The Pricing of Options and Corporate Liabilities,”
Journal of Political Economy 81 (1973): 637–659.
2 Fischer Black, Emanual Derman, and William Toy, “A One-Factor Model of Interest Rates
and Its Application to Treasury Bond Options,” Financial Analysts Journal 46, No. 1 (1990):
33–39.
3 David Heath, Robert Jarrow, and Andrew Morton, “Bond Pricing and the Term Structure of
Interest Rates: A Discrete Time Approximation,” Journal of Financial and Quantitative
Analysis 25 (1990): 419–440.
176
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
The Black-Scholes Model
(Modified for Dividends)
The Black–Scholes formula for pricing a European call (C) with fixed The Black–Scholes
and known dividends is
model was a
[
(
C = e - rT ´ F ´ N (d )- K ´ N d - s T
where
)]
é
ù
F = Price Forward = ê P - å Dti e - rti ú ´ e rT ,
i
ë
û
2
æFö s
lnç ÷ +
´T
K
2
d= è ø
,
s T
N(d) is the cumulative normal distribution up to d,
P is the current price of the security,
T is the term of the option,
K is the strike price of the option,
Dti is the dividend at time ti ,
r is the short-term interest rate, and
s is the standard deviation of forward prices.
breakthrough in
pricing European
equity options (in
closed form)
It has limited value
for fixed income
because it assumes
constant interest
rates and cannot
account for the drift
towards par (nonconstant price
volatility)
It is based on a
number of fairly
restrictive
assumptions
The Black–Scholes option-valuation formula was derived using certain
assumptions and an arbitrage-free requirement between a continuously
adjusted hedge portfolio and the option itself. Relaxing any of these
assumptions prevents closed-form solutions:
•
•
•
•
•
•
Returns are log-normally distributed and independent over time
The security has constant risk or standard deviation of return
Interest rates are constant over time
No instantaneous price jumps—continuous and infinitely divisible
trading
No early exercise
No transaction costs or taxes
177
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Pricing Options with a
Binary Interest Rate Tree
Fixed-income
options are often
priced using a
binary interest rate
tree centered
around risk-neutral
expectations for
forward short rates
with current market
volatilities
The model does not
depend on
individual views or
expectations; it can
be shown that as
long as asset prices
evolve continuously
(no jumps), no
derivative prices
depend on the
perception of risk
A common binary
tree is called the
Black– Derman –Toy
(BDT) model; it
constructs a onefactor tree, similar
to the one shown,
that matches the
input yield and
volatility curve
Zero-Coupon
Term
(Years)
Annual
Yield (%)
Implied Yield
Volatility (%)
1
6.08
18
2
6.50
17
3
6.70
16
4
6.81
15
In this tree, the initial 1-year rate is known
to be 6.08%. Given the 6.08% rate, the
1-year rate one year forward (T2) has a
50% chance of being 5.77%. This
generalizes, so that from any node, there is
a 50% chance of each of the 1-year rates
occurring. The 50%/50% up/down
probability is constant in this tree;
however, it need not always be the case.
This tree is recombining, which means that in each period the number of
nodes increases by just one. This structure greatly increases the
tractability of the model.
Q1: What is the average price for a 3-year zero, given equal weightings
for the four possible interest rate scenarios?
Q2: Is this consistent with our initial table?
Q3: If our tree shows that interest rates could rise 4.32% from 6.08% to
10.40%, why doesn’t it also show them falling 4.32% to 1.76%?
Q4: Price a 4-year 71/2% bond callable at par after one year.
Q5: Price a 4-year 6.50% cap.
178
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Pricing a Callable Bond
Four-Year 71/2% Bond Callable at Par After One Year
Price this bond by
stepping backwards
through the tree,
carrying back the
lesser of 1) par and
2) the present value
of future cash flows
at each node
1 æ 100.00% + 7.5% 98.46% + 7.5% ö
´ç
+
÷
2 è 1 + 6.98%
1 + 9.44% ø
1 æ 107.5%
107.5% ö
´ç
+
÷
2 è 1 + 8.00% 1 + 10.40% ø
Each node’s present
value is the average
of 1) the “up” node
value plus a coupon
discounted one
period at the “up”
rate and 2) the
“down” node value
plus a coupon
discounted one
period at the
“down” rate;
however, each
node’s present
value is never more
than par (after one
year)
The callable bond is
worth 101.26% (the
same bond without
the call option is
worth 102.44%, so
the option is worth
1.18%)
179
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Pricing a Cap
Four-Year Cap with 6.50% Strike
A cap is the right to
receive any excess
of the target rate
over the strike rate
in each period
A cap can also be
valued using a
Black– Derman –Toy
binary tree
1.60% +
1 æ 1.17%
5.40% ö
´ç
+
÷
2 è 1 + 6.98% 1 + 9.44% ø
2.94% +
1 æ 1.50%
3.90% ö
´ç
+
÷
2 è 1 + 8.00% 1 + 10.40% ø
180
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Building the Black–Derman–Toy Tree
Way Advanced
Zero-Coupon SemiTerm
Annual
(Years)
Yield (%)
Annual
Yield (%)
Yield
Volatility
s·s (%)
Forward
Volatility
s·f (%)
1
5.99
6.08
18.00
N/A
2
6.40
6.50
17.00
17.00
3
6.59
6.70
16.00
15.05
4
6.70
6.81
15.00
13.09
r2h = r2l ´ e
s 2f
pq
This section shows how to build the
Black–Derman–Toy tree used in the prior
examples. This tree has annual stages, so we
will use the annually compounded zero yields.
Building the tree is vastly simpler if the
forward volatilities are already known. The
forward volatility is the variation in the yield
over one stage of the tree. At any given point
in time, for any given node, there are two potential outcomes for the
short rate one year forward. Call p the probability of the higher rate (r•h)
and q = 1 – p the probability of the lower rate (r•l). Under the log-normal
distribution, their mean and volatility are
m = p ´ ln(r· h ) + q ´ ln(r·l )
s·f =
(
)
(
p ´ ln(r· h )- m + q ´ ln(r· l )- m
2
)=
2
ær ö
pq ´ lnç · h ÷
è r· l ø
Each time period in
the tree is called a
stage; each rate
within a stage
corresponds to a
state
The structure of the
tree guarantees
that, for a given
stage, the short rate
in successive states
of the tree is a
constant multiple of
the short rate in the
prior state
That constant
depends solely on
the forward
volatility and the
up/down
probabilities
In this case, we already know the forward volatility and the up/down
probabilities. It is, therefore, more useful to express the higher rate at
any branching point in terms of the lower rate:
s· f
r· h = r·l ´ e
pq
By induction, and because of recombination, the rate for any state is a
function of the lowest rate for that stage and the forward volatility.
181
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Building the Black–Derman–Toy Tree
(Continued)
Way Advanced
To price securities
with the
Black–Derman–Toy
tree, we used
backward induction
To simplify the building of the tree, we will make use of state prices. A
state price is the value today of one dollar in that, and only that, state.
One dollar in every state within a stage provides a certain cash flow, so
the total of all state prices within a stage needs to be the price of the zerocoupon bond in order for the tree to be consistent with its inputs.
To build the tree,
we will use forward
induction, which
saves us from
needing the entire
tree to extend rates
for a stage; instead,
we just need the
state prices from
the prior stage
The value of a dollar in one year (at 6.08%) is 94.27%. This is a state
price. Successive state prices are found using forward induction. The only
way to reach the first (bottom) state of T2 is if rates branch lower from
6.08%. This event, given a short rate in T1 of 6.08%, has probability q. So
the value of a security that pays a dollar in state one of T2, and zero in any
other state, is q ´ 94.27% discounted one period at r2l.
To progress, we need to define p. Using p = q = 50 % and given the
2-year forward volatility of 17%, we seek r2l such that
Price2-Year Zero =
100%
(1 + 6.50%)
2
=
1 é 94.27%
94.27% ù
´ê
+
ú
2 ë 1 + r2l
1 + r2l ´ e 2 ´17% û
This can actually be solved in closed form as a quadratic, but that will
not be possible for later stages. However, Newton–Raphson can be used
to solve for r2l = 5.77%.
182
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Building the Black–Derman–Toy Tree
(Continued)
Way Advanced
Later stages of the tree are calculated in exactly the same manner. Using
forward induction, there are two ways to reach the second state of T3: the
up branch from state one of T2 and the down branch from state two of
T2. The value of a security that pays a dollar in state two of T3, and zero
in any other state of T3 is the sum of the previously defined values in
each state of T2 that leads to state two of T3 (44.56% and 43.60%),
multiplied by the probability of the T2 state branching to state two of T3
(p and q), discounted by the short rate for state two of T3.
The same
methodology is
used to determine
the state prices and
short rates at each
stage of the tree
For stage 3, given p = q = 50% and a forward volatility of 15.05%, the
rate r3l must be chosen so that
Price3-Year Zero =
100%
(1 + 6.70%)
3
=
ù
1 é 44.56% 44.56% + 43.60%
43.60%
+
´ê
+
2 ´ 15.05%
2 ´ 2 ´ 15.05% ú
2 ë 1 + r3l
1 + r3l ´ e
1 + r3l ´ e
û
Newton–Raphson gives r3l = 5.17%.
183
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Building the Black–Derman–Toy Tree
(Continued)
Way Advanced
Forward volatilities
are not always
immediately
available
In the original
presentation of the
Black– Derman –Toy
model, the forward
volatilities come
from spot volatilities
of the zero-coupon
bonds
If the forward volatilities are not already known, they must be inferred
from other market data. In the original presentation of the model, there
are one-year volatilities of the various zero-coupon securities that can be
observed in the market. Since the one-year short rate is known today, its
volatility has no effect on the construction of the tree. At the first
branching (in this example, after one year), the yield of the 2-year zero
(with one year remaining) is 8.10% in the up scenario and 5.77% in the
down scenario. Since the forward yield of the 2-year is also the short
rate one year from now, these rates satisfy the equation:
s 2f = s 2s = 17% =
1
æ 8.10% ö
´ lnç
÷
è 5.77% ø
2
In order to proceed, we will need to determine the yield of the 3-year
zero in one year (with two years remaining) under both the up and down
scenarios.
184
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Building the Black–Derman–Toy Tree
(Continued)
Way Advanced
From the state prices, we can determine that the yields would satisfy the The volatilities of
following equations:
the zero-coupon
Calculating y3h
100%
(1 + y3h )
2
=
Calculating y3l
ù
1 é 92.50%
92.50%
´ê
+
2´ s 3 f
2´ 2´ s 3 f ú
2 ëê 1 + r3l ´ e
1 + r3l ´ e
ûú
100%
(1 + y3l )
2
ß
y3h =
=
1 é 94.55%
94.55% ù
´ê
+
ú
2´ s
2 ëê 1 + r3l
1 + r3l ´ e 3 f ûú
bonds all satisfy the
same log-normal
volatility equation
ß
100%
-1
é
ù
1
92.50%
92.50%
´ê
+
ú
2 ëê 1 + r3l ´ e 2´s3 f 1 + r3l ´ e 2´2´s3 f úû
y 3l =
100%
-1
é
1 94.55%
94.55% ù
´ê
+
ú
2´ s
2 ëê 1 + r3l
1 + r3l ´ e 3 f ûú
Just as the 1-year zero in one year satisfied the log-normal volatility
equation, we would like the yield of the 3-year zero under a 50%/50%
tree (in one year, with two years remaining) to satisfy the equation
s 3 s = 16% =
æy ö
1
´ lnçç 3 h ÷÷
2
è y 3l ø
We then have two equations in two unknowns, which we can proceed to
solve (using two-dimensional Newton–Raphson):
s 3s
æ
ç
ç
ç
ç
1
= 16% = ´ lnç
2
ç
ç
ç
ç
è
Price3-YearZero =
ö
- 1÷
÷
ù
1 é 92.50%
92.50%
´ê
+
÷
2´ s 3 f
2´ 2´ s 3 f ú
2 ëê 1 + r3l ´ e
1 + r3l ´ e
÷
ûú
÷
100%
÷
-1
÷
1 é 94.55%
94.55% ù
÷
´ê
+
ú
2´ s
÷
2 ëê 1 + r3l
1 + r3l ´ e 3 f ûú
ø
100%
ù
100%
1 é 44.56% 44.56% + 43.60%
43.60%
+
= ´ê
+
3
2´s 3f
2´2´s 3f ú
2 êë 1 + r3l
(1 + 6.70%)
1 + r3l ´ e
1 + r3l ´ e
ûú
The reader can verify that r3l = 5.17% and s 3 f = 15.05% satisfy these
equations.
185
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
The Heath-Jarrow-Morton (HJM)
Interest Rate Model
The Heath –Jarrow–
Morton model is a
multi-factor model
with non-constant
(but deterministic)
volatility that
generates the entire
interest rate term
structure at every
stage
The Heath–Jarrow–Morton (HJM) interest rate model is a multi-factor
model that generates the entire yield curve at each point in time (stage).
This is in contrast to the BDT model, which only generates the short
rates. Any yield curve can be represented in the BDT model using the
appropriate series of short rates, but it is time-consuming and difficult to
control the curves that actually appear along the BDT paths for observed
yield-curve correlations.
HJM is non-recombining and does not require a tree. At every stage,
there are infinitely many potential yield curves. In order to evaluate
derivatives with the model, the current yield curve is adjusted to
determine the yield curve at the first stage. That yield curve is used to
evaluate any payments on the derivative at the first stage, and the
payments are then present-valued to the initial settlement date. The next
stage’s yield curve is then determined by adjusting the yield curve in the
first stage, and so on. This methodology produces the value of the
derivative under that specific interest rate path.
There are a number of considerations in implementing the
Heath–Jarrow–Morton model: 1) the expected value of any security on a
future date should be the arbitrage-free forward price of the security, 2)
the model should accurately price various derivatives, including calls,
floors, and swaptions, and 3) the model should not produce negative
yields or forward rates. There is a further enhancement to absolutely
prevent negative yields or rates, but as a first step, we can apply the
model to forward rates instead of yields. This helps because the forwardrate curve and the spot yield curve unambiguously describe each other,
and if the forward rates are all positive, then the yield curve will be all
positive. The converse is not true.
186
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participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
HJM Interest Rate Model (Continued)
Spot Rates and Forward Rates
The spot-yield curve
implies a forward
curve, and the
forward curve
implies a spot-yield
curve
There are many allpositive yield curves
that imply negative
forward rates, so
evolving the yield
curve is dangerous
An all-positive
forward curve
cannot imply
negative spot rates
HJM describes the
evolution of
forward rates to
more easily restrict
the space of yield
and forward rate
curves to be
positive
187
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
HJM Interest Rate Model (Continued)
Evolving the Forward-Rate Curve
The Heath –Jarrow–
Morton model
evolves the yield
curve by adding (or
subtracting) random
amounts of various
predefined curves
each period
HJM evolves the forward-rate curve by adding random (positive or
negative) amounts of various “primary functions” to the prior yield
curve at each stage. The functions, multiplied by the random amounts,
tell how much to change each forward rate across the curve; thus, if the
primary function has the same height for two different dates, both
forward rates will be changed by the same amount. For example, the
two-year primary has its predominant effect on bonds shorter than five
years and is responsible for the vast majority of yield-curve changes in
the under-three-year sector. The two- and three-year rates will thus
display strong correlation. The primary functions have their maximum
effect in different sectors of the curve. Examples of primary functions:
The primary curves are constructed empirically to provide observed or
desired correlations when independent, random amounts of each curve
are added at each stage. Therefore, there can be no correlation other than
that visible in the primary-curve structure. The distribution of amounts
of the curves added controls the observed volatility. Over the long run,
each curve will provide the same amount of positive and negative
change, but this is not true for any one particular path.
188
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
HJM Interest Rate Model (Continued)
Potential Curve Evolutions
These graphs show
three potential oneyear forward-curve
shifts, as well as the
associated forward
curves and yield
curves
The evolutions were
constructed by
adding random
amounts of the
primary functions
and are unbiased:
the average of the
forward-scenario
curves is the actual
forward curve
The following
year’s curve shifts
would be applied to
this year’s curve for
that scenario; thus,
the shifts
“compound”
189
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
HJM Interest Rate Model (Continued)
Constructing Arbitrage-Free Forward Rates
The HJM model
specifies a drift for
forward rates that
constrains the
expected forward
price for any asset,
given the range of
possibilities, to be
the forward price of
that asset
Our methodology up until this point does not generate forward-rate
curves which are arbitrage-free. In order for a curve to be arbitrage-free,
the expected value on any date of a riskless financial instrument without
any embedded options (like a zero-coupon bond, for example) must be
the arbitrage-free forward price of the security on that date.
Since the forward rates are random, there are a range of yields for pricing
each instrument. The critical constraint is that the average price of each
instrument over the range of possibilities must equal its arbitrage-free
forward price, which is not the same as requiring the average yield of
each instrument over the range of possibilities to equal its arbitrage-free
forward yield. The difference is due to the convexity of the security.
When the forward yield rises, the price of the security declines by less
than the price rises when the forward yield declines. In order to force the
expected forward price to the arbitrage-free forward price, a positivedrift curve must be added to the forward-rate curve. The magnitude of
the drift depends on the expected volatility: the lower the volatility, the
less drift is required, and the higher the volatility, the more drift is
required. Generally, the drift can only be calculated one year at a time,
because it depends on the (random) market conditions in the prior year.
190
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
HJM Interest Rate Model (Continued)
Matching Observed Spot and Forward Prices
The drifts we needed to develop our arbitrage-free forward curve
depended on the volatility in the factors. The volatility, in turn, is
calculated to match observed prices for caps and swaptions. However,
there is one problem. The cap and swaption prices also depend on the
forward-rate curve, including drifts. There is, therefore, circular logic if
either the volatilities or the drifts are calculated by themselves. To avoid
the circular logic, the drifts and the volatilities must be simultaneously
derived to accurately price the zero-coupon bonds to their arbitrage-free
forward prices and the caps, swaps, and swaptions to their market prices.
If the benchmark securities are “overspecified,” in that there are too
many similarities between benchmark securities, the model may not
have enough flexibility to price all of them exactly. In that case, the
volatilities and drifts would be chosen to minimize the total pricing error
over all the benchmarks, which is a complicated optimization.
In practice, the drift
and volatility
parameters over
time are chosen to
price a range of
liquid securities,
both with and
without embedded
options, as well as
possible
As specified thus far, there is no constraint that would prevent forward
rates from becoming negative. The model adds random amounts of the
primary curves, and adding a large enough negative amount would cause
the forward-rate curve to cross the threshold. One way of dealing with
this issue is to model the evolution of forward rates as a percent of their
current value. This is called a log-normal model. However, this approach
becomes unstable as the time gradations between stages gets shorter and
shorter. A current area of research is the hunt for model structures that
avoid negative forward rates and yet maintain stability.
191
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Simulation
Unlike the Black–
Derman –Toy tree,
the Heath –Jarrow–
Morton model
cannot be evaluated
by backward
induction
The goal of simulation in the fixed-income context is to evaluate
securities by “randomly” generating prices or yields to represent their
potential evolution over time. At each point in time, the investment
would be analyzed to determine which options, if any, would be
exercised. After enough iterations, a broad range of factor combinations
would have occurred, so the average of the scenario values would
approximate fair value for the investment.
This technique for
evaluating the
value of the option
or security is called
simulation or Monte
Carlo simulation
The inputs used in a simulation are rarely truly random. Truly random
numbers are hard to come by, and even if they were available, there
would be noticeable differences in the prices of securities from one run
to another. Values which cannot be replicated tend to make investors
nervous, so the random numbers are usually retained and reused from
analysis to analysis. These are called pseudo-random numbers. The
downside of this technique is that if your pseudo-random numbers are
poor, the results of your analysis will consistently be poor.
The number of sample paths in a simulation is a critical factor in the
accuracy of its results. Generally, 500 is considered to be a bare
minimum, with some complicated analyses requiring many more to fully
cover the range of possibilities. Unfortunately, many analyses are done
using far fewer, and it is worthwhile to learn about which applications
perform more iterations and have stronger results.
Often, simulation takes advantage of techniques called variance
reduction. In variance reduction, the “random” numbers are constrained
to attempt to ensure that they evenly cover the range of possibilities.
Because there is less clumping, the results of the simulation are more
stable with fewer iterations. Again, if your random-number algorithm is
poor, you will have poor accuracy. One simple way of accomplishing
variance reduction (antithetic variates) is to step through time randomly
and then again along the mirror-image path.
192
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Path-Dependent Options
Path-dependent options add complexity to security valuation. In a path- Some options are
dependent option, either the probability of exercise or the gain or loss on path-dependent: the
probability of
exercise depends on history.
In a simulation, one steps forward through time, and so, at any time, one
knows the prior course of history. Path-dependent options do not add
significant complexity to using the HJM model, because the primary
method of evaluation is simulation. The backward-induction technique
on the BDT tree, however, does not take into account the variety of paths
that lead to any particular state. A simulation can be run using the BDT
tree by stepping forward through the tree one stage at a time.
An example of a path-dependent option is the prepayment options in a
pool of mortgages. When interest rates decline, many homeowners will
find a loan with a lower interest rate and prepay their mortgage.
However, some homeowners will not elect to exercise their prepayment
option. If rates subsequently rise and then decline to the same level,
many of those homeowners will still not prepay, having already
demonstrated an insensitivity to interest rates. Therefore, an important
characteristic in predicting prepayment in a mortgage pool at a given rate
is whether and when it has passed through a similar environment before.
exercise depends
on the prior course
of history
This type of option
eliminates the
ability to price by
stepping backward
through the tree
and increases
computational
complexity
Another example is a knock-in option. A knock-in option becomes
vested once a given threshold is met. Therefore, the ability to exercise
is dependent on the prior course of yields or prices.
193
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Hedging Options
Strike Price = Current Price = 100
Options are
frequently hedged
using the
underlying security
according to the
slope of the optionprice curve at the
current price of the
asset; this is known
as delta-hedging
Delta corresponds
roughly to the
probability of
exercise
The most significant component of valuing a call or put option is the
price of the underlying asset. Delta measures how the price of the option
(C) changes when the price of the underlying asset changes. Thus, an
initial hedge for a $100 long call position would be to sell $100 × Delta
of the security.
Options that are at-the-money generally have a delta of about 50%, which
means that the option can be hedged with half the face amount of the
underlying asset. Options that are out-of-the-money have lower deltas, and
options that are in-the-money have higher deltas. Delta roughly correlates
to the probability that the option will be exercised. When an option is deep
in-the-money, its delta approaches one, and, conversely, when an option is
far out-of-the-money, its delta approaches zero.
194
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Hedging Options (Continued)
Hedge and Option Equivalency
Hedging an option requires constant adjustments, because the option’s
delta and other sensitivities vary as the market moves. Delta, for
example, increases the more an option trades in-the-money. Therefore,
a trader hedging a call option must buy more of the underlier when prices
rise. Conversely, delta decreases as the option trades more out-of-themoney, and the trader would sell some of the underlier. Option hedging,
therefore, requires buying high and selling low. If implied volatility
becomes reality, the expected loss from hedging will equal the cost of the
option. Of course, the bid/ask spread on the option provides an
opportunity to trade it for a more attractive price than the expected
hedging costs.
The ability to hedge continuously is the underpinning of most of the
options models used by traders. The appropriate price for the option
assumes that the trader intends to hedge in this manner. If the model
assumptions (including volatility) hold and the trader does hedge
continuously, the profit will be the same regardless of market direction.
This is why derivatives prices are independent of risk tolerance (under
the assumption that prices move continuously). Any risk the trader
elects to take by hedging less frequently, as well as the transaction costs
of hedging, are not reflected in the model, although the trader may
account for them by increasing the implied volatility in the model.
Likewise, the model’s price is not changed by the trader’s ability to
immediately and risklessly offset the position, although the trader may
feel more comfortable with a lower profit margin under these
circumstances.
Option traders
generally seek to
hedge the
important options
sensitivities of their
trading portfolios
The cost of an
option (before the
bid/ask spread)
equals the expected
cost of hedging it in
an arbitrage-free
market
Often, different options positions in a portfolio partially offset each
other. In this case, traders would hedge the residual risk of the portfolio,
which is less costly and risky than hedging each position independently.
195
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Hedging Options (Continued)
Other Option Sensitivities
Complicated option
positions have
many sensitivities
other than delta
Option traders manage complicated positions and must control risk from
any of the factors that affect option valuation.
The Curve
The value of an option depends not only on the yield
of the underlying security, but also on the short rate
until the term of the option. Furthermore, for a security
with periodic payments, the option value will depend
on the yields to the various payment dates. Therefore,
an option trader will seek to hedge the sensitivity of the
options portfolio to the entire yield curve.
Gamma
Gamma measures the change in delta. One way to
hedge gamma is to continually adjust the hedge as the
prices of the underliers change. This strategy has an
expected cost in that it forces the hedger to buy high
and sell low; the higher the gamma, the higher the
expected cost of hedging. An alternative hedge would
be to take an offsetting option position on the same
underlier (with offsetting gamma) and crystallize the
cost.
Option traders must
manage the
sensitivity of their
position to all of
these factors
¶ 2C
g = 2
¶P
Kappa
k=
¶C
¶s
Theta
q=
¶C
¶t
Kappa measures an option value’s sensitivity to
volatility s. Options increase in value when volatility
rises and decrease in value when volatility declines.
Volatility is most frequently hedged by offsetting with
options; however, it can also be hedged by offsetting
with option-free assets with large positive convexity.
Option values (usually) decline with the passage of
time. As with kappa, this decline can be hedged by
offsetting the position with other options; if unhedged,
the time decay offsets the value of any expected
volatility during that time period.
196
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Option-Adjusted Duration
and Option-Adjusted Convexity
Option-valuation techniques can also illustrate the riskiness of a security
with embedded options. When interest rates change, the present value of
future cash flows changes, but the value of the option changes as well.
For most securities with embedded options, it is impossible to determine
duration and convexity in closed form.
An option model
can calculate
option-adjusted
duration and
convexity
An option-valuation model can be used to empirically estimate duration
and convexity by calculating price and duration changes for small
changes in yields. The Black–Derman–Toy tree and the Heath–Jarrow–
Morton model were constructed to match the initial yield curve; the
procedure could be repeated for a yield curve that was increased or
decreased by a slight amount. The option or security could then be
repriced, providing the estimate of duration:
Duration @ -
DPV
PV
Dy
Convexity measures the second-order sensitivity of present value to a
change in yields. It can be estimated given three prices: the base-case
(middle) price, the price when yields increase by Dy, and the price when
yields decline by Dy.
Convexity @
(PriceHigh + PriceLow - 2 ´ PriceMiddle ) PV
(Dy )
Middle
2
For securities with embedded options, convexity can actually change
quite quickly as interest rates change. There are two ways of handling
this: including higher-order terms in the Taylor series or empirically
measuring convexity over a larger interest rate change (Dy).
197
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Correlation Derivatives
Some derivatives
depend on the
degree of
correlation between
different assets
These have
additional
complexity because
the correlations
tend to be less
stable over time
than volatilities of a
single asset
Some derivatives depend on several rates and prices that may be
partially related or almost completely unrelated. There are special issues
concerning pricing and analyzing these types of derivatives.
One example of a correlation derivative is a spread option: a call on the
price differential of two securities. Buying a put on the low-priced asset
and a call on the high-priced asset could also provide the same payments
as the spread option; however, the put and call also pay off in other
scenarios (boxed below) and, therefore, cost more:
High-Priced
Asset
Low-Priced
Asset
­
­
¯
­
­
¯
¯
¯
Spread Call
Call Option
Option
on High-Priced
Put Option
on Low-Priced
­
­­
­
­
­
The higher the correlation between the securities, the less expensive the
spread option. This can be seen by:
s H2 - L = s H2 + s L2 - 2 ´ r ´ s H ´ s L
Trading these options requires an estimate of the correlation of the
underlying asset prices. While volatility and yield tend to revert to the
mean, correlations do so to a much lesser extent, if at all. Therefore,
correlations tend to be unstable, which adds to the difficulty in pricing
these options. The market in these options is also relatively thin, so there
are few comparables to gauge and hedge the market’s implied
correlation. Many risks can be addressed more efficiently using this type
of derivative, so this market is likely to grow over time.
198
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Future Volatility and Historical Volatility
An estimate of future volatility is a critical component of valuing
options. Because the future volatility is not known, investors often
attempt to use current volatility as a benchmark. Unfortunately, current
volatility is not known either; it requires a series of prices to calculate
volatility, and the series of prices must extend into the past. Therefore,
investors actually use historical volatility as a benchmark for future
volatility.
There is a tension between the desire to use more data to get a more
accurate volatility estimate and the desire to make sure that the data used
is as relevant to the future as possible. The crux of the issue is how to
ascertain whether an unusually big market movement is a surprising, but
possible, event drawn from a distribution with historical volatility or a
common event reflecting an increase in volatility. One way to address
this issue is to weight recent observations more heavily than older
observations.
Investors often use
historical volatility
as a measure of
potential future
volatility
Q: If the market has
a strong trend, will
the measured
historical volatility
be higher or lower
than if it trades
“sideways,” i.e.,
ends where it
started?
The annual volatility implies different absolute volatilities for different
periods of time. If the absolute volatility for some period T1 is v1, then
the volatility v2 for a different period T2 is
v2 = v1 ´
T2
T1
For example, 10% annual volatility implies monthly volatility of
10% ´
1
= 2.89%
12
199
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Estimating Historical Volatility
Historical volatility
is defined as the
standard deviation
of past log-return or
log-yield changes
The most straightforward measurement of historical volatility is the
sample standard deviation:
There are several
methods for
estimating historical
volatility
The 1 – a confidence interval for s 2 (if the underlying distribution of
ln x is normal) is
The most common is
the sample
standard deviation
There is another
methodology for
estimating volatility
using the high and
low prices
Both techniques are
estimates of
volatility; the true
volatility at any
point in time can
never be known
n
ˆ =
s
å( x
i =1
i
- x )2
n-1
y
ö
; xi = lnæç i
÷
y
i -1 ø
è
æ (n - 1) ´ s
$ 2 (n - 1) ´ s
$ 2ö
çç 2
÷
,
c a2 2,n - 1 ÷ø
è c 1-a 2,n - 1
There is another method for estimating s, which involves using the high
and low data for the series over a period of time:4
s$ ¢ =
n
2
1
ln(Highi )- ln(Lowi )
å
n ´ 4 ´ ln(2) i =1
(
)
The confidence interval for s$ ¢ is about 50% narrower than for s$ .
There are similar techniques using the price at some fixed time every day
(i.e., open or close), which can estimate volatility even more accurately.
If the distribution of returns or yields is log-normal, then the mean and
standard deviation fully specify the distribution. If the returns or yields
follow another distribution, the standard deviation will be correct, but
cannot be used as the volatility in a log-normal valuation model. The
range estimate of volatility depends explicitly on a log-normal
distribution, and does not even estimate the true standard deviation if this
assumption is not met. Therefore, use it cautiously.
4
Parkinson, Michael, “The Extreme Value Method for Estimating the Variance of the Rate of
Return,” Journal of Business 53, no. 1 (1980): 61-65.
200
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Implied Volatility
Implied volatility is obtained by using different volatilities in the options
model to see which one comes up with the observed prices for the most
liquid, frequently traded options. The Newton–Raphson technique is
useful for solving this problem. Implied volatility is a measure of market
expectations for future volatility.
Implied volatility tends to be higher than historical volatility. There are
several factors that contribute to this phenomenon:
•
Most option models assume log-normal price or yield changes. This
assumption is consistent with 5% of the percentage price moves
exceeding two standard deviations from the mean. In fact, large price
moves are more common than predicted, and so we say that the
actual price change distribution has “fat tails” or is lepto-kurtotic.
The option receives value from one of those tails, but is unaffected
by the other tail.
•
Measurement of historical volatility is inaccurate, and option prices
are convex to volatility: They rise more when volatility increases
than they fall when volatility decreases. (This property holds in the
current environment for near-the-money options.)
•
Volatility follows its own random process, which adds value given
the option’s volatility convexity.
•
Discontinuities in the market eliminate the ability to continuously
hedge.
•
Transaction costs reduce traders’ willingness to continuously hedge.
Historical volatility
is based on changes
in price observed in
the past, while
implied volatility
embodies
expectations about
future price risks
and the specific
model used to
evaluate options
Different models
will produce
different implied
volatilities
In order to price these risks in a model that assumes “normal” tails and
known volatility, implied volatility must be increased. These effects are
more important the more an option is out-of-the-money (because it is
more leveraged), resulting in the “volatility smile.”
201
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 7
Futures
This chapter is an excerpt from A Morgan Stanley Guide to Fixed Income Analysis by
Andrew R. Young, ©2003 Morgan Stanley & Co. Incorporated.
203
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
In This Chapter, You Will Learn...
•
Why Futures Exist
•
How Futures Relate to Forwards
•
Attributes of Bond Futures
•
Delivery Economics
•
About the Cheapest-to-Deliver Option
•
How to Calculate the Implied Repo Rate
•
About the Financing and Wild Card Options
•
About Basis
•
How to Hedge with Futures
•
How to Calculate the Rate of Return on Futures
204
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
What Are Futures?
Futures are exchange-traded contracts with standardized terms that
provide exposure to a market or a segment of a market. The federally
designated exchange in which trading is conducted establishes the terms
and conditions of the contracts and trading. There are a wide variety of
futures contracts covering a range of domestic and international fixedincome, currency, equity, and commodity products.
Futures are
standardized,
exchange-traded
financial contracts
that provide
exposure to a
market or a
segment of a
market
A clearing corporation acts as the counterparty to every futures
transaction, so the creditworthiness of other traders is irrelevant. A
trader who sold a futures contract can offset the sale by purchasing a
futures contract from a third party, leaving a flat position (containing no Futures only add
value to the extent
contracts).
Some futures contracts are cash-settled; others require delivery of an
underlying instrument. The closing prices for cash-settled contracts are
determined by reference to some index or price (i.e., S&P 500 futures).
Deliverable contracts which remain open at expiration require the
physical transfer of securities or commodities (i.e., bond futures). Many
futures investors will offset their positions prior to this time; however,
pricing should always account for the economics of the delivery process.
that they are more
liquid or more
tradeable than the
underlying
securities or
commodities
Futures are leveraged; they are bought by posting collateral (initial
margin), and thereafter, the investor pays or receives the daily change in
the value of the futures (variation margin). The initial margin is meant
to protect the exchange from a relatively severe one-day market move.
The exchanges have rules to maintain a fair market, including setting
limits on maximum positions, prohibiting market manipulation, and
requiring most trades to be done by open outcry in the designated pit.
Open outcry guarantees execution at a price if the futures trade through
that price, but not necessarily if the futures trade at that price.
205
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Convergence in the Futures Market
Financial futures
tend to have
positive
convergence, or
backwardation,
while commodities
futures tend to have
negative
convergence
On the delivery
date, the price of
the futures should
approximate the
price of the
underlying
instrument
On their delivery date, futures should trade at the same price as the
underlying instrument. However, prior to (expected) delivery, there may
be a difference in price. The deviation of the futures price from the price
of the underlying asset decreases as the time remaining until delivery
decreases. This phenomenon is called convergence.
Most futures, based on commodities or metals, trade above the
underlying price. This occurs because the futures price reflects the
negative carry associated with buying and then storing and financing a
non-income-producing asset. A fairly valued futures price creates
indifference between
–
buying and accepting delivery on futures, and
–
buying and then financing and storing the commodity until the
futures delivery date.
With these types of futures it is important to consider the cost of delivery
and who is required to pay for it under the contract.
Fixed-income futures, on the other hand, usually trade below the
underlying price. This phenomenon is called backwardation. It occurs
because the underlying instrument generally bears a higher rate of
interest than the short-term financing cost. Since there is no cost of
storage or delivery, the fair futures price creates indifference between
buying the future and buying the underlying asset, receiving its coupon,
and paying financing costs.
206
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Structuring a Futures Contract
Considerations in structuring a futures contract:
•
A broad range of deliverable securities to reduce the risk of a squeeze
(inability to purchase bonds for delivery)
•
A precise definition of grading to delineate the quality of the
deliverable
•
A high-credit counterparty to each contract (clearing corporation)
•
Attention to fairness of the market: brokers act as agents, price
discovery by open outcry in trading pits (with some after-hours
electronic trading), and position limits
•
Circuit breakers to prevent a market meltdown
Every element in the
design of a futures
contract is meant to
add liquidity to the
market
However, the
details of the
contract add
complexity to the
behavior of futures
prices
Futures open interest (outstanding contracts) in the 5-, 10-, and
30-year contracts is roughly one-eighth of the U.S. Treasury
outstandings in these sectors. Despite this, the trading on the futures
dwarfs trading in the actual Treasury bonds, demonstrating higher
liquidity.
207
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Futures vs Forward Contracts
Futures and
forward contracts
both provide
exposure to price
movements without
requiring the full
up-front investment
Futures and
forwards are not
identical; forwards
are more valuable,
but less liquid
Futures and forward contracts are very similar. They both provide
exposure to an underlying instrument, and neither requires the current
purchase of that instrument. Both can be used for hedging or speculation
(although the futures are more liquid). In both cases, the long position
agrees to buy an underlying security on the delivery date; the short
agrees to sell it.
However, futures and forwards are not identical. Forward contracts are
written on a specific security with a definite settlement date. On the other
hand, some futures have a range of deliverable securities, and the seller
can be expected to deliver the least attractive one (substitution option).
If a futures contract has a substitution option, it will be worth less than a
forward contract (although the value of the substitution option declines
as the contract nears expiration). Another difference is that forwards
trade over the counter (OTC), not on an exchange, and since there is no
clearing corporation to act as counterparty, forwards have counterparty
credit risk.
A more complex difference between futures and forwards is the effect of
margin. Because long and short rates tend to be correlated, an increase
in long-term interest rates, and therefore a decline in futures prices,
would typically be accompanied by an increase in short-term interest
rates. When the futures price falls, its owner needs to make a same-day
variation margin payment (mark-to-market), which would be borrowed
in a higher-interest-rate environment. Alternatively, when the futures
price rises because of a decline in yields, the owner would receive
variation margin, which could only be invested at a lower interest rate.
Since futures buyers expect volatility, they expect to be forced to borrow
for variation margin at a higher rate than their investment rate for
variation margin received. This decreases the value of futures relative to
forwards, since forwards usually have no margin requirement. This
difference also declines in magnitude as the contract nears expiration.
208
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Bond Futures
Bond futures have the following characteristics:
Bond futures are
the most commonly,
but by no means
the only, traded
financial futures
•
The contract is for $100,000 par amount of Treasury bonds.
•
The contract requires an initial margin, or collateral, of 1.75% of
notional amount (the exchange’s estimate of how much the market
might move on a volatile day) and daily variation margin (mark-tomarket). The exchange can modify the initial margin requirement The rest of this
chapter focuses on
depending on market conditions.
•
Any U.S. Treasury security is deliverable if it is longer than 15 years
as of the first day of the delivery month (measured to the earlier of Q1: How does
maturity or first call date).
each of the bond
Factors that approximately equalize the attractiveness of delivering futures’
any of the bonds subject to the contract (otherwise, the holder would characteristics
increase liquidity?
always deliver the lowest-priced bond).
•
bond futures
•
Contracts expire every March, June, September, and December.
•
The seller may elect delivery on any business day during the
expiration month by:
– Notifying the exchange (through the broker) prior to 8:00 PM CST
on the Tender Date, which is two business days prior to delivery.
Q2: Do you see
any hidden options
in the contract
description?
– Notifying the exchange (through the broker) which security will
be delivered prior to 2:00 PM CST on the Notice Date, which is
one business day prior to delivery.
– Delivering $100,000 par amount of valid securities on the
delivery date in exchange for the amount of cash equal to 1) the
closing futures price at 2:00 PM CST on the Tender Date, 2)
multiplied by $1,000, 3) multiplied by the factor for the security,
4) plus accrued interest. The last day of trading is the eighth
business day prior to the end of the delivery month; any contracts
tendered subsequent to that date will use that day’s closing price.
209
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Bond Futures Factors
Each bond’s factor
is roughly the price
of the bond at an
8% yield
In theory, if each
bond’s yield was
8%, there would be
no preference for
delivering any
particular bond
Since there is a
range of possible
delivery dates, the
contract specifies a
somewhat arbitrary
settlement date for
the calculation
Q: Does this
arbitrary date
introduce bias
toward delivering
any particular
security?
Factors “equalize” various bonds with different prices so that, if every
bond eligible for delivery (in the basket) yielded 8%, there would be no
preference for delivering any particular bond. On delivery, the owner of
a futures contract will pay the invoice price (the price of the futures
multiplied by the factor of the delivery bond plus accrued interest) in
exchange for the bonds.
A bond’s factor is the price of the bond, in decimal (rounded to four
places), given the following parameters:
Maturity:
The bond’s maturity date
Settlement: 1) The first business day on or after 2) the earliest
quarterly anniversary of the bond’s maturity date
that 3) falls on or after the first day of the futures’ delivery
month
Coupon:
The bond’s coupon
Yield:
8%
Example: For the September 1996 bond futures contract, the 6¼% of
August 15, 2023 (noncallable) would have a factor computed using a
settlement date of November 15, 1996, which is a Friday. The price of
the bond to that settlement at an 8% yield is 80.792950%, so the factor
is 0.8079. Note that if the settlement date had been in the middle of the
delivery month (on September 16, 1996), the factor would have been
slightly lower, 0.8077.
210
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Calculating and Using Factors
September 1996 Bond Contract
Factor
Coupon
(%)
Maturity
11.250
2/15/15
8.000
11/15/21
6.000
2/15/26
Expiration
Settlement
Factor
Date
Yield (%)
Factor
Contract
Price (%) at
Expiration
Factor Yield
Assume that on the futures delivery date, the September 1996 futures
price is 100 and that the investor is short one contract. The investor must
buy a bond in the market to deliver against the futures. Assume that all
bonds are priced to yield 8%. For the purposes of this calculation, we
can ignore accrued interest since the investor would pay it when
purchasing the bonds, but receive it when delivering the bonds against
the futures.
Coupon
(%)
Maturity
11.250
2/15/15
8.000
11/15/21
6.000
2/15/26
Investor
Investor
Receives ($)
Pays ($)
The factor is the
price, in decimal
(rounded to four
places), of a bond
at an 8% yield
using the settlement
date that is 1) the
first business day
on or after 2) the
earliest quarterly
anniversary of the
bond’s maturity
date that 3) falls on
or after the first day
of the delivery
month
Profit & Loss ($)
211
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Calculating and Using Factors (Continued)
September 1996 Bond Contract
When all yields are
8%, the lowestfactor bond will be
the cheapest-todeliver (CTD)
because of the bias
implied by the
selection of the
factor settlement
date
The factors are all
slightly closer to
1.0000 than they
“should” be
Factor
Coupon
Expiration
Settlement
Factor
Contract
Price (%) at
Factor Yield
(%)
Maturity
Date
Yield (%)
Factor
Expiration
11.250
2/15/15
11/15/96
8.000
1.3089
9/30/96
130.992
8.000
11/15/21
11/15/96
8.000
1.0000
9/30/96
99.985
6.000
2/15/26
11/15/96
8.000
0.7751
9/30/96
77.485
Assume that on the futures delivery date, the September 1996 futures
price is 100 and that the investor is short one contract. The investor must
buy a bond in the market to deliver against the futures. Assume that all
bonds are priced to yield 8%. For the purposes of this calculation, we
can ignore accrued interest since the investor would pay it when
purchasing the bonds, but receive it when delivering the bonds against
the futures.
Coupon
Investor
Investor
(%)
Maturity
Receives ($)
Pays ($)
11.250
2/15/15
130,890
130,992
–102
8.000
11/15/21
100,000
99,985
15
6.000
2/15/26
77,510
77,485
25
Profit & Loss ($)
212
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Delivery Economics
September 1996 Bond Contract
Coupon
Assumed
Assumed
9/30/96
9/30/96
(%)
Maturity
Factor
Duration
Yield (%)
Price (%)
11.250
2/15/15
1.3089
Low
7.191
141.015
8.000
11/15/21
1.0000
Medium
7.252
108.579
6.000
2/15/26
0.7751
High
7.089
86.613
Coupon
Investor
Investor
Profit
Break-Even
Pays
($)1
Futures
Maturity
Receives
($000)1
& Loss
(%)
($)
Price (%)
11.250
2/15/15
?
?
?
?
8.000
11/15/21
?
?
?
?
6.000
2/15/26
?
?
?
?
Profit & Loss ($)
Coupon
Futures
Futures
Futures
Futures
Price
Price
Price
Price
(%)
Maturity
90
100
110
???
11.250
2/15/15
?
?
?
?
8.000
11/15/21
?
?
?
?
6.000
2/15/26
?
?
?
?
Assume the investor
is considering
shorting one futures
contract on the
delivery date
At the arbitragefree futures price on
the last day of
delivery, an
investor should
break even by
selling the futures,
buying the
cheapest-to-deliver,
and immediately
delivering it
Which bond should
the investor deliver?
Why?
1 Excluding accrued interest
213
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Delivery Economics (Continued)
September 1996 Bond Contract
The break-even
futures price is
about 107-24,
slightly higher than
the actual futures
price of 107-05 on
June 25, 1996 (due
to convergence,
which is discussed
later)
The 111/4% due
February 15, 2015
is the cheapest-todeliver
The fact that the
111/4%’s factor is
slightly lower than
it “should” be due
to the arbitrary
factor-computation
method somewhat
offsets this delivery
preference
Coupon
Assumed
Assumed
9/30/96
9/30/96
(%)
Maturity
Factor
Duration
Yield (%)
Price (%)
11.250
2/15/15
1.3089
Low
7.191
141.015
8.000
11/15/21
1.0000
Medium
7.252
108.579
6.000
2/15/26
0.7751
High
7.089
86.613
Coupon
Investor
Investor
Profit
Break-Even
Pays
($)2
& Loss
Futures
($)
Price (%)
(%)
Maturity
Receives
($000)2
11.250
2/15/15
P × 1.3089
141,015
P × 1,308.9 – 141,015
107-24
8.000
11/15/21
P × 1.0000
108,579
P × 1,000.0 – 108,579
108-19
6.000
2/15/26
P × 0.7751
86,613
P × 775.1 – 86,613
111-24
Profit & Loss ($)
Coupon
Futures
Futures
Futures
Futures
Price
Price
Price
Price
(%)
Maturity
90
100
110
107-24
11.250
2/15/15
–23,214
–10,125
2,964
0
8.000
11/15/21
–18,579
–8,579
1,421
–843
6.000
2/15/26
–16,854
–9,103
–1,352
–3,107
The 111/4% bond, with the lowest break-even futures price, is the
cheapest-to-deliver (CTD). On delivery, any price other than 107-24
would provide arbitrage for one of the counterparties.
2 Excluding accrued interest
214
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Delivery Options
In Order of Importance
Substitution: This option is also called the quality option; it describes
the futures seller’s option to deliver the bond that
maximizes profit (or minimizes loss). Each deliverable
bond has a different duration. When rates are 8%, each
bond is almost equally cheap to deliver. When rates are
below 8%, the CTD will be a low-duration bond. On the
other hand, when rates are above 8%, the CTD will be a
high-duration bond. Even around 8%, some bonds will
trade rich and others will trade cheap, providing for a
yield-spread option (independent of the overall level of
interest rates). The cheapest-to-deliver typically has a
limited amount outstanding compared to open interest on
futures and can become rich relative to the market.
Financing:
The structural
details of the bond
futures contract
provide for several
interesting options
The most valuable
option is called the
substitution or
quality option
The “cash and carry” trade consists of a long position in a
financed bond (usually the CTD) hedged with a short
futures position. As long as the CTD bond’s coupon is
higher than the cost of financing it (repo rate × present
value), the investor who is short the futures will delay
delivery. If short rates increase or the CTD changes such
that the financing cost is higher than the coupon, the
investor would prefer to deliver the bond as soon as
possible.
Wild Card: The delay between futures close at 2:00 PM CST and the
notification deadline at 8:00 PM CST provides the
opportunity to deliver based upon the closing price even if
prices have subsequently changed.
Switching:
The special case of substitution after the close on the last
day of futures trading, when the futures price is locked in,
but before the Notice Date.
215
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Delivery Security Selection
The Factor-Adjusted Prices of Various Bond Deliverables
The optimal security
to deliver depends
on the intrinsic
richness or
cheapness of the
security, the yield
environment, and
the shape of the
yield curve
The cheapest-todeliver tends to be
a low-duration
bond when yields
decline and a highduration bond
when yields rise
While the
underlying bonds
have convexity, the
futures have very
little, or even
negative, convexity
Note that futures always trade below the cheapest-to-deliver because
when you buy futures, you simultaneously short several delivery
options, including the valuable substitution option. This option is worth
the most when yields are near 8%, so the difference between futures and
the CTD is greatest there. When yields are far from 8%, the CTD
becomes more entrenched and the right to substitute is not worth much.
There are several different ways for the cheapest-to-deliver security to
change. This change can occur with changes in the overall level of
yields or with changes in yield spreads among the deliverable securities.
For bond futures, the option on the change in yield levels is more
significant because the deliverables have significantly different
durations; this is in contrast to 5-year note futures, where the
deliverables all have similar durations and the option on changes in the
yield relationships of the deliverables is more significant.
216
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Non-parallel Shifts
The yield curve does not always shift in a parallel fashion. In fact, many
market participants believe that, when yields decline, the curve will
steepen and that when yields rise, the curve will flatten or invert. The
history of the past 10 years has lent credence to this hypothesis. In
addition, the curve can steepen or flatten without the level of interest
rates changing.
Low-duration deliverables are located at the short end of the basket,
while high-duration securities reside at the long end. When yields fall,
low-duration securities become more attractive to deliver. However, if
the curve steepens as yields fall, the yields of the lower-duration
deliverables will fall by more than under a parallel shift, which will
somewhat offset the incentive to deliver them.
The historical
negative correlation
between the level of
interest rates and
the steepness of the
yield curve tends to
reduce the value of
the substitution
option, as does the
cheapest-to-deliver
liquidity effect
When yields rise, the high-duration securities become more attractive to
deliver. However, if the curve flattens as yields rise, this will again
offset the incentive to deliver the high-duration security.
There is a significant economic penalty for delivering a bond other than
the cheapest-to-deliver. Therefore, the cheapest-to-deliver tends to trade
a little rich because investors buy it to hold against a short futures
position. The corollary is that the CTD tends to trade special in the repo
market. Occasionally, although rarely, short-term demand for the CTD
has caused it to have a negative repo rate.
As other securities near deliverability, the premium on the cheapest-todeliver decreases because there are close substitutes. This cheapening
slightly extends the CTD’s reign.
217
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Determining the Cheapest-to-Deliver
The implied breakeven repo rate is
the financing rate
that would cause
the following
transaction to have
exactly zero present
value:
1) selling the
futures,
2) buying the bond,
3) financing it to the
delivery date, and
4) delivering it
against the futures
The bond with the
highest implied
repo rate is the
cheapest-to-deliver;
its implied repo rate
is near (but usually
below) its market
repo rate
Given the futures price, there is a relatively simple way to determine the
bond that is cheapest-to-deliver: it has the highest implied repo rate. The
implied or “break-even” repo rate is the financing rate at which there
would be no gain or loss from selling the futures, buying a deliverable
security, financing it, and delivering it on the assumed delivery date. If
there were no options embedded in the futures, then the implied repo rate
for the CTD should be its market repo rate. The implied repo rate
satisfies the following equation:
é
DateDelivery - DateSpot ö ù
æ
÷ ú
ê PriceSpot + AccruedSpot ´ ç 1 + RepoImplied ´
360
è
ø ú
ê
PriceFutures ´ Factor = ê
ú
DateDelivery - DateCouponi öú
Coupon k æ
ê- Accrued
´ Sç 1+ RepoImplied ´
÷ú
Delivery ê
2
360
i =1 è
øû
ë
(
)
This can be solved for the implied repo rate:
RepoImplied =
(
Coupon
´k
2
DateDelivery - DateCouponi
)
PriceFutures ´ FactorBond + AccruedDelivery - Price Spot + AccruedSpot +
(PriceSpot + AccruedSpot )´
DateDelivery - DateSpot
360
-
Coupon k
´å
2
i =1
360
where k is the number of coupons paid prior to delivery.
At any financing rate below the implied repo rate, the transaction will
guarantee a profit to the seller of the futures. Even at a financing rate
above the implied repo rate, the transaction may be profitable because of
the various options embedded in futures. However, the security with the
highest implied repo rate will provide the seller of the futures the greatest
profit at any market repo rate and, therefore, would appear to be the most
attractive to deliver.
218
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Determining the Cheapest-to-Deliver
(Continued)
September 1996 Bond Futures (Trade Date June 25, 1996)
Coupon (%)
Maturity
Price (%)
Yield (%)
Factor
11.250
2/15/15
141-09+
?
?
8.000
11/15/21
108-19+
?
?
6.000
2/15/26
86-18+
?
?
Assume parallel yield shifts on June 25, 1996 of up and down 200 bp.
Which bond would be the cheapest-to-deliver in each scenario?
First Step: Current CTD
Futures Price: 107-05
Number of
6/25/96
9/30/96
Coupons
Implied
Maturity
Accrued (%)
Accrued (%)
Paid (k)
Repo (%)
11.250
2/15/15
?
?
?
?
8.000
11/15/21
?
?
?
?
6.000
2/15/26
?
?
?
?
Coupon (%)
Scenario: Down 200 bp
CTD Repo:
Futures Price:
Implied
Coupon (%)
Maturity
Yield (%)
Price (%)
Repo (%)
11.250
2/15/15
?
?
?
8.000
11/15/21
?
?
?
6.000
2/15/26
?
?
?
This example uses
the implied repo
rate to determine
the cheapest-todeliver in several
interest rate
scenarios
Assume the implied
repo rate of
whichever bond is
the CTD in each
scenario is 1) the
implied repo rate of
the current CTD plus
2) the scenario yield
shift
The next page
provides important
clues for finding the
futures price and
the CTD in each
scenario
Scenario: Up 200 bp
CTD Repo:
Futures Price:
Implied
Coupon (%)
Maturity
Yield (%)
Price (%)
Repo (%)
11.250
2/15/15
?
?
?
8.000
11/15/21
?
?
?
6.000
2/15/26
?
?
?
219
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Determining the Cheapest-to-Deliver
(Continued)
September 1996 Bond Futures (Trade Date June 25, 1996)
On June 25, 1996,
the 111/4% of
February 15, 2015
had the highest
implied repo rate
and was the
cheapest-to-deliver
We can calculate
the futures price
that would cause
each bond to have
that implied repo
rate; the lowest
such futures price is
the estimate of the
scenario futures
price
No investor should
pay more than that
lowest price,
because a futures
seller could create
arbitrage by selling
the futures, buying
the CTD, financing
it, and delivering it
against the futures
Coupon (%)
Maturity
Price (%)
Yield (%)
Factor
11.250
2/15/15
141-09+
7.191
1.3089
8.000
11/15/21
108-19+
7.252
1.0000
6.000
2/15/26
86-18+
7.089
0.7751
Assume parallel yield shifts on June 25, 1996 of up and down 200 bp.
Which bond would be the cheapest-to-deliver in each scenario?
First Step: Current CTD
Futures Price: 107-05
Number of
6/25/96
9/30/96
Coupons
Implied
Maturity
Accrued (%)
Accrued (%)
Paid (k)
Repo (%)
11.250
2/15/15
4.080
1.406
1
5.02
8.000
11/15/21
0.913
3.000
0
2.17
6.000
2/15/26
2.176
0.750
1
–8.36
Coupon (%)
The 11¼% of February 15, 2015 (assuming delivery on September 30,
1996, the latest date possible) would have an implied repo of 5.02%, as
seen in the following equation:
é
96 ö
46 ö ù
æ
æ
107 .156250% ´ 1.3089 = ê145.376545% ´ ç 1 + RepoImplied ´
÷ - 1.406250% - 5.625% ´ ç 1 + RepoImplied ´
÷
è
è
360 ø
360 ø úû
ë
Now, if we only had the futures price in the up and down scenarios, we
could calculate the implied repo rates and determine the CTD. We will
use the following assumption to estimate the scenario futures prices:
the implied repo rate of 5.02% on the cheapest-to-deliver maintains a
consistent relationship to short-term rates under the other scenarios, so it
changes according to the scenario yield-curve shift.
220
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Determining the Cheapest-to-Deliver
(Continued)
September 1996 Bond Futures (Trade Date June 25, 1996)
Scenario: Down 200 bp
CTD Implied Repo: 3.02%
Futures Price
Coupon (%)
Maturity
Yield (%)
Price (%)
if CTD (%)
11.250
2/15/15
5.191
171-25+
130-02
8.000
11/15/21
5.252
138-087
137-10
6.000
2/15/26
5.089
113-27+
146-02
Maturity
Yield (%)
Price (%)
Repo (%)
11.250
2/15/15
5.191
171-25+
3.02
8.000
11/15/21
5.252
138-087
–16.51
6.000
2/15/26
5.089
113-27+
–37.55
Futures Price: 130-02
Implied
Coupon (%)
Scenario: Up 200 bp
CTD Implied Repo: 7.02%
Futures Price
Coupon (%)
Maturity
Yield (%)
Price (%)
if CTD (%)
11.250
2/15/15
9.191
118-056
89-24
8.000
11/15/21
9.252
87-26
87-12
6.000
2/15/26
9.089
68-141
87-30
Maturity
Yield (%)
Price (%)
11.250
2/15/15
9.191
118-056
8.000
11/15/21
9.252
87-26
6.973
6.000
2/15/26
9.089
68-141
4.65
Futures Price: 87-12
Implied
Coupon (%)
Repo (%)
–2.71
The implied repo for
the CTD also shifts
in each scenario
We can calculate
the futures price
that would cause
each bond to have
that implied repo
rate; the lowest
such futures price is
the estimate of the
scenario futures
price
No investor should
pay more than that
lowest price,
because a futures
seller could create
arbitrage by selling
the futures, buying
the CTD, financing
it, and delivering it
against the futures
Due to its rich yield,
the 6% bond does
not become the CTD
in the up scenario
despite its longer
duration
3 Discrepancy due to rounding the futures price to the nearest 32nd
221
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Deliverable Bonds
September 1996 Bond Futures at 107-05
(Settlement June 26, 1996)
This is the complete
range of securities
that are deliverable
against the
September 1996
bond futures
There are four
potential delivery
candidates; no
other bond even
comes close
Coupon (%)
Maturity
Factor
11.250
2/15/15
10.625
Price
Implied
Duration
Repo (%)
Price (%)
Yield (%)
1.3089
141-09+
7.191
9.346
5.02
8/15/15
1.2525
135-05+
7.207
9.559
5.03
9.875
11/15/15
1.1816
127-16
7.217
9.743
4.93
9.250
2/15/16
1.1215
121-00+
7.228
9.917
4.87
7.250
5/15/16
0.9266
100-04+
7.235
10.457
3.87
7.500
11/15/16
0.9505
102-22+
7.243
10.498
4.00
8.750
5/15/17
1.0750
116-02
7.245
10.300
4.53
8.875
8/15/17
1.0877
117-15+
7.245
10.328
4.42
9.125
5/15/18
1.1146
120-13+
7.249
10.432
4.31
9.000
11/15/18
1.1027
119-06+
7.252
10.558
4.07
8.875
2/15/19
1.0901
117-28
7.254
10.633
3.99
8.125
8/15/19
1.0128
109-20+
7.258
10.900
3.45
8.500
2/15/20
1.0522
113-29+
7.258
10.904
3.45
8.750
5/15/20
1.0789
116-26
7.257
10.894
3.44
8.750
8/15/20
1.0790
116-28
7.257
10.940
3.31
7.875
2/15/21
0.9865
107-01+
7.256
11.234
2.53
8.125
5/15/21
1.0133
109-29+
7.257
11.213
2.64
8.125
8/15/21
1.0132
109-31+
7.254
11.257
2.43
8.000
11/15/21
1.0000
108-19+
7.252
11.330
2.17
7.250
8/15/22
0.9185
100-00+
7.248
11.654
1.15
7.625
11/15/22
0.9592
104-14+
7.243
11.590
1.14
7.125
2/15/23
0.9044
98-21+
7.237
11.774
0.41
6.250
8/15/23
0.8079
88-14+
7.226
12.146
–1.02
7.500
11/15/24
0.9445
103-16
7.208
11.937
–1.20
7.625
2/15/25
0.9580
105-07+
7.190
11.953
–2.04
6.875
8/15/25
0.8740
96-19+
7.152
12.263
–4.43
6.000
2/15/26
0.7751
86-18+
7.089
12.687
–8.36
222
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Financing Option
Choosing the Date of Delivery
When the financing cost is less than the coupon accretion on the
cheapest-to-deliver, there will be an incentive to deliver as late in the
month as possible. An investor could buy bonds, finance them until late
in the expiration month, and sell the futures. If the curve flattens enough
(by short rates rising), the investor could “repurchase” the financing at a
discount and deliver early in the month instead.
For example, on September 2, 1996, assume the 111/4% due February 15,
2015 was the cheapest-to-deliver, priced at 141-09+ (7.191% yield). A
futures seller who had a financed position in the bond would earn
0.856% of accrued interest for the 29 days between September 2 and
September 30. At a 5% repo rate, the financing for the same period
would be 0.571%, so the futures seller would hold the bond, continue to
finance it, and make delivery at the end of the month. If repo rates rose
to 8%, the cost of the financing would rise to 0.913%, so the futures
seller would deliver the bond on September 2 rather than continuing to
finance it. If the bond were already financed to September 30, the
futures seller would borrow the bond and deliver it. The futures seller
would then earn the spread between the two repo rates. At the end of the
month, the seller would receive the bond from the original financing and
return it to the bond lender.
The seller of a
futures contract also
has the right to time
the delivery within
the exercise month
The optimal time to
deliver depends on
market conditions
for financing
deliverable bonds
The futures seller, by making delivery, would lose the ability to exercise
any of the embedded options. The seller, therefore, has a natural bias
towards delivering late in the month. If the financing option indicates
better economics for early delivery, the seller still needs to ascertain if
the cost of financing the bond position for another day is greater than the
decay in the value of the options. The seller will deliver early only when
this condition is met.
223
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
The Wild Card Option
An Intra-Day Option During the Delivery Month
The wild card
option provides
intra-day exposure
to market
changes between
2:00 PM and
8:00 PM CST during
the delivery month
This option can
provide value in a
sharp market
decline when the
cheapest-to-deliver
has a factor greater
than one, or in a
sharp market rally
when the cheapestto-deliver has a
factor less than one
Consider an investor with a short futures position when the cheapest-todeliver has a factor of 1.5000 and a price of 150-08 during the delivery
month and the price of the futures has closed at 100-00. This bond trades
1
/4 point rich to the futures, predominantly because of the difference
between the bond’s coupon and its cost of financing. Suppose that the
price of the bond falls by 11/2 points after the close of futures, but before
8:00 PM CST. When trading begins the next day, the price of the futures
would be 99-00, so the investor would expect to receive a one-point
variation margin. On the other hand, the investor could buy the bonds in
the market for 148-24 and give notice of delivery based on the closing
futures price of 100-00. This would create a gain of 11/4 points, 1/4 point
better than waiting until the next day: a payoff under the wild card
option.
Alternatively, suppose the cheapest-to-deliver has a factor of 0.7500 and
a price of 75-08 during the delivery month and that the price of the
futures has again closed at 100-00. This bond also trades 1/4 point rich to
the futures because of the value of delaying delivery. In this case,
suppose that the price of the bond rises by 11/2 points after the close of
futures, but before 8:00 PM CST. The opening price of futures the next
day would be 102-00, so the investor would expect to post a two-point
variation margin. On the other hand, the investor could buy the bonds in
the market for 76-24 and give notice of delivery based on the closing
futures price of 100-00. This would reduce the loss to 13/4, which is
1
/4 point better than waiting until the next day.
Note that exercising this option required sacrificing the 1/4-point value of
postponing delivery, but it was worth it. All of the value of the wild card
option comes from the time difference between the close of futures at
2:00 PM CST and the deadline for electing delivery at 8:00 PM CST.
224
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
The Basis
Basis (for the CTD) is the primary number by which many participants The basis of a bond
sum up a futures contract. Basis is defined as
is the combined
Basis = PriceBond - PriceFutures ´ FactorBond
The components of basis include carry (the difference between the
bond’s accrual rate and financing cost), the value of the substitution
option and other options inherent in the delivery process, and arbitrage.
Since both carry and the value of the substitution option decrease as the
contract nears expiration, we can expect the basis to approach zero
(convergence) as the contract ages.
value of all of the
elements that
differentiate the
price of the bond
from the proceeds if
that bond was
delivered against
the futures
immediately (even
prior to the delivery
period)
Basis can also be described as the sum of carry and the basis net of carry If no bond is
(BNOC); the BNOC comprises the market’s valuation of the delivery mentioned, the
options and arbitrage:
basis refers to the
Basis=ValueDAccrued Plus FV of Coupons Paid - ValueFinancing on Price+Accrued + ValueOptions + Arbitrage
= ValueCarry + BNOC
é
DateDelivery - DateCouponi ö ù
Coupon k æ
´ å ç 1 + RepoActual ´
÷ú
ê Accrued Delivery - AccruedSpot +
2
360
øú
i =1 è
ê
Basis = ê
ú
DateDelivery - DateSpot
ê
ú
+ BNOC
ê- PriceSpot + AccruedSpot ´ RepoActual ´
ú
360
ë
û
(
)
cheapest-to-deliver
“Long the basis”
means long the
bond and short the
futures; “Short the
basis” means short
the bond and long
the futures
Implementing a basis trade (speculating on an increase or decrease in the
BNOC) involves transacting in three securities, each with a bid-ask
spread: the cash security, the cash security financing (repo), and the
futures. For example, going long the basis involves buying a bond (at the
offered price), financing it (at the bid rate), and selling the futures (at the
bid side).
225
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Specific-Bond Basis When Yields Change
The basis of a
specific bond will
resemble an option
strategy
The basis of a lowduration security
replicates a put
option: its value
increases when
rates rise
The basis of a highduration security
replicates a call
option: its value
increases when
rates decline
The basis of a midduration security
replicates a straddle
Basis for 11.250%
Due February 15, 2015
Basis of Low-Duration
Bond
Put Option
Basis for 6.000%
Due February 15, 2026
Basis of High-Duration
Bond
Call Option
Basis for 8.000%
Due November 15, 2021
Basis of Mid-Duration
Bond
Straddle
226
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Hedging with Futures
Futures Are Commonly Used to Hedge Positions
If there were no chance that the cheapest-to-deliver or the basis would
change, the price of the futures would change by roughly the same
percentage as the price of the cheapest-to-deliver. The deviation will be
due to the basis, which is small relative to the price of the bond. If the
basis were zero, then: PriceBond = PriceFutures ´ FactorBond, and
DPriceCheapest-to-Deliver PriceCheapest-to-Deliver
Dy
=
DPriceFutures PriceFutures
Dy
The percentage
change in the
futures price is
roughly the same
as the percentage
change in the price
of the cheapest-todeliver; therefore,
they have nearly
the same price
duration
Recall that this is the price duration of the cheapest-to-deliver. Since the
basis is not zero, duration can be calculated more precisely. However, the
basis includes the value of carry and options, which is also sensitive to
changes in long-term and short-term interest rates.
The dollar duration
The substitution option implies that the duration of the futures should
approximate the modified price duration of each of the possible
deliverables, weighted by the probability of that bond’s being delivered.
As of June 25, 1996, the price durations of deliverable bonds varied from
9.35 to 12.69, a significant range. However, there is very low probability
of a high-duration bond being delivered in a low-yield environment, so
the actual duration of the futures is very near 9.35.
of the futures
equals $100,000
multiplied by the
futures price (as a
percent) multiplied
by the modified
price duration of the
cheapest-to-deliver
On June 25, 1996, the cheapest-to-deliver, the 11¼% of February 15,
2015, had a price of 141-09+ and a price duration of 9.35. The long
bond, the 6% of February 15, 2026, had a price of 86-18+ and a price
duration of 12.69. The futures had a price of 107-05. The dollar
duration of $100,000,000 long bonds was $1,098,000,000. To offset the
duration using futures would require shorting
$1,098 ,000 ,000
= $109,700 ,000
107 .156% ´ 9.35
or 1,097 contracts. This hedge would be subject to basis risk from
steepening or flattening in the yield curve.
227
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Margin and Rate of Return on Futures
Trade Date June 25, 1996
Futures, like any
other asset, have a
rate of return
Futures are often held in a portfolio for speculative or hedging reasons.
Therefore, it is important to be able to calculate their total rate of return,
like any other asset, to get an accurate measure of portfolio rate of return.
Assume the cheapest-to-deliver is the 111/4% of February 15, 2015. The
bond’s price is 141-09+, and the September 1996 futures price is 107-05.
On September 16, 1996, the bond rallies by two points, remaining the
cheapest-to-deliver. The short-term rate is 5% and remains unchanged.
What is the annualized bond-equivalent return of the bond and the future
if bought on June 25, 1996?
Note: There is gradual convergence (decrease in basis) on the future
during the holding period. Assume that this convergence all takes place
on September 16, 1996 for ease of calculation and that the BNOC (value
of options and any residual arbitrage) remains unchanged.
Hints: What is the settlement date?
What are the cash flows?
What is the factor?
228
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Margin and Rate of Return on Futures
(Continued)
Trade Date June 25, 1996
Assume the cheapest-to-deliver is the 111/4% of February 15, 2015. The
bond’s price is 141-09+, and the September 1996 futures price is 107-05.
On September 16, 1996, the bond rallies by two points, remaining the
cheapest-to-deliver. The short-term rate is 5% and remains unchanged.
What is the annualized bond-equivalent return of the bond and the
futures if bought on June 25, 1996?
Bond
Futures
Date of Initial
Investment
June 26, 1996
June 25, 1996
Initial Investment (%)
145.377
Price + Accrued
1.750
Initial Margin
Factor
1.3089
Initial Basis (%)
1.040
Basis = Price Bond - Price Futures ´ Factor Bond
BNOC (%)
–0.01
BNOC = Basis - ValueCarry
Final Basis (%)
The leverage
intrinsic to a futures
contract provides
the opportunity for
very high (and low)
rates of return on
capital invested
For many
applications, the
dollar change in
value is the critical
quantity; percent
return can be
misleading when
the initial
investment is small
or the time period is
short
0.122
Basis = ValueCarry + BNOC
Final Futures Price (%)
109-12
PriceFutures =
Variation Margin (%)
PriceBond - Basis
FactorBond
2.219
September 16, 1996
Value (%)
149.956
Price + Accrued +
Reinvested Coupon
3.969
Initial Margin + Variation Margin
Annualized BondEquivalent Rate
of Return (%)
14.16
990.60
229
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Expected Rate of Return
Trade Date June 25, 1996
Calculating the
expected rate of
return on futures
requires assessing
the cheapest-todeliver and the
basis in each
scenario
On June 25, 1996, the September 1996 futures price was 107-05.
Assume the following deliverable bonds:
Coupon (%)
Maturity
Price (%)
11.250
2/15/15
141-09+
8.000
11/15/21
108-19+
6.000
2/15/26
86-18+
Accrued (%)
Yield (%)
Factor
Further assume the following parallel-yield scenarios on July 25, 1996,
with all convergence occurring on that date. What is the expected rate
of return on the futures?
Probability
Cheapest-to-
Scenario
(%)
Deliver
+200 bp
30
0
40
–-200 bp
30
Basis (%)
Futures
Variation
Price (%)
Margin (%)
Probability-Weighted Average Variation Margin:
Average Bond-Equivalent Rate of Return on Futures:
Hints: How will you determine which bond is the cheapest-to-deliver?
What assumptions do you need to make to estimate horizon
basis?
What is the embedded option value (BNOC)?
230
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Expected Rate of Return (Continued)
Trade Date June 25, 1996
On June 25, 1996, the September 1996 futures price was 107-05. In order to calculate
Assume the following deliverable bonds:
the basis at horizon,
Coupon (%)
Maturity
Price (%)
Accrued (%)
Yield (%)
Factor
11.250
2/15/15
141-09+
4.080
7.191
1.3089
8.000
11/15/21
108-19+
0.913
7.252
1.0000
6.000
2/15/26
86-18+
2.176
7.089
0.7751
Further assume the following parallel-yield scenarios on July 25, 1996,
with all convergence occurring on that date. What is the expected rate
of return on the futures?
The cheapest-to-deliver in each scenario can be determined from the
earlier example, or using similar reasoning, the BNOC (–0.009% in this
case) can be held constant when the market moves. The basis and
implied futures price can then be calculated assuming each bond, in turn,
is the cheapest-to-deliver. The bond which implies the lowest futures
price is the actual cheapest-to-deliver; no investor should pay a higher
price for the futures knowing that they could be delivered that bond.
Probability
Cheapest-to-
Futures
Variation
Scenario
(%)
Deliver
Basis (%)
Price (%)
Margin (%)
+200 bp
30
11/15/21
0.279
87-17
–19.625
0
40
2/15/15
0.717
107-11
0.188
–200 bp
30
2/15/15
1.047
130-10
23.156
Probability-Weighted Average Variation Margin:
1.135%
Average Bond-Equivalent Rate of Return on Futures:
3621%
you need a reporate assumption
and an assumption
regarding the value
of any options
Q1: Why does the
February 2026 not
become the
cheapest-to-deliver
in the interest-rateup scenario?
Q2: Why is the
expected variation
margin positive
even though the
rate changes are
symmetrical?
231
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Valuing Options Embedded in Futures
Like any security
with embedded
options, the random
future cash flows
from a futures
position can be
discounted at an
appropriate rate to
derive an
embedded option
value
Because the
evaluation needs
both long- and
short-term rates at
each stage,
Heath–Jarrow–
Morton would be a
logical choice
The options embedded in futures can be valued using simulation. For
each date along each path, the model would project short-term rates and
prices for the deliverable bonds, allowing for normal variation between
bonds. A futures contract could then be priced using that day’s cheapestto-deliver bond, its factor, and its price and basis for that day.
The basis, in turn, depends on a valuation of the carry and the options
embedded in the futures. The carry depends on short-term rates, the
coupon and price of the cheapest-to-deliver, and an assessment of
whether there is a preference for delivering at the beginning or the end
of the month. The valuation of the options depends on volatility and how
close the options are to being at-the-money. Once the futures are priced,
the change in price from the prior day leads to that day’s variation
margin.
Each day’s variation margin would be discounted back to the settlement
date using the appropriate short-term rate for that path. The average
value of these payments would be compared to the average value of the
payments on a futures contract with no variability as to the deliverable
bond or the delivery timing.
There is another, simpler approach to calculating fair value for the
futures contract. The first step is to generate a distribution for yields of
the current 30-year on the delivery date using a binary tree. Every bond
is then assigned a spread to the current 30-year so that the bond’s average
probability-weighted price using the tree equals the bond’s arbitrage-free
forward price (using that bond’s specific repo rate). Then, the economics
of delivery are priced at each node and valued back to settlement to
obtain a fair value for the futures. This methodology does not account
for the volatility of the spreads between issues, the volatility of the shape
of the curve (financing option), or the timing of margin payments.
232
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Increasing Futures Liquidity
Although each of the features of the bond futures contract was designed
to increase liquidity, the complexity of the futures may have some
liquidity-reducing effects. Offsetting these effects is the attraction of
basis trading to speculators. Clearly, there is no hurry to tamper with the
formula of a successful contract. However, liquidity-enhancing
suggestions might include:
•
Limiting the range of deliverable securities. The average amount
outstanding per bond is larger now than when the bond futures were
developed. Therefore, it should be possible to place stronger limits
on maturity or place a range on duration to reduce the value and
importance of the delivery option. Potential consequences of the
limit might be less hedging of bonds that are no longer deliverable or
a more complicated description of the contract. On the other hand,
the duration of the current deliverable, the 111/4% due February 15,
2015, is closer to the duration of a 10-year note than to the duration
of the long bond. It is thus arguable how relevant the futures
currently are for long-bond hedging.
•
Delivery mechanics could be simplified to reduce complexity.
Notice of delivery could include security identification and could be
required prior to the close of futures. This would eliminate the wild
card option, but might provide more incentive for price manipulation
after notice.
•
The period during which delivery can be made could be curtailed.
This would reduce the value of the financing option, but would
constrain holders of short futures positions to make a decision more
quickly. Under the current contract, futures owners must be ready to
accept bonds anytime during the delivery period.
Reducing the value
of the futures
contract’s
embedded options
could possibly
increase liquidity;
however, an ability
to trade the basis
attracts speculators,
which increases
liquidity
233
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 7 Exercises
Prices as of June 25, 1996
1. The June 25, 1996 September 1996 futures price was 107-05.
Assume that there were three deliverable bonds: the 11¼% of
February 15, 2015 (priced at 141-09+), the 8% of November 15,
2021 (priced at 108-19+), and the 6% of February 15, 2026 (priced at
86-18+). Assume the short-term rate is 5%. Which bond would be
the cheapest-to-deliver if interest rates increased by 70 bp?
2. The September 1996 futures price on June 25, 1996 was 107-05.
How much would the cheapest-to-deliver, the 11¼% of February 15,
2015 (priced at 141-09+) have to richen before the 8% of November
15, 2021 (priced at 108-19+) becomes the cheapest-to-deliver?
3a. The June 25, 1996 price of the September 1996 futures was 107-05.
The cheapest-to-deliver was the 11¼% of February 15, 2015 (priced
at 141-09+). You have an outstanding liability of $100,000,000 with
a duration of 10. How many futures would you buy to hedge it?
b. How big an interest rate move would it take to exhaust the initial
margin?
c. Assume a short-term rate of 5% and unchanged option values. What
would be the one-month bond-equivalent rate of return on the futures
if prices do not change? If yields do not change?
234
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 8
Corporate Bonds
This chapter is an excerpt from A Morgan Stanley Guide to Fixed Income Analysis by
Andrew R. Young, ©2003 Morgan Stanley & Co. Incorporated.
235
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
In This Chapter, You Will Learn...
•
The Characteristics of Corporate Bonds
•
How Corporate Bonds Are Priced
•
Option-Adjusted Spread Models
•
About Credit Risk
•
About Types of Corporate Bonds
– Sinking-Fund Bonds
– Floating-Rate Notes
– Adjustable-Rate Preferred Stock
– Credit Derivatives
– Brady Bonds
•
Corporate Debt Retirement Analysis
236
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Overview
Corporate bonds have many different structures. Some of the differences
relate to the interest rate profiles of the bonds and some relate to the level
of credit riskiness of the obligor. Investors strive to fairly compare
different securities and determine value or to accurately hedge the value
of the securities.
Corporate bonds are frequently callable. By calling the bonds, the issuer
leaves the investor to reinvest at a lower yield. Often, the bonds can be
called only after some future date, perhaps at a premium to par, to offer
investors call protection. There are other types of options that can also
appear in a corporate bond, including puts, extension options (the
issuer’s right to leave the bonds outstanding longer than the original
term), and sinking-fund options. Using an option-valuation model,
investors seek to normalize corporate bonds for the value of any
embedded options.
Different bonds
with different
structures are not
directly comparable
By removing the
economics of any
embedded options
and normalizing for
credit, we can
construct a
framework for
comparing relative
value
Another factor in comparing corporate bonds is credit. Investors expect
to earn a higher spread when investing in riskier assets, both to cover the
expected loss from defaults and to compensate them for taking the extra
risk. Different bonds have different types and amounts of collateral, and
investors may seek to isolate and evaluate the returns attributable to each
separate component.
Ultimately, investors want to be able to compare different securities by
stripping away as much of the structure as possible so that the remaining
exposures are similar. Even then, evaluating the fairness of different
spreads for different issuers or different segments of the yield curve is a
challenging assignment.
237
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Quoting Corporate Bonds
Corporate bonds
are generally
quoted on a
spread-to-maturity
or a spread-to-call
basis
This is the simplest
way to quote the
bonds and makes it
easy to compute the
price, but it is
unsatisfactory for
analysis
Many corporate
bonds are callable
prior to maturity,
often at a premium
to par
“True” spread,
duration, and
convexity can be
calculated with the
same techniques
used to price
options
Corporate bonds are generally quoted as a spread to U.S. Treasuries. For
example, a trader could quote a corporate bond as “80 off the 30-year”
or as “80 off the curve.” The yield-to-maturity of the bond would then be
80 bp plus either the yield of the 30-year benchmark Treasury or the
yield of the closest-maturity Treasury, respectively. This yield would
define the price of the bond using the usual coupon-bond-pricing
formula from Chapter 2, rounded to three decimal places. Corporate
bond yield quotations are usually semi-annual, regardless of the payment
frequency, and use a 30/360 calendar.
The fact that a bond is callable does not affect the methodology of
computing its price; traders include the value of the short call option
when they quote the spread. Therefore, the spread on a callable bond
will generally be wider than the spread on a noncallable bond, leading to
a lower price.
When the issuer is likely to call the bonds, traders will sometimes quote
a bond on a yield-to-call basis (possibly as a spread over a Treasury
maturing near the call date). In the pricing formula, the call date is used
for the maturity, and the call price, including premium, is used for the
redemption value to compute the price of a bond trading to call.
Some bonds have sinking funds, which require the issuer to partially
redeem prior to maturity. These bonds sometimes trade on a yield-toaverage-life basis (or as a spread over a Treasury maturing near the
average life). The average life is the average time until (or date of)
principal repayment.
238
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Sources of Corporate Bond Prices
The S&P 500 comprises 500 individual stocks. These same 500
corporations have issued over 10,000 bonds. Each bond has its own
individual maturity and coupon; many of them are callable and have
different option characteristics. The bonds may be secured by different
collateral. Furthermore, some bonds have esoteric structures that add
complexity to the evaluation process.
One methodology for pricing corporate bonds is called matrix pricing.
Matrix pricing designates a relatively small number of bonds as anchors,
which are priced frequently. The other bonds are priced relative to the
anchors, with relationships that are updated periodically. One problem
with matrix pricing is that if an anchor undergoes company-specific
pressures, the new price for that anchor can cause a mispricing of all the
other bonds whose prices depend on it. Another problem is the different
embedded options, which will cause the bonds to reprice differently as
interest rates change. This change may not be picked up quickly in the
matrix.
Accurate corporate
prices are very
difficult to obtain
Often, corporate
bonds are matrix
priced or priced as
some aggregation
of other prices
Another approach is for a third party to obtain prices from several firms.
Then an algorithm could identify and reject possible mispricings and
outliers. The prices would not necessarily reflect actual transactions and
would only be as good as the diligence of the contributors.
The lack of a public transaction record definitely impairs liquidity in the
corporate market. In the equity market, traders can buy or sell a portfolio
based on the individual stocks’ closing prices and the portfolio’s
statistical characteristics without knowing the identity of the individual
stocks. This type of transaction has been a major source of liquidity in
the equity market, but it is impossible in fixed income due to the lack of
reliable closing prices in bonds.
239
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Credit Risk and Defaults
Corporate bonds
have an element of
credit risk — they
can default
In default, the
assets of the
corporation pay off
creditors in order of
seniority, subject to
bankruptcy court
procedures and
review
Rating agencies
assess the
probability of
default for various
issuers
The capital structure of a corporation comprises debt and equity. The
equity is the most junior piece of the capital structure and only has a right
to receive cash as long as the company appears able to pay bondholders.
All bondholders are not equal: some have access to specific assets of the
corporation as collateral, others are unsecured, and still others are
subordinated to other lenders. If a corporation defaults on any of its
debt, it usually defaults on all of it, due to cross-default provisions found
in most bond indentures. However, the more senior the debt, the more
the bondholder can expect to recover through liquidation and the lower
the bond’s market spread.
The rating agencies make assessments of a corporation’s financial
structure, management, and prospects and assign a credit rating. The
credit rating implies a safety level or likelihood of default.
Rating-Implied Default Probabilities (Based on Historical Experience)1
Moody’s Rating
Average 1-Year
Default Rate
Cumulative 10-Year
Default Rate
Aaa
0.00%
0.74%
Aa
0.03%
1.13%
A
0.01%
1.73%
Baa
0.12%
4.61%
Ba
1.36%
20.94%
B
7.27%
44.31%
The cumulative 10-year default rates are so big because, for example,
Baa or better companies rarely default while so rated, but they can
deteriorate, get downgraded, and default later.
1 Global Credit Research, “Historical Default Rates of Corporate Bond Issuers, 1920–1996.”
Moody’s Investors Service, New York, January, 1997.
240
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
A Callable Bond’s Price vs Yield
ABC 81/8% Maturing April 15, 2016, Yielding 8.28%,
Callable at 103.28% in 1997
As yields fall, the
price of a callable
bond begins to
“cushion” and
increases much
more slowly
In the extreme,
high-coupon bonds
that are currently
callable experience
no price
appreciation as
yields decline
At high yields, a callable bond performs like a noncallable bond.
However, as yields decline, a callable bond’s embedded call option will
move in-the-money (i.e., the issuer is more likely to call the high-coupon
debt and replace it with the lower, market, interest rate). Therefore, there
is a region where a callable bond’s price will be negatively convex.
The duration of a callable bond ranges between zero and the duration of
a noncallable bond. Under many circumstances, when yields decline, so
does the duration. When yields are above the coupon, the duration
approaches the duration of the noncallable bond.
Being long a
callable bond is
equivalent to
owning a
noncallable bond
and being short a
call option
Properly valuing the
embedded options
leads to a better
understanding of a
security’s value and
risk
At very low yields, a bond that is nearly certain to be retired after a noncallable period (unlike this ABC bond, which is currently callable) can
have positive convexity again. The positive convexity is due to the
bond’s resemblance to a noncallable bond maturing on the call date.
241
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Option-Adjusted Spread (OAS)
Market convention
is to quote a spread
to Treasuries that
gives a yield-tomaturity on a
corporate bond
Option-adjusted
spread is a measure
of the spread to
Treasuries of a noncallable bond of the
same issuer
An investor who owns a callable bond should not expect to earn the
quoted (static) spread over Treasuries because, under some interest rate
scenarios, the bond will be called away to the investor’s detriment.
However, the investor does expect to earn some positive spread to
Treasuries to compensate for the credit risk and lower liquidity of the
corporate bond. This spread is called option-adjusted spread (OAS).
0 £ SpreadOption- Adjusted £ SpreadStatic
In a binary-tree model with constant option-adjusted spread, the OAS is
added to the short rate at each node in the tree. The modified short rates
(risk-free rates plus OAS) are used to discount future cash flows and
determine option exercise. Because the option will be exercised if it is
to the disadvantage of the bondholder, reducing the effective yield on the
security, the OAS is effectively the static spread less the value of the
option.
Given an OAS, it is possible to price the bond, either by stepping
backward through the tree or, potentially, by simulating future interest
rates if the bond has embedded path-dependent options. OAS analysis
can also provide option-adjusted duration (OAD), option-adjusted
convexity (OAC), and option value. OAD and OAC are usually shown
with respect to a change in the underlying yield curve, but showing the
sensitivity to a change in OAS is also possible.
Given a price, the OAS can be determined using the Newton–Raphson
method. If the prices are being determined using simulation, it is
important to reuse the same random numbers and paths through the tree.
Otherwise, the algorithm might not converge on an answer because the
random changes in price (caused by the random variations in tree paths)
could cause the OAS to wobble around its true value.
242
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Option-Adjusted Spread (Continued)
Example
The interest rate tree below is the four-period tree derived in Chapter 6.
It states that the one-period zero-coupon annually compounded rate
today equals 6.08%. If we are analyzing a 23-year bond with a 7%
annual coupon that can be called at par in three years only, and we
assume the OAS is 40 bp, what is the price of the bond? Assume that if
the bond is not called, its yield-to-maturity in three years will be the oneyear rate two years forward.
There are two
methodologies for
evaluating this
bond: explicit
enumeration and
backward induction
Q1: If the observed market
price is 92%, is the actual OAS
higher or lower than the
assumption?
Q2: If the bond were noncallable and the OAS were
40 bp, what would be the price
of the bond?
Hints: What are the bond
values at the end of three years
in each scenario? There are four
possible paths for interest rates
during the three-year period.
243
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Option-Adjusted Spread (Continued)
Example: Explicit Enumeration
One methodology
for determining the
prices under the
interest rate tree is
to enumerate all
interest rate
possibilities
The terminal price is
either par or the
price of a 20-year
noncallable bond
with a 7% coupon
yielding the short
rate two years
forward, whichever
is less
The discounting
yields are the shortterm rates in the
tree plus the OAS
Example 1 — Callable Bond
3-Year
Fwd Price (%)
Period 3
Yield (%)
Period 2
Yield (%)
Period 1
Yield (%)
Scenario
PV (%)
Up/Up
75.555
9.84
8.50
6.48
77.689
Up/Down
96.090
7.38
8.50
6.48
95.732
Down/Up
96.090
7.38
6.17
6.48
97.689
100.000
5.57
6.17
6.48
102.421
Scenario
Down/Down
Average
93.383
This price of 93.383% is greater than 92%, so if the bond is priced at
92%, its OAS is more than 40 bp.
Example 2 — Noncallable Bond
Scenario
3-Year
Fwd Price (%)
Period 3
Yield (%)
Period 2
Yield (%)
Period 1
Yield (%)
Scenario
PV (%)
Up/Up
75.555
9.84
8.50
6.48
77.689
Up/Down
96.090
7.38
8.50
6.48
95.732
Down/Up
96.090
7.38
6.17
6.48
97.689
116.990
5.57
6.17
6.48
116.657
Down/Down
Average
96.942
244
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Option-Adjusted Spread (Continued)
Example: Backward Induction
The other, more
elegant,
methodology is to
set up a tree with
the OAS built in and
use backward
induction
245
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
The Constant-OAS Model
Zero-Coupon Yields as of June 25, 1996
The most commonly
used OAS model,
constant OAS, is
inconsistent with
market information
The constant-OAS
curve crosses the
credit curves
between which it
should lie
The UST zerocoupon curve is
more convenient
than the par-coupon
curve for pricing
individual cash
flows; the curves
are entirely
consistent
descriptions of the
market
This model is
unrealistic in that all
volatility is in the
risk-free rate; none
is in the OAS
Assume ABC Corporation has outstanding an 81/8% bond due April 15,
2016, priced at a constant OAS of 57 bp. If the company is rated A– by
S&P, each of its cash flows should be discounted at a yield between the
A-rated curve and the BBB-rated curve. Under the constant-OAS model,
the short cash flows are discounted at too high a rate, and the long flows
are discounted at too low a rate.
246
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Better Models for Corporate Bond OAS
Constant-Proportion OAS
Zero-Coupon Yields as of June 26, 1996
This OAS (zerocoupon) curve is a
fixed percentage of
the difference
between
surrounding credit
curves and provides
spreads in this
pricing model
The solution is the
fixed percentage,
found iteratively
through an option
model, that
properly prices the
bond
This model finds the OAS zero-coupon curve that prices the bond and
lies a fixed percentage of the distance between two surrounding credit
curves. One of the features of this model is that it provides a means for
comparing bonds with different structures and terms. For example, if
there were two investment alternatives, a 2-year ABC bond priced at 30
over Treasuries and a 10-year ABC bond priced at 60 over Treasuries,
constant-OAS analysis would show that the spread on the longer bond
was higher. Since spreads generally increase with maturity, this result
does not lead to a relative-value conclusion. If the constant-proportion
OAS model determined that the 2-year was priced on the curve 55% of
the way between A and BBB, while the 10-year was priced 48% of the
way, there would be some evidence that the 2-year was priced more
cheaply. The major assumption underlying this analysis is that the credit
curve for ABC is similar to the general A and BBB curves.
The OAS zerocoupon curve is
equivalent to a parcoupon-bond curve;
the entire curve can
be identified by
either curve’s
spread to Treasuries
at any point
247
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Better Models for
Corporate Bond OAS (Continued)
Issuer-Specific OAS
Zero-Coupon Yields as of June 26, 1996
An even better
model would be
based only on
bonds of
comparable quality;
however, this
methodology is
subjective and much
more laborintensive
Possible extension:
A two-factor model
to account for
volatility in rates
and spreads
When there is doubt about how the shape of a corporation’s credit curve
compares to the shape of the market as a whole, or when the magnitude
of the investment decision demands the utmost accuracy, it may be
worthwhile to develop an issuer-specific OAS curve. This curve would
be the one that did the “best” job pricing ABC bonds or, alternatively, the
curve that did the best job pricing a relevant subset of the market. The
specific bond would then be priced as a constant spread to that curve. A
positive spread would imply relative cheapness, and a negative spread
would imply relative richness. Because the issuer-specific OAS curve is
customized, it is a more demanding analysis than the constant- or
constant-proportion-OAS model.
248
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Normalizing for Structure and Credit
There are many different types of bond structures in the market.
Investors analyze them to gain insight into the best investment
opportunities. One of the more difficult problems is understanding how
the corporations evaluate their options, to better predict how the
corporations will behave. In addition to standard callable bonds, various
important structures (explained in more detail in the following pages)
include:
The various OAS
models can be used
to value the options
embedded in
securities
The securities can
then be compared
on an option-free
basis
Sinking-Fund
Bonds:
How the corporation chooses to meet its sinkingfund obligations and whether it exercises its right to
sink faster than necessary are important factors in
The risk profile of
analyzing sinking-fund bonds.
Floating-Rate
Notes:
Although new-issue floating-rate notes are
generally priced at par, as the credit of the issuer
changes, the price of the bond will change.
Floating-rate notes priced at a discount can have a
negative duration.
Adjustable-Rate
Preferred Stock:
Adjustable-rate preferred stock has a broad Further adjustments
range of types of embedded options, which makes can be made for
performing a valuation analysis on it more difficult. credit, but they are
Another element of structure is credit risk. To compare bonds with
different credit quality, it is often helpful to start by normalizing the
spreads for expected defaults. There are other structures that provide
more complex exposures to credit risk. These structures include:
the bond is the risk
profile of the
option-free bond
plus the risk profile
of the embedded
options
more subjective
Credit Derivatives: Allow investors to hedge or speculate on credit risk.
Brady Bonds:
Have emerging-markets exposure, combined with
higher-quality collateral. The value and impact of
the collateral has to be removed in order to be able
to analyze the emerging-markets component.
249
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Other Types of Securities
Sinking-Fund Bonds
A sinking-fund
bond repays a
portion of its
principal prior to
maturity
Sinking-fund bonds call for the redemption of bonds prior to maturity by
delivering to the bond trustee (the institution responsible for enforcing
covenants, collecting payments, and disbursing payments to the
bondholders) either cash or bonds to redeem principal. Sinking-fund
bonds have an average life that is shorter than their maturity.
Many sinking-fund
bonds, with doubleup or triple-up
options, embed a
complicated series
of interrelated,
path-dependent
options
Sinking-fund bonds are complicated due to several embedded options:
Investors need to
understand how the
issuer is likely to
behave in order to
properly adjust for
the impact of the
option
•
Most sinking-fund bonds have a standard embedded call option.
•
The issuer has the option of meeting any sinking-fund payment with
either cash or the same principal amount of the bond. If the bond
trades at a premium, the issuer will deliver cash; if it trades at a
discount, the issuer will buy it in the open market (returning the
bondholders less than par). If bondholders can cooperate, however,
they can refuse to sell and thus force the issuer to pay par.
•
Many sinking-fund bonds have double-up or triple-up options, which
provide the issuer with the possibility of redeeming a multiple of the
required sink amount at par in a low-interest-rate environment. The
issuer can thus redeem bonds either prior to the call date or at a
discount to the call price. Sinking-fund bonds are path-dependent,
because the amount of bonds available to call at any point in time
depends on the prior course of interest rates.
The corporation’s decision to double or triple up is easy if the
premium call is uneconomic and the sink option is economic. The
analysis is more difficult when both the premium call and the sink are
economic. The issuer must then compare the economics of gradually
“calling” the bonds at par through the sink options with the
economics of paying a call premium to redeem the bonds
immediately.
250
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Other Types of Securities
Floating-Rate Notes
Some corporate bonds have coupons that are not fixed; rather, they float
at a spread off an index. The most common index is the London InterBank Offered Rate (LIBOR). A floating-rate bond with a stable credit
should always trade near par, regardless of the interest rate environment.
Slight deviations from par are possible because of the length of time until
the coupon resets, during which time the rate is fixed. Thus, a par
floating-rate bond has a duration equal to the next reset date.
The credit on a floating-rate note can change after it has been issued.
Therefore, corporate floaters are often priced above or below par.
Floaters can be quoted on a discount margin basis. Discount margin is
the spread to the bond’s index that discounts the bond’s future cash flows
back to the bond’s actual present value. Since the future index levels are
not known, the discount factors and future cash flows are not known.
However, the swap market (Chapter 9) can give the fair fixed rate that is
equivalent to the index; spread and discount margin would be added to
that rate. Discount margin is roughly comparable to a fixed-rate bond’s
spread to Treasuries, except that it is a spread to the index instead.
A floating-rate note
priced at par has a
duration equal to
the next reset date
However, a floater
priced at a discount
can have a negative
duration
A floating-rate bond not priced at par can be thought of as a combination
of two securities: 1) a par floating-rate bond with a coupon spread equal
to its discount margin (with a duration equal to the next reset date) and
2) an annuity of the difference between the coupon spread and the
discount margin for the life of the bond (also with positive duration).
An estimate of the bond’s duration is the average of the durations of the
two components, weighted by their respective market values. The
duration of a discount floater can be negative, because it consists of a par
floater with low duration and a short position in a positive duration
annuity. From another perspective, the coupon changes by a greater
percentage amount when rates change (because of its small spread) than
the yield (because of its larger spread). The duration of a premium floater
is positive.
251
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Other Types of Securities (Continued)
Adjustable-Rate Preferred Stock
Adjustable-rate
preferred stock is
an example of a
complicated product
with many
embedded options
The typical adjustable-rate preferred stock has a coupon that resets to a
spread plus a multiple of the highest yield of various Treasuries
(for example, the 3-month bill, the 10-year note, and the 30-year bond).
The coupon can be subject to a cap and a floor, and the security can be
callable. This security illustrates many valuation issues:
•
The fundamental reset of the coupon based on different rates across
the curve embeds a correlation option. Heath–Jarrow–Morton is a
good framework for analyzing this aspect of the security because its
interest rate paths conform to various yield-curve correlations and
because it generates the entire yield curve at every stage.
•
The multiple can be less than one. If so, this contributes duration
because the coupon does not rise as quickly as interest rates, causing
a loss of value in a rising-rate environment.
•
The cap and floor add duration because there are interest rate
scenarios under which the coupon becomes fixed.
•
The call reduces the value and duration contribution of the floor,
because if interest rates fall far enough below the floor, the issuer
may wish to call the bond. Many of the issuers of this type of
security have somewhat lower credit quality, so another potential
reason for call exercise is the issuer’s improving credit.
Heath–Jarrow–Morton sometimes has trouble with American calls of
this type and does not have spread (i.e., credit) as an additional
random factor.
252
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Other Types of Securities (Continued)
Credit Derivatives
Most investment in derivatives is due to a desire to either hedge or Investors can
speculate on interest rates. There is a smaller market for derivatives that increase the
proportion of their
are driven predominantly by credit risk.
One structure provides investors with the ability to increase credit
exposure to a given subset of the market. The structure permits the
issuer, at the end of some period of time, to exchange its bond for any
corporate bond from a pre-agreed list. Like futures, the bonds are given
normalization factors to equalize them (at issue) for differences in
coupon, maturity, or credit. Changes in relative credit during the holding
period provide the issuer with an incentive to deliver the worst one of
those bonds. Investors would receive a significantly increased coupon
during the holding period to compensate for this risk.
investment exposed
to credit by
purchasing lowerrated securities
Credit derivatives
also provide an
opportunity to
increase or hedge
exposure to credit
Another structure, which investors can use to reduce credit exposure, is
a derivative contact that allows the investor, at the end of some period of
time, to exchange the worst bond on a pre-agreed list for the best bond
on that list. The investor is then assured of improving credit quality, but
pays a premium for the option.
Traders can hedge this type of option by buying or selling each bond on
the list in varying amounts so that the sensitivity of the hedge portfolio
to changes in prices of any bond in the portfolio matches the change in
value of the credit derivative. However, this hedge assumes that spreads
evolve smoothly. In practice, individual bond spreads can jump
dramatically, and a trader would be unable to fully hedge this risk
without entering into offsetting credit derivative positions.
253
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Other Types of Securities (Continued)
Stripped Yield for Brady Bonds
Brady bonds
(certain emergingmarkets
governmental
issues) have
varying degrees of
principal and
interest collateral
Many emerging-markets sovereign borrowers restructured bank loans in
the early 1990s. The new securities that were spawned in the
restructuring process are called Brady bonds.
Most Brady bonds have credit-enhancing collateral. The collateral
usually comprises long-duration STRIPS (to provide for the principal on
the bond and, therefore, called principal collateral) and short-term AArated or better investments to cover from six to 18 months of interest
(called interest collateral). Some Brady bonds have a floating interest
It is possible to back rate; the amount of interest collateral is fixed, so in a high-interest-rate
out the value of the
environment, the collateral could cover a shorter period of time.
collateral to
calculate a yield
attributable to the
credit-risky
component of the
bond
The value of the
principal collateral
is its price in the
market; its value
and cash-flow
contribution can be
subtracted from the
bond to leave an
annuity with a
lower price
The credit on a Brady bond is thus a blend of emerging markets, AA, and
Treasury credit. Investors naturally want to compare the yield on the
emerging-markets investment portion to that of other Brady issues as
well as other collateralized and uncollateralized issues. A stripped
yield—the yield on only the emerging-markets portion of a Brady
bond—permits this comparison. The stripped yield generally accounts
only for collateral and does not take into account options that may be
embedded in the bond (call options, oil-price options, etc.)
If a bond had only principal collateral, calculating the stripped yield
would be relatively easy: the risk-free (collateralized) principal payment
would be valued back to settlement at the applicable STRIPS rate. The
stripped yield would be the yield that equates the present value of the
remaining coupon annuity to the cost of the Brady bond less the present
value of the principal payment.
For example, an 8% 30-year Brady bond is priced at 80% (yielding
10.14%). The principal collateral is worth roughly 12.69%, so the
emerging-markets annuity (no principal) is worth 67.31%. The stripped
yield that discounts the 8% annuity to a present value of 67.31% is 11.47%.
254
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Other Types of Securities (Continued)
Stripped Yield for Brady Bonds
Valuing interest collateral is slightly trickier: If the issuer defaulted
tomorrow, the collateral would have a high value, and if the issuer never
defaulted, the collateral would have a lower value. The interest collateral
pays for the coupons immediately following default. At any point the
issuer defaults, therefore, the next coupons have the same credit as the
interest collateral. But what is the probability of the issuer defaulting at
any given time?
One estimate of the default probability in each year is the stripped yield’s
spread to Treasuries. Since we do not yet know the stripped yield, we
estimate it using an iterative process. Of course, there are other reasons
for spread and thus other, lower, estimates of default in any given year.
The probability of first default occurring in that year is the product of
1) the probability the issuer has not defaulted up to that time and 2) the
probability (constant for every year) the issuer defaults in that year.
If the issuer has not defaulted prior to the last few payments, then the
interest collateral can be applied to them, valuing them according to the
credit of the interest collateral. The probability of first default on each
payment date, plus the probability of never defaulting, is 100%.
The value of the
interest collateral is
the probabilityweighted chance
that it is used for
each interest
payment,
discounted at a rate
appropriate for the
credit of the interest
collateral
The value of the interest collateral would then be its value at each point
in time, weighted by the probability that the issuer first defaults at that
time. Then, as for the principal collateral, the stripped yield would
discount the future cash flows (reduced by the chance that they are paid
out of interest collateral) to the cost of the bond less the value of the
interest collateral.
For example, if the same 8% 30-year bond, worth 67.31% stripped of
principal, had 18 months of interest collateral, the value of the future
payments made by the interest collateral would be 5.22%. The stripped
price of the bond is then 62.09%. A stripped yield of 12.55% discounts
the remaining future cash flows (now less than 8% per year) to that price.
255
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
How Corporations Analyze
Debt Retirement
Corporations are
not always able to
exercise their call
options optimally
(from the capital
market’s
perspective)
Whenever an issuer has a callable bond outstanding, the issuer needs to
decide whether to exercise the option. By exercising, the issuer
sacrifices time value, but can refinance at a lower rate. By not
exercising, the borrower continues to pay the high interest rate, and the
option’s intrinsic value declines as the bond amortizes toward par. As a
rule of thumb, corporations call when intrinsic value exceeds 80% of
option value.
Bonds that, at first
blush, seem likely
to be called may be
cheap if there is
reason to believe
the issuer will not
exercise the call
In order to call bonds, a corporation needs to obtain funds. One obvious
source of those funds is to issue new bonds. However, many bonds have
a cash call period, during which they may be retired with excess cash,
but not refunded with the proceeds of a new debt financing. Therefore,
corporations may elect not to call some bonds that trade at a premium
because their sources of cash — issuing stock and pulling money out of
their ongoing business — may be more expensive than the bonds.
A thorough analysis
of all the factors
affecting potential
exercise will lead to
the value of an
embedded call
option
The corporation may be able to issue a limited amount at one time. It
may, therefore, elect to apply the proceeds to the most onerous debt first.
Sometimes, that debt may not be currently callable, and the company
may stockpile cash until it can pay it off. Additionally, the corporation
may not be able to borrow with as long a maturity as that of its current
debt, so it must evaluate the impact of shortening its liabilities.
The call premium provides special problems for some issuers. The call
premium is usually accounted for as a loss in the current period; some
corporations may be unwilling to take that loss for accounting,
regulatory, or ratings purposes. Furthermore, some regulated
companies, particularly utilities, have no incentive to reduce financing
cost, and may not even be able to do so without the approval of regulators.
Finally, the senior creditors often impose restrictions that prevent the
corporation from refunding its highest-cost, subordinated, debt.
256
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
New-Issuance Analysis
Issuers usually pay an underwriting commission to the firms that help
raise capital. This commission is called a gross spread and is quoted as
a percent of face amount. It is paid upon the completion of the
underwriting.
Issuers often compute an all-in cost of financing. This measure of yield
is the reoffered yield, or the yield at which the original investors
purchase the bond, plus an additional cost for the gross spread. The best
way to compute the all-in yield is to compute the yield of the bond, with
all its actual characteristics, at a price equal to the net proceeds of its
original-issue price less the gross spread. This methodology
automatically amortizes the gross spread according to the effectiveinterest method; the amortization of gross spread period by period can be
tabulated using the all-in yield, coupon, and net proceeds.
The yield at which
an investor buys a
bond is not the
same yield at which
the issuer accounts
for it
Different issuers treat the gross spread differently for tax purposes.
Under the effective-interest method, the gross spread would be deducted
as the net issuance price (reoffered price less gross spread) accretes
toward par. This methodology results in lower deductions up front and
higher deductions in the future. However some issuers can deduct the
gross spread evenly over the life of the bond. This is advantageous
treatment compared to the effective-interest method, because the same
total gross spread is taken as a deduction earlier. Analyzing the effective
after-tax all-in yield under these circumstances is slightly more
complicated.
257
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 8 Exercises
1. Price a 25-year 10% semi-annual-pay bond callable in five years at a
price of 102.500% that sinks 5% each of the last 10 years
(to maturity, call, and average life). Use a spread of 100 bp over the
Treasury curve. What is the risk of misinterpretation for a
$10 million block?
2. What is the price of a 25-year 10% bond callable at par in five years
if the spread is 100 bp over the Treasury curve? What about if the
bond is putable in five years? What about if the bond is both callable
and putable?
3. A zero slash bond pays a coupon that starts in the future, so it has a
zero-coupon component and a normal bond component. If an
issuer’s yield is 8.500%, what is the coupon of a semi-annual-pay
par-priced 20-year bond with a 5-year zero-coupon period?
What is the duration?
4. A step-up bond pays a coupon that steps up after a period of time.
The step-up date often coincides with a call date. What is the yieldto-call and yield-to-maturity of a 20-year 8% semi-annual-pay
coupon bond priced at 102%, with a coupon step-up after 10 years to
10% and a 10-year par call date? How would you hedge this bond?
5. Estimate the duration of a 10-year LIBOR-flat semi-annual-pay,
semi-annual-reset floater priced at 90%. Assume 6-month LIBOR
semi-annually swaps to 8% fixed semi-annually (i.e., an investor
would be indifferent between receiving LIBOR for 10 years and
receiving 8% for 10 years).
6. What is the price of the ABC 81/8% of April 15, 2016 for settlement
June 26, 1996 if the trader quotes the spread as 80 off the old bond?
Eighty off the curve?
258
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 9
Swaps
This chapter is an excerpt from A Morgan Stanley Guide to Fixed Income Analysis by
Andrew R. Young, ©2003 Morgan Stanley & Co. Incorporated.
259
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
In This Chapter, You Will Learn...
•
Why Investors Use Swaps
•
How to Hedge Floating-Rate Payments Using Eurodollar Futures
•
How to Build a Swaps Zero Curve
•
How to Adjust the Swaps Curve for Convexity in Eurodollar Futures
•
How to Price and Unwind Swaps
•
How to Calculate Forward Rates
•
How to Extend the Swaps Curve to 30 Years
•
About Idiosyncrasies of This Swaps Curve Methodology
•
About Foreign Exchange Equilibrium
260
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Swaps Overview
Swaps allow an investor to exchange one set of payments for another by
entering into a contractual agreement. Frequently, at least one side of the
swap pays based on future market rates not known at the swap’s
inception. In a broader framework, the investor may have or grant the
right, but not the obligation, to start, cancel, or modify the swap.
Furthermore, the payments themselves may not be symmetrical to
changes in the market observable.
Swaps are
derivatives that
allow investors to
modify the structure
of their assets or
liabilities
In order to find a counterparty, the value of the swap payments must
equal or exceed the value of the swap receipts. However, if the
payments’ value exceeds the receipts’ value, the investor will not agree,
so the value of the two sides of the swap must be equal. More precisely,
both parties must perceive greater value to their receipts than to their
payments.
The most common swaps have a floating-rate leg dependent on future
market observations of LIBOR. In order to value these swaps, we will
build a curve, called the LIBOR zero-coupon curve, which we can use to
value any set of future payments. The LIBOR zero curve prices any
floating-rate bond with a LIBOR-flat coupon at par, so it is a fair set of
discount rates for any set of cash flows promised to be paid by a
corporation that can issue LIBOR-flat floaters at par.
Swaps that have asymmetrical or optional payments (i.e., caps, floors,
and other specialized products) need to be valued according to a
methodology that allows for uncertainty in interest rates. The same
techniques we applied in Chapter 6 can be adapted to LIBOR rates to
provide for this uncertainty. This methodology should produce the same
price for swaps with symmetrical exposure as that obtained using the
LIBOR zero curve.
261
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Motivations for Swapping Interest Rates
Speculate on the Market
A speculator who
enters a fixedpayer swap is
betting rates will
rise; if they do, the
speculator will earn
a profit
Alternatively, a
hedger exposed to
risk when rates fall
could enter the
opposite side of the
same swap to
reduce risk
Each party to a
swap is taking the
opposite bet
regarding the
direction of rates
The fixed-rate payer on a swap has the economic position of a short
position in a fixed-rate bond. When interest rates rise and the price of a
bond declines, the short position has a profit. A speculator (A) who is
convinced that the 10-year segment of the market is about to drop in
price can pay fixed on a 10-year swap. Suppose the current market rate
for a 10-year fixed/LIBOR swap is 7.00%:
One 10-Year Swap
If yields rise 25 bp, the market rate for such swaps will be 7.25%; the
speculator could enter another transaction to receive 7.25% fixed vs. the
same LIBOR.
Two 10-Year Swaps
262
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Motivations for Swapping Interest Rates
(Continued)
Speculate on the Market
The speculator can either lock in a profit of 0.25% per year for 10 years
or can unwind the 7.00% swap and collect the present value of all the
profit today. The correct methodology for calculating the unwind value
of a swap is presented later in this chapter, after a discussion about how
to value individual cash flows at LIBOR.
The speculator can
recognize the profit
on the swap either
today or over time
Two 10-Year Swaps
or
Lump Sum
263
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Motivations for Swapping Interest Rates
(Continued)
Lower Borrowing Costs
Two parties may
enter into a swap
because each may
have a particular
advantage
borrowing at a
particular type of
interest rate
Suppose that Firm A wishes to borrow at a floating rate, and Firm B
wishes to borrow at a fixed rate. Suppose further that Firms A and B have
the following rates available to them: A can borrow at 10% fixed or
LIBOR + 0.30% floating; B can borrow at 11.20% fixed or LIBOR +
1.00% floating. Firm B is clearly riskier, since its borrowing costs are
higher in both fixed and floating rates.
In a relative sense, B has more of an advantage borrowing floating than it
does fixed (it pays only a 70-bp premium over the floating rate available
to A as opposed to the 120-bp premium over the fixed rate available to A).
If A wanted to borrow fixed, or if B wanted to borrow floating, then at
least one of the firms would borrow in the segment of the market in which
it has a relative advantage and would have no incentive to swap. The
swap, therefore, only occurs since A wants to borrow floating and B
wants to borrow fixed.
If, contrary to their desires, A borrowed at a fixed rate and B borrowed at
a floating rate, the total borrowing cost would be LIBOR + 11.00%.
Since A wants to borrow at a floating rate and B at a fixed rate, the
combined cost would be LIBOR + 11.50%, 50 bp more expensive. The
firms could enter into the following transactions to achieve this
borrowing profile and save the 50 bp:
•
Firm A borrows at a 10% fixed rate, and Firm B borrows at LIBOR
plus 1.00%;
•
Firms A and B agree to swap fixed and floating payments, independent
of their previous borrowing obligations, negotiated to allow each of
them to benefit.
264
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Motivations for Swapping Interest Rates
(Continued)
Lower Borrowing Costs
Let A borrow fixed at 10%, B borrow floating at LIBOR + 1.00%, and In this case, each
let A agree to pay B LIBOR flat, and B agree to pay A 9.95% fixed:
firm saves 25 bp
per year, although
in general, the
savings are not
necessarily split
equally
The Net Payment by A Is
The Net Payment by B Is
Pay Out
10.00%
Pay Out
LIBOR + 1.00%
Receive
9.95%
Receive
LIBOR
Pay Out
LIBOR
Pay Out
9.95%
SWAP
Net Floater
LIBOR + 0.05%
Net Fixed
10.95%
Direct Floater
LIBOR + 0.30%
Direct Fixed
11.20%
25 bp
Net Savings
25 bp
Net Savings
Firm A effectively ends up borrowing floating at LIBOR + 0.05%, which
is 25 bp less than the floating rates it normally faces; Firm B effectively
ends up borrowing fixed at 10.95%, also 25 bp less than the fixed rate it
normally faces. The swap has, therefore, proven to be mutually
beneficial.
As a practical matter, Firms A and B usually rely on a dealer to
intermediate the trade. A dealer intermediates swaps by providing other
parties with their desired exposure and managing its own risk through
offsetting swaps, cash securities, or the attenuation of risk in the dealer’s
diversified portfolio.
265
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Motivations for Swapping Interest Rates
(Continued)
Asset-Swap a Bond
Some leveraged
investors with
floating-rate
funding swap all
investments back to
floating to hedge
interest rate risk
Suppose an investor is considering purchasing an ABC Corporation
6.625% coupon bond that matures on 8/15/02 at par. Suppose further
that the investor can purchase an ABC floating-rate bond with a coupon
of 3-month LIBOR and a maturity of 8/15/02. If the current fixed rate
on a swap out to 8/15/02 is 6.50%, then the investor can buy the fixedrate bond and pay fixed on the swap. The investor’s net position would
then be to receive 3-month LIBOR quarterly actual/360 in addition to
12.5 bp semi-annually 30/360. This cash flow is superior to that of the
floating-rate bond.
6.500% Semi-Annually 30/360
Dealer
Investor
3-Month LIBOR Quarterly Actual/360
6.625% Semi-Annually 30/360
ABC
Bond
266
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Swap Fundamentals
Example: A 1-Year $100MM Notional Fixed/Floating Swap
Starting 6/18/96
Party A pays Party B
$100MM ´ 6.15% ´
180 Days
= $3,075,000
360 Days
In a basic fixed/
floating interest rate
swap, one party
agrees to pay
another party a
fixed rate in
exchange for
payments
determined relative
to a floating-rate
index
at the end of every semi-annual period. In return, Party B pays Party A
$100MM ´ LIBOR 3-month ´
Actual Days in Period
360 Days
at the end of every quarterly period. In this example, the first quarterly
period runs for 92 days from 6/18/96 to 9/18/96. If 3-month LIBOR is
set at 5.5625% on 6/18/96, then B will pay A:
$100MM ´ 5.5625% ´
92 Days
= $1,421,527.78
360 Days
on 9/18/96 (the end of the quarterly period). At this point, we cannot
determine any of the other future payments B will make to A because we
do not know what 3-month LIBOR will be in the future.
The notional amount on a swap ($100MM in this example) is never
exchanged between the two parties, but is used in the formulas for
determining the payments they make to each other.
267
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Swap Fundamentals (Continued)
Known Cash Flows as of 6/18/96
The cash flows on
the fixed leg are
known at the outset
The cash flows on
the floating leg are
uncertain
Number
Number
Period
of Days
of Days
Ending
(30/360)
(Actual/360)
A Pays B ($)
B Pays A ($)
Start Date
6/18/1996
9/18/1996
90
92
0
1,421,528
12/18/1996
90
91
3,075,000
?
3/18/1997
90
90
0
?
6/18/1997
90
92
3,075,000
?
The payments A makes to B are called “fixed” payments since they are
determined by a fixed rate (6.15% semi-annually 30/360).
The payments B makes to A are called “floating” payments since they
are determined by a floating rate that varies over time (3-month LIBOR
quarterly actual/360).
In order for this trade to make economic sense for A, the present value
of the payments it makes to B must be the same or less than the expected
present value of the payments it receives from B. Unfortunately, the last
three payments B is going to make to A are unknown at the start of the
transaction. To deal with this uncertainty, A could hedge these payments
by buying Eurodollar futures.
268
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Eurodollar Futures
Example: September 1996 Eurodollar Futures
The September 1996 Eurodollar futures is a contract on an investment that
pays interest at 3-month LIBOR from September 1996 to December 1996.
Each contract corresponds to a $1,000,000-par-amount investment. The
price of the contract is determined by
100 ´ (1 - Expectation of LIBOR3-month at Expiration)
As the market expectation of 3-month LIBOR at contract expiration
increases, the futures price decreases. This makes a Eurodollar futures
behave like a bond; as interest rates go up, its price goes down.
Eurodollar futures
are contracts on
3-month (90-day)
LIBOR investments
The change in
future interest
earned on a
$1 million
investment at LIBOR
for a one-bp
change is
Suppose that the market currently expects 3-month LIBOR to be 5.84%
when the September 1996 Eurodollar futures contract expires (in
September). Then the futures will trade at a price of 94.16 (in decimal, not $1,000,000 × 0.01% ×
32nds). If the market expectation of 3-month LIBOR in September declines = $25
from 5.84% to 5.83%, then the futures price will rise to 94.17.
As defined in the contract, a trader will realize a profit of $25 by buying one
contract at 94.16 and later selling it at 94.17. This profit offsets the loss the
trader would realize on a future 90-day investment of $1,000,000 at
LIBOR. The terminal value of the 90-day investment is
æ
90 Days ö
ValueTerminal = $1,000,000 ´ ç 1 + r ´
÷
360 Days ø
è
At a rate of 5.84%, the investor would pay $1,000,000 today and receive
$1,014,600 in 90 days.
If LIBOR fell to 5.83%, the investor would pay $1,000,000 today and
receive $1,014,575 in 90 days. The proceeds of the 5.83% investment are
$25 less than the proceeds of the 5.84% investment; the gain on the
futures (received today) offsets the lost income (missed later).
269
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
1
4
Hedging Swap Payments with
Eurodollar Futures
An uncertain
floating-rate cash
flow can be hedged
or locked in by
buying Eurodollar
futures
Eurodollar futures
contracts expire on
the third
Wednesday of
every March, June,
September, and
December
Like other futures
contracts, Eurodollar
futures require
initial margin and
have daily variation
margin payments in
either direction
Example: Hedging the Floating-Rate Cash Flow
9/18/96 to 12/18/96
Suppose the market expects that 3-month LIBOR will be 5.84% on
9/18/96. Based on this assumption, A can expect to receive a payment
on the floating-rate leg of the swap on 12/18/96 of
$100MM ´ 5.84% ´
91 Days
= $1,476,222.22
360 Days
If this assumption turns out to be wrong by one bp and 3-month LIBOR
ends up at 5.83%, then A will receive a payment of
$100MM ´ 5.83% ´
91 Days
= $1,473,694.44
360 Days
a loss of $2,527.78.
To hedge this risk, A can buy (ignoring the effects of discounting for now)
$2,527.78 91 Days
=
´ $100 @ 101
$25
90 Days
September 1996 Eurodollar contracts. If the market expects 3-month
LIBOR to be 5.84% on 9/18/96, then A can buy the contracts at a price
of 94.16. If 3-month LIBOR ends down at 5.83%, then the September
1996 contract will expire at 94.17, and A will make a $25 profit on each
contract. A’s gain on 101 contracts will be $2,525, which almost exactly
offsets the reduction in the December 18, 1996 swap cash flow.
The number of Eurodollar contracts needed to hedge the uncertainty of
a future floating-rate cash flow (still ignoring the effects of discounting)
is determined by
Payment Floating- Rate + 0.01% - Payment Floating- Rate
$25
By hedging each floating-rate cash flow, A effectively locks in its value.
270
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Hedging Swap Payments with
Eurodollar Futures (Continued)
Hedging all the Floating-Rate Cash Flows
Start Date 6/18/96
Using the same technique as before, A can hedge the floating-rate cash
flow from 12/18/96 to 3/18/97 with December 1996 Eurodollar futures
contracts and the floating-rate cash flow from 3/18/97 to 6/18/97 with
March 1997 Eurodollar futures contracts. Three-month LIBOR for the
period 6/18/96 to 9/18/96 is already known to be 5.5625%, so there is no
uncertainty in the first floating cash flow that needs hedging.
The party receiving
uncertain floatingrate payments can
hedge the
uncertainty by
buying Eurodollar
futures
Number of
Expected
Number
Period End
Eurodollar LIBOR Implied
Expected
Eurodollar
Floating
Contracts A
of Days
Futures
by Futures
Payment from
Uses to
(Actual/360)
Price (%)
Price (%)
B to A ($)
Hedge
5.5625
1,421,528
6/18/1996
9/18/1996
92
12/18/1996
91
94.16
5.8400
1,476,222
101
3/18/1997
90
93.80
6.2000
1,550,000
100
6/18/1997
92
93.61
6.3900
1,633,000
102
Access to Eurodollar futures allows entities to put on swaps
synthetically, by re-creating all the payments of the swap. However,
because of the high cost of building a swap pricing-and-payment system
and continually hedging the position, it is usually more efficient to use
swaps dealers. Furthermore, if the floating-rate payments do not reset on
the futures expiration cycle, there will be basis risk between the swap
and the hedge.
The party making
uncertain floatingrate payments can
hedge by selling
Eurodollar futures
The high liquidity of
the Eurodollar
futures makes them
a desirable
benchmark for
beginning the
construction of a
LIBOR zero curve
Since Eurodollar futures contracts provide a mechanism for swap
counterparties to lock in future LIBOR payments, they can be used to
start building a LIBOR zero curve that prices any LIBOR-flat floatingrate bond at par. This curve can then be used to value any swap payments
on a consistent basis.
271
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Generating a Swaps Curve
Swaps Curve: The First Five Years
As of June 25, 1996
Eurodollar futures
can be used to build
the first five years
of the swaps curve
These zero rates
price LIBOR-flat
floaters at par; they
also can be used to
derive the market’s
expectation of
future LIBOR
The zero rate is also
the fixed rate for an
at-market fixed/
floating swap
where the fixed
interest is all paid
at maturity
The zero rates are
useful for valuing
other swaps where
one side is LIBORbased
We will use the first 20 Eurodollar futures contracts to build the first five
years of the swaps curve. The front 20 contracts have a high degree of
liquidity; longer contracts exist, but we will avoid them because of their
illiquidity. The maturity date of the underlying investment is always
exactly three months after contract expiration. The futures prices are as
of June 25, 1996; regular settlement on a swap is T+2.
Investment
Futures
Forward
Forward
Zero Price
Zero Rate
Maturity
Price (%)
Rate (%)
Price (%)
(%)
(%)
Settlement
6/27/1996
9/18/1996
5.563
12/18/1996
94.16
3/19/1997
93.80
6/18/1997
93.61
9/17/1997
93.40
12/17/1997
93.24
3/18/1998
93.10
6/17/1998
93.06
9/16/1998
93.00
12/16/1998
92.95
3/17/1999
92.86
6/16/1999
92.85
9/15/1999
92.80
12/15/1999
92.75
3/15/2000
92.67
6/21/2000
92.67
9/20/2000
92.63
12/20/2000
92.58
3/21/2001
92.50
6/20/2001
92.50
9/19/2001
92.46
272
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Generating a Swaps Curve (Continued)
Swaps Curve: The First Five Years
Step 1: Eurodollar futures price to forward rate
Step 1: Determine
forward rates from
Eurodollar futures
prices
Consider the 91-day period from 9/18/96 to 12/18/96. This is the period
between expiration of the September 1996 and December 1996
Eurodollar contracts. The September 1996 Eurodollar futures are trading
at 94.16, which implies that the market currently expects 3-month Step 2: Determine
LIBOR to be 5.84% on 9/18/96. In general, the formula linking the forward discount
Eurodollar futures price to the forward rate is
factors from
Rate Forward
100 - Price Eurodollar Futures
100 - 94.16
=
=
= 5.84%
100
100
There is a subtle convexity adjustment to the forward rates, which arises
from the effect of discounting on the hedge (ignored thus far). That
refinement to the methodology of projecting forward rates is described
shortly.
Step 2: Forward rate to forward discount factor
Define the forward discount factor to be the present value, at the
beginning of the period, of $1 at the end of the period, using the forward
rate as the discount rate. In this case, the forward discount factor is the
value, on 9/18/96, of $1 on 12/18/96:
FactorForward Discount =
=
forward rates
Step 3: Determine
zero prices from
forward discount
factors
Step 4: Determine
zero rates from
zero prices
100%
Days in Period
1 + Forward Rate ´
360 Days
100%
= 98.545%
91 Days
1 + 5.84% ´
360 Days
A special case is the first forward discount factor, which uses the same
formula, with the spot rate instead of the forward rate, and produces a
discount factor of 98.734% from 6/27/96 to 9/18/96.
273
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Generating a Swaps Curve (Continued)
Swaps Curve: The First Five Years
Step 1: Determine
forward rates from
Eurodollar futures
prices
Step 2: Determine
forward discount
factors from
forward rates
Step 3: Determine
zero prices from
forward discount
factors
Step 4: Determine
zero rates from
zero prices
Step 3: Forward discount factor to zero price
The zero price for 12/18/96 is the present value, on 6/27/96, of $1 on
12/18/96, using the forward rates as discount rates. The discount factor
from 12/18/96 back to 9/18/96 is 98.545% and the discount factor from
9/18/96 back to 6/27/96 is 98.734%. This means that the present value
of $1 on 12/18/96 is 98.545% × 98.734% = 97.297%. In general, the
zero price for any future maturity date is the product of all the forward
discount factors from that future date back to today, taking care to ensure
the discount factors precisely cover the discounting period.
Step 4: Zero price to zero rate
The zero rate for 12/18/96 is the semi-annually compounded 30/360
yield of a zero-coupon bond with a price today equal to the zero price
from Step 3. The zero rate is determined by
PriceZero =
100%
y ö
æ
ç 1 + Zero ÷
è
2 ø
n+1-x
In this case, there are 171 days between settlement (6/27/96) and
12/18/96 according to the 30/360 day-count convention. A zero price of
97.297% on settlement implies that the zero rate must satisfy the
equation
100%
97.297% =
Þ y Zero = 5.852%
171 180
y Zero ö
æ
ç1 +
÷
è
2 ø
We only use this method of building the swaps curve for the first five
years, where Eurodollar futures are liquid. There is another
methodology, discussed later, for using other liquid benchmarks to
extend the zero curve out to 30 years.
274
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Generating a Swaps Curve (Continued)
Swaps Curve: The First Five Years
As of June 25, 1996
We will use the first 20 Eurodollar futures contracts to build the first five
years of the swaps curve. The front 20 contracts have a high degree of
liquidity; longer contracts exist, but we will avoid them because of their
illiquidity. The maturity date of the underlying investment is always
exactly three months after contract expiration. The futures prices are as
of June 25, 1996; regular settlement on swaps is T+2.
Investment
Futures
Forward
Forward
Zero Price
Zero Rate
Maturity
Price (%)
Rate (%)
Price (%)
(%)
(%)
Settlement
6/27/1996
9/18/1996
5.563
98.734
98.734
5.745
12/18/1996
94.16
5.840
98.545
97.297
5.852
3/19/1997
93.80
6.200
98.457
95.796
5.989
6/18/1997
93.61
6.390
98.410
94.273
6.141
9/17/1997
93.40
6.600
98.359
92.726
6.275
12/17/1997
93.24
6.760
98.320
91.169
6.380
3/18/1998
93.10
6.900
98.286
89.606
6.465
6/17/1998
93.06
6.940
98.276
88.061
6.552
9/16/1998
93.00
7.000
98.261
86.530
6.626
12/16/1998
92.95
7.050
98.249
85.015
6.683
3/17/1999
92.86
7.140
98.227
83.508
6.732
6/16/1999
92.85
7.150
98.225
82.025
6.785
9/15/1999
92.80
7.200
98.213
80.559
6.835
12/15/1999
92.75
7.250
98.200
79.109
6.875
3/15/2000
92.67
7.330
98.181
77.670
6.916
6/21/2000
92.67
7.330
98.044
76.150
6.958
9/20/2000
92.63
7.370
98.171
74.758
6.996
12/20/2000
92.58
7.420
98.159
73.381
7.028
3/21/2001
92.50
7.500
98.139
72.016
7.057
6/20/2001
92.50
7.500
98.139
70.676
7.091
9/19/2001
92.46
7.540
98.130
69.354
7.124
Eurodollar futures
can be used to build
the first five years
of the swaps curve
These zero rates
price LIBOR-flat
floaters at par; they
can also be used to
derive the market’s
expectation of
future LIBOR
The zero rate is also
the fixed rate for an
at-market fixed/
floating swap
where the fixed
interest is all paid
at maturity
The zero rates are
useful for valuing
other swaps where
one side is LIBORbased
275
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Convexity Adjustment
Sensitivity Analysis: Yields Unchanged
As of June 25, 1996
There is a difference
between receiving a
known fixed-rate
forward and
hedging floating
forward to fixed
using Eurodollar
futures
The difference
arises from the cash
settlement feature
of the futures
There are
adjustments that
can provide better
information on the
market’s
expectations of
forward rates than
Eurodollar futures
alone
The LIBOR zero curve that we built assumed that the LIBOR rates
implied by Eurodollar futures were the market’s expectations. However,
the future LIBOR payments under the swap are forward contracts, and
there is a subtle difference between forward and futures contracts. There
is an adjustment to bring the rate implied by the futures in line with the
fair rate for forwards.
Suppose on 6/25/96 an investor wants to receive fixed semiannual 30/360 and pay 3-month LIBOR quarterly actual/360 on
$106.7MM for the 91-day period from 6/18/97 to 9/17/97. Given the
swaps curve below, suppose a dealer quotes a fixed rate of 6.600%.
Investment
Futures
Forward
Forward
Zero Price
Zero Rate
Maturity
Price (%)
Rate (%)
Price (%)
(%)
(%)
Settlement
6/27/1996
9/18/1996
5.563
98.734
98.734
5.745
12/18/1996
94.16
5.840
98.545
97.297
5.852
3/19/1997
93.80
6.200
98.457
95.796
5.989
6/18/1997
93.61
6.390
98.410
94.273
6.141
9/17/1997
93.40
6.600
98.359
92.726
6.275
The floating payment the investor will pay on 9/17/97 is calculated using
$106.7MM ´ LIBOR6/18/97 ´
91 Days
= $26,971,388.89 ´ LIBOR6/18/97
360 Days
The investor wants to hedge the present value of the floating-rate
payment. Now the effect of discounting is important. Given the current
market expectation of 6.600%, the expected payment is $1,780,111.67.
For every basis-point increase in LIBOR, the payment increases by
$2,697.14. The present value of this change in the payment is 92.726%
× $2,697.14 = $2,500.96. To hedge the present value of this future
floating payment, the investor can sell $2,500.96/$25 @ 100 June 1997
Eurodollar contracts.
276
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Convexity Adjustment (Continued)
Sensitivity Analysis: Yields Up
As of June 25, 1996
Now suppose the market expectation of 3-month LIBOR on 6/18/97 In an increasingincreases 100 bp to 7.600%. If no other rates changed, then the swaps yield environment,
the present value of
curve will change to the one shown below:
Maturity
Futures
Forward
Forward
Zero Price
Zero Rate
Price (%)
Rate (%)
Price (%)
(%)
(%)
Settlement
6/27/1996
9/18/1996
5.563
98.734
98.734
5.745
12/18/1996
94.16
5.840
98.545
97.297
5.852
3/19/1997
93.80
6.200
98.457
95.796
5.989
6/18/1997
93.61
6.390
98.410
94.273
6.141
9/17/1997
92.40
7.600
98.115
92.496
6.485
future floating-rate
payments rises by
less than the value
of the short futures
hedge position
The expected floating-rate payment the investor will make is now
$106.7MM ´ 7.600% ´
91 Days
= $2,049,825.56
360 Days
an increase of $269,713.89. Using the new swaps curve, the present
value of this increase is 92.496% × $269,713.89 = $249,475.64.
Meanwhile, the hedging short position of 100 June 1997 Eurodollar
futures has resulted in a gain of $250,000.00.
Overall, the investor has made $524.36. The gain arises because the
mark-to-market gain on the Eurodollar futures would be received today,
while an offsetting loss on paying a higher forward LIBOR would be due
in the future. Due to the rate increase, that loss is now discounted at a
higher rate and is less significant in present-value terms. If the shorter
rates rose as well, then the differential would be even greater.
277
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Convexity Adjustment (Continued)
Sensitivity Analysis: Yields Down
As of June 25, 1996
In a decreasingyield environment,
the present value of
future floating-rate
payments declines
by more than the
value of the short
futures hedge
position
Instead, suppose the market expectation of 3-month LIBOR on 6/18/97
decreases 100 bp to 5.600%. If no other rates changed, then the swaps
curve will change to the one shown below.
Investment
Futures
Forward
Forward
Zero Price
Zero Rate
Maturity
Price (%)
Rate (%)
Price (%)
(%)
(%)
Settlement
6/27/1996
9/18/1996
5.563
98.734
98.734
5.745
12/18/1996
94.16
5.840
98.545
97.297
5.852
3/19/1997
93.80
6.200
98.457
95.796
5.989
6/18/1997
93.61
6.390
98.410
94.273
6.141
9/17/1997
94.40
5.600
98.604
92.957
6.065
The investor will now expect to pay
$106.7MM ´ 5.600% ´
91 Days
= $1,510,397.78
360 Days
a decrease of $269,713.89. Using the new swaps curve, the present
value of this decrease is 92.957% × $269,713.89 = $250,719.27.
Meanwhile, the hedging short position of 100 June 1997 Eurodollar
futures has resulted in a loss of $250,000.00 to the investor.
Overall, the investor has made $719.27. A mark-to-market loss on the
Eurodollar futures would be paid today, while an offsetting gain due to
paying a lower forward LIBOR would come in the future. The net gain
arises because that gain due to paying a lower forward LIBOR is now
discounted at a lower rate.
278
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Convexity Adjustment (Continued)
The investor makes money both when rates rise and when rates fall
because of positive convexity. As the fixed-rate receiver in the swap, the
investor has the exposure of a long fixed-rate bond and short floatingrate bond position. On the Eurodollar hedge, the investor is short futures
contracts.
Investor
Investor
Security
Convexity
Position
Convexity
Fixed-Rate Bond
Positive
Long
Positive
Floating-Rate Bond
None
Short
None
Eurodollar Contracts
None
Short
None
The fixed-rate
payer demands to
pay a rate that is
slightly lower than
the fixed rate
previously
calculated to make
up for the negative
convexity
Overall, the investor’s position has positive convexity:
•
The long position in the fixed-rate bond loses less and less money for
each basis-point rise in rates, while making more and more money
for each basis-point decline in rates,
•
The short position in the floating-rate bond does not change in value
as rates rise or fall, and
•
The short position in Eurodollar contracts makes money at a constant
rate as rates rise and loses money at a constant rate as rates fall.
Because of this, the fixed-rate payer on a swap typically demands to pay
a rate that is slightly lower than the fixed rate implied by the swaps curve
as previously constructed. This adjustment in rates is called the
convexity adjustment.
The convexity adjustment is greater for longer Eurodollar contracts.
This added valuation twist is one reason why liquidity in Eurodollar
futures declines for longer maturities.
279
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Convexity Adjustment (Continued)
An enhancement to
the swaps curve
modifies the
forward rates
implied by futures
by adding a
negative convexity
adjustment
The convexity
adjustments are
empirically derived
to accurately price
liquid swaps
shorter than five
years
Step 1 for generating the swaps curve is modified to take the convexity
adjustment into account. Previously, we calculated the forward rate by
using the formula:
RateForward =
100 - PriceEurodollar Futures
100
Now we modify the formula to:
RateForward =
(
100 - PriceEurodollar Futures + AdjustmentConvexity 100
100
)
The convexity adjustments, which are negative, result in lower forward
rates, which result in lower fixed rates. The adjusted forward rates
should accurately price the liquid swaps (that do not mark-to-market)
that are shorter than five years. Alternatively, interest rate simulations
can estimate the value of the convexity for each futures contract.
The swaps zero curve out to five years with convexity adjustments is
shown on the next page. Given the zero curve, we can price and hedge
any (option-free) swap out to five years.
280
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Swaps Curve with
Convexity Adjustments
As of June 25, 1996
Settlement T+2
Convexity
Investment
Futures
Adjustment
Forward
Forward
Zero Price
Zero Rate
Maturity
Price (%)
(bp)
Rate (%)
Price (%)
(%)
(%)
Settlement
6/27/1996
9/18/1996
5.563
98.734
98.734
5.745
12/18/1996
94.16
0.00
5.840
98.545
97.297
5.852
3/19/1997
93.80
– 0.30
6.197
98.458
95.797
5.988
6/18/1997
93.61
– 0.60
6.384
98.412
94.275
6.138
9/17/1997
93.40
– 1.00
6.590
98.361
92.731
6.271
12/17/1997
93.24
– 1.50
6.745
98.324
91.176
6.374
3/18/1998
93.10
– 2.00
6.880
98.291
89.618
6.457
6/17/1998
93.06
– 3.00
6.910
98.283
88.079
6.541
9/16/1998
93.00
– 3.70
6.963
98.270
86.556
6.612
12/16/1998
92.95
– 4.30
7.007
98.260
85.049
6.666
3/17/1999
92.86
– 4.90
7.091
98.239
83.552
6.712
6/16/1999
92.85
– 5.90
7.091
98.239
82.080
6.762
9/15/1999
92.80
– 6.90
7.131
98.229
80.627
6.808
12/15/1999
92.75
– 7.90
7.171
98.220
79.192
6.844
3/15/2000
92.67
– 9.20
7.238
98.203
77.769
6.881
6/21/2000
92.67
– 9.40
7.236
98.068
76.266
6.919
9/20/2000
92.63
– 11.60
7.254
98.199
74.893
6.952
12/20/2000
92.58
– 12.90
7.291
98.190
73.538
6.979
3/21/2001
92.50
– 14.10
7.359
98.174
72.195
7.003
6/20/2001
92.50
– 15.60
7.344
98.177
70.879
7.032
9/19/2001
92.46
– 17.00
7.370
98.171
69.583
7.059
This is it: a LIBOR
zero-rate curve out
to five years that is
consistent with the
methodology used
by many dealers
281
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Collapsing the Floating-Rate Leg
An Alternative View of Swaps
When combined
with a hypothetical
principal payment
at maturity, using
the LIBOR zero
curve to value all
the floating-rate
cash flows after the
next reset date
collapses them to
one payment of par
(notional) on the
next reset date
When combined
with the
hypothetical
principal payment,
one side of the
swap resembles a
fixed-rate bond and
the other, a
floating-rate bond
There is an alternative view of swaps that is very useful because, in some
situations, it can collapse the entire floating-rate side of the swap to one
payment. Valuing the swap is then simply a matter of valuing fixed cash
flows using the LIBOR zero curve.
The method begins by observing that, although the notional principal
payment is not exchanged, the economics of the transaction would not
change if the two parties were to exchange that payment. With that
payment added to the schedule, the fixed-rate side of the swap resembles
a fixed-rate bond, and the floating-rate side resembles a floating-rate
bond.
On any payment date, the value of the future floating-rate payments plus
their hypothetical principal payment is par (notional amount). For
example, on June 25, 1996, Eurodollar futures implied a LIBOR rate
(after adjusting for convexity) of 5.84% for the period from 9/18/96 to
12/18/96. The floating-rate leg of a $100MM swap that matures on
12/18/96 would pay $100MM + $100MM × 5.84% × 91/360 = $100MM ×
(1 + 5.84% × 91/360). The forward price of 98.545% used to build the
LIBOR zero curve was derived as
100%
1 + 5.84% ´ 91
360
The value of the fixed payment plus hypothetical principal on 9/18/96 is
(
)
100%
$100 MM ´ 1 + 5.84% ´ 91 ´
= $100 MM
360 1 + 5.84% ´ 91
360
By induction, the floating-rate payments, including the hypothetical
principal payment, are worth the notional amount on any payment date.
282
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Pricing Swaps
Example: Forward Swap from 1/1/97 to 1/1/99
Settlement: June 27, 1996
Swaps do not necessarily have to start today. Swaps that start in the
future are called forward swaps, and valuing them follows the same
methodology as valuing swaps starting today. In this example, consider
a swap from 1/1/97 to 1/1/99 on $100MM where a client wants to pay a
semi-annual 30/360 fixed rate and receive 3-month LIBOR quarterly
actual/360. Given the swaps curve, what is the fair fixed rate?
Step 1: Find the present value on T+2 (6/27/96) of $100MM on 1/1/97.
Interpolating between the zero rates on 12/18/96 (5.852%) and 3/19/97
(5.988%), we get a zero rate of 5.873% for 1/1/97. The corresponding
zero price is 97.085%. This means $100MM on 1/1/97 has a present
value of $97.085MM on settlement.
We can use the
LIBOR zero curve to
calculate today’s
price for a
fixed/floating swap
that starts accruing
payments at some
forward date
(called a forwardstarting swap)
Step 2: The present value on the swap start date (1/1/97) of the floating
payments plus the hypothetical $100MM principal is par. The present
value of all the fixed payments plus the hypothetical $100MM principal
repayment is also par. From Step 1 we know that the present value on
6/27/96 of $100MM paid on 1/1/97 is $97.085MM. Therefore, the
fixed-leg cash flows must have a present value of $97.085MM on
settlement (6/27/96).
Step 3: Generate the zero prices for all the cash-flow dates. Using the
swaps curve from the previous page, we can interpolate zero rates and
then calculate zero prices for all the dates where fixed-rate payments are
made.
Note that if the swap is at-market (has zero value) on its start date, then
its value will be zero on any date before its start date.
283
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Pricing Swaps (Continued)
Example: Forward Swap from 1/1/97 to 1/1/99
Settlement: June 27, 1996
Within this
framework, there is
a closed-form
solution for the
fixed rate for the
forward swap
Step 4: Solve for a fixed rate such that the present value today of all the
fixed-leg cash flows is $97.085MM.
Date
Zero Rate (%)
Zero Price (%)
Nominal Cash Flow ($)
Settlement
6/27/1996
1/1/1997
5.873
97.085
7/1/1997
6.157
94.052
100MM × Fixed Rate × (180 days/360 days)
1/1/1998
6.388
90.936
100MM × Fixed Rate × (180 days/360 days)
7/1/1998
6.552
87.840
100MM × Fixed Rate × (180 days/360 days)
1/1/1999
6.674
84.800
100MM × Fixed Rate × (180 days/360 days)
1/1/1999
6.674
84.800
100MM
We need a fixed rate that satisfies the following equation:
$97.085MM = 94.052% × 100MM × Fixed Rate × (180 days/360 days) +
90.936% × 100MM × Fixed Rate × (180 days/360 days) +
87.840% × 100MM × Fixed Rate × (180 days/360 days) +
84.800% × 100MM × Fixed Rate × (180 days/360 days) +
84.800% × 100MM
n
PVBond = PVZero + å PVZero ´ RateFixed ´ Ti
n
RateFixed =
i
i =1
PVBond - PVZero
n
å PVZero
i =1
i
n
´ Ti
Given the previous swaps curve, the fair fixed rate is 6.870%.
284
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Pricing Swaps (Continued)
Example: Unwinding a Swap (on June 25, 1996)
Approach #1
Suppose on 4/1/95 an investor entered into a $50MM swap transaction
to pay 6.60% semi-annual 30/360 and receive 3-month LIBOR quarterly
actual/360 until 4/1/98. Now, on 6/25/96, the investor wants to unwind
the swap by taking or paying a lump sum on 6/27/96. The investor would
then be discharged from all future obligations. What is the fair unwind
value? Assume that 3-month LIBOR on 4/1/96 was 5.465%, and that
4-day LIBOR is 5.375%
We will show three
alternatives for
pricing a swap that
is not currently atmarket using the
LIBOR zero curve;
here is the first one
Step 1: Project the future cash flows from the existing swap.
Step 1: Project
future cash flows
from existing swap
Number of
3-Month
Investor
Investor
Days
LIBOR (%)
Receives ($)
Pays ($)
7/1/1996
91
5.465
690,715
10/1/1996
92
L1
50MM × L1 × (92/360)
1/1/1997
92
L2
50MM × L2 × (92/360)
4/1/1997
90
L3
50MM × L3 × (90/360)
7/1/1997
91
L4
50MM × L4 × (91/360)
10/1/1997
92
L5
50MM × L5 × (92/360)
1/1/1998
92
L6
50MM × L6 × (92/360)
4/1/1998
90
L7
50MM × L7 × (90/360)
Date
4/1/1996
1,650,000
Step 2: Determine
the current market
fixed rate on a new
swap
1,650,000
Steps 3: Combine
both floating-rate
legs
1,650,000
Step 4: Combine
both fixed-rate legs
1,650,000
For the payment due on 7/1/96, 3-month LIBOR was set at the beginning
of the period, 4/1/96, to 5.465%. All the future LIBORs are unknown on
6/25/96.
Step 5: Combine
results from Steps 3
and 4
285
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Pricing Swaps (Continued)
Example: Unwinding a Swap (on June 25, 1996)
Approach #1
Step 1: Project
future cash flows of
existing swap
Step 2: Determine
the current market
fixed rate on a new
swap
Steps 3: Combine
both floating-rate
legs
Step 4: Combine
both fixed-rate legs
Step 2: Determine the semi-annual 30/360 fixed rate that the investor
could currently receive by paying 3-month LIBOR quarterly actual/360
from settlement (6/27/96) to 4/1/98 on $50MM. Most of the LIBOR
payments would then offset, reducing the valuation problem to valuing
the residual fixed-rate payments. This step follows the same
methodology as the previous example on pricing a forward swap. On the
fixed leg, we will assume a short first period from 6/27/96 to 7/1/96. The
table below illustrates that a fixed rate of 6.450% creates cash flows on
the fixed leg that have a present value of par.
Fixed Side
Date
Zero Rate
Zero Price
Cash Flow
PV
(%)
(%)
($MM)
($MM)
Settlement
6/27/1996
Step 5: Combine
results from Steps 3
and 4
10/1/1996
5.760
98.528
0.842
0.830
4/1/1997
6.010
95.593
1.612
1.541
10/1/1997
6.287
92.490
1.612
1.491
4/1/1998
6.470
89.393
1.612
1.441
4/1/1998
6.470
89.393
50.000
44.696
Total
50.000
Suppose the investor enters into this swap. At this instant, the swap has
a market value of $0 because we determined the fixed rate based on
current market conditions. Entering into this swap does not change the
economic value of the investor’s position. If market conditions change,
however, this new swap will no longer have a market value of $0.
286
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Pricing Swaps (Continued)
Example: Unwinding a Swap (on June 25, 1996)
Approach #1
Step 3: Combine the cash flows of the floating legs of both swaps. On
the floating leg of the new swap, we will assume a short first period of
four days (from 6/27/96 to 7/1/96). Since we are at the beginning of the
short period, we need to set LIBOR today to calculate the floating
payment that will be due at the end of the period (7/1/96). Given that
4-day LIBOR is currently 5.375%, then the payment due at the end of
the period is
4 Days
$50MM ´ 5.375% ´
= $29,861.11
360 Days
Floating Side
Number
Investor
Investor
Period
of
Receives on the
Pays on the
Ending
Days
Existing Swap ($)
New Swap ($)
7/1/1996
91
690,715.28
29,861.11
10/1/1996
92
L1
50MM × L1 × (92/360)
50MM × L1 × (92/360)
1/1/1997
92
L2
50MM × L2 × (92/360)
50MM × L2 × (92/360)
4/1/1997
90
L3
50MM × L3 × (90/360)
50MM × L3 × (90/360)
7/1/1997
91
L4
50MM × L4 × (91/360)
50MM × L4 × (91/360)
10/1/1997
92
L5
50MM × L5 × (92/360)
50MM × L5 × (92/360)
1/1/1998
92
L6
50MM × L6 × (92/360)
50MM × L6 × (92/360)
4/1/1998
90
L7
50MM × L7 × (90/360)
50MM × L7 × (90/360)
LIBOR
Step 1: Project
future cash flows of
existing swap
Step 2: Determine
current market
fixed rate on a new
swap
Steps 3: Combine
both floating-rate
legs
Step 4: Combine
both fixed-rate legs
Step 5: Combine
results from Steps 3
and 4
Notice that all the cash flows after 7/1/96 cancel out. The net result is
that the investor receives $690,715.28 – $29,861.11 = $660,854.17 on
7/1/96. Using a 4-day LIBOR of 5.375%, the present value at settlement
(6/27/96) is
$660,854.17
= $660,459.73
1 + 5.375% ´ (4 Days 360 Days )
287
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Pricing Swaps (Continued)
Example: Unwinding a Swap (on June 25, 1996)
Approach #1
Step 1: Project
future cash flows of
existing swap
Step 2: Determine
current market
fixed rate on a new
swap
Steps 3: Combine
both floating-rate
legs
Step 4: Combine
both fixed-rate legs
Step 5: Combine
results from Steps 3
and 4
Step 4: Combine the cash flows of fixed legs of both swaps.
Date
Investor
Investor
Pays on
Receives
Investor’s
PV of
the
on the
Net
Net
Zero Rate
Zero Price
Existing
New Swap
Payment
Payment
(%)
(%)
Swap ($MM)
($MM)
($MM)
($MM)
Settlement
6/27/1996
10/1/1996
5.760
98.528
1.650
0.842
0.808
0.796
4/1/1997
6.010
95.593
1.650
1.612
0.038
0.036
10/1/1997
6.287
92.490
1.650
1.612
0.038
0.035
4/1/1998
6.470
89.393
1.650
1.612
0.038
0.034
Total
0.900
The present value for settlement on 6/27/96 of the combined fixed-leg
cash flows is a payment of $900,079.54 by the investor.
Step 5: Combine the results of Step 3 and Step 4. In Step 3, the investor
receives $660,459.73. In Step 4, the investor pays $900,336.82. Adding
these quantities together results in a net payment of $239,877.09 by the
investor to unwind the swap.
The LIBOR legs cancel out after 7/1/96. The fixed legs form an annuity
which, including the net payment on 7/1/96, costs the investor
$239,877.09 in present value on 6/27/96.
288
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Pricing Swaps (Continued)
Example: Unwinding a Swap (on June 25, 1996)
Approach #2
There is an alternative method for pricing swaps. Include the principal
payment of $50,000,000 at maturity on both the fixed and the floating
legs of the swap. Then the value of the remaining unknown floating
payments, plus the notional principal of the floating leg, is worth
$50,000,000 on July 1, 1996, the next reset date. The floating-rate
interest payment due on July 1, 1996 is $690,715.28 (given LIBOR on
4/1/96 of 5.465%) and would not be included in the value of the bond on
that date. The total value of the floating-rate bond is then
$50,690,715.28, and the floating leg of the transaction collapses to one
payment on the next reset date.
Investor
Date
Investor Pays
Receives
Investor’s
PV of
(Including
(Including
Net
Net
Zero Rate
Zero Price
Principal)
Principal)
Payment
Payment
(%)
(%)
($MM)
($MM)
($MM)
($MM)
– 50.691
– 50.660
There is an
alternative method
for pricing a swap
that utilizes the fact
that, on any
floating-rate reset
and payment date,
the present value of
the floating-rate
payments, plus the
present value of the
notional amount at
maturity, is the
notional amount
Settlement
6/27/1996
7/1/1996
5.435
99.940
50.691
10/1/1996
5.760
98.528
1.650
1.650
1.626
4/1/1997
6.010
95.593
1.650
1.650
1.577
10/1/1997
6.287
92.490
1.650
1.650
1.526
4/1/1998
6.470
89.393
51.650
51.650
46.171
Total
0.240
This methodology also results in the investor owing $239,877.09 on
6/27/96.
289
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Pricing Swaps (Continued)
Example: Unwinding a Swap (on June 25, 1996)
Approach #3
Alternative unwind
valuation
methodology:
Step 1: Determine
implied forward
floating rates from
new LIBOR zero
curve
Step 2: Calculate
and present value
net cash flows using
new LIBOR zero
curve
There is a third methodology for unwinding swaps. We have already
computed a LIBOR zero curve. We can use that curve to calculate
implied forward LIBOR rates. We can then use those rates to project
future floating-rate payments under the swap and value those payments
back to today. Recall that LIBOR was 5.465% on April 1, 1996.
Date
Investor
Investor
Investor’s
Number
LIBOR
Forward
Receives
Pays
PV of Net
of
Zero Yield
LIBOR
Floating
Fixed
Payment
Days
(%)
(%)
($MM)
($MM)
($MM)
Settlement
6/27/1996
7/1/1996
91
5.454
5.465
0.691
0.000
– 0.690
10/1/1996
92
5.760
?
?
1.650
?
1/1/1997
92
5.873
?
?
0.000
?
4/1/1997
90
6.010
?
?
1.650
?
7/1/1997
91
6.157
?
?
0.000
?
10/1/1997
92
6.287
?
?
1.650
?
1/1/1998
92
6.388
?
?
0.000
?
4/1/1998
90
6.470
?
?
1.650
?
Total
290
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Pricing Swaps (Continued)
Example: Unwinding a Swap (on June 25, 1996)
Approach #3
Step 1: Calculate the forward LIBOR rate.
We are interested in the forward rate between two dates: T1 and T2. If we
know the spot (zero-coupon) rates for those maturities, we can calculate
the respective spot present values, PV1 and PV2.
Given two zerocoupon yields or
prices, it is possible
to determine a
forward rate for the
period between the
maturities of the
zero-coupon bonds
Investing $K in a zero-coupon instrument that costs PV provides
K ´ 1 PV at maturity. Under market-expectations pricing, a zero-coupon
forward investment settling on T1 and maturing on T2 would be priced
such that investing to T1 and purchasing the forward investment
The same effective
(for PV1,2) provides the same amount of money on T2 as investing for the
forward rate can be
longer term directly.
expressed as
Mathematically,
1
PV2 (1 + y1 2)
1
1
1
K´
´
=K´
Þ PV1,2 =
=
PV1 PV1,2
PV2
PV1 (1 + y2 2)2 ´T2
2 ´T
different rates using
different
conventions
Given the market-expectations cost of the forward investment, its
forward yield can be calculated using any convention. For example:
Simple-interest:
PV1 ,2 =
æ
çç 1 + rf ´
è
1
Days Actual, T
2
- T1
360
ö
÷÷
ø
Þ rf =
360
Days Actual, T
2
- T1
ö
æ PV
´ ç 1 - 1÷
ø
è PV2
Bond-equivalent:
PV1,2
where
rf ö
æ
= 1 ç1 + ÷
2ø
è
n+1- x
1
é
ù
n
+
1
æ PV1 ö - x
ê
- 1úú
Þ r f = 2 ´ êç
÷
è PV2 ø
êë
úû
n is the number of whole semi-annual periods, and
x is the actual/actual length of the accrual period 0 £ x < 1
291
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Pricing Swaps (Continued)
Example: Unwinding a Swap (on June 25, 1996)
Approach #3
Observed LIBOR
rates can be used to
compute LIBOR
rates for future
periods
The value of a
swap is the future
market-expectationpredicted cash
flows discounted at
the current LIBOR
spot rate for that
cash flow’s term
Market LIBOR Spot (Zero-Coupon) Rates
(30/360 Semi-Annually Compounded)
5.454%
5.760%
5.873%
6/27/1996
4/1/1996
7/1/1996
5.465%
Current Coupon
10/1/1996
f
1
Next Reset
1/1/1997
f
2
Second Reset
By applying the formula on the prior page, we can project LIBOR for the
forward periods.
94 180
ù
360 é (1 + 5.760% 2)
ê
ú = 5.608%
1
f1 =
92 ê (1 + 5.454% 2)4 180
úû
ë
1+ 4 180
ù
360 é (1 + 5.873% 2)
ê
ú = 5.817%
1
f2 =
92 ê (1 + 5.760% 2)94 180
úû
ë
292
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Pricing Swaps (Continued)
Example: Unwinding a Swap (on June 25, 1996)
Approach #3
Step 2: Work out the present value of the swap’s projected cash Alternative unwind
flows.
valuation
Date
Investor
Investor
Investor’s
Pays
Net
Number
LIBOR
Forward
Receives
of
Zero Yield
LIBOR
Floating
Fixed
PV
Days
(%)
(%)
($MM)
($MM)
($MM)
Settlement
6/27/1996
7/1/1996
91
5.454
5.465
0.691
0.000
0.690
10/1/1996
92
5.760
5.608
0.717
1.650
– 0.920
1/1/1997
92
5.873
5.817
0.743
0.000
0.722
4/1/1997
90
6.010
6.241
0.780
1.650
– 0.832
7/1/1997
91
6.157
6.482
0.819
0.000
0.771
10/1/1997
92
6.287
6.608
0.844
1.650
– 0.745
1/1/1998
92
6.388
6.689
0.855
0.000
0.777
4/1/1998
90
6.470
6.906
0.863
1.650
– 0.703
Total
methodology:
Step 1: Determine
implied forward
floating rates from
new LIBOR zero
curve
Step 2: Calculate
and present value
net cash flows using
new LIBOR zero
curve
– 0.240
This approach also values the swap at a $239,877.09 payment by the
investor.
Because the first two approaches pair off almost all the floating-rate cash
flows, they are simpler computationally. This approach, however, has
more straightforward reasoning, in that it does not add the hypothetical
principal payment or involve entering into a new transaction.
293
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Market Quotes on Swaps
As of June 25, 1996
There is a liquid,
publicly visible
market in certain
swaps
Beyond five years,
these swaps are
more liquid than
Eurodollar futures
We can use this
information to
extend our LIBOR
zero curve for
pricing other swaps
that are less liquid
or not publicly
visible
When a dealer bids
a swap, the dealer
pays fixed (at a
lower spread to the
offered yield on
Treasuries); an offer
would be to receive
fixed (at a higher
spread to the bid
yield on Treasuries)
Longer-maturity U.S. dollar swaps are typically quoted as a spread to
Treasuries. On June 25, 1996, the yields for the on-the-run 5-year,
10-year, and 30-year were 6.712%, 6.933%, and 7.086%, respectively.
All other Treasury points are linearly interpolated. For example, the 20year Treasury is calculated by
y20-Year =
1
1
1
1
´ y10-Year + ´ y30-Year = ´ 6.933% + ´ 7.086% = 7.010%
2
2
2
2
According to the table below, the mid-market quote for the 20-year swap
is 49 bp (over the mid-market interpolated Treasury yield). Thus, the
mid-market fixed-rate quote is 7.500% for 20 years. Swap spread quotes
are unusual because the bid spread is always lower than the offered
spread. For a swap bid, the bid spread is added to the (lower) offered
yield of Treasuries, while for a swap offering the offered spread is added
to the (higher) bid yield of Treasuries. Even though the nomenclature is
reversed, market participants still want to pay a lower rate than they
receive.
Maturity
Mid-market
Interpolated Treasury (%)1
Mid-market
Quoted Spread (bp)
Mid-market
Fixed Rate (%)
6-Year
6.756
30.5
7.061
7-Year
6.800
32.5
7.125
10-Year
6.933
36.0
7.293
12-Year
6.948
43.0
7.378
15-Year
6.971
48.5
7.456
20-Year
7.010
49.0
7.500
30-Year
7.086
39.0
7.476
In other currencies, there may not be such liquid benchmarks, but there
can still be a swaps yield curve. The fixed rate, not the spread, is the
important economic quantity.
1 The quotes for 10-year and 30-year swaps are spread off the most recently issued, not the
interpolated, Treasuries.
294
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
LIBOR Swap Rates vs Treasury Yields
As of June 25, 1996
LIBOR may be
thought of as a
yield near which a
AA bank could
borrow
LIBOR is a generic rate that is approximately where an AA-rated
London-based bank can borrow for the short term. Why, then, are the
longer-term swaps rates so much lower than AA-rated bank rates? The
high-quality banks that can borrow at LIBOR change over time. For a
longer horizon, therefore, LIBOR is exposed to the overall credit of the
bank sector. Any individual bond is exposed to the prospects for that
particular company only. There is a chance that the credit quality of that
company could deteriorate, in which case it could no longer borrow at
LIBOR. The swaps curve relates to less risky short-term borrowings by
a pool only of credits that should be relatively stable, so the swap fixed
rates are lower than the bond yields.
Because LIBOR is renewable, it is essentially risk free. Swaps have low
credit risk due to netting and offsetting (covered later), so LIBOR is a
good approach to discounting their payments.
295
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Extending the Swaps Curve
As of June 25, 1996
Settlement: June 27, 1996
We are looking for
LIBOR zero rates to
12/27/01 and
6/27/02 that will
value the fixed-rate
cash flows of a
6-year swap,
including the
hypothetical
principal payment,
to par today
Remember that the
floating-rate cash
flows, including the
hypothetical
principal payment,
are always worth
par
According to the swaps quote sheet, the fixed rate for a
6-year swap is 7.061%. The cash flows from the fixed leg on a $100MM
notional are listed below. If we add the hypothetical principal payment
of $100MM at maturity, then the cash flows should have a present value
of $100MM.
Date
Nominal
Present Value
Zero Rate
Zero Price
Cash Flow
Cash Flow
(%)
(%)
($MM)
($MM)
Settlement
6/27/1996
12/27/1996
5.865
97.120
3.531
3.429
6/27/1997
6.152
94.089
3.531
3.322
12/27/1997
6.383
90.974
3.531
3.212
6/27/1998
6.549
87.877
3.531
3.103
12/27/1998
6.672
84.837
3.531
2.995
6/27/1999
6.767
81.870
3.531
2.891
12/27/1999
6.849
78.972
3.531
2.788
6/27/2000
6.921
76.145
3.531
2.688
12/27/2000
6.981
73.406
3.531
2.592
6/27/2001
7.033
70.750
3.531
2.498
12/27/2001
?
?
3.531
?
6/27/2002
?
?
103.531
?
Total
100.000
Recall that earlier we constructed zero rates out to 9/19/01 using the first
20 Eurodollar futures. To find the zero rate for 12/27/96, we interpolate
between the zero rates for 12/18/96 (5.852%) and 3/19/97 (5.988%) to
get 5.865%. Using the interpolated zero rate, we determine a zero price
of 97.120% and present value of $3.429MM. This technique works for
all the cash flows out to 6/27/01.
296
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Extending the Swaps Curve (Continued)
As of June 25, 1996
Settlement: June 27, 1996
To extend the curve out to 6/27/02, we make the assumption that zero
rates change linearly from 9/19/01 to 6/27/02 such that the present value
of the cash flows is $100MM. The zero rate for 9/19/01 was 7.059%.
The constant rate of change that correctly prices the swap can be found
iteratively using Newton–Raphson. The constant-rate-of-change
assumption gives us zero rates of 7.075% for the 12/27/01 maturity and
7.104% for the 6/27/02 maturity.
Date
Nominal
Present Value
Zero Rate
Zero Price
Cash Flow
Cash Flow
(%)
(%)
($MM)
($MM)
Settlement
6/27/1996
12/27/1996
5.865
97.120
3.531
3.429
6/27/1997
6.152
94.089
3.531
3.322
12/27/1997
6.383
90.974
3.531
3.212
6/27/1998
6.549
87.877
3.531
3.103
12/27/1998
6.672
84.837
3.531
2.995
6/27/1999
6.767
81.870
3.531
2.891
12/27/1999
6.849
78.972
3.531
2.788
6/27/2000
6.921
76.145
3.531
2.688
12/27/2000
6.981
73.406
3.531
2.592
6/27/2001
7.033
70.750
3.531
2.498
12/27/2001
7.075
68.196
3.531
2.408
6/27/2002
7.104
65.754
103.531
68.075
Total
The critical
assumption for
extending the
swaps curve is that
LIBOR zero rates
change at a
constant rate
between market
swap observations
Given this
assumption, there is
only one possible
curve that will
correctly price the
benchmark swap
Making this
assumption for
successive swap
points allows us to
extend the swaps
curve out to
30 years
100.000
To extend the curve out to 6/27/03, we use the same technique. We
assume that 1) zero rates change at a constant rate from 6/27/02 to
6/27/03, and 2) that the zero prices implied by the zero rates present
value the cash flows from the fixed leg of a 7-year swap (with
hypothetical principal repayment at maturity) to par.
297
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
A Swaps Curve Out to 30 Years
As of June 25, 1996
Settlement: June 27, 1996
This is the complete
specification of the
swaps curve as of
June 25, 1996
Convexity
Investment
Futures
Adjustment
Forward
Forward
Maturity
Price (%)
(bp)
Rate (%)
Price (%)
Zero Price Zero Rate
(%)
(%)
Settlement
6/27/1996
9/18/1996
5.563
98.734
98.734
5.745
12/18/1996
94.16
0.00
5.840
98.545
97.297
5.852
3/19/1997
93.80
– 0.30
6.197
98.458
95.797
5.988
6/18/1997
93.61
– 0.60
6.384
98.412
94.275
6.138
9/17/1997
93.40
– 1.00
6.590
98.361
92.731
6.271
12/17/1997
93.24
– 1.50
6.745
98.324
91.176
6.374
3/18/1998
93.10
– 2.00
6.880
98.291
89.618
6.457
6/17/1998
93.06
– 3.00
6.910
98.283
88.079
6.541
9/16/1998
93.00
– 3.70
6.963
98.270
86.556
6.612
12/16/1998
92.95
– 4.30
7.007
98.260
85.049
6.666
3/17/1999
92.86
– 4.90
7.091
98.239
83.552
6.712
6/16/1999
92.85
– 5.90
7.091
98.239
82.080
6.762
9/15/1999
92.80
– 6.90
7.131
98.229
80.627
6.808
12/15/1999
92.75
– 7.90
7.171
98.220
79.192
6.844
3/15/2000
92.67
– 9.20
7.238
98.203
77.769
6.881
6/21/2000
92.67
– 9.40
7.236
98.068
76.266
6.919
9/20/2000
92.63
– 11.60
7.254
98.199
74.893
6.952
12/20/2000
92.58
– 12.90
7.291
98.190
73.538
6.979
3/21/2001
92.50
– 14.10
7.359
98.174
72.195
7.003
6/20/2001
92.50
– 15.60
7.344
98.177
70.879
7.032
9/19/2001
92.46
– 17.00
7.370
98.171
69.583
7.059
6/27/2002
65.754
7.104
6/27/2003
61.012
7.179
6/27/2006
48.396
7.387
6/27/2008
41.315
7.500
6/27/2011
32.621
7.606
6/27/2016
22.254
7.654
6/27/2026
10.859
7.538
298
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
LIBOR Zero Rates vs STRIPS Yields
As of June 25, 1996
Eurodollar futures
and LIBOR swaps
define a LIBOR
zero-coupon curve
that can be used to
value individual
cash flows at LIBOR
flat
The LIBOR zero
curve is the
fundamental
building block for
pricing swaps
The LIBOR zero curve beyond five years is a linear interpolation of
LIBOR zero yields between observed LIBOR swap points (6, 7, 10, 12,
15, 20, and 30 years) that accurately prices those observed LIBOR
swaps.
Combined with an
option model, the
LIBOR zero curve
can be used to price
swaptions, caps,
floors, etc.
The forward rates are the rates that equilibrate the surrounding LIBOR
zero prices. That is, the forward rate is the rate that would make an
investor indifferent between 1) investing to the forward start date and
rolling over into the forward and 2) investing directly to the forward
maturity.
299
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
LIBOR Forward Rates vs
Short-Term UST Forward Rates
As of June 25, 1996
This LIBOR zero
curve implies some
unrealistic spreads
between LIBOR
forwards and UST
forwards,
particularly
between 10 and 30
years
A different
approach to
building a LIBOR
zero curve can
eliminate this
problem and still fit
the swap
benchmarks
The previously described methodology produces forward estimates of
LIBOR that are lower than the forward estimates of Treasuries for some
maturities. This is counterintuitive; LIBOR should always be higher
than Treasuries. Furthermore, since Treasuries are the most liquid
securities in the market, it would be advantageous to use them as a
benchmark for building the LIBOR curve.
The alternative methodology shown here is to assume the LIBOR spread
over Treasuries changes at a constant rate, instead of LIBOR zero yields
changing at a constant rate. The advantage of this alternative method is
that it is consistent with the information embedded in Treasury rates.
The disadvantage is that this technique cannot be replicated in many
foreign currencies that lack such strong, liquid, and low-risk
benchmarks. The former approach can be followed consistently in any
currency with Eurodollar futures and a swaps curve.
300
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Credit Risk on Swaps
At the inception of a swap, there is no direct credit exposure because the
fixed and floating legs of the swap each have the same present value.
Credit is further controlled through “netting,” which means that there is
not an exchange of the full fixed or floating interest. Rather, a net cash
flow is calculated, and the owing party sends that net amount.
Furthermore, under standard swaps documentation constructed by the
International Swaps and Derivatives Association (ISDA), in the event of
a bankruptcy, agreements with positive present value will offset
agreements with negative present value to create a net exposure that can
be pursued through the bankruptcy system. This is a much more
favorable treatment than having to make payments to the bankrupt
company on some swaps while receiving no payments in return.
While swaps have
a credit-risk
component, it is
much less important
than for the related
bonds
When interest rates move after inception, a swap will have either
positive or negative present value, exposing one party to the other’s
credit for the variation. Some swap agreements call for mark-to-market
payments if the exposure exceeds a limit. Any expected exposures that
are not offset or collateralized have the same credit risk as any obligation
issued by the counterparty. Using statistical techniques, it is possible to
determine the likelihood of a given swap or group of swaps having a
credit exposure in excess of a threshold, which can illustrate the credit
risk present in a swaps portfolio.
Even if rates do not move, there can be credit exposure because
•
in an upward-sloping-yield-curve environment, the fixed payments
are higher than the first several floating payments, so the fixed-rate
payer winds up making net payments and expecting to receive net
payments later, and
•
the floating payments are often paid out more frequently than the
fixed payments.
301
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Swaps on Other Indices
There are many
indices other than
LIBOR which could
be used as the
floating-rate index
of a swap
In addition to 3-month LIBOR, there are several other indices often used
for the floating-rate leg of a swap: 1-month LIBOR, 6-month LIBOR,
1-year LIBOR, Fed Funds, Prime Rate, Commercial Paper Composite
Index, Constant Maturity Treasury (CMT) of various maturities (5-, 7-,
and 10-years), 11th District Cost of Funds Index (COFI), and the PSA
Municipal Swap Index.
Sometimes, these contracts are fixed/floating swaps. However, there are
also swaps between one floating rate and another; these are called basis
swaps. Often, investors will use basis swaps to convert one floating-rate
index that may have been generated from their assets into a more
common benchmark for assets, such as LIBOR.
The CMT indices deserve special attention because, although they
represent yields on relatively long-term assets, they are nevertheless
often the basis for the floating-rate leg of a swap. The yield of the CMT
varies as the yield of that sector of the curve varies. Since, in the usual
steep yield curve, the various CMT indices are significantly higher than
3-month LIBOR, the spread to Treasuries for the fixed-rate leg of the
swap would be significantly higher than generic LIBOR swap spreads.
There are also liquid swaps based on LIBOR and other indices in foreign
currencies such as yen, pound sterling, Deutsche mark, French franc,
Swiss franc, etc.
302
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Caps, Floors, and Swaptions
An interest rate cap on quarterly actual/360 3-month LIBOR with a
strike of 7%, on a notional amount of $100MM, from 1/1/97 to 1/1/98 is
a series of options that pays the holder in any period in which 3-month
LIBOR exceeds 7%. If 3-month LIBOR is below 7%, then the option
pays nothing. In general, the payoff for the 3-month “caplet” in each
period is determined by
There are a variety
of options which
are frequently
embedded in swaps
contracts
é æ
Days in Period ö ù
Payoff = Max ê0,ç Notional ´ (LIBOR - Strike ) ´
÷ú
360 Days ø û
ë è
Typically, the payoff is determined at the beginning of each period and
paid at the end of that period.
An interest rate floor is similar to a cap except that it pays only if the
index is below the strike.
A swaption is an option that entitles the holder to enter into a swap
whose terms are determined at the time the option is sold. For example,
an investor can purchase a swaption on 6/25/96 for $200,000. The
swaption entitles the investor to enter into a forward swap on or before
7/1/97 (the expiration) to pay quarterly actual/360 3-month LIBOR and
receive 8.00% on $50MM from 1/1/99 to 1/1/02. Between now
and 7/1/97, the investor has the right but not the obligation to enter into
this swap.
303
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Cross-Currency Swaps
Generally, an
institution that
borrows at LIBOR in
one market will
borrow at LIBOR in
any other market
Once we have built
a LIBOR curve in
different markets,
we can value any
currency exchange
or cross-currency
interest rate swap
Forward exchange rates are calculated assuming an arbitrage-free model.
An investor should not be able to earn a risk-free excess return over an
n-year investment in dollars, for example, by taking those same dollars,
exchanging them into sterling, investing in a sterling security of the same
issuer and term, and then exchanging back into dollars at maturity at
today’s forward rate. Therefore, if we know the current exchange rate
and interest rates in two currencies for the same term of the same quality
(LIBOR), we can calculate the forward exchange rate that would
eliminate arbitrage:
rHome ö
æ
ç1 +
÷
è
2 ø
2´ n
=
1
s Spot
rForeign ö
æ
´ ç1 +
÷
2 ø
è
2´ n
´ s Forward
ß
s Forward = s Spot ´
rHome ö
æ
ç1 +
÷
è
2 ø
2´ n
rForeign ö
æ
ç1 +
÷
2 ø
è
2´ n
where the exchange rate, s, is in terms of home currency/foreign
currency.
Just as a LIBOR zero curve gives all the information needed to price
single-currency swaps, LIBOR zero curves in different currencies give
all the information needed to price cross-currency swaps (without
embedded options).
304
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 9 Exercises
1. Given the market data below, construct a swaps curve out to 30 years. Only one of these
Assume trade date is 7/22/96 and settlement date is 7/24/96 (swap numbers is difficult
to compute
rates next page).
Convexity
Maturity
Futures
Adjustment
Forward
Forward
Price (%)
(bp)
Rate (%)
Price (%)
Zero Price Zero Rate
(%)
(%)
Settlement
7/24/1996
9/18/1996
5.505
12/18/1996
94.21
0.00
3/19/1997
93.89
– 0.30
6/18/1997
93.72
– 0.60
9/17/1997
93.57
– 1.00
12/17/1997
93.43
– 1.50
3/18/1998
93.29
– 2.00
6/17/1998
93.26
– 3.00
9/16/1998
93.19
– 3.70
12/16/1998
93.13
– 4.30
3/17/1999
93.04
– 4.90
6/16/1999
93.02
– 5.90
9/15/1999
92.96
– 6.90
12/15/1999
92.90
– 7.90
3/15/2000
92.81
– 9.20
6/21/2000
92.81
– 9.40
9/20/2000
92.76
–11.60
12/20/2000
92.70
–12.90
3/21/2001
92.61
–14.10
6/20/2001
92.61
–15.60
9/19/2001
92.56
–17.00
7/24/2002
7/24/2003
61.341
7/24/2006
48.790
7/24/2008
41.891
7/24/2011
33.121
7/24/2016
22.666
7/24/2026
11.030
305
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 9 Exercises (Continued)
1. (Continued from previous page)
Maturity
5-Years
Interpolated
Mid-market
Mid-market
Quoted
Mid-market
Treasury (%)
Spread (bp)
Fixed Rate (%)
6.609
6-Years
30.5
7-Years
34.0
10-Years
6.836
37.0
12-Years
41.5
15-Years
47.5
20-Years
48.0
30-Years
7.010
38.5
306
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 9 Exercises (Continued)
2. Given the swaps curve from question 1, quote an unwind price for the
following swap. An investor is currently paying 7.40% semiannually 30/360 to receive 3-month LIBOR quarterly actual/360 on
a $400MM notional from 12/1/93 to 12/1/03. The notional is
amortizing according to the following schedule:
Period
Notional
Notional
Ending
Outstanding ($MM)
Maturing ($MM)
12/1/96
400
0
12/1/97
400
0
12/1/98
400
0
12/1/99
400
0
12/1/00
300
100
12/1/01
200
100
12/1/02
100
100
12/1/03
0
100
Assume 3-month LIBOR was 5.50% on 6/1/96 and assume LIBOR
from 7/24/96 to 9/1/96 is currently 5.4375%.
3. Is a 6-month swap paying 6-month LIBOR (set at inception to 6%)
and receiving 6% fixed semi-annually at-market? Why?
307
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 10
Mortgages
This chapter is an excerpt from A Morgan Stanley Guide to Fixed Income Analysis by
Andrew R. Young, ©2003 Morgan Stanley & Co. Incorporated.
309
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
In This Chapter, You Will Learn...
• What a Mortgage Is
• About Mortgage Pools
• About Prepayments and Their Effects on
- Fixed-Rate Mortgage Pools
- Adjustable-Rate Mortgage Pools
- Collateralized Mortgage Obligations (CMOs)
• Mortgage Market Conventions
• About Dollar Rolls
• About Mortgages with Credit Risk
- Whole Loans
- Commercial MBS
310
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Mortgage Overview
Mortgages are loans to individuals or corporations for the purpose of
buying real estate, with that real estate used as collateral for the loan.
There are two key issues in analyzing mortgages: prepayments and
credit. In residential mortgages, the borrower generally has the right to
prepay the mortgage (to call it at par in whole or in part) for any reason
at any time. In commercial mortgages, if the mortgagor defaults, the only
recourse the lender may have is the value of the collateral.
Prepayments would be no more challenging than other embedded call
options except for two items: mortgages have no call protection, and yet
their prices assume some reduced efficiency of exercise resulting from
the variety of financial circumstances faced by homeowners. Thus, if the
homeowners prepay more quickly than expected, the mortgage may have
been a bad investment.
There are two fundamental developments in the mortgage market to deal
with prepayments. The first is pooling, where mortgages are bundled
together to provide some amount of diversification. The independent
actions of pool members lead to a statistical understanding of
prepayments, including their path-dependence. The second is the
collateralized mortgage obligation (CMO) market, where mortgage
pools are “sliced and diced” to produce tranches, which have either
reduced or concentrated risk to better appeal to certain investors.
A mortgage is a
loan collateralized
by real estate
Mortgages are
valued using the
same concepts as
other securities
Mortgage
complexity arises
from the variety of
loans, the difficulty
in assessing the
prepayment
behavior of
homeowners, and
the way the loans
are packaged
Credit risk requires analysis of the strength of the borrower and the
property, and has many subjective facets. Thus, it is dealt with
incompletely in this book. However, it promises to have a growing
impact on the mortgage market in the future.
311
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Contractual Cash Flows on a Mortgage
If a residential
mortgage never
prepaid, every
payment would be
the same size, and
would be a mix of
interest and
principal
Absent prepayments, a residential mortgage is a level-pay (amortizing)
instrument. However, with a realistic prepayment assumption, expected
cash flows would be much shorter than the contractual cash flows.
The first contractual payments are predominantly composed of interest;
the last payments, of principal.
The first payments
are predominantly
interest, and the
last payments are
predominantly
principal
Recall that for an annuity:
PV =
PV -
n
PMT
PMT
+å
i
æ
y ö i=2 æ
ö
y
ç1 + ÷
ç1 + ÷
fø
è
fø
è
n
PV
PMT
PMT
=å
i +
n+1
æ
y ö i=2 æ
æ
ö
yö
y
1
+
ç
÷
ç1 + ÷
ç1 + ÷
fø
è
fø
è
fø
è
1
PV
PMT
PMT
1=
n
n+1
1
+
y
f
(
)
æ
yö æ
yö æ
ö
PV = PMT ´
ç1 + ÷ ç1 + ÷ ç1 + y ÷
y f
fø è
fø è
è
fø
Note that it is easy to convert between present value, PV, and periodic
payment, PMT, using the lower-right equation.
312
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Mortgage Pools
Pools or pass-through certificates represent a pro-rata share of interest
and principal payments, minus a servicing fee, from underlying
residential mortgages. Pass-throughs were first issued by the
Government National Mortgage Association (GNMA) in 1970 and are
valuable to investors because they simplify the mortgage investment
process compared to the alternative whole-loan market.
Different pass-through issuers have different credit quality and
structures. Agency pass-throughs guarantee the timely payment of
interest and principal (except for FHLMC 75-day pools, which merely
guarantee the eventual repayment of principal) and payoff in the event
the mortgagor defaults. GNMA is an agency of the federal government
that guarantees GNMA pools. The Federal National Mortgage
Association (FNMA) and the Federal Home Loan Mortgage Corporation
(FHLMC) are government-sponsored enterprises (GSEs), and their
pools only carry the guarantee of the issuing agency. However, these
agencies have lines of credit with the Treasury, which provides a large
measure of security. There is also a minority of pools sponsored by
private issuers with no explicit or implicit government guarantee.
Differences between the agencies’ underwriting standards and security
structures and the demographics of the agencies’ constituent borrowers
can affect the value and performance of their pass-throughs.
From an investor’s perspective, owning a pool of mortgages can be more
desirable than owning a single loan because a pool is diversified, is not
subject to the whims of a single homeowner, and relieves the investor of
the responsibility and expense of servicing the loans.
The first step in
aggregating
individual mortgage
loans is the pooling
process
Pooling may
involve a
government agency
or governmentsponsored
enterprise
supporting the
creditworthiness of
the pool
The largest such
entities are the
Federal National
Mortgage
Association (FNMA),
the Federal Home
Loan Mortgage
Corporation
(FHLMC), and the
Government
National Mortgage
Association (GNMA)
313
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Prepayments
Prepayments are a
critical factor in
mortgage
evaluation
Each individual
residential
mortgagor
generally has the
option to prepay
their mortgage in
full without penalty;
therefore, a
mortgage owner is
short a series of
options
Mortgagors do not
always seem to
exercise this option
rationally
For agency mortgages, any early return of principal is classified as a
prepayment. Prepayments can arise out of mortgagors moving,
refinancing their current mortgages or paying down debt. Another
source of prepayments is borrower defaults, in which case the guarantor
will buy the mortgage from the pool at par, and the pool will return the
proceeds to the investor as a principal prepayment.
Prepayments are a critical factor in evaluating mortgages. Since each
individual residential mortgagor has the option to prepay their mortgage
in full without penalty, a mortgage owner is short a series of call options.
Mortgagors, however, often do not exercise this option efficiently from
the perspective of the investor. Consequently, prepayment models have
a large behavioral component. The uncertain exercise reduces the value
of options embedded in mortgages compared to options held by parties
with more predictable exercise practices.
Prepayments are also dependent on frictional factors (moving, etc.), the
refinancing incentive, previous refinancing incentives, borrower
demographics, seasonality, and other factors. Because pool prepayments
can depend on the prior course of prepayments and interest rates,
mortgage models are path-dependent, which adds complexity to the
evaluation process. In making mortgage-investment decisions, the
levels of these factors and any other information that gives insight into
the probable future behavior of mortgagors can significantly affect the
value of individual securities.
314
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Types of Prepayments
Payoffs vs Curtailments
Total Payoff of 20% in Year 10
The remaining payments are proportional to pre-payoff amounts. The
maturity of the mortgage is unchanged.
A prepayment can
come in two forms:
a total payoff,
when a mortgage
in a pool is
refinanced or a
homeowner moves,
or a curtailment,
when a homeowner
has extra cash and
pays off a portion
of a mortgage
Most prepayments
are total payoffs
Curtailment of 20% in Year 10
Payments continue at the same level, but the maturity is accelerated.
It has been difficult
to estimate
curtailments from
prepayment data;
however, some
agencies have
recently begun
releasing weightedaverage-maturity
updates, which lead
to an assessment of
historical
curtailment
315
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Mortgage Prepayment Conventions
The same
prepayment view
may be quoted as
an annual Constant
Prepayment Rate
(CPR), a Single
Month Mortality
rate (SMM), or a
percent of the
Public Securities
Association (PSA)
table
All mortgage
calculations are
actually done in
terms of SMM
The Constant Prepayment Rate (CPR) is an annual measure of
prepayments. The Single Month Mortality (SMM), or Constant Monthly
Prepayment (CMP), is a monthly measure of prepayment and is not
annualized. Because these rates both apply to a decreasing function, the
proper conversion functions are
1
SMM =1-(1-CPR)12
CPR=1-(1- SMM )
12
Note the similarities to the yield-compounding functions:
1 - CPR = (1 - SMM ) , while
12
yMonthly ö
æ
1 + y Annual = ç 1 +
÷
12 ø
è
12
The only differences are the lack of annualization and the sign reversals.
SMM and CPR do not have a linear relationship. When averaging
prepayments across pools, the average SMM correctly describes the
average principal prepayment. The average CPR does not.
316
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Mortgage Prepayment Conventions
(Continued)
The Public Securities Association’s model is widely used to describe
prepayments. The PSA benchmark begins at 0.2% CPR in the first
month after origination and increases by 0.2% CPR every month
thereafter, leveling off at a CPR of 6.0% 30 months after the origination
of the mortgage. Prepayments are often quoted as a percent of this
curve; a 200% PSA implies a starting point of 0.4%, increasing by 0.4%
every month until leveling off at 12% CPR in month 30.
The PSA’s
prepayment
schedule is a
standard method of
quoting
prepayment speeds
Another, related, methodology for quoting prepayments is a Projected
Prepayment Curve (PPC), which gives the starting level of prepayments
and some future level of prepayments. The prepayments before the
future date are interpolated; after the future date, the prepayment rate is
held constant. Like PSA-based quotations, prepayments can be quoted as
a percent of PPC.
317
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
The Effects of Prepayments
Total Cash Flows of a Pool Under Different Prepayment Scenarios
Even though
mortgages are
contractually 30year instruments,
their effective
maturities are much
shorter; very high
prepayments can
dramatically reduce
the term of
mortgage securities
Prepayments also
limit the price
appreciation of
pass-through
securities above par
Prepayments imbue
mortgage passthroughs with
negative convexity:
as rates fall, faster
prepayments force
more reinvestment
at lower rates; as
rates rise, slower
prepayments lead
to less reinvestment
when yields are
higher
318
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Modeling Prepayments
An Illustrative Model for FNMA 30-Year Fixed-Rate Mortgages
Prepayments play an enormous role in the timing and amount of cash
flow received by the holder of a mortgage-backed security (MBS). It is,
therefore, critical to project accurate prepayment rates when examining
and evaluating MBS. Prepayment models can be estimated from
historical mortgage prepayments over time in a variety of situations.
The challenge is to determine when anomalous prepayment behavior
indicates a deficiency in the model. We detail a model that projects the
prepayment rate based upon many variables, such as loan age, gross
weighted-average coupon, and even the month of the year.
A dependable
prepayment model
is critical in the
selling and trading
of MBS
One major
complication of
modeling
prepayments is that
homeowner
behavior, market
conditions, and the
data availability
change over time,
which leads to the
need to periodically
re-estimate the
prepayment model
This model expresses prepayments in CPR units and assumes that the
CPR of any given mortgage pool is composed of two primary parts —
refinancing and turnover. Refinancing refers to the prepayment
phenomena that can be attributed to interest rate conditions. Turnover
describes all other sources of prepayments—for example, defaults,
catastrophes, and relocations. The graphs on the following pages
illustrate some relationships between different variables and
prepayments for FNMA 30-year fixed-rate mortgages. The effect of any
given variable on the refinancing or turnover component of prepayment
is represented by a factor; the factors are then multiplied together to Turnover and
refinancing are the
obtain the two components of prepayments. The prepayment model can
two components of
be stated as:
CPRTotal = CPRTurnover + CPRRefinancing, where
prepayment in this
model
CPRTurnover = a Turnover ´ xTurnover ´ yTurnover ´ zTurnover ´L and
CPRRefinancing = a Refinancing ´ xRefinancing ´ yRefinancing ´ zRefinancing ´L where
aTurnover and aRefinancing are constants, and x, y, and z are variables (factors).
319
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Modeling Prepayments (Continued)
An Illustrative Model for FNMA 30-Year Fixed-Rate Mortgages
The factors of this
prepayment model
are estimated by
creating splines
which, in
aggregate,
efficiently predict
prepayments
aTurnover, aRefinancing and the factor curves will be the same for all pools
with the same issuer, structure, and initial maturity. For example,
FNMA 30-year 6.50% mortgages will have the same model as FNMA
30-year 8.00% mortgages.
The a multiplier is essentially an average of the component CPR for a
given pass-through type and allows the individual factors to be
normalized near one.
The x’s, y’s, z’s are factors that measure the impact of a given variable
on the average component CPR.
If a turnover factor for a particular pool is two, then that variable will
double the turnover prepayment rate for that pool relative to the average
CPR for that pass-through type. Similarly, a factor of one indicates that
that variable will have no effect on the component CPR for that pool
relative to the average.
The factor curves are estimated using kernel smoothing, more
specifically the general additive model. Each factor curve that you are
about to see is a spline. The fitting technique, using historical
prepayment data, adjusts 1) the number of knot points in each curve,
2) the locations of the knot points, and 3) the parameters of the splines
in order to minimize the error in predicting CPR without adding
unnecessary parameters to the model. The technique is constrained by
the types and amount of data that are available. A decision must be made
about how much (or what sections) of the data are still relevant to today’s
market conditions. A much better model could be developed based on
loan-level data, but for agency pools, that information is not yet widely
available. Stay tuned.
320
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Turnover Factors for FNMA 30
Weighted-Average-Maturity (WAM) Factor
WAM is simply the
weighted-average
maturity (in
months) of a pool
of mortgages
It is the most
significant
component of
turnover
prepayments
Generally, as the
mortgage pool
ages, the turnover
component of
prepayment
increases
WAM is the weighted-average maturity (in months) of a pool of
mortgages. For example, a $1 million pool containing two loans, a
$250,000 loan maturing in 120 months and a $750,000 loan maturing in
360 months, has a WAM of:
WAM =
$250,000
$750,000
´ 120 +
´ 360 = 300 months
$1,000,000
$1,000,000
The decrease in turnover for pools with ages between 8 and 15 years
occurs because the most mobile borrowers have already prepaid.
As the pool ages beyond 15 years, turnover increases again because
mortgagors gradually pay off their mortgages to decrease their overall
debt burden (curtailment).
321
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Turnover Factors for FNMA 30
(Continued)
Refinancing-Incentive Factor
A low-cost
mortgage is an
incentive not to
move
When a homeowner has a low-cost mortgage, there is a disincentive to
relocate because the new mortgage would have a higher rate, which
increases cost. A measure of the cheapness of the current mortgage is the
ratio of the Gross Weighted Average Coupon (GWAC) to the new
mortgage rate (FRM).
Gross coupon is the interest rate paid by the homeowner, which, when
weighted by loan balance, is more relevant in predicting prepayments
than the mortgage coupon, which is paid net of servicing fees.
This factor does not account for any incentive to move or refinance due
to a high-rate mortgage; that incentive is included in the refinancing
factors.
322
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Turnover Factors for FNMA 30
(Continued)
Other Factors
Seasonality and
home sales have
subtle, but
noticeable, effects
on the turnover
portion of
prepayments
323
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Refinancing Factors for FNMA 30
Refinancing-Incentive Factor
The most critical
component of
prepayments is the
desire of
homeowners to
refinance if they can
reduce their interest
cost
Refinancing activity
increases
significantly as the
refinancing option
gets deeper in-themoney
The higher the gross weighed average coupon, the higher the incentive
to refinance. This incentive nearly triples when the GWAC is 30% over
the current coupon (about 240 bp when new mortgages are available at
8%). When the GWAC is less than the new-issue rate, this factor sets to
zero, which effectively shuts off the refinancing component of the
prepayment model since all the refinancing factors are multiplied
together.
324
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Refinancing Factors for FNMA 30
(Continued)
Burnout Factor
Burnout is the
second most
important factor in
predicting
refinancings
Burnout is measured as the cumulative product of 1 – SMMRefinancing, the
amount not refinanced, on a monthly basis:
n
Õ (1 - SMM Refinancing, i )
i=1
This provides the fraction of the original pool that has not refinanced.
Note that the fraction of the pool that has prepaid due to turnover does
not affect this statistic and is not considered burnout.
Burnout captures
the notion that
remaining
mortgages in a
high-prepayment
pool have already
had the opportunity
to refinance, but
declined to do so,
and so are less
likely to do so in
the future
The remaining
mortgagors may
not be able to
refinance or may
simply not care
about the value of
the refinancing
option
The assumption is that homeowners who elected not to prepay in a
favorable environment must have poor credit, low equity, or ignorance
of or indifference to the benefits of refinancing in a low-interest-rate
environment.
325
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Refinancing Factors for FNMA 30
(Continued)
Other Factors
Two other factors
affecting
prepayments are
the WAM of the
pool and the
market level of
refinancing fees
The WAM and
burnout effects on
the refinancing
component of
prepayments are
highly correlated
After a certain
point, the
mortgagors left in
the pool cannot or
will not refinance
326
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Refinancing Factors for FNMA 30
(Continued)
Other Factors
The trend and
shape of the yield
curve are also
factors in predicting
prepayments
A significant
decrease in rates
temporarily slows
prepayments
When short rates
are near or above
long rates,
refinancing into an
ARM, balloon, or
15-year is less
attractive
327
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Refinancing Factors for FNMA 30
(Continued)
Other Factors
There are two other
factors that exhibit
correlation to
prepayments; they
are the least
significant and are
difficult to project
328
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Predicting Prepayment
Using the Model to Predict CPR
FNMA 30 6.00% 1993
Value
Actual CPR
Factor
FNMA 30 8.50% 1996
Value
5.000
Factor
10.900
Estimated CPR
Turnover
aTurnover
WAM
10.383
10.383
316.000
355.000
0.810
1.090
Seasonality
11.000
11.000
Home Sales
3.970
3.970
Refinancing Incentive
Turnover CPR
Refinancing
aRefinancing
10.908
10.908
Refinancing Incentive
0.810
1.090
Burnout
1.000
0.980
316.000
355.000
WAM
Fees
1.600
1.600
FRM Trend
0.990
0.990
Yield Curve Shape
1.170
1.170
Home Sales
3.970
3.970
Housing Returns
1.080
0.960
11.000
11.000
Seasonality1
Use the preceding
graphs to compute
the CPR for the
given MBS
Hint: Look out for
discount and
premium coupons
Note the different
housing return
values, due to the
different year of
origination
1.042
Refinancing CPR
1 Graph not shown
329
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Predicting Prepayments (Continued)
Using the Model to Predict CPR
The FNMA 6.00%
mortgages are
discounts, so they
have no
contribution to CPR
from the financing
component
The FNMA 8.50%
mortgages are
premiums, so the
mortgagors in that
pool have an
incentive to exercise
their refinancing
option, but because
they are newly
issued, there is very
low turnover
As you can see,
there is a limit to
the accuracy
possible in
predicting
prepayments
FNMA 30 6.00% 1993
Value
Factor
FNMA 30 8.50% 1996
Value
Factor
Actual CPR
5.000
10.900
Estimated CPR
6.320
8.485
Turnover
a Turnover
WAM
Refinancing Incentive
10.383
316.000
0.532
10.383
355.000
0.136
0.810
1.012
1.090
1.177
Seasonality
11.000
1.009
11.000
1.009
Home Sales
3.970
1.121
3.970
1.121
Turnover CPR
6.320
1.878
Refinancing
aRefinancing
Refinancing Incentive
Burnout
10.908
0.810
0.000
10.908
1.090
0.308
1.000
0.980
1.869
316.000
355.000
0.798
Fees
1.600
1.600
1.407
FRM Trend
0.990
0.990
1.006
Yield Curve Shape
1.170
1.170
0.977
Home Sales
3.970
3.970
1.047
WAM
Housing Returns
Seasonality2
Refinancing CPR
1.080
0.960
0.873
11.000
11.000
1.042
0.000
6.606
2 Graph not shown.
330
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
The Value of Seasoned Pools
A seasoned pool is one that is aged (usually more than 12 months, Seasoned pools
depending on the coupon). Given two otherwise identical pools with the provide value
same initial term, the pool with the shorter WAM will be more seasoned. relative to newer
pools
Seasoned pools tend to cost more than comparable unseasoned pools,
regardless of the coupon. Seasoning provides value to premium and
discount pools for different reasons. Pools that have been seasoned for
three years can be worth up to ½ point above new issuance; pools with
10 years of seasoning can be worth two full points above new issuance.
The value of seasoning lies in the prepayment behavior of older pools.
Both discount and
premium bonds
benefit from
seasoning, but for
different reasons
Premium pools are those that have a coupon greater than the prevailing
market rates. As the pool ages, those homeowners who have not already
refinanced are less likely to do so in the future, perhaps due to credit
problems or indifference to the refinancing option. This phenomenon is
called burnout, and its effects were illustrated in the preceding graphs.
The slowdown in prepayments due to burnout provides value because
the bondholder receives a relatively high coupon payment for a longer
period of time.
Discount pools are those that have a coupon that is less than prevailing
market rates. The WAM effect on turnover increases prepayments for
the seasoned pools. So, as turnovers and, hence, prepayments increase
with the age of the pool, bondholders receive principal more quickly and
are able to reinvest at higher yields.
331
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Pool Structure and Prepayments
Residential Mortgages
The specific details
of the various
mortgage programs
can have a
significant effect on
prepayments
Ginnie Mae (GNMA)
GNMA pools are composed of assumable mortgages guaranteed by the
Federal Housing Authority (FHA) or the Veterans Administration (VA).
Because these mortgages are assumable (a home seller can permit a
home buyer to take over the mortgage), turnover need not result in
prepayment, so turnover would have reduced significance for GNMA
mortgages. The GNMA program has the smallest maximum balance of
any of the agencies, so GNMA pools represent a different demographic
that would tend to have slower prepayment behavior. There is a high
degree of homogeneity in GNMA I pools: all the mortgages in a pool
must be of the same type and must be less than 12 months old, and 90%
of the pooled mortgages backing the 30-year pass-through must have
original maturities of 20 years or more. GNMA II pools may be less
homogenous because they can have multiple lenders. The government
guarantees the timely payment of interest and principal on GNMA passthroughs, so there is absolutely no credit risk.
Fannie Mae (FNMA)
FNMA also guarantees timely payment of interest and principal, but it is
not a government guarantee. FNMA- (and FHLMC-) eligible residential
loans are called conforming, and have a higher maximum balance than
GNMA loans. FNMA loans have the following initial loan-to-value
requirement (LTV), with any excess over 80% guaranteed by mortgage
insurance (MI):
Type
Maximum LTV
Single-Family
95%
Two-Family
90%
Three- to Four-Family
80%
332
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Pool Structure and Prepayments
(Continued)
Residential Mortgages
Fannie Mae (continued)
Pool Type
Collateral
FNMA CL
Level-pay amortizing in 16–30 years
FNMA CI
Conventional mortgages 8–15 years
FNMA CX
7-year balloons
FNMA GL
FHA/VA 30-year loans
There may be more than one originator in a pool, and the pools may be
new-origination or seasoned mortgages. The underlying mortgage rates
may be up to 200 bp above the pass-through rate, although lower
deviations are much more common. All these factors, particularly the
demographics, the equity requirement, and the difference between the
gross coupon and the pass-through coupon, affect prepayments. For a
fee, FNMA allows investors to exchange small, older pools with the
same coupons for a single “MEGA” certificate in amounts of $10
million and up to increase liquidity and simplify bookkeeping.
Freddie Mac (FHLMC)
FHLMC’s Gold Participation Certificate (PC) program pools fixed- and
adjustable-rate as well as non-assumable FHA and VA loans. The interest
rates on the underlying loans may be up to 250 bp greater than the passthrough coupon rate. Loans may be of any age. Smaller and older loans
can be repackaged in a single “GIANT” pool. Unlike other programs,
FHLMC’s 75-day program guarantees the payment of principal within
one year instead of timely payment of principal.
There is no new issuance of 75-day pools, and they may be exchanged
for Gold pools.
333
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Pool Structure and Prepayments
(Continued)
Characteristics of GNMA, FNMA, and FHLMC Pass-Throughs
Data current as of
February 1997
GNMA
FNMA
Allowed Servicing (bp)
GNMA I:
50
GNMA II: 50–150
Stated Delay3
GNMA I:
GNMA II:
Actual Delay
GNMA I:
GNMA II:
Maximum Loan Size
(First Lien)
Assumability
FHA:
VA:
FHLMC
25–250
Gold:
75-Day:
250
25–75
45
50
55
Gold:
75-Day:
45
75
14
19
24
Gold:
75-Day:
14
44
$81,548
$203,000
Yes
SF: $214,600
No
SF:
$203,150
No
3 Measured from the day before the beginning of the actual period. Some software programs
start counting delay a day later and, thus, report stated delay as a day shorter, so be certain of
the convention. There is no ambiguity about actual delay. The impact of stated and actual
delay is discussed later in this chapter.
334
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Fixed-Rate Mortgages
30-Year
The most common type of mortgage, 30-year mortgages require level
payments of interest and principal over the life of the mortgage.
Balloons
Balloons are fixed-rate mortgages that, at the end of either five or seven
years, repay the principal outstanding to the MBS investor. The
homeowner has the option to convert the balloon to a 23- or 25-year
mortgage at maturity, which the issuer would buy from the pool at par;
the proceeds of the buy-out would still be a pool payment. Balloons are
attractive to homeowners because, in upward-sloping-yield-curve
environments, they offer rates that are significantly lower than generic
30-year mortgages. Investors are attracted to balloons because the
shorter maturities offer greater stability. Although premium balloons
exhibit comparable prepayment behavior to generic 30-year mortgages,
discount balloons tend to prepay faster, possibly evidencing that balloon
borrowers may foresee moving soon when taking out a mortgage.
The most common
type of residential
mortgage is a
fixed-rate 30-year
mortgage
Other variations on
the mortgage theme
include balloons,
midgets, and
dwarfs
Dwarfs, Midgets, and Gold 15s
Dwarfs are 15-year fixed-rate mortgages sponsored by FNMA,
15-year GNMAs are called midgets, and 15-year FHLMCs are called
gold 15s. As a rule of thumb, 15-year mortgages have similar
prepayment behavior to the comparable 30-year mortgage with a coupon
50 bp higher. In other words, dwarf 6.50%s and FNMA 7.00%s are
comparable in terms of prepayments.
335
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Adjustable-Rate Mortgages
ARM Characteristics
Many mortgage
payments are
determined using a
rate that floats
relative to an index
Adjustable-Rate
Mortgages (ARMs)
have additional
complexity due to
index behavior,
introductory teaser
rates, and both
periodic and
lifetime rate caps
and floors, as well
as payment
adjustment
limitations
Like fixed-rate mortgages, ARMs pay principal and interest. However,
unlike fixed-rates, ARMs undergo a periodic adjustment of the
borrower’s interest rate (and payment amortization schedule) to a level
based on the value of an index, for example, 6-month LIBOR, 11th
District Cost of Funds (COFI), or 1-year CMT.
The actual coupon paid by the borrower would be a function of the index
value, the lookback (number of days prior to reset date used to determine
index value), the margin or spread, periodic collars (limits on periodic
rate increases or decreases) and lifetime caps and floors.
New-production ARMs frequently offer initial coupons that are below
the fully indexed levels. These low rates are called teaser coupons and
are used to attract more borrowers, but may lead to higher prepayments
as the interest rate indexes in.
ARMs that are making payments that do not cover the accrual of interest
may experience negative amortization, or “neg-am.” If rates are rising,
the interest a borrower owes would be greater than the interest the
borrower pays, although the total payment of interest and principal
would remain constant. If rates rise sufficiently that the interest owed
actually exceeds those constant payments, the excess would be added to
the balance of the mortgage, resulting in neg-am.
The caps, floors, teaser rate, and index behavior can increase the
duration of ARMs. Assessing the importance of these rate constraints
requires an option model. Prepayments will tend to increase where the
rate floor takes effect, as well as when the index lags the market in a
declining-rate environment.
336
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Collateralized Mortgage Obligations
Collateralized Mortgage Obligations (CMOs) are bonds that are backed
by mortgage collateral, including agency mortgage pass-through
securities, whole-loan mortgages and other CMOs. The cash flows
generated from the collateral are used to first pay down interest and then
principal to the CMO holders.
CMOs are generally composed of many different classes, or tranches,
each with a different structure. The primary structural difference
between a CMO and a pass-through is the process by which cash flows
are allocated. With a pass-through, all bondholders receive a prorata
share of any interest or principal payments made by the mortgagor; a
pass-through holder receives some interest and some principal each
month, with complete return of principal not occurring until the final
mortgage in the pool has been fully paid. CMO tranches have different
coupon rates and bondholders receive payments based upon a principal
paydown or sinking-fund schedule according to predetermined rules
detailed in the prospectus rather than receiving a pro-rata share of the
collateral’s cash flows.
A tranche is a
“slice” of a CMO
deal
GNMA, FNMA and
FHLMC will, for a
fee, wrap their own
collateral into CMOs
The total principal
value of all the
tranches of a CMO
deal sum to the
principal value of
the original
mortgage loan
collateral
CMOs are an important innovation because they allow issuers to design
specific securities to meet the risk tolerances and yield requirements of
a wide range of investors. Some of the structures minimize prepayment
risk, and others concentrate it. Some products have very high durations,
some are insensitive to interest rates, and some even have negative
durations. Depending on the characteristics of a tranche, the yield may
be higher or lower than the yield of the underlying mortgages.
337
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Types of CMO Tranches
Sequential tranches
pay down in order
from shortest to
longest average life
Z bonds are pay-inkind (PIK) CMOs
VADM tranches
benefit from the
accrual feature of
Z bonds
Sequential
Sequential-pay bonds are structured such that after interest payments
have been made on every tranche, all remaining cash goes toward
repaying the principal on the shortest remaining tranche. Once a tranche
has been retired, the next tranche becomes the exclusive recipient of
principal payments, and so on until the longest tranche is retired. The
sequential structure enables issuers to structure bonds with different
average-life characteristics from the collateral. The riskiest sequential is
often an intermediate tranche. It has significant duration and, thus,
interest rate sensitivity, but is still susceptible to prepayment.
Z (Accrual)
Z bonds, which were the first innovation to follow the creation of the
CMO, are a type of pay-in-kind (PIK) bond; they pay their stated coupon
with more bonds, not cash. The outstanding principal amount of the Z
tranche grows at a compound rate as the interest that would otherwise be
paid on them is used to pay principal on other traches. Once all tranches
preceding the Z tranche are paid down, the Z bond begins receiving
interest and principal cash payments. Z bonds have a longer duration
than their average lives suggest because of their accrual feature, but are
not necessarily highly exposed to prepayment risk.
Very Accurately Defined Maturity (VADM)
Also known as accretion-directed tranches, VADM tranches are paid
down from the accretion of Z bonds. Because Z-bond accretion is not
adversely affected by zero prepayments, VADM tranches have no risk of
extension, or longer average lives. On the other hand, an extremely fast
prepayment scenario may result in shortening, or shorter average lives.
338
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Types of CMO Tranches (Continued)
Planned Amortization Class (PAC)
PAC bonds pay according to a predetermined sinking-fund schedule as
long as prepayment rates stay within a predefined range (PAC band).
The enhanced certainty of cash flows from PAC bonds comes at the
expense of companion or support tranches, which absorb some of the
uncertainty in principal paydown.
Targeted Amortization Class (TAC)
PAC bonds have a
defined sinkingfund schedule if
prepayments stay
within the PAC
band
TAC bonds are
similar to PAC
bonds except there
is only one speed at
which the sinkingfund schedule is
defined
TAC bonds are similar to PAC bonds except that the sinking-fund
schedule is only met at one prepayment rate, rather than at a band of
rates. TACs provide some cash-flow stability, but not as much as PACs.
A TAC’s performance will largely depend on its priority in the deal
structure. If there are both PACs and TACs, the TACs will act more like
companion bonds because they provide less protection than the PACs.
Companion bonds
Companion
Also known as support tranches, companion bonds absorb the
uncertainty and negative convexity of the pool’s principal cash flows,
allowing the higher-priority tranches to pay on schedule. The primary
factor in a companion bond’s cash-flow uncertainty is the percent of
bonds in the deal with higher priority. Virtually any type of CMO can be
a companion bond. For example, many TAC bonds support PAC bonds,
while in turn being supported by other lower-priority tranches.
protect the highertranche bondholder
from the
uncertainties of the
collateral’s cash
flows
Residual
Residual-interest tranches are composed of the collateral’s excess cash
flows over regular interest flows. Residuals have very different tax
characteristics than other fixed-income securities. They represent the
equity portion of a CMO deal.
339
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Types of CMO Tranches (Continued)
POs are MBS
principal strips
IOs are useful for
hedging because of
their unusual priceyield relationship
Principal Only (PO)
POs are zero-coupon mortgage-backed securities. They are created by
stripping the coupon interest from the underlying mortgages to create the
PO and the corresponding IO (interest only) security. Since POs pay no
interest, they are sold at a deep discount, with the principal being
returned in the form of scheduled amortization and prepayments. In a
lower-rate environment, higher prepayments lead to a higher return. The
value of a PO is further enhanced by the lower discounting factor. POs
can, therefore, have very long durations. POs also usually have positive
convexity, which is reflected in low market yields. However, at very low
yields, POs have negative convexity because their durations go to zero.
Interest Only (IO)
IOs represent the interest payments from an underlying pool of
mortgages. Since the IO and PO together add up to a pass-through, the
IO also trades at a discount. A notional amount of principal is used to
calculate the amount of coupon interest due. In a low-rate environment,
as the notional principal amount prepays, the IO payments decline,
decreasing the value of the IO. If prepayments are extremely high, an
investor can receive less cash flow over the life of the asset than the
amount invested. This relationship between rates and IO values gives
the IO a negative duration, and hence they are useful hedging tools. The
IO and PO together have the duration of the underlying security, which
is why owners of mortgages servicing, which resembles an IO, often
hedge by buying POs. IOs generally have negative convexity, which is
reflected in high market yields.
340
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Types of CMO Tranches (Continued)
Floater
Floater Rate =
Floating-rate CMOs usually pay a rate that periodically resets to a spread Index + Margin
over a specified index. Although these tranches pay a floating rate, they
Inverse Floater Rate
are usually backed by fixed-rate collateral. In order to ensure there are = Nominal Rate –
no shortfalls of cash flow, the floating rates are capped. The caps can be (Index Factor) ´ k
set at a higher rate than the collateral coupon, if there are inverse floaters
to absorb the shortfall.
Super Floater Rate
= (Index Factor) ´ k
Inverse Floater
– Constant
Inverse floaters have coupons that move inversely with the index,
usually with some multiplier effect. Inverse floaters pay based on some Usually, k is greater
nominal rate, less a multiple of the index. A floating-rate payment than one
stream has negative duration since combining it with a zero-coupon
bond with positive duration produces a floating-rate bond with zero
duration. Analyzing an inverse floater as a fixed-rate note (with a
positive duration) less a floating-rate payment stream (with a negative
duration) shows that inverse floaters have a long duration. The interest
rate has a floor of zero. Therefore, if the inverse floaters are created out
of a structure, other floating-rate bonds in the structure would need to
have interest rate caps. Otherwise, in a high-interest-rate environment,
the coupon on the floaters would grow beyond what the deal can support.
An inverse floater is similar to a financed position in a fixed-rate bond,
with a cap on the floating financing rate.
Super Floater
Super floaters are similar to floaters except that the coupon is leveraged
on its index. A one-basis-point increase in LIBOR will cause the coupon
of a super floater to increase by more than one basis point. Because a
floating-rate bond has low duration and an extra floating-rate payment
stream has negative duration, the sum of these two structures, the super
floater, has a negative duration.
341
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Modeling CMOs
All these CMO tranches are created using mortgage pools as collateral.
Whenever the pools can be created and sold for more than the cost of the
underlying mortgages, there is potential for new CMO issuance.
Dividing the mortgages into tranches is not an easy process. There are
many different types of tranches, and many of them have different
parameters that control the exposure to interest rate risk. It is hard to
know in advance how these parameters will affect the marketability of
the tranches. Efficient structuring of CMOs requires insight into this
process, as well as a talent for visualizing how the parts add up to the
whole.
In the past, the entire CMO structuring process was driven by one buyer
purchasing one illiquid piece (like an inverse floater), allowing the
creation of a large amount of liquid CMOs (like floaters) with little
valuation uncertainty. An ability to place those difficult pieces has had a
dramatic impact on an underwriter’s profile in the mortgage market.
Pricing CMOs requires an understanding of the structure and priority of
the tranches. While some larger deals, and most agency deals, are
widely modelled, many private-label tranches are not widely known, and
may not even be public issues. Private CMOs can be less liquid because
of the work required to analyze the structure.
342
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Effective Duration and Hedging
It is easy to hedge a long position in a bond (with a predetermined
payment schedule) with a short position in a bond with a similar
modified duration. This method of hedging is acceptable because
modified duration is a good measure of many bonds’ interest rate
sensitivity (if there are no embedded options).
However, most mortgage-backed securities do not follow a fixed
payment schedule. Because movements in interest rates will affect
prepayments, which will cause a different pattern of cash flows, it is
impossible to derive a closed-form formula for the interest rate
sensitivity of most MBS. Therefore, an empirical measurement of
duration is used for hedging.
Modified duration
(to maturity) is not
a useful hedging
tool for most MBS
because the cash
flows depend on
interest rates
Option-adjusted
duration is a more
viable method of
measuring interest
rate sensitivity
Effective duration is one such measurement. It can be used for hedging
because it shows a bond’s interest rate sensitivity. It is constructed by
statistically analyzing the security’s price history against the history of
appropriate interest rates.
Another alternative is option-adjusted duration (OAD). OAD can be
derived by fixing the OAS of a bond, shocking the base-case interest rate
supporting the interest rate model and recomputing the price. The ratio
of the difference of the new prices (as a percent of the current price) to
the difference of the base-case interest rates is the bond’s OAD.
Option-adjusted convexity (OAC) can be determined the same way using
three different interest rate scenarios. Because mortgages often have
non-constant convexity, a wider range of interest rates will provide a
better understanding of risk in a significant market move.
343
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Mortgage Conventions
Mortgage Yield
Mortgages usually
pay monthly,
although some pay
less frequently, and
some pay even
more frequently
Mortgage yields
and spreads can be
quoted on either a
monthly or a bondequivalent basis, so
it is important to
communicate clearly
Convention:
coupons paid
monthly, yield
quoted on a bondequivalent basis
Yield Conversions:
2
y
y
(1 + y1 ) = æçè 1 + 22 ö÷ø = æçè 1 + 1212 ö÷ø
12
1
ù
é
y2 ö 6
æ
ê
y12 = 12 ´ ç 1 + ÷ - 1ú
2ø
ú
êè
û
ë
6
é
ù
y
y2 = 2 ´ êæç 1 + 12 ö÷ - 1ú
12 ø
ëè
û
Mortgage-yield quotes are almost always bond-equivalent, regardless of
the payment frequency.
Quoted
Yield (%)
Quoted
Coupon (%)
Effective
Semi-Annual
Coupon (%)
Pick-up (bp)
5.000
5.000
5.052
5.2
6.000
6.000
6.076
7.6
7.000
7.000
7.103
10.3
8.000
8.000
8.135
13.5
9.000
9.000
9.170
17.0
10.000
10.000
10.211
21.1
Q: If an 8.50% yield is quoted on a mortgage, what would be the error
in price if the yield were misinterpreted as monthly and the security had
a duration of 5?
A: 15.25 bp (interpolated from above table) ´ 5 = 0.7625%, or
over ¾ point.
344
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Mortgage Conventions (Continued)
Delay and Factor in the Mortgage Market
Pass-through securities make payments after a specified delay:
Pool Type
Stated Delay
Actual Delay
GNMA I
45
14
GNMA II
50
19
FHMLC Gold
45
14
FNMA
55
24
FHLMC 75-Day
75
44
For FNMA 30
6.00%s with a 357
WAM, the extra 10
days of delay vs.
GNMAs translates
to three basis points
of yield
Notice the difference between stated delay and actual delay. For
example, a FNMA with a 55-day delay accrues interest from August 1.
If it paid interest in advance, on August 1, that would be called a one-day
delay. Normally, a monthly-pay security would pay August’s coupon on
September 1, which would be called a 30-day delay. The actual delay of
24 days signifies that FNMA would actually pay interest and principal
for August on the 25th of September, 24 days later than “normal.” The
August payment will be paid to the holder as of September 1, regardless
of subsequent sale.
The factor of a pool of mortgages is the ratio of current principal balance
to the original principal balance. Because the principal balance of a
mortgage declines over time, the factor will decrease over time (except
for ARMs, which can have negative amortization).
The agencies release factors once a month. FHLMC releases factors on
the first business day of every month, and FNMA releases factors on the
fifth business day of every month. On the other hand, GNMA releases
factors in three stages—preliminary, intermediate, and final. GNMA
final factors are released on the fifteenth business day of the month. It is
impossible to settle a mortgage without knowing the factor, because it
defines the remaining balance.
345
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Mortgage Conventions (Continued)
Trading and Settlement Conventions
TBA trading makes
agency pools
fungible, thereby
creating greater
liquidity
Agency mortgage-backed securities have trading and settlement
procedures that differ from those for government and corporate
securities. This is primarily due to the nature of agency collateral, which
allows the trading of mortgages on a generic basis and provides latitude
in delivery for the benefit of smaller participants.
Since each agency pool has unique characteristics, pass-throughs mostly
trade on a TBA (to be announced) basis. With TBA trades, bonds with
matching coupons and issuing agencies must be delivered and the
counterparty must be notified of the specific pools two days prior to
settlement (known as a 48-hour day) by 3:00 PM EST. The Public
Securities Association’s other “good delivery” guidelines that determine
how TBA trades should be filled are as follows:
•
TBA pools are allocated in “good million” dollar lots.
•
Each TBA trade with a coupon less than 11% must be filled with
fewer than three pools per million dollars of current par, and TBA
trades with a coupon greater than 11% must be filled with fewer than
five pools per million dollars of current par.
•
The pools assigned to each “good million” must sum to be within
1.0% of $1,000,000.
•
Additional pools cannot be added once the other pools sum to within
1.0% of $1,000,000.
346
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Mortgage Conventions (Continued)
Trading and Settlement Conventions
While TBA trading makes agency pools fungible, trades that specify
certain characteristics frequently occur. Characteristics that can be
stipulated, or “stipped,” include specific pool, weighted-average
maturity, geographic composition of the pool, originator, servicer,
weighted-average coupon, pool size, and variance.
Agency pass-through securities can trade on any business day, but
settlement of most trades takes place only once a month to make life
easier for the operations departments of the counterparties and to allow
TBA trading. The PSA releases a schedule that divides pass-through
securities into six groups, each settling on a different day.
Most pass-throughs accrue interest from the beginning of the month; the
owner of the pool at the end of the month receives that month’s principal
and interest, even if the pool has been sold and settled before the end of
the delay.
347
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Dollar Rolls
Factors Affecting the Cost of Rolls
The dollar roll is a
financing
transaction similar
to a repo financing
for Treasuries
The significant
calculation
differences are the
principal paydown
and the effect of
delay
The significant
economic difference
is that the exact
securities returned
may be different
than the securities
initially provided as
collateral
Like repo, the roll
can be used to
finance inventory or
cover a short
position
There is a repo transaction defined for mortgage pass-through securities.
However, it is not very common. Much more common is a transaction
called a dollar roll. There are two minor differences in its calculation:
•
In addition to interest, principal pays down in the form of scheduled
amortization and expected prepayments.
•
Because of delay, the “future” value of principal and interest can
include the coupons that are earned, but have not yet been paid,
discounted to the forward date.
Today
$100 invested
in a dollar roll
agreement
Equal
Roll


$100 × 1+ rd 

360 
$100 invested in
a mortgage pool
Investment
Equal
The proceeds of selling the
remainder of the pool at the
prearranged forward price
plus the future value of any
coupons and principal paid
on the collateral security
Forward Date
An economic difference is that the lender can return any similar pools
within the ±1% variance and keep the intervening payments; because the
lender will return the worst securities available and utilize the variance, the
borrower demands to pay a lower interest rate on the roll than on the repo.
348
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Dollar Rolls (Continued)
Arbitrage-Free Conditions
As in repo, in order to prevent arbitrage, the value, on the forward date, Using arbitrage-free
of an investment in the roll must be the same as the value, on the forward pricing, we can
compute the implied
date, of buying and holding the pools:
(Price
Spot
rd ö
æ
+ Accrued Spot ´ ç 1% +
÷
è
360 ø
)
rate on a dollar roll,
given a price drop
= (100% - Paydown ) ´ (Price Forward + Accrued Forward )+ FVCoupons + FV Principal
Dollar rolls are often quoted as a drop, which tells how much the forward
price is below the spot price. The arbitrage-free equation is then solved
for the implied repo rate represented by that drop. For simplicity, a
money rate is assumed for valuing any coupons and principal on the
forward date, although this equation could also be solved for an implied
repo rate that is also the money rate.
rImplied =
ù
360 é (100% - Paydown) ´ (Price Forward + Accrued Forward ) + FVCoupons + FVPrincipal
´ê
- 1ú
d
Price
Accrued
+
Spot
Spot
êë
úû
Example: A new-issue GNMA 8.00% pool (45-day stated delay, 14-day
actual delay) with 360 months remaining until maturity has its original
settlement on June 19, 1996, and is trading at 102-14. The drop is quoted
at 9+. Assume a money rate of 5% and prepayments of 3.4% CPR.
Further assume no servicing expense. What is the implied roll rate to the
next GNMA settlement date of July 22, 1996?
349
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Dollar Rolls (Continued)
Calculating the Implied Repo Rate
Using the
information on the
previous page, we
can calculate the
implied roll rate
•
The transaction will be in place from June 19, 1996 until July 22,
1996 (d = 33).
•
A 360-month, 8.00% coupon mortgage has monthly payments of
PMT =
1-
8% / 12
1
= 0.734%
(1 + 8% / 12)360
•
The interest due in the first month is 0.667%, and so the scheduled
principal is 0.067%.
•
A CPR of 3.4% translates into an SMM = 1 - (1 - CPR) = 0.288%
The principal prepayment (which applies to principal remaining after
the scheduled principal payment) is then
1 12
PrincipalPrepaid = (100% - PrincipalScheduled )´ SMM = (1 - 0.067%) ´ 0.288% = 0.288%
PrincipalPaydown = 0.355%
•
Since agency mortgages accrue from the beginning of the month,
AccruedSpot =
•
18 8%
´
= 0.400%
30 12
Accrued Forward =
21 8%
´
= 0.467%
30 12
The principal and interest is paid on July 15, 1996, and is reinvested
for seven days:
7 ö
æ
FVCoupons + FVPrincipal = (0.667% + 0.067% + 0.288%) ´ ç 1 + 5% ´
÷ = 1.022%
è
360 ø
•
The forward price is
•
Plugging these variables into the formula,
rImplied =
PriceForward = PriceSpot - Drop = 102-14 - 9 + = 102 - 04+
ù
360 é (100% - 0.355%) ´ (102.141% + 0.467%) + 1.022%
´ê
- 1ú = 4.542%
33 ë
102.438% + 0.400%
û
350
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Whole Loans
Mortgages in FNMA, GNMA, and FHLMC pools must conform to
certain requirements. The agencies’ criteria are primarily based on three
factors—minimum debt-servicing ratio, maximum loan-to-value ratio,
and a maximum loan amount. If the specific mortgage does not meet the
conformity requirements, or if the seller perceives better economics, the
mortgage can be pooled and securitized by a private entity or it can be
sold as a whole loan.
Whole loans do not need to conform to agency specifications; they also
do not carry the guarantee of timely payment of interest and principal.
Credit, therefore, becomes an additional dimension requiring analysis.
Debt-servicing ratio, loan-to-value ratio, and the loan amount are all
critical quantitative factors in evaluating a whole loan. Another
important factor to consider is the characteristics of the property relative
to the other properties in nearby markets including geography, quality,
and rents. For example, if rents are much higher on any given property,
the tenant is not likely to renew the lease. It is not atypical for some
properties to be physically examined before they are purchased. Finally,
because whole loans are sometimes bought with the intention of
securitization, the “fit” of the whole loan’s cash flows with the cash
flows of the other mortgages in the planned pool is also important.
Whole loans can trade either servicing retained or servicing released.
The quality of the servicing has a large impact on the frequency and
severity of defaults. When servicing is retained, the buyer will receive
the gross coupon instead of the net coupon, but will have all the
responsibility of managing the individual loans.
Whole loans are
sold without any
type of credit
support from the
government
Whole loans,
therefore, have the
added dimension of
credit exposure to
evaluate
Whole loans are
frequently nonconforming, which
can reduce the
information
available for
prepayment
modeling; on the
other hand, the
servicer or seller
will sometimes
provide more
detailed information
than the agencies
Finally, investors can get a great deal of information about a specific
pool of whole loans, but not as much about historical prepayments across
the general whole-loan market.
351
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Commercial Mortgage-Backed Securities
Credit Risk
Commercial Mortgage-Backed Securities, or CMBS, are securities that
are collateralized with commercial mortgage loans—mortgages on
office buildings, shopping centers, multifamily houses, etc. The CMBS
market is composed of both agency and non-agency issuers. Agency
issuers account for about 35% of the CMBS market and are limited by
their charter to multifamily housing. Principal is returned at par upon
default for agency CMBS; therefore, credit concerns are really limited to
prepayment risk. However, for non-agency issuers, there is no guarantee
of full return of principal upon default, so investors are subject to
prepayment and principal risk.
Property type is the most important factor in determining the credit
quality of a commercial mortgage pool. For example, hotels have
historically been riskier credits than regional malls. Quantitative
statistics such as debt-service-coverage ratio, loan-to-value ratio,
borrower concentration, average loan size, rate structure, and timing of
losses and prepayments are also very important. Debt-service coverage,
a measure of the property’s ability to service its overall debt burden, and
loan-to-value, which measures what portion of the property’s value is
mortgaged, are viewed by rating agencies as probably the most
important factors after property type in predicting default.
Qualitative factors such as underwriting quality, information quality,
geography, servicing quality, environmental risk and management
quality also play a role in defaults. These factors pose a challenge to
modeling default risk because they are so subjective.
352
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 10 Exercises
1. Derive the formula for the monthly payment of a mortgage, assuming
a fixed-rate n-year mortgage with a c% rate and a starting balance B0.
2. Fill in the following chart out to six months for a new-origination
30-year mortgage pool with an 8.00% rate, 0.50% servicing fee, and
8% CPR, assuming that all prepayments are total payoffs.
Starting
Month
1
Monthly
Net
Balance ($) Payment ($) Int ($)
Servicing
Sched
Prepay
Ending
Fee ($)
Prin ($)
Prin ($)
Balance ($)
1,000,000.00
2
3
4
5
6
3. Do the same exercise, but assume that 50% of the prepayments are
curtailments. How would this affect the life of the mortgage pool?
Starting
Month
1
Monthly
Net
Balance ($) Payment ($) Int ($)
Servicing
Sched
Prepay
Ending
Fee ($)
Prin ($)
Prin ($)
Balance ($)
1,000,000.00
2
3
4
5
6
353
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 11
Portfolio Theory
and Market
Dynamics
This chapter is an excerpt from A Morgan Stanley Guide to Fixed Income Analysis by
Andrew R. Young, ©2003 Morgan Stanley & Co. Incorporated.
355
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
In This Chapter, You Will Learn...
•
Portfolio Theory
– Asset Allocation
– Mean-Variance Optimization
– Capital Asset Pricing Model
•
Risk Management
– Market Dynamics
– Interest Rate Processes
– Market Pressures and Tactics
356
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Overview
Up until now, we have largely focused on evaluating and analyzing
individual assets. In many cases, that evaluation depended on an option
model, which, in turn, depended on its assumptions.
This chapter has two main thrusts: How do investors allocate their scarce
capital among competing investments, and how realistic are the
assumptions underlying our valuation framework?
Investors seek portfolios that meet their risk-return criteria. The first
example shows the strictest reduction of risk: matching individual
contractual cash flows. However, there are many other ways of
understanding and managing risk, most of which take into account the
correlation among the various assets in a portfolio.
Option models depend on price evolution of the underlying assets
following a Brownian motion (covered later). When the assumptions
underlying Brownian motion do not hold, the valuation model does not
reflect the actual value of the option, and for illiquid options,
determining an appropriate price requires some level of subjectivity.
Finally, to take advantage of any deviation from the assumptions, there
are many strategies investors may follow to try to get the better of the
market, which we cover briefly.
357
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Portfolio-Optimization Methods
Minimum-Cost Defeasance
Linear
programming is
often used to solve
portfolio problems
Typical solutions
include defeasances
or escrows, which
are portfolios
designed to
generate sufficient
cash to precisely
meet liabilities
Other solutions
include yield or
return
maximizations,
subject to
constraints on risk
and diversification
Q: What if only
$20MM of the zero
were available?
Suppose you have two liabilities: one in six months for $2 million and
another in one year for $102 million. You wish to purchase a Treasury
portfolio that will generate sufficient cash flow to meet these liabilities,
regardless of future reinvestment rates. There are two Treasury securities
that you are considering: a 1-year 8% coupon bond (priced at 103%) and
a 1-year zero-coupon bond (priced at 80%).
Define P0 and P1 as the par amount of the zero and the coupon bond,
respectively. You cannot short either bond (P0 ³ 0, P1 ³ 0) . Your
investments must generate enough cash to meet the first obligation
(4% ´ P1 ³ $2MM Þ P1 ³ $50MM) . Finally, the investments must
generate enough cash to meet the two liabilities together
(108 % ´ P1 + P0 ³ $104MM – 108 % ´ P1 ).
The objective of this analysis is to minimize cost. Given a cost C,
C = 103% + 80% ´ P0 Þ P0 =
Note that there is a family of cost lines and
that they are parallel (have the same slope).
The methodology for solving the problem is to
create a graph, map the constraints, and find
the lowest-cost line that passes through the
permissable (feasible) area.
Par Amount of Zero (P0)
($MM)
100
C - 103% ´ P1
80%
P1 ≥ $50MM
80
Optimum Solution on "Convex Hull"
Minimum Cost Line
60
P0 ≥ $104MM – 108% × P1
40
20
0
40
50
60
70
80
90
100
Par Amount of 8% Coupon Bond (P1) ($MM)
358
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Portfolio-Optimization Methods
(Continued)
Portfolio-Optimization Process
The portfolio-optimization process is heavily dependent on its inputs.
Every security must be priced simultaneously so that all the prices reflect
the same market conditions. The optimization also relies on an adequate
number of new securities to replace any sold investments. The optimizer
will recommend the purchase of securities that are underpriced in the
reinvestment set and the sale of securities that are overpriced in the
current portfolio. So, the first analysis of any optimization results should
be to confirm the accuracy of pricing inputs. Once any pricing errors
have been fixed, the optimizer will suggest another trade that may lead
to the discovery of more pricing errors.
The portfolio-optimization process will typically produce extreme
solutions that cannot be implemented for practical reasons. For
example, when maximizing IRR (usually approximated by dollarduration- weighted yield), the optimizer will typically recommend an
extreme barbell because it has a higher IRR than a bullet portfolio. The
short end of the barbell consists of short-duration securities that will
mature quickly. Their proceeds will be reinvested at prevailing yields,
exposing the portfolio to reinvestment risk. The IRR will only be
realized if the average reinvestment rate over time equals the IRR. In
this stage of the process, extra constraints are added to reduce the
optimizer’s ability to choose unreasonable solutions.
A portfolio
optimization is only
as good as its
inputs
Experience has
shown that valid
solutions are only
possible when you
already have a
good idea of the
general structure of
the optimal solution
There are other
types of portfolio
analyses that can
answer more
relevant questions
than defeasance or
yield maximization
can
By the time pricing errors have been exorcised and the portfolio problem
has been properly constrained, the solution will represent a potential reallife transaction. However, too often the portfolio merely reflects a
preconceived view of optimality. More advanced optimization techniques
can better handle real-world objectives and constraints without imposing
this preconceived view. Properly framed, optimization can provide insight
into optimal strategies and combinations of securities.
359
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Asset Allocation
Assume an Investor Who Cares About Mean and Variance Only
Asset-allocation
theory (often called
the Markowitz
model (1952) and
also known as
modern portfolio
theory) shows how
to construct an
efficient frontier of
portfolios, based on
the expected
returns, volatility of
returns, and
correlations of
returns of various
assets
There is no portfolio
which is
unconditionally
better (same return
at lower risk) than
a portfolio on the
efficient frontier
Q1: An investor has a choice between bond A, with an expected return
of 10% and a standard deviation of return of 10%, and bond B, with an
expected return of 20% and a standard deviation of return of 20%.
Which bond should the investor buy?
A1: We do not have enough information. Bond B offers a higher return,
but is riskier. It is not necessarily better.
Q2: What if bond B had a standard deviation of return of 10%, and the
investor can only select one security?
A2: Bond B is clearly better. It offers higher expected return with no
incremental risk.
Q3: Assume bond B does have a standard deviation of return of 20%.
What if there was a bond C with an expected return of 0% and a standard
deviation of return of 20%, but its return was perfectly negatively
correlated with bond B’s return?
A3: Bond B and bond C together offer a superior investment alternative
to bond A. A portfolio of 50% bond B and 50% bond C would have an
expected return of 10%, with a standard deviation of 0%. The portfolio
offers the same expected return as bond A with less risk.
360
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Optimal Risky Portfolios
Minimum Variance for Any Return
(Portfolio Does Not Include the Risk-Free Asset)
The efficient frontier
is based on
expected returns,
standard deviations
of returns, and
covariances among
returns
The results are
highly sensitive to
the inputs, and, in
particular, to the
estimates of
correlations
This analysis assumes either that investors’ utility functions depend only
on the mean and variance (but not skewness or kurtosis) of returns, or
that returns are normally distributed (more generally, elliptically
distributed).
Q1: Why are the utility curves shaped the way they are?
Utility curves, which
show different
levels of risk and
return that a
particular investor
views as
equivalent, are also
completely
subjective
A1: Investors need the prospect of additional return in order to take
additional risk. As risk increases, the required return premium accelerates.
Investor One is more risk-averse than Investor Two and, unsurprisingly,
In an efficient
selects a portfolio with lower risk and lower expected return.
Q2: Why is the efficient frontier shaped the way it is?
A2: Except in a degenerate case, even the least-risky asset can be
combined with another (higher-risk and higher-return) asset to provide a
portfolio with less risk and higher return. Eventually, however, this is no
longer possible, and increased expected return can only come by
accepting higher risk.
market, every asset
must be included in
at least one
portfolio on the
efficient frontier
361
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Optimal Risky Portfolios (Continued)
Minimum Variance for Any Return
(Portfolio Includes the Risk-Free Asset)
When investors can
lend or borrow at
the risk-free rate,
the portfolio that
provides the highest
utility will be a
blend of the riskfree asset and the
optimal portfolio;
the risk-free asset
can be present in a
positive amount
(investment) or
negative amount
(leverage)
In the presence of a risk-free asset, with the added assumption that
investors can lend and borrow at the same risk-free rate, there is one
optimal portfolio. The investment decision is then how much to invest in
the optimal portfolio and how much to invest in the risk-free asset. The
portfolios including the risk-free asset allow each investor to increase
utility (move to a higher curve) while maintaining feasibility.
For less risk-averse investors, the optimal strategy may be to invest even
more in the optimal portfolio and finance that investment by borrowing
at the risk-free rate. However, most investors cannot borrow at the riskfree rate. The higher the borrowing cost, the closer the optimal strategy
for Investor Two gets to the optimal portfolio prior to the addition of the
risk-free asset.
362
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Capital Asset Pricing Model (CAPM)
Define rf as the risk-free rate, E(rM) as the expected return on a market The Capital Asset
portfolio, and E(rS) as the expected rate of return on security S. The risk Pricing Model is
(variance, not volatility) of security S relative to the market is attributable to
bS =
Covariance(rS ,rM )
s M2
In other words, security S is bS times as risky as the overall market. The
CAPM states that in an efficient market, the expected rate of return on
security S can be expressed as E(rS) = rf +bS ´ (E(rM) – rf ): any security’s
excess return over the risk-free rate will be the market’s return over the
risk-free rate multiplied by that security’s riskiness relative to the market.
If the security were uncorrelated to the market, then it should return the
risk-free rate, and if the security were inversely correlated to the market,
then its return should lie below the risk-free rate. In practice, an estimate
of bS is easily determined using linear regression.
Sharpe (1964),
Lintner (1965), and
Mossin (1966)
It states that any
asset’s expected
excess return over
the risk-free rate
should be the
market’s excess
return multiplied by
that security’s
riskiness relative to
the market (bs)
The CAPM is predicated on some fairly strong assumptions:
• There are many investors; none of their actions affect the market
• All investors have the same holding period
• All investments are public, freely available, securities; there is also
a risk-free asset and an ability to borrow at the risk-free rate
• There are no taxes or transaction costs
• All investors use the same mean/variance model and have otherwise
similar perspectives, training, information, and expectations
• Returns are normally distributed
Any return in excess of that predicted by the CAPM is called alpha (a).
The concept of alpha is often extended to investment managers with a
perceived ability to consistently outperform the market. This is an
impossibility under the conditions of the CAPM (and very difficult in the
real world, over the long run, without a different asset allocation).
363
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Fixed-Income Analogue to CAPM
Credit Risk and Diversification
Portfolio theory is
also applicable to
bond portfolios
In an optimal
portfolio, every
asset has the same
marginal risk
contribution to the
portfolio
Under portfolio
theory, in an
efficient market,
investors are only
compensated for
non-diversifiable
(systematic) risk
The application of portfolio theory to the fixed-income market requires
a correlation matrix for bond returns. Unfortunately, bond prices are not
accurately measured on a regular basis. There is clearly a high
correlation between a corporate bond and a similar-duration Treasury,
but the inaccuracy of fixed-income prices masks potential uncorrelated
or negatively correlated spreads that could lead to a diverse portfolio
across products.
There is a theoretical argument that equity is a call option on the assets
of a company. In this context, if the assets decline in value, the equity
could become worthless, but it never has a negative value. Conversely,
the equity has all the upside in the value of the assets. Carrying this
analysis a step further, a company’s economic leverage (measured by the
market value of debt and the market value of equity) and the market
volatility of its equity imply a volatility of assets. Based on that
volatility, there is an imputed probability of default: the lower the stock
price, the higher the default risk, and the higher the stock volatility, the
higher the default risk.
Because equity prices are usually observable, it is possible to develop a
correlation matrix for them; a correlation in equity prices implies a
correlation in asset values and a correlation in default risk. This
framework suggests a methodology for measuring, controlling, and
diversifying credit risk in a bond or loan portfolio. The analysis is
performed using standard assumptions about price evolution (log-normal
distribution, deterministic volatility, continuous sample path); to the
extent the assumptions are not credible, the model may produce
inappropriate solutions.
364
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Factor Models
One technique that is often applied to securities analysis is factor
analysis. The factors are determined mathematically and are somewhat
abstract, but they are usually visualized as sub-portfolios with a common
theme; in the equity market, the price of a basket of oil company stocks
would be a potential factor in the performance of other sectors of the
market, rather than the price of oil itself. In this way, the analysis is
constructed completely within the context of the market, rather than
guessing the specific external prices that affect the market.
Factor models, also
called Arbitrage
Pricing Theory
(Ross, 1976),
extend the Capital
Asset Pricing Model
to account for more
factors in security
returns
In fixed income, factors often relate to the level of interest rates, the
steepness of the yield curve, credit spreads, and other secondary
variables. These factors can illustrate a portfolio’s sensitivity to more
environmental changes than just the level of interest rates. For example,
a bullet and a barbell portfolio with the same duration will respond the
same way to a parallel shift in rates. However, they will behave very
differently if the yield curve steepens or flattens. The portfolios could
act as a proxy for two factors for analyzing the performance of any
portfolio under different market conditions. Factor analysis provides an
opportunity to compare different portfolios’ responses to the most
significant factors in the market.
CAPM is a onefactor special case
of APT
Factor models are
often extended to
fixed income, for
example, with the
factors roughly
relating to the level
of rates, the
steepness of the
yield curve, etc.
Factor analysis provides a framework for tilting a portfolio toward or
against factors, based on investor views, while maintaining maximum
diversification.
365
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Risk Management
Investment banks
use modern
portfolio theory as
one tool in an
ongoing effort to
assess risk of loss
from trading
operations
Modern portfolio
theory is only a
tool: it provides a
framework for
assessing relative
risk as long as its
weaknesses are not
exploited, but it is
prone to
understatement of
risk when
correlations
undergo a sudden
change
In the past, investors measured risk by aggregating it across their
portfolios. For example, a mortgage position with the same dollar
duration as $1 billion long bonds (long-bond risk-equivalent units) and a
short Treasury position of $1 billion long bonds would aggregate to
$2 billion long-bond risk-equivalent units. The risk in this situation is
significantly less than if both positions were long, so this methodology
did not provide very useful or consistent risk information.
Many investors have begun to modify their risk-analysis techniques to
use some of the underpinnings of modern portfolio theory. This theory
examines riskiness on a broader scale, using historical correlations
among various asset classes. Constructing the correlation matrix is quite
difficult, because it takes a reasonable amount of data to obtain a viable
estimate of correlations and include a cross-section of market phases, but
the market can undergo significant shifts in correlation very quickly, and
the correlation matrix needs to be sensitive to that. These techniques
also have difficulty with convex assets (caps and options) over large
interest rate movements.
The results of risk analysis are often encapsulated in a Value at Risk
(VAR). One way to quote VAR is the largest daily loss that the investor
would expect each year, given the current position and assumptions. The
analysis can also provide a probability distribution of returns.
Other techniques use an asset-pricing framework with respect to various
factors and scenarios. For example, an investor could focus on the gain
or loss on the portfolio under a given interest rate scenario.
These techniques are no substitute for effective corporate management,
because they fail to capture every potential market situation. However,
they do provide an improved methodology for day-to-day comparison of
riskiness.
366
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Efficient Market Hypothesis
There are three forms of the efficient market hypothesis:
Weak Form:
Current prices reflect all information that can be
derived by examining past prices.
Semi-strong
Form:
All publicly available information is reflected in prices.
Strong Form:
All public and private information is reflected in prices.
The efficient market
hypothesis is due to
Fama (1965)
It has historically
had adherents in
academia, but is not
given much
credence on the
“street” or by other
market participants
The efficient market hypothesis is just that, a hypothesis. There is
evidence that supports it and evidence that rejects it. The strong-form
hypothesis is almost impossible to believe. There are several conditions
that could lead to the validity of the semi-strong-form hypothesis:
A large number of independent players in the market
•
none of whom has undue size or impact,
•
all of whom analyze the market in the same way,
•
all of whom participate in a market with low transaction costs, and
•
all of whom exploit any investment opportunities offering excess
return.
Note that even under the efficient market hypothesis, some investors will
obtain excess returns.
There are several arguments against any form of the efficient market
hypothesis, which derive from violations of the above conditions.
367
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Random Walks (Brownian Motion)
A buy-and-hold
investor’s primary
concerns would be
mean expected
return and standard
deviation of return
However, option
traders (or option
replicators) are very
concerned with the
precise evolution of
prices, because they
need to dynamically
hedge continuously
Brownian motion,
also called a
random walk, is
one model for the
evolution of prices
Consider a coin-toss experiment measuring the number of heads less the
number of tails. The binomial (coin-toss) distribution representing this
experiment converges to the normal distribution after a large number of
samples. A similar process measuring heads over tails for an infinite
number of miniscule flips converges to Brownian motion. If you take a
small slice out of Brownian motion and magnify it, you would find that
it is a scaled Brownian motion process itself; without labels, you cannot
tell anything about the period of time or the size of the changes observed.
Try comparing a two-month daily price history to a one-year weekly
history without referring to the axes! This is the definition of a fractal—
it is scale-independent. There are enhancements to Brownian motion to
add a drift component, but the volatility is constant over time.
Clearly, if the market truly followed a Brownian motion process, the
efficient market hypothesis would be true, because information would
have no value.
There is a log-normal analogue to Brownian motion: Geometric
Brownian motion. This process is described by the running product of
log-normal random steps. The sum of the log of the steps is a Brownian
motion process. Geometric Brownian motion is the underpinning for
some important option-valuation models, including Black–Scholes.
The technical definition of Brownian motion is a process with
independent, identically distributed (IID) increments, infinite
divisibility, and a continuous sample path (starting at zero), where the
increments are normal, with mean and variance proportional to the
length of the interval.
368
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Example of Brownian Motion
Heads – Tails at a Rate of Three Flips per Day
This example shows
the excess of heads
over tails over time,
which converges to
Brownian motion as
the number of flips
increases and the
value of each flip
relative to the total
declines
369
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Mean Reversion
There is some
evidence of mean
reversion in interest
rates
Mean reversion
reduces the
probability of
extremely high or
extremely low
interest rates
Under geometric Brownian motion with market volatility as of June 25,
1996, there is roughly a 2% chance of bond yields rising to 20% over the
next 30 years. Many practitioners believe this probability is too high.
On the other hand, that level of rates has not been unheard of around the
world, and just because yields have stayed under control recently does
not mean that the situation will continue. The past 15 years may have
been a statistically possible aberration from the kind of randomness
anticipated with prevailing market volatility.
Many models incorporate some mean reversion. Mean reversion
specifies a drift parameter for the Brownian motion to reduce the
probability of very large deviations from the mean. The drift parameter
would have to be dependent on the level of interest rates.
One simple and common method of incorporating mean reversion is to
model rates as drifting toward the mean by some fixed percentage of the
difference between the rate and the mean at every point in time.
Obviously, this methodology is predicated on deciding what the long-run
mean is. The drift in the process does not overcome the randomness of
the process, it only tilts it toward the mean. One study has shown that a
reversion of 30% for short-term rates over one year fits the data,
although with a low predictive value. This type of study is very sensitive
to the exact period covered by the data.
370
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Deviations from (Log-)Normality
So, do the underlying price or yield processes follow Brownian motion, Actual return
as most option-valuation models assume? There are several deviations distributions have
“fatter tails” than
from the definition of Brownian motion.
would be expected
under the normal or
log-normal
distributions
Empirical research shows that large price (or yield) movements occur
much more often than predicted under the normal distribution. This is
one rationale for using the log-normal distribution, which has a greater
probability of large increases than the normal distribution. However,
The fat tails indicate
even the log-normal distribution does not adequately explain the
that extreme events
frequency or severity of large moves, particularly large declines.
Most option-valuation models are based on log-normal return
distributions. The fact that actual tails are fatter increases the value of all
options, because the added probability of a large, favorable move pays
off handsomely, while the added probability of a large, adverse move has
no impact.
happen more often
than would be
predicted by the
normal model
Traders price out-of-the-money options with higher implied volatilities
than at-the-money options to compensate for the greater leverage an outof-the-money option has to large moves. A gigantic move will provide
nearly the same payoff for all options, so the value of that event is nearly
the same for all options. However, that value is more significant relative
to the lower premium on a deep out-of-the-money option. It takes a
higher implied volatility to produce that additional value in the model.
Another possible reason for using a higher implied volatility for out-ofthe-money options is uncertainty in measuring volatility. Option prices
are usually convex to volatility; the price increases more when volatility
rises than it declines when volatility falls. Again, the out-of-the-money
option, with its smaller proceeds, has more risk from this uncertainty
than the at-the-money option.
371
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Processes for Volatility
The market implies
different volatilities
for different terms
In actuality,
volatility comes
from a random
process
In fixed-income markets, it is readily apparent that there is a term
structure for volatility. Implied volatility generally decreases as the term
of the option increases. One possible cause of this phenomenon is mean
reversion.
Black–Scholes and other models that can be reduced to closed form
require a deterministic volatility assumption, European exercise, and
deterministic interest rates. Many other models, including
Most volatility in the Black–Derman–Toy and Heath–Jarrow–Morton, are designed to account
for volatility in interest rates, a deterministic term structure of volatility,
Treasury market is
and other exercise possibilities. Typically, they do not allow for
centered around
economic news
randomness in the volatility process.
releases that are
scheduled well in
advance
A more difficult problem than a term structure for interest rate volatility
is the random evolution of volatility. New information entering the
market can sharply increase volatility, while a lack of information
changes volatility gradually. Unfortunately, it is difficult to know in
advance what information the market will consider significant. Some
research has shown a degree of autocorrelation in volatility time-series;
that is, volatility depends on recent historical volatility and yield levels.
Volatility is not just random; it is discontinuous. There are short periods
of time during which most of the market’s volatility is concentrated.
Often, these periods can be identified well in advance. Assuming a
50-hour trading week, a one-hour option should be worth roughly oneseventh of a one-week option if volatility were smooth. (Why?) The
option for 8:00–9:00 AM on the first Friday of the month, when the
monthly employment report is released, can be worth approximately
three times an option on any other business hour.
372
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Jump Processes
Black–Scholes, Black–Derman–Toy, and Heath–Jarrow–Morton are all
representations of a Brownian motion process; they are predicated on a
continuous sample path for prices. However, this is inconsistent with the
way information is released into the market.
Most market jumps occur due to newly available information. However,
in an illiquid market, buying or selling can itself be significant
information and can cause prices to jump. These jumps frequently
overshoot the new equilibrium, which provides trading opportunities.
There are two
components of
market movements:
random movements
caused by the
accumulation of buy
and sell orders for
reasons unrelated
to the market and
responses to new
information in the
market
When the market gaps (changes discontinuously), option prices and
deltas change instantly. This can result in a bigger loss than would have
been incurred had the trader been able to continuously adjust the deltahedge, as assumed by the option-valuation theory. As usual, the practical These movements
way to adjust for the probability of added losses is to increase implied are frequently
volatility.
discontinuous;
One example of a jump process is the future value of stock or bonds
issued by a tobacco company. Imagine that there is an envelope, sealed
for the next five years, containing the results of ongoing tobacco
litigation. In the absence of new information on the litigation, the
securities trade relatively stably, albeit at a discount to expected value to
compensate for the extra risk. Pricing long-term options would clearly
be difficult with a continuous price process. Additionally, there would
be a large price difference between the 5-year option and the 4-year
364-day option. The 5-year option would be impossible to delta-hedge.
Another example is the occurrence of catastrophes, such as earthquakes
and hurricanes. There is a nascent market in securities that provide
coverage for these risks. The catastrophes happen with little or no
warning, and when they do occur, the expected loss on the security
jumps instantly. One distribution that often arises in the analysis of jump
processes is the Poisson distribution for rare events.
therefore, the price
process is a jump
process
Most option models
are predicated on
an ability to
continuously hedge,
which you cannot
do with a jump
process; there is
then a wide range
of valuations
consistent with the
arbitrage-free
condition
373
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
The Effect of Transaction Costs
An investor who is
motivated to do a
transaction usually
has to pay
transaction costs in
the form of either
commission or bidask spread
At any given point
in time, the true
price for a security
lies between the
best bid and the
best offer
Investors view the bid-ask spread quoted by broker-dealers as a
transaction cost. Investment banks have to earn a profit to be willing to
bear the cost of committing capital, taking risk, and paying
compensation and overhead. There are also additional costs of trading,
including operations.
Transaction costs have a number of effects on the market. One is that,
except at the precise moment a transaction takes place, it is impossible
to know the exact level of the market. The true price may be close to the
bid price, and it may be close to the offer price. Sometimes, the price of
a security may change, as evidenced by the prices on the screen, without
a transaction even taking place. The uncertainty as to the true price
implies that there is a range of prices that are arbitrage-free; there is a
wider range of arbitrage-free prices for derivatives that depend on
several underlying securities or investments.
Another effect of transaction costs is that even if prices evolved along a
continuous sample path, option hedgers could not continuously hedge.
As prices move, a trader would have to make microscopic hedge
adjustments all the time. The transaction costs from hedging
continuously would be infinite; so, as a practical matter, traders usually
adjust their hedges only when rates change more than some hurdle.
374
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
The Screens
Screens in the fixed-income markets are run by inter-dealer brokers.
There are several in each market, adding varying degrees of liquidity.
The prices shown on a fixed-income trading screen are the highest bid
for a security, with its size specified, and the lowest offer for the security,
with its size specified. Prices on the screens usually represent primary
dealers; customers do not generally have the ability to post prices.
The size shown may reflect the amount the trader wants to transact, or
the trader may be illustrating a price without showing the full scale of the
transaction anticipated. Once either the bid or offer is exhausted, one of
two things will happen: another (or the same) trader will step in with the
same price, or the price will move to the next-best bid or offer. A large
enough transaction will move the price significantly, even in the absence
of other information, before other investors realize that there is an
opportunity. Investors represent a dramatically larger pool of capital
than dealers, so large customer trading can have a big impact.
When the market is stable, most screens will provide a realistic picture
of market conditions because any time an offer is lower than a bid, the
market will quickly arbitrage away the discrepancy. However, when the
market is moving quickly, there can be discrepancies between the
screens and the market. This occurs because a trader may be prepared to
trade at a worse price to achieve the desired transaction size
immediately, whereas, in a calmer market, the trader would start with the
most advantageous price and gradually move on.
Most fixed-income
trading screens
show the “inside
market”: the best
bid and offer, with
sizes, for securities
Other markets also
show second-best,
or worse bids and
offers, which can
provide more
information about
the expected impact
of a larger
transaction
A trader is the best
source of
information about
current market
conditions
375
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Making Trades
Trading as Principal
Traders have
developed a
shorthand jargon
for communicating
quickly and
precisely
•
Dealers open the price negotiation, whether solicited or not, with a
bid (to buy securities) or an offer (to sell securities).
•
Indicative modifies bid or offer to designate that it is for pricing
purposes only and is not executable.
•
A two-way market is a simultaneous bid and offer by the same trader.
It discloses the bid/ask spread. A locked market is a bid and offer at
the same price. If requested and given, it must be acted upon.
Traders will sometimes make a locked market to generate activity.
•
Firm modifies bid or offer to designate that the price is good for
execution. It may also include a requested size. Firm bids or offers
require an immediate response.
•
Subject or out, indicates a withdrawal of the firm bid or offer.
•
If an investor proposes a size for the transaction, the trader may
accept or reject. The total size of a transaction may affect the price.
Sizing the market means that the investor wants to know how much
the trader is willing to execute. In olden days, this left the dealer the
option to execute any amount of the transaction.
•
Done indicates unconditional, irrevocable acceptance of a bid or
offer and is an oral contract. Bids are hit, and offers are lifted or
taken.
376
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Types of Orders
Trading as Principal or Agent
Basic Order Types
Transactions may
be executed as
principal or as
agent
•
A market order is executed immediately at the best price available. A
market-on-open order is executed at the best price possible as soon as
the market opens, and a market-on-close order is executed at the best
Agency orders are
price possible just before the market closes.
•
A limit order is executed at a level no worse than that specified.
•
A stop order becomes a market order when the market trades through
the stop.
•
A stop-limit order becomes a limit order when the market trades
through the stop.
executed for an
agreed commission
when a
counterparty is
identified
Qualifiers
•
A fill-or-kill order must be executed immediately in its entirety or not
at all.
•
An all-or-none order must either be executed in its entirety or not at all.
•
An immediate-or-cancel order must be executed immediately or not
at all, but may be executed partially.
•
A day order must be executed on the day given.
•
A good-’til-canceled (GTC) order may be executed at any time until
canceled, although it is a good idea to periodically refresh the order.
•
A not-held order may be executed at any time at the market at the
discretion of the salesman. The order must clearly specify the
amount, direction, and security for the transaction.
377
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Technical Analysis
Technical analysis is
the art of trying to
prove the weak
form of the efficientmarket hypothesis
wrong: trading
based on past
prices
One type of technical analysis, known as charting, involves creating
graphs of price or yield history and looking for patterns like head-andshoulders, trend lines, and price-to-moving-average relationships. The
investor reads the chart and forecasts the direction of the market. While
indefensible on theoretical grounds, if other investors believe that the
patterns have meaning, then the market will respond to the pattern.
Thus, by observing the pattern, you may gain information about the
future evolution of prices. In many ways, the market is a self-fulfilling
prophecy. Investors examine the market, decide it is cheap, and try to
buy. The buying drives up prices, and everybody congratulates
themselves on how smart they are.
Another concept frequently mentioned is support or resistance levels.
Support is a price level that the market should have difficulty dropping
below, and resistance is a price level the market should have difficulty
rising above. An economic rationale for the existence of these levels is
limit orders (actual or planned) placed by investors. If investors believe
that at a certain level a security is a buy, and are willing to commit in
size, it will take a lot of selling to pass through that level. There is,
therefore, real economic support for prices at that level. Sometimes the
support or resistance is tax-driven. For example, if prices rise
immediately after a new issue, and later decline, investors may rationally
be more willing to sell at or below the issuance level than above it,
because there would be no capital gains.
Seasonality is another effect that has been observed in the market. For a
long period of time, small stocks tended to perform well during early
January. Investors do have seasonal concerns and pressures, and it is
quite possible that supply and demand could balance at a different price
one month than another, even with no change in underlying market
expectations.
378
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Market-Participant Expectations
If there is a tight consensus among market participants about an
economic release, their expectations are almost always priced into the
market. If the economic release is in line with expectations, there should
be almost no movement in prices, because market participants tend to be
“forward looking,” rather than “backward looking.” If the economic
release differs significantly from expectations, market participants will
analyze the meaning of the release and rapidly reassess the appropriate
level for the market.
If, on the other hand, there is a wide variation of opinions about an
economic release, there is the potential for much greater volatility in the
market. Whatever the release says, it will be different from the views of
some market participants. They can be expected to react to it.
Additionally, there tends to be the greatest difference of opinion when
the economy may be bottoming out or cresting, which adds to the
general level of uncertainty. The Fed may also be awaiting uncertain
numbers to determine future monetary policy. The release of the
numbers may then have the added impact of changing the market’s
perception of future Fed policy. Investor opinions and expectations
about future Fed actions are perhaps the most significant influences in
shaping the development of market prices.
Economists try to
estimate economic
statistics using prior
economic releases
and other
information about
the economy
Market prices
usually discount the
“consensus”
forecast
The variation of
different
economists’
estimates can
provide information
about the risk of
future market
changes
379
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Behavioral Analysis
Most institutional
investors are subject
to review
by superiors,
regulators, or
shareholders; these
investors may
prefer to be wrong
in a herd than
right alone
Some investors are largely driven by the desire to “fit in.” Why buy the
latest high-tech start-up at stratospheric levels? Everyone else is, and if
it continues to go up, and you do not own any, you may have to explain
yourself. Why not buy inverse floaters when the market has been
trashed? After their negative publicity, they just do not look prudent on
the portfolio statement. The trend toward measuring investors against an
index accelerates this effect, because any deviation from the index
represents a gamble.
What starts the chain of events in a market crash? Some researchers have
appealed to chaos theory. Chaos theory seeks to describe dynamic
systems, systems with feedback. The weather is one example: What is
the inconsequential event that starts a thunderstorm, and how does it
cascade into such a violent and coherent event? Why does a pile of sand
sit undisturbed until one grain of sand falls on it and starts a landslide?
Why did the market coast along in October 1987 (at, according to some
pundits at the time, uncomfortably high levels) and suddenly break away?
Often, dramatic reversals are buying (or selling) opportunities. An old
truism is to “buy the rumor, sell the news,” meaning that the rumor of
important information has an effect on the market, while the
confirmation of the information is often less significant. After many
economic releases with market impact, prices overshoot the equilibrium
that will hold later in the day. Of course, the trick is divining when the
market has overshot. Sometimes, the market will have begun a new
long-term trend and never reverse.
380
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
The Gorilla in the Market
How do Fed actions affect the market? By adding or removing a
relatively minor amount of money. If another market participant
undertook the same transaction, it might have no effect whatsoever.
However, unlike any other market participant, the Fed has huge
resources at its disposal, including the ability to print money. Therefore,
if market participants view the Fed’s actions as having policy
implications, they will generally accept the Fed’s view (unless the view
has already been anticipated, in which case the lack of Fed action would
be the telling event). The Fed has other powerful tools at its disposal that
other market participants do not have. For example, only the Fed can
provide emergency funds to banks (through the discount window); the
Fed sets the rate for those loans. And only the Fed can change banks’
reserve requirements.
Some market
participants,
including the Fed
and other central
banks, act to set
policy
Other times, action
by marquee-name
investors can be
market-influencing
information by itself
The Fed and the other central banks rely heavily on credibility to do their
jobs since they are minuscule compared to the combined weight of
investors in the market. In 1993, many central banks in Europe tried to
defend weak currencies by buying while traders sold. Eventually, the
banks used up available reserves and the currency found its new level
anyway, at great cost to the central banks. Following these fiascos, many
resolved to let their currencies float more freely.
Individual investors with great credibility can often influence the market
as well. When a high-profile investor has bought or sold, other investors
may be induced to pile on. Of course, that piling on comes after the
original transaction. The resulting market movement proves the original
investor correct and redoubles the investor’s prestige.
381
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Constrained Behavior
Investors may have
barriers to trading,
which can reduce
the float, or
accessible bonds,
reduce liquidity for
a security, and
cause apparent
unclaimed arbitrage
These constraints
reduce overall
liquidity in the
market, increasing
the market’s
sensitivity to
information or
supply/demand
imbalances
An investor is studying a trade: sell security A, buy security B with the
exact same structure and a higher credit, and pick up 5 bp. The investor
declines. Why?
If the investor is a taxpayer, and security A is at a gain (worth more in
the market than its value on the tax books), the investor would have to
pay taxes this year on any gain from selling the security. This tax will
exceed the present value of the taxes due from continuing to hold the
security. This phenomenon reduces the available float of the security.
Alternatively, if the investor’s capital is subject to regulatory or rating
agency review, and security A is at a loss (worth less than its value on the
customer’s GAAP or regulatory books), the investor could suffer an
apparent loss of surplus. This loss could constrain the investor’s ability to
add new business, affect the price of equity, or cost the investor its rating.
Every organization and investor will be measured and compensated
differently. Often, “mysterious” behavior can be better understood in
this context. For example, an investor with expensive funding will
measure investments based on current yield, because that allows the
investor to pay down debt as quickly as possible. Investors with a lower
cost of funds are more driven by total rate of return. Additionally, some
investors’ performance is measured based on pre-tax returns, and others
based on after-tax returns. This difference can cause different analyses
of the same transaction by different investors.
382
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
The Shape of the Yield Curve Revisited
Theories Regarding the Shape of the
Yield and Expected-Return Curve
Pure
Expectations
The expected return of holding any riskless security Long-term yields
over a holding period will be the same: the risk-free have historically
tended to exceed
rate for the term of the holding period.
short-term yields
Risk (Term)
Premium
Investors demand a higher return for holding longerThere are many
duration securities, because longer securities have more
theories for why
price risk.
this is true;
Preferred
Habitat
Different groups of investors tend to congregate in
different segments of the yield curve. Different
segments are, therefore, partially subject to different
supply/demand equilibria. For example, life insurance
companies and pension funds would tend to own more
long-duration assets than the average investor. Note
that only 14% of Treasury outstandings are longer than
10 years.
Relative Risk
Premium
Pursuant to the Capital Asset Pricing Model, investors
demand increased return the more risk an asset has
relative to the overall market.
General
Equilibrium
Hedge
Assets that perform poorly during a recession or other
difficult times require an additional risk premium,
while assets that perform well during those times
require a smaller risk premium.
however, the issue
remains open
Market volatility
and investor
holding periods are
important issues in
understanding the
term structure
Remember that long-duration assets have expected returns in excess of
their yield due to convexity. If the one-year expected-return curve were
flat, the yield curve would be inverted because long-term investors
would receive some of their expected return from the value of convexity.
383
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Liquidity and Market Impact
A security is liquid if
there are many
buyers and sellers
of that security
There are issues
with executing
large transactions in
less-liquid markets
A security is liquid if there are many buyers and sellers of that security.
The result is a narrow bid-ask spread that can support significant trading
volume. Liquid securities are easy to trade. Conversely, illiquid
securities have few buyers or sellers and may have a wide spread. This
makes them more difficult to trade.
Markets for illiquid securities are more likely to gap when a huge buyer
or seller enters the market because the other side of the market is quickly
exhausted. This is much less likely in a liquid market. However, even
a liquid market, such as the Treasury market, can become illiquid during
The less liquid the
a crisis or right around the release of economic data. Future market
market, the greater
movements follow different patterns depending on the source of the
the potential market current movement: True information usually has a long-term effect on
impact of a
the market, while market movements caused by supply or demand
transaction, and the
imbalances tend to revert (unless there is a perceived information
stronger the
component to the imbalance).
argument for
executing a
negotiated
transaction
In an illiquid market, a smoothly executed negotiated transaction can
provide more efficient pricing for an investor than a competitive
transaction. This is because the dealer can help the seller disguise the
nature of the transaction from the market and retain control on the
seller’s behalf:
384
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Bidding Strategies
In the U.S. Treasury market, where most players have a good estimate of The optimal bid
value, there is little risk to participating in a competitive bid.
level depends on
In a bid for an asset of more debatable value, competition has a more
complicated effect. If you have better information or certainty about the
value of the asset, and that information leads you to arrive at a higher
value, you may increase your bid as the number of bidders increases.
Your objective would be to purchase the asset at a relatively cheap level,
while clearing the highest bid of your competitors, who have less
information.
the strength of
competition, the
absolute certainty
of valuation, and
the relative
certainty of
valuation
However, if all bidders have access to the same information, and
valuations are not certain, it is likely that the winner of a broad bid will
be the bidder who most overvalues the asset. This is called the winner’s
curse. There is actually an argument for reducing your bid as the number
of bidders increases. If all bidders reduce their bids, the winning bidder
may actually earn a return on the asset.
Often bidders get caught up in “bid fever” and bid at or above the top of
their valuation range. Organizations can feel pressure to add assets to
achieve growth and amortize sunk costs in evaluating the assets subject to
bid. Economically, the best thing to do in those situations is to walk away.
Analysis of the number and profile of other bidders early in the process
can help screen for realistic opportunities to focus resources and
maximize chances of success.
385
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
The Prisoner’s Dilemma
The prisoner’s
dilemma is the
classic game-theory
problem in decisionmaking
Two suspects have been accused of committing a crime together. The
police are interviewing them separately. The suspects know that if they
admit nothing, they will be found guilty of a lesser crime and will go to
jail for only six months. On the other hand, either one could accuse the
other of the more serious crime. If only one accuses, that suspect gets a
plea bargain and the other (if he remains silent) goes to jail for 10 years.
If they accuse each other, they both go to jail for nine years. What would
you do?
Payoff Diagram (Years in Jail)
The best overall outcome would be if neither prisoner made an
accusation. However, both suspects know that, whatever the other
suspect chooses, they will be better off making the accusation.
Therefore, without collusion, each will accuse the other and suffer a
markedly worse outcome than by saying nothing. The solution to the
prisoner’s dilemma is trust. If both parties trust each other, they can be
confident in taking the best course of action.
Real-world situations with a structure similar to the prisoner’s dilemma
are common. Bidders face it: if they collude, they could buy an asset
cheaply, but they know each has an incentive to pay a little more than
agreed to guarantee purchase of the asset, so the bid price rises to the
point of indifference. Offerers also face it, e.g., price fixing vs. price
wars. Antitrust laws are designed to preserve a competitive market by
limiting the ability to collude.
386
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Auction Strategies
Treasury
Auction
In Treasury auctions, bidders may submit only one bid.
Small bidders may submit a “non-competitive” bid, which
is guaranteed to be filled at the average price of the
accepted competitive bids. Competitive bids are then filled
from highest to lowest price, with a prorated fill at the
lowest bid level accepted to meet the size of the intended
issuance. The Treasury reports the average bid, the “tail,”
or the difference between the average and the lowest bid
level, and the percent filled at the lowest bid level. A high
percent of bids filled at the tail implies a weak auction.
Dutch
Auction
In a Dutch auction, all winning bidders buy at the same
price, the lowest bid level accepted. Bids are taken from
highest to lowest with a prorated fill at the lowest level
accepted. By reducing the risk of buying at a level richer
than other bidders, a Dutch auction can encourage all
bidders to pay more and achieve a better result. The
Treasury experimented with Dutch auctions on the 2- and 5year note auctions and found no concrete improvement in
execution, although they continue to auction these
securities in this format.
Open
Outcry
The objective of an open outcry auction is to generate a
fever pitch of competition among bidders. Sellers hope that
when bidders see their competitors raising the price, they
will be willing to pay a little more. Each increment can be
a relatively small amount, which can encourage bidders to
pay a little more to win. For example, if a security is worth
101-01, it is probably worth 101-02. Of course, your
competitor may then raise the price to 101-03. The
Resolution Trust Corporation (RTC) had excellent results
with this type of auction.
There are a variety
of different auction
strategies used by
different market
participants
387
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Glossary
This chapter is an excerpt from A Morgan Stanley Guide to Fixed Income Analysis by
Andrew R. Young, ©2003 Morgan Stanley & Co. Incorporated.
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Glossary
Accretion
The increasing of the carrying value of a security
Accrued Interest The interest up until settlement earned but not yet
paid since the prior coupon payment date or the
dated date, whichever is later
Actual/Actual
A calendar convention where the numerator for
accrued interest is the actual number of days elapsed
in the accrual period and the denominator is the
actual number of days in the full coupon period
Actual/360
A calendar convention where the numerator is the
actual number of days elapsed and the denominator
is 360 days
All-In Cost
The cost of issuing a bond, including the amortized
cost of the gross spread
American
Option
An option that can be exercised at any time up to and
including the expiration date
Amortization
The decline in the carrying value of a security
Anchor
A frequently priced benchmark for other bonds
using matrix pricing
Annualized
Yield
Nominal yield that may compound to produce a
higher effective annual yield
Annual Yield
Yield that compounds once per year
APT
See Arbitrage Pricing Theory
Arbitrage
An investment strategy that may make money and is
certain not to lose it
490
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Glossary
Arbitrage
Pricing Theory
A factor model that provides a more general
approach to portfolio management than CAPM
ARM
Adjustable-Rate Mortgage; a mortgage whose
coupon rate resets periodically
Arrears
An obligation to pay at the end of a payment period
Ask
Price at which a dealer is willing to sell a bond
At-the-Money
An option with a strike price equal to the current
price (alternatively, the forward price) of the
underlying asset
Average Life
The average time until (or the average date of)
principal repayment
Backward
Induction
A tree evaluation methodology that starts with
the future value in any state and discounts the
expected cash flows back, combining the states as
the tree narrows
Balloon
Fixed-rate mortgage that returns the entire principal
outstanding to the MBS investor before fully
amortizing
Barbell
The longer- and shorter-duration securities in a
butterfly hedge or portfolio
Basis
The difference between the market value of a bond
and the value that a futures seller would receive by
immediately delivering that bond
Basis Net of
Carry (BNOC)
Comprises the market’s valuation of the delivery
options and arbitrage of a futures contract
491
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Glossary
Basis Point
Equal to one-hundredth of one percentage point, or
0.01%. For example, 5.22% and 5.23% differ by one
basis point (1 bp)
Basis Risk
Variance between the hedge and the hedged asset
BEY
See Bond-Equivalent Yield
Bid
Price at which a dealer is willing to buy a bond
Binary Tree
A model of the future where each state leads to two
possible states in the next period (stage)
Black–Derman–
Toy
An option-valuation model that describes a binary
interest rate tree
Black–Scholes
An option-valuation model that provides a closedform solution for pricing a European option (with
several strong assumptions)
Bond-Equivalent The semi-annual actual/actual yield that equates the
Yield (BEY)
discounted value of a bond’s actual future cash flows
with the bond’s present value in the market
BP
See Basis Point
Brady Bond
Sovereign bond issued by a developing country to
restructure defaulted commercial bank debt, usually
structured with principal and interest collateral
Break-Even
The point of indifference between two alternatives
Brownian Motion A random process often used as a model for the
evolution of security prices
492
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Glossary
Bullet
The position of the mid-duration security in a
butterfly hedge
Butterfly
A proceeds-neutral, duration-neutral three-bond
trade where a bond is hedged with both a longerduration and a shorter-duration bond
Call
An option granting the holder the right to buy the
underlying asset on (or before) a specified date at a
specified (strike) price
Callable
A bond with an embedded call option; the issuer can
redeem the bond prior to maturity
Cap
A series of options that pays out the excess of a
given rate above the cap rate over time
Capital Asset
Pricing Model
(CAPM)
A model describing any asset’s expected excess
return over the risk-free rate as the market’s excess
return multiplied by that security’s riskiness relative
to the market
Carrying Value
The value of a security on a company’s books, which
starts at the acquisition price and drifts toward par
over the life of the bond
Cash (Settlement) Same-day settlement
Cash-Callable
May not be refunded with lower-cost debt, but may
be called if the issuer has cash available
Cash-on-Cash
Return
See Current Yield
493
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Glossary
Cheapest-toDeliver (CTD)
The deliverable bond that satisfies a futures contract
at the lowest total cost
CMO
Collateralized Mortgage Obligation
CMT
Constant-Maturity Treasury
Compounding
A method of calculating interest where interest earns
interest; compounding increases the investor’s
effective yield on unpaid principal of outstanding
investments
Compound
Interest
Interest earned on interest
Convergence
The decrease in the deviation of the futures price
adjusted by factor from the price of the underlying
asset as the time remaining until delivery decreases
Convexity
A measure of the curvature in the price/yield
relationship; the rate of change in duration
Coupon
The contractual rate of interest on a bond
Covered Call
A short call position against which the writer owns
the underlying asset
CPR
Constant prepayment rate; the annual percent of
balance projected to be prepaid
CTD
See Cheapest-to-Deliver
Current Yield
Also known as cash-on-cash return; the annual cash
flow of an investment as a percentage of the amount
(sometimes price) invested
494
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Glossary
Curtailment
A prepayment made by a homeowner who has extra
cash to pay off a portion of the mortgage but does
not fully prepay
CUSIP
A nine-digit code that uniquely represents a security
Dated Date
The date on which a security begins accruing
interest; if not on the coupon cycle, the security will
have an odd first coupon
Day Count
The convention for measuring partial periods for
accrued interest and discounting
Defeasance
Also known as an escrow; a portfolio constructed to
provide sufficient cash to precisely meet liabilities
Delivery
(Futures)
The period of time during which delivery or
settlement must be made
Delta-Hedging
Hedging an option with a percentage of the
underlying asset
Denomination
The minimum incremental face amount of a security
that can be traded. For most corporate and
government bonds, the denomination is $1,000
Discount Bond
A bond whose price is currently below par; a bond
whose coupon rate is less than its yield
Discounting
Reducing future cash flows to their present value
Discount Margin The spread to a floating-rate bond’s index that
discounts the bond’s expected future cash flows to
the bond’s actual present value
495
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Glossary
Dollar Duration
The change in value for a small change in yield
Dollar Value of
a Basis Point
(DV01)
See Present Value of a Basis Point
Duration
A linear measurement of the interest rate sensitivity
of the value (or price) of a security; the change in
value, as a percent of value, for a small change in
yield; also known as modified duration or presentvalue duration
DV01
See Present Value of a Basis Point
Effective Interest The accretion or amortization of a bond toward par
over time according to the yield of the security
Efficient Frontier The curve showing the lowest-risk portfolio for any
level of expected return
Embedded
Option
Any option that is contained in the structure of a
security
Escrow
See Defeasance
Eurodollar
Futures
Contracts on 3-month CDs that pay interest at
LIBOR
European Option An option that can only be exercised on the
expiration date
Exercise Date
Also known as expiration date; the last day on which
an option can be exercised
Expiration Date
See Exercise Date
496
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Glossary
FHLMC
Federal Home Loan Mortgage Corporation
Financing
Option
An option arising out of a futures seller’s ability to
time delivery to maximize positive carry
Fitted-Yield
Curve
A hypothetical smoothed yield curve that
minimizes the error between the hypothetical fitted
prices and actual prices
Fixed Coupon
A coupon rate that is constant over the life of a bond
Floating Coupon A coupon rate that changes over the life of a bond
Floor
A series of options that pays out the shortfall of a
given rate below the floor rate over time
Forward
Discount Factor
The value, at the beginning of a future period, of
$1 at the end of that period
Forward
Settlement
A transaction-settlement
settlement
Forward Yield
The yield of a security for forward settlement
FRM
The new-issue mortgage rate
Futures
Standardized, exchange-traded contracts for the
purchase or sale of an asset in the future
Future Value
The value of a payment or stream of payments on a
specified future date
General
Collateral
A repo transaction where the borrower can supply
any Treasury securities as collateral
GNMA
Government National Mortgage Association
date
after
regular
497
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Glossary
Gross Spread
The commission paid to the underwriter
GWAC
Gross Weighted Average Coupon; the average
interest rate currently being paid by a pool of
mortgages
Handle
The integer part of a security’s price, when
expressed in percent
Heath–Jarrow–
Morton
An option-valuation model that evolves the entire
forward-rate curve at every stage
Hedge
To limit financial risk by entering an offsetting
transaction
Holding Period
The period over which return is to be measured
Horizon
The end of the period over which return is to be
measured
Immediate Pay
Bonds that pay at the beginning of each payment
period
Implied Repo
Rate
The financing rate that creates indifference to
selling the futures and buying a deliverable security,
financing it, and delivering it on the futures
Index
A cross-section of the market that provides a
benchmark against which many investors are
measured
Initial Margin
The initial collateral for a futures contract
Internal Rate of
Return (IRR)
The annualized yield that would result
in a zero net present value for an investment
(including its cost)
498
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Glossary
In-the-Money
An option that would result in a gain for the long
position if it were exercised immediately
Intrinsic Value
The amount of gain that would result if an option
were exercised immediately
IRR
See Internal Rate of Return
Issuer
An entity wishing to raise capital in a financial
market
LIBOR
London Inter-Bank Offered Rate; the deposit rate
offered among leading international banks
Liquidity
The ease and efficiency of purchasing or selling a
security
Loan-to-Value
The ratio of the amount of a loan to the value of the
collateral
Log-Normal
The exponentiation of a normally distributed
random variable
Long Position
The position of the buyer of a security
Macaulay
Duration
The present-value-weighted time to payment
of a security’s cash flows
MarketExpectations
Forward Yield
The forward yield implicit in the market, assuming
that securities are priced so that investors are
indifferent between buying a longer security and
buying a shorter security and rolling over to the
longer term
499
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Glossary
Mark-to-Market Addition or subtraction of margin due to the change
in the market price of an investment
Matrix Pricing
A pricing method that prices most bonds relative to
a few anchor bonds
MBS
Mortgage-backed security
Mean Reversion
A level-dependent parameter added to Brownian
motion to prevent the price (or interest rate) path
from deviating too far from the mean
Mid-market
The average of the bid and ask price
Modified
Duration
See Duration
Monte Carlo
Simulation
A technique of modeling many possible paths of
future events to ascertain a complex security’s value
Newton–Raphson A method of solving an equation using an initial
guess and iteratively refining the guesses based upon
the error and slope of the curve at each successive
point
Notice Date
The day after the tender date, on which the futures
seller must identify which security will be delivered
Notional Amount The base amount for calculating the fixed and
floating payments on a swap
OAC
See Option-Adjusted Convexity
500
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Glossary
OAD
See Option-Adjusted Duration
OAS
See Option-Adjusted Spread
OAY
See Option-Adjusted Yield
Offer
Price at which a dealer is willing to sell a bond
Off-the-Run
All Treasury issues that are not on-the-run
On-the-Run
The most recently auctioned Treasury issue for each
maturity
Open Interest
The number of futures contracts outstanding
Option-Adjusted The measure of a bond’s convexity that takes into
Convexity (OAC) account the effect of any embedded options
Option-Adjusted The measure of a bond’s duration that takes into
Duration (OAD) account the effect of any embedded options
Option-Adjusted The expected spread to Treasuries for a bond with
Spread (OAS)
embedded options
Option-Adjusted The yield that discounts future cash flows (assuming
Yield (OAY)
no option exercise) to the value of the bond plus the
value of the embedded option
Out-of-theMoney
An option that would result in a loss for the long
position if it were exercised immediately
Par
Refers to the principal or face value of a bond; a
bond whose price is par has a dollar price of 100%
of face value
501
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Glossary
Par Bond
A bond that is priced at par (100% of face value); a
bond whose yield equals its coupon
Pass-Through
A security that represents a pro-rata share of cash
flows, minus a servicing fee, generated from
underlying mortgages; a pool
Path-Dependent
An option that depends on the prior history of option
exercise, yields, or prices
PIK
Pay-in-kind; receiving payments in the form of more
bonds
Premium
The cost of purchasing an option
Premium Bond
A bond whose price is currently above par; a bond
whose coupon rate exceeds its yield
Prepayment
The early redemption of all or a portion of a
mortgage
Present Value
The value on settlement of a payment or a stream of
payments due and payable at some specified future
date(s), discounted at some interest or discount rate
(yield); the act of calculating present value; the cost
(price plus accrued) of purchasing a security
Present-Value
Duration
See Duration
Present Value
of a Basis Point
(PV01)
Also known as dollar value of a basis point
(DV01); the change in value of a security due to a
one-basis-point change in yield
502
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Glossary
Price Duration
The change in value, as a percent of price, for a
small change in yield
PSA
Public Securities Association; the prepayment
model proposed by the PSA
Pseudo-Coupon
Date
A date on which a generic coupon bond with the
same maturity and conventions would pay a coupon,
but on which a specific bond does not pay a coupon
Pseudo-Random A number that is used in simulation as a random
Number
number, but in fact is generated systematically and
can be regenerated at will; furthermore, a series of
pseudo-random numbers may be chosen to eliminate
the clumping that is likely to occur with truly
random numbers
Put
An option granting the holder the right to sell the
underlying asset on (or before) a specified date at a
specified price
Putable
A bond with an embedded call option; the investor
can force the issuer to redeem the bond prior to
maturity
Put/Call Parity
The relationship between the value of a European
call option and a European put option with the same
underlying asset, expiration date, and strike price
PV01
See Present Value of a Basis Point
Quality Option
See Substitution Option
503
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Glossary
Regular
(Settlement)
The usual settlement for a security’s market
Reoffered Yield
The yield at which a security’s original investors
purchase it
Repo
Repurchase agreement; a contract where the system
(dealer community) sells a security and
simultaneously agrees to repurchase it at a later date,
which is equivalent to a collateralized financing
ROR
See Total Rate of Return
Seasoned Pools
A mortgage pool that is aged, usually more than 12
months
Settlement Date
The actual date on which cash is exchanged for a
security
Short Position
The position of the seller of a security
Simple Interest
A linear method of calculating interest without
compounding
Simulation
See Monte Carlo Simulation
Sinking-Fund
Bond
A bond that requires the retirement of debt
according to a predetermined schedule throughout
its life
Skip Day
For Treasuries, settlement in two business days (one
more than regular)
SMM
Single Month Mortality; the monthly percent of
balance projected to be prepaid
504
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Glossary
Specific
Collateral
A repo transaction where the borrower must supply
as collateral the specific Treasury security requested
by the lender
Stage
The time period
State
The interest rate environment
Straddle
The option strategy of buying a put and a call on the
same asset with the same strike price and expiration
date
Strike
The price at which the asset underlying an option
can be bought or sold
Stripped Yield
The yield of the emerging-markets portion of a
Brady bond, after removing the effects of collateral
STRIPS
Separate Trading of Registered Interest and
Principal Securities; zero-coupon Treasury
securities that are coupon or principal payments
separated from Treasury coupon bonds
Substitution
Option
Also known as the quality option; the futures seller’s
option of delivering the least-attractive qualifying
security
Swap
A transaction that allows an investor to exchange
one set of payments for another
Swaps Curve
A curve of market fixed rates that can be swapped
for LIBOR
Swaption
An option that entitles the holder to enter into a swap
with predefined terms
505
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Glossary
Switching Option A special case of the substitution option after the
close on the last day for trading a futures contract
Tender Date
The day the futures seller announces intent to deliver
in two business days
30/360
A calendar convention where every month is
assumed to have 30 days; the Securities Industry
Association (SIA) has published the rules for
calculating the number of days of accrued interest
using the 30/360 calendar
Tick
Equal to one thirty-second of a percent of par for
U.S. Treasury (UST) bonds; some other markets
define a tick as 0.01% or 0.05% of par
Time Value
An assessment of the value of an option attributable
to time; defined as the excess of the premium over
intrinsic value
Total Rate of
Return (ROR)
Uses all the information available to calculate the
return of all projected cash flows from a security
over a fixed time period, including cash flow
reinvestment
Tracking Error
Deviation of actual portfolio returns from index
returns
Tranche
A single security issued as part of a CMO structure
Turnover
Prepayments that occur independent of refinancings
(for example, due to defaults, catastrophes and
relocations)
506
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Glossary
Unwind
The cancellation of a swap contract before maturity
by paying a lump sum
Value at Risk
(VAR)
A measure of potential loss
Variation Margin The amount an investor pays or receives from the
daily change in the value of a futures contract
Volatility
Variability (standard deviation) of the price, yields
or return of a security
WAM
Weighted-average maturity (in months) of a pool of
mortgages
Wild Card
Option
An option arising during the delivery month from
the difference between the futures closing price at
2:00 PM CST and the deadline for providing notice
of intent to deliver at 8:00 PM CST
Writer
The seller of an option
Yield
The rate of interest earned on an investment
Yield Curve
The curve that shows the relationship between yields
and maturities
Yield-to-Call
(YTC)
The yield that discounts the cash flows of a bond,
assuming that it is called on a particular call date, to
the present value of the bond in the market
Yield-to-Maturity The yield that discounts the cash flows of a bond
(YTM)
to the present value of the bond in the market
507
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Glossary
Yield-to-Worst
(YTW)
The lowest of YTM and all YTCs of different call
dates
Zero-Coupon
Bond
A bond that pays no periodic interest
508
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Equation
Reference
This chapter is an excerpt from A Morgan Stanley Guide to Fixed Income Analysis by
Andrew R. Young, ©2003 Morgan Stanley & Co. Incorporated.
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Bond Variable Definitions
v
is the par amount, also known as redemption value (usually
100%)
y
is the yield, quoted on a compound basis
f
is the compounding frequency (also the payment frequency for
coupon bonds)
n
is the number of whole compounding periods between the next
coupon (or pseudo-coupon) date and maturity
x
is the length of the accrual period, using the appropriate
calendar, 0#x<1
n +1- x
f
is the number of years remaining until maturity
t
is the number of years until maturity for continuous
compounding
r
is a short-term rate
510
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
General
Compounding:
yf ö
æ
1 + Annual Yield = ç 1 + ÷
f ø
è
f
f
æ
yö
lim ç 1 + ÷ = e y
f ®¥ è
fø
Fundamental Theorem of Fixed Income:
Present Value = Price + Accrued
Newton–Raphson Iterative Formula:
Solving for x such that f (y) = x
yi + 1 = yi -
f (yi ) - x
df (y )
dy y = y
= yi +
Price i - Price Actual
DurationDollar,i
i
Averaging:
Arithmetic:
1
´ å wi xi
w
å i i
n
Geometric:
å wi
i =1
Õ (x
n
i =1
i
i
wi
)
511
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
General (Continued)
Solving for the Coupon:
100%
æ
ç Price n +1- x
æ
y n+1-x ö
ç
ç 1+
÷
ç
f ø
è
c= f ´ç n
1
ç å
i +1- x - x
ç i =0 æ
ö
y
çç
ç 1+ i +1- x ÷
f ø
è
è
ö
÷
÷
÷
÷
÷
÷
÷÷
ø
Foreign Exchange Equilibrium:
æ
ö
r
ç 1 + Home ÷
fH ø
è
fH ´ n
=
1
sSpot
rForeign ö
æ
´ ç1 +
÷
fF ø
è
fF ´ n
´ sForward Þ sForward = sSpot ´
æ
ö
r
ç 1 + Home ÷
fH ø
è
fH ´ n
rForeign ö
æ
ç1 +
÷
fF ø
è
fF ´ n
Barbell Portfolio Weights:
ParBarbell - Short =
ParBarbell - Long =
(
ParBullet ´ PVBullet ´ DBarbell - Long - DBullet
(
PVBarbell - Short ´ DBarbell - Long - DBarbell - Short
)
)
ParBullet ´ PVBullet ´ (DBullet - DBarbell - Short )
(
PVBarbell - Long ´ DBarbell - Long - DBarbell - Short
)
512
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Duration and Convexity Definitions
Different Methods of Quoting Duration and Convexity:
DurationDollar = Duration ´ PV = DurationModified PV ´ PV = DurationModified Price ´ Price =
DurationModified PV = Duration =
DurationMacaulay
´ PV
y
1+
f
DurationDollar
Price DurationMacaulay
= DurationModified Price ´
=
PV
PV
y
1+
f
ConvexityDollar = Convexity ´ PV = 2 ´ ConvexityGain ´ PV
Using Duration and Convexity (Taylor Series):
P1 = P0 +
dP
1 d 2P
2
y1 - y0 ) +
y - y0 ) + L
(
2 ( 1
dy
2 dy
P1 @ P0 - PV0 ´ Duration ´ (y1 - y0 ) + PV0 ´
Convexity
2
´ (y1 - y0 )
2
Weighting Duration and Convexity:
Duration =
1
1
´ å PVi ´ Durationi =
´ å PVi ´ Durationi
PV
i
å PVi i
i
Other Methods of Quoting Duration:
DurationDollar =
DP PV 01 1 32
=
=
Dy 0.01% YV 32
Calculating Convexity from Three Equally Spaced Observations:
Convexity @
PriceHigh + PriceLow - 2 ´ PriceMiddle
PVMiddle ´ ( Dy)
2
513
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Zero-Coupon Bonds
Price
Dollar
Duration
Definition
Differential
(for small
Dy)
Compound
Formula
v
æ
yö
ç1 + ÷
fø
è
Continuous
Formula
v´
n+1- x
e - yt
Modified
Duration
Dollar
Convexity
dP
d 2P
dy 2
-
dP
dy
-
-
DP
Dy
-
n + 1- x
f
æ
yö
ç1 + ÷
fø
è
n+ 2- x
te - yt
P
dy
DP
P
Dy
n + 1- x
f
æ
yö
ç1 + ÷
fø
è
t
-
v´
DDurationDollar
Dy
n + 1- x n + 2 - x
´
f
f
æ
yö
ç1 + ÷
fø
è
n+3- x
t 2 e - yt
Convexity
d 2P
P
dy 2
-
DDurationDollar
Dy
P
n + 1- x n + 2 - x
´
f
f
æ
yö
ç1 + ÷
fø
è
2
t2
514
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Coupon Bonds
Series Formula
Present
Value (PV)
v
æ
ö
y
ç 1 + n +1- x ÷
f ø
è
n +1- x
Accrued
Interest
+
c
f
x´
Price
n
å
i =0
i +1- x
c
f
PV - x ´
Modified
PV
Duration
1
æ
ö
y
ç 1 + i +1- x ÷
f ø
è
These formulas
apply when the
maturity lies on the
coupon cycle, the
first coupon is
regular, and the
security is not in its
last coupon period
c
f
ö
æ
÷
ç
n
i+1- x ÷
1 çv
(n + 1 - x ) + c ´
´ç ´
å
n+2- x
i+2- x ÷
PV ç f æ
f 2 i =0 æ
yö
yö
÷
ç1 + ÷
ç1 + ÷
÷
ç
fø
fø
è
è
ø
è
Convexity
ö
æ
÷
ç
n
i + 1 - x ) ´ (i + 2 - x )÷
1 ç v (n + 1 - x ) ´ (n + 2 - x ) c
(
´ç 2 ´
+
´
÷
å
i + 3- x
n + 3- x
PV ç f
f 3 i =0
æ
æ
yö
yö
÷
ç1 + ÷
ç1 + ÷
÷
ç
fø
fø
è
è
ø
è
Closed-Form Formula
æ
yö
vy - c
cç 1 + ÷ +
n
fø æ
è
yö
ç1 + ÷
fø
è
Present
Value (PV)
æ
yö
yç1 + ÷
fø
è
Accrued
Interest
Price
Modified
PV
Duration
x´
1- x
c
f
PV - x ´
1
æ
yö
PV y ç 1 + ÷
fø
è
2
Convexity
1
æ
yö
PV y 3 ç 1 + ÷
fø
è
2- x
1- x
c
f
é y 2 v(n + 1 - x ) æ
ù
ö
y
ê
ú
- cç 1 + (n + 2 - x )÷
f
f
æ
ö
è
ø
y
ê
ú
1
1
´ê
+
+
c
x
(
)
ç
÷
n +1
f
è
øú
æ
yö
ê
ú
ç1 + ÷
ê
ú
fø
è
ë
û
é y 3 v(n + 1 - x )(n + 2 - x ) æ
æ
öö
y
y
- cç 2 + (n + 3 - x )ç 2 + (n + 2 - x )÷ ÷
ê
f
f2
f
è
øø
è
ê
n +1
ê
æ
ö
y
ê
ç1 + ÷
´ê
fø
è
ê
ê æ
ö
êcç 2 + y (2 - x )æç 2 + y (1 - x )ö÷ ÷
ê è
f
f
è
ø
ø
ë
ù
ú
ú
+ú
ú
ú
ú
ú
ú
ú
û
515
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Short-Term Investments
Using Discount Yield:
æ Par - Price ö æ 360 ö
Yield Discount = ç
÷ ´ç
÷
è
ø è d ø
Par
Price=Par -
Yield Discount ´ Par ´ d
360
T-Bill BEY:
d £ 182:
d > 182:
YBEY =
YBEY
Par - Price 365
´
Price
d
182 .5
é
ù
d
Par
Price
æ
ö
ê
= 2 ç1 +
- 1ú
÷
êè
ú
Price ø
úû
ëê
Simple-Interest Investments (Actual/360):
rd ö
PV ´ æç 1 +
÷ = FV
è
360 ø
Price =
Par
1+
d
´ Yield Simple-Interest
360
516
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Calendar Conventions
•
To determine the number of 30/360 days between two dates, Date1
(prior coupon date) and Date2 (settlement), where Date1 is earlier :
360 × (Year2 – Year1)
+ 30 × (Month2 – Month1)
+ DDays (from the following table)
= 30/360 days between Date1 and Date2
Day1
Not End of Month
End of Month
End of Month
Except:
End of Month (Excluding February)
•
Day2
DDays
Not End of Month
End of Month
Day2 – Day1
Day2 – 30
0
End of February
Day2 – 30
The denominator always has 180 days for a semi-annual bond.
More generally, it has 360 f days.
517
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Forward Pricing
Arbitrage-Free Forward Price (Traditional Convention):
(Price
Spot
rd ö
c
æ
+ Accrued Spot ´ ç 1 +
÷ = PriceForward + Accrued Forward +
è
360 ø
f
)
æ
ri d i ö
å çè 1 + 360 ÷ø
i
Arbitrage-Free Forward Price (Alternative Convention):
PriceSpot + Accrued Spot =
PriceForward + Accrued Forward c
+
rd ö
f
æ
ç1 +
÷
è
360 ø
1
r (d - d i )ö
ç1 +
÷
360 ø
è
åæ
i
Forward Yield Estimation:
Dy »
(y - r ) ´ t
Duration Modified PV
Market-Expectations Forward Yields:
y0 ,m ö
æ
ç1 +
÷
f ø
è
f ´m
y
æ
ö
´ ç 1 + m ,m+ n ÷
f ø
è
f ´n
y
æ
ö
= ç 1 + 0 ,m+ n ÷
f ø
è
f ´(m+ n )
Forward Rates:
RateSimple- Interest Forward =
RateBEY Forward
360
Days Actual between T
1 and T2
æ (1 + y2 f )f ´T2
ö
ç
÷
1
ç 1 + y f f ´T1
÷
)
è(
ø
1
1
é
ù
êæ (1 + y f )f ´T2 ö Actual / Actual Semi- Annual Periods between T1 and T2
ú
2
÷
ú
= 2 ´ êçç
1
f ´T
êè (1 + y1 f ) 1 ÷ø
ú
ê
ú
ë
û
518
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Measures of Return
Dollar-Duration-Weighted Yield:
å Par ´ PV ´ Dur ´ y
Dollar-Duration-Weighted Yield =
å Par ´ PV ´ Dur
i
i
i
i
i
i
i
i
i
Market-Value-Weighted Yield:
å Par ´ PV ´ y
Market-Value-Weighted Yield =
å Par ´ PV
i
i
i
i
i
i
i
Rate-of-Return Calculations:
RORBond-Equivalent ö
æ
Initial Investment ´ ç 1+
÷
2
è
ø
2 ´ Holding Period (Years)
= Future Value
1
æ
ö
2
´
Holding
P
eriod (Years)
ç æ Future Value ö
÷
RORBond-Equivalent = 2 ´ ç ç
- 1÷
÷
ç è Initial Investment ø
÷
è
ø
519
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Distributional Definitions
Log-Normal Distribution:
Log-Normal = e Normal
m Log - Normal = e
s Log - Normal = e
m Normal +
m Normal +
s2Normal
2
s2Normal
2
2
´ e sNormal - 1
520
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Options Models
Black–Scholes:
[
(
C = e - rT ´ F ´ N (d )- K ´ N d - s T
where
[
F = Forward Price = P - å Dti e
- rti
]´ e
)]
rT
æFö s
lnç ÷ +
´T
Kø 2
è
d=
s T
2
Black–Derman–Toy:
s· f
at any node
r·h = r·l ´ e
pq
521
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Measuring Volatility
Normalizing Volatility:
T2
T1
vT2 = vT1 ´
Sample Standard Deviation:
n
ˆ =
s
å (x - x )
2
i =1
i
n-1
Chi-Squared Confidence Interval for Standard Deviation:
æ (n - 1)´ s
ˆ2
(n - 1)´ sˆ 2 ö÷ = 1 - a
Prç 2
£s£
ç c
c a2 2,n-1 ÷ø
è 1-a 2,n-1
Parkinson’s Extreme Value Method for Estimating Standard Deviation:
ˆ =
s
n
1
(ln(Highi )- ln(Lowi ))2
å
n ´ 4 ´ ln (2) i=1
522
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Futures and the Basis
Invoice Price:
PriceInvoice = PriceFuture ´ Factor + Accrued
Futures Basis:
Basis = PriceBond - PriceFutures ´ FactorBond
Basis=ValueCoupon Accrual Plus FV of Coupons Paid - ValueFinancing on Price+Accrued + ValueOptions + Arbitrage
é
DateDelivery - DateCoupon ö ù
Coupon k æ
i
÷÷ ú
´ å çç 1 + REPO Actual ´
ê Accrued Delivery - Accrued Spot +
2
360
ê
i =1 è
ø ú
Basis = ê
ú
ê
ú
DateDelivery - DateSpot ö
æ
ê- PriceSpot + Accrued Spot ´ ç REPOActual ´
÷ + ValueOptions + Arbitrageú
360
êë
úû
è
ø
(
)
Implied REPO Rate:
PriceFutures
éæ
ù
DateDelivery - DateSpot ö
æ
÷ - Accrued Delivery ú
êçè PriceSpot + Accrued Spot ö÷ø ´ ç 1 + REPO Implied ´
360
è
ø
ê
ú
ú
´ Factor = ê
ê Coupon k æ
ú
DateDelivery - DateCoupon ö
i
÷
êú
´ å çç 1 + REPO Implied ´
÷
2
360
êë
úû
i =1 è
ø
ß
Repo Im plied =
(
)
Coupon
´k
2
Date Delivery - DateCouponi
Price Futures ´ Factor Bond + Accrued Delivery - Price Spot + Accrued Spot +
(Price
)
+ Accrued Spot ´
Spot
Date Delivery - Date Spot
360
-
Coupon k
´å
2
i =1
360
where k is the number of coupons paid prior to delivery.
523
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Mortgages
Balance-to-Payments Calculation:
1Balance = Payment ´
1
(1 + c f )Term ´ f
c
f
CPR to SMM Calculation:
(1 - CPR)= (1 - SMM )
12
CPR = 1 - (1 - SMM )
12
1
12
SMM = 1 - (1 - CPR)
Dollar Rolls:
(Price
Spot
rd ö
æ
+ Accrued Spot ´ ç 1 +
÷
è
360 ø
)
= (1 - Paydown) ´ (Price Forward + Accrued Forward )+ FVCoupons + FVPrincipal
Balance at end of nth period (no prepayments):
é (1 + c f )Term´ f - (1 + c f
Bn = B0 ´ ê
êë
(1 + c f )Term´ f - 1
)n ùú
úû
Payment, given prepayments (with a percentage that are curtailments):
æ
(1 - Curtailments) ´ Prepaymenti -1 ö
Paymenti = Paymenti -1 ´ ç 1 ÷
Bi -1 + Prepaymenti -1
è
ø
524
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Exercise Solutions
This chapter is an excerpt from A Morgan Stanley Guide to Fixed Income Analysis by
Andrew R. Young, ©2003 Morgan Stanley & Co. Incorporated.
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 1 Exercise Solutions
1. Calculate the present value, modified duration, dollar duration,
and convexity of these two Treasury STRIPS (zero-coupon
bonds).
Maturity
(Years)
Yield
(%)
Present
Value
(%)
5 Years
6.75
71.754
4.837
347.056
25.734
25 Years
7.50
15.871
24.096
382.425
592.249
Modified
Duration
Dollar
Duration (%)
Convexity
Define n to be the number of periods until maturity, and y to be the yield.
STRIPS compound semi-annually, so f=2.
PV =
100%
æ
ç1 +
è
yö
÷
2ø
DurationModified =
n
n/2
æç 1 + y ö÷
è
2ø
DurationDollar = PV ´ DurationModified
Convexity =
n( n + 1)
yö
æ
4ç 1 + ÷
è
2ø
2
390
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 1 Exercise Solutions (Continued)
2. What is the dollar duration of a 1-year STRIPS yielding 5%?
What is its modified duration? What is the dollar duration of a
30-year STRIPS yielding 8%? What is its modified duration?
The price/yield relationship for STRIPS (yields on STRIPS are quoted
on a semi-annually compounded basis):
PV =
100%
æ
ç1 +
è
yö
÷
2ø
n
so the dollar duration and modified duration formulas are:
DurationDollar = -
dP 100% ´ n / 2
=
n+1
dy
yö
æ
ç1 + ÷
è
2ø
DurationModified = -
dP / P
n/2
=
dy
æç 1 + y ö÷
è
2ø
For a 1-year STRIPS at 5% yield:
DurationDollar = -
dP 100% ´ 2 / 2
=
2+1 = 92.860%
dy æ
5% ö
ç1 +
÷
è
2 ø
DurationModified = -
dP / P
2/ 2
=
= 0.976
5% ö
dy
æ
ç1 +
÷
è
2 ø
391
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 1 Exercise Solutions (Continued)
For a 30-year STRIPS at 8% yield:
DurationDollar = -
dP 100% ´ 60 / 2
=
60+1 = 274 .213%
dy æ
8% ö
ç1 +
÷
è
2 ø
DurationModified = -
60 / 2
dP / P
=
= 28.846
8% ö
dy
æ
ç1 +
÷
è
2 ø
3. What are the price, modified duration, and convexity of a 30year STRIPS at a 7% and a 7½% yield? How do these numbers
all fit together?
Price =
100%
yö
æ
ç1 + ÷
è
2ø
DurationModified =
60
60 / 2
æç 1 + y ö÷
è
2ø
Convexity =
60(60 + 1)
yö
æ
4ç 1 + ÷
è
2ø
2
At 7.00%: P = 12.69% ; DurationModified = 28.99 ;Convexity = 854.16
At 7.50%: P = 10.98% ; DurationModified = 28.92;Convexity = 850.05
Py =7 .50% @ Py =7 .00% + Py =7 .00% ´ Dy =7 .00% ´ (7.00% - 7.50%)
1
2
´ Py =7 .00% ´ C y =7 .00% ´ (7.00% - 7.50%)
2
@ 12.69% + 12.69% ´ 28.99 ´ (7.00% - 7.50%)
+
1
2
´ 12.69% ´ 854.16 ´ (7.00% - 7.50%)
2
@ 10.99%
+
392
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 1 Exercise Solutions (Continued)
4. A pension fund manager has a $23 million liability due in five
years. How much needs to be invested today if the manager can
lock in an annual interest rate of 6.75% for five years? How
much if the rate compounds semi-annually?
These are simple present-value calculations with two different
conventions for quoting interest rates. At a 6.75% annual yield:
PV =
$23,000 ,000
(1 + 6 .75%)5
= $16 ,591,606
At a 6.75% bond-equivalent yield:
PV =
$23,000,000
6.75% ö
æ
ç1 +
÷
è
2 ø
= $16,503,369
2´5
5. What is the semi-annually compounded yield of a Treasury
STRIPS that matures in 20 years and is priced at 23.111%?
PV = 23.111% =
100%
æ
ç1 +
è
yö
÷
2ø
40
1
y ö æ 100% ö 40
æ
ç1 + ÷ = ç
÷ = 1.0373
è
2 ø è 23.111% ø
y = 7.46%
393
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 1 Exercise Solutions (Continued)
6. If Manhattan was worth $24 in trade goods 360 years ago, what
has been the annual total rate of return on the investment if the
island is worth $100 billion today?
We are looking for an annual rate of return given a present and future
value. If y is the annual yield, then:
360
$24 ´ (1 + y)
= $100,000,000,000
1
æ $100,000,000,000 ö 360
y=ç
-1
÷
è
ø
$24
y = 6 .35%
394
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 1 Exercise Solutions (Continued)
7. If a corporation expects to pay $100 million in the year 2020
(24 years from now) to its pension beneficiaries, what is the
present value of this liability at an annual discount rate of
7.25%? If rates decline by 100 bp, what is the new value of the
liability? What is the error if we estimate the new liability value
using duration?
Assuming an annual discount rate of 7.25%,
PVActual , y =7 .25% =
$100 ,000 ,000
(1 + 7 .25%)24
= $18 ,640 ,810
At 6.25%,
PVActual , y =6 .25% =
$100 ,000 ,000
(1 + 6 .25%)24
= $23,340,248
PVEstimate , y = 6 .25% = PV7 .25% - PV7 .25% ´ D7 .25% ´ (6 .25% - 7 .25%)
= $18,640,810 - $18,640,810 ´
24
´ (6 .25% - 7 .25%) = $22,812,180
1+7.25%
(
)
Error = $22,812,180 – $23,340,248 = –$528,068 (about 2.3% of the
cost at an annual rate of 6.25%)
395
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 1 Exercise Solutions (Continued)
8. A security that promises to pay $10,000 five years from now can
be purchased for $7,175.38 today. What is its semi-annually
compounded yield? If there is a secondary market for this
security, how will its market yield change as the credit quality of
the issuer deteriorates?
$7,175.38 =
$10,000
yö
æ
ç 1+ ÷
è
2ø
5´ 2
1
é
ù
5´ 2
$10,000
æ
ö
ê
y=2´ ç
- 1ú = 6 .75%
êè $7,175.38 ÷ø
ú
êë
úû
If the credit quality of the issuer deteriorated, the value of the bond
would fall because there would be a greater chance of not receiving
future cash flows. If the price of the bond falls, its yield will rise. The
difference between the market yield on this bond and the yield on
comparable-maturity Treasuries is commonly called the spread. As
credit quality deteriorates, the spread “widens” (increases).
396
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 1 Exercise Solutions (Continued)
9. Should you pay $6 million today for a bond that promises to pay
$9 million in five years if you need to earn an 8.00% annual
return?
Here, we just need to compare the present value of the bond at our
required rate of return to the market value of the bond.
PV =
$9,000,000
(1 + 8.00%)5
= $6,125,249
So, buy the bond. It is less expensive than we would have been willing
to pay. Alternatively, we could calculate the yield on the bond and
compare it to our required return.
$6,000,000 =
$9,000,000
(1 + y)5
1
5
æ $9 ,000 ,000 ö
y=ç
÷ - 1 = 8.45%
è $6 ,000 ,000 ø
The annual yield is 8.45%, which is higher than the 8.00% return
requirement.
.
397
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 1 Exercise Solutions (Continued)
10. A municipality has a $10 million liability payable July 15, 2020.
To satisfy the liability, the municipality must either set aside
$10 million cash today (June 26, 1996) or buy U.S. Treasury
securities disbursing $10 million to ensure that the debt will be
paid. If the following zero-coupon Treasury securities are
available, what must the municipality pay today to satisfy this
liability, assuming short rates rarely fall below 3%?
Maturity
Price (%)
Yield (%)
2/15/20
17.828
7.43
5/15/20
17.507
7.43
8/15/20
17.269
7.41
11/15/20
17.040
7.39
To satisfy the liability you must have $10 million in cash on June 15,
2020. If you buy $10 million face of February 15, 2020 STRIPS for
$1,782,800, then you will have $10 million in cash on February 15,
2020. You will be able to reinvest that money until June 15, 2020, but
you do not know what the rate will be, so you can only safely assume
that it will be 0%. For that reason, it is cheaper to buy $10 million May
15, 2020 STRIPS for $1,750,700.
Alternatively, you could purchase the August or November STRIPS, but
they will not have matured by June 15, 2020. Although you could sell
them before June 15, 2020, in a high-interest-rate environment, you
could never be certain that they would be worth the $10 million you
need.
398
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 1 Exercise Solutions (Continued)
11. Derive a simple formula for convexity of a zero-coupon bond in
terms of its duration and yield.
P=
v
æ
yö
ç1 + ÷
fø
è
n
D = DurationModified =
n/ f
æ
yö
ç1+ ÷
fø
è
C = Convexity =
n(n + 1)
æ
yö
f ç1+ ÷
fø
è
2
2
nö
æ
çn + ÷
n(n + 1)
è
n/ f
nø
C = Convexity =
´
2 =
æ
æ
yö
yö
æ
yö
ç1 + ÷ f ç1 + ÷
f 2 ç1 + ÷
fø
fø
è
è
fø
è
= D´
n f
1ö
1ö
æ
æ
´ ç 1 + ÷ = D2 ´ ç 1 + ÷
è
nø
nø
æ
yö è
ç1 + ÷
fø
è
Note that for large n, C @ D 2 .
399
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
159
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 2 Exercise Solutions
1. Calculate the present value, modified duration, dollar duration,
and convexity of these Treasury STRIPS for settlement on June
26, 1996.
BondEquivalent
Maturity
Yield (%) n
Present
Modified
Dollar
1–x
Value (%)
Duration
Duration (%)
Convexity
11/15/99
6.63
6
0.771739
80.184
3.277
262.782
12.326
11/15/22
7.38
52
0.771739
14.775
25.447
375.987
659.814
02/15/23
7.36
53
0.274725
14.583
25.692
374.671
672.464
Even though these STRIPS do not have coupons, to value them we still
need to understand the day-count conventions. This is because we need
to know how to calculate the fraction of a period until the next “coupon.”
While the calculation of the fraction may appear to be arbitrary and so
lead to arbitrary prices, the market actually determines the price. The
quoted yield must mesh with the actual value of the bonds. Since cash
is exchanged on price, not yield, it is the price that ultimately matters.
The formulas on the next page hold for semi-annual zero-coupon bonds.
401
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 2 Exercise Solutions (Continued)
Formulas for semiannual zero-coupon
bonds
PV =
100%
yö
÷
2ø
æ
ç1 +
è
DurationDollar =
100% ´ (n + 1 - x) / 2
æ
ç1 +
è
DurationModified =
ConvexityDollar =
n+1- x
yö
÷
2ø
n+2-x
(n + 1 - x)/ 2
yö
æ
ç1 + ÷
è
2ø
100% ´ (n + 1 - x) ´ (n + 2 - x)
Convexity =
yö
æ
4ç 1 + ÷
è
2ø
n+3- x
(n + 1 - x) ´ (n + 2 - x)
yö
æ
4ç 1 + ÷
è
2ø
ConvexityGain =
2
Convexity
2
where n is the number of full semi-annual periods between settlement
and maturity, and x is the accrual period between the prior coupon date
and settlement, using an actual/actual calendar (0# x <1). More
rigorously:
x=
Actual Number of Days Between the Previous Coupon Date and Settlement
Actual Number of Days Between the Previous Coupon Date and the Next Coupon Date
402
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 2 Exercise Solutions (Continued)
2. Using the bond price formula, what is the price of a 10-year 7%
coupon bond at an 8% bond-equivalent yield?
7%
100%
2
Price = å
+
i
20
8% ö
i =1 æ
8% ö
æ
ç1 +
÷
÷
ç1 +
è
è
2 ø
2 ø
20
7% æ
8% ö
7% / 2
´ ç1 +
÷19
2 è
2 ø æ
8% ö
ç1 +
÷
è
100%
2 ø
=
+
20
8% æ
8% ö
8%
æ
ö
´ ç1 +
÷
ç1 +
÷
2 è
2 ø
è
2 ø
7% =
7% +
=
7%
8% ö
æ
ç1 +
÷
è
2 ø
8%
20
+
100%
8% ö
æ
ç1 +
÷
è
2 ø
20
100% ´ 8% - 7%
8% ö
æ
ç1 +
÷
è
2 ø
8%
20
= 93.205%
Note that n=19; x=0
403
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 2 Exercise Solutions (Continued)
3. What is the price of an 8% semi-annual pay coupon bond that
matures in exactly 15 years if the required bond-equivalent yieldto-maturity is 6%?
Since it matures in exactly 15 years, we are on a coupon date so accrued
interest=x=0 and Price=PV.
8%
100%
2
Price = å
+
30
i
6% ö
6% ö
æ
i =1 æ
ç1 +
÷
ç1 +
÷
è
è
2 ø
2 ø
30
8% æ
6% ö
8% / 2
´ ç1 +
÷29
2 è
2 ø æ
6% ö
ç1 +
÷
è
100%
2 ø
=
+
30
6% æ
6% ö
6% ö
æ
´ ç1 +
÷
ç1 +
÷
2 è
2 ø
è
2 ø
8% =
8% +
=
8%
6% ö
æ
ç1 +
÷
è
2 ø
6%
30
+
100%
6% ö
æ
ç1 +
÷
è
2 ø
30
100% ´ 6% - 8%
6% ö
æ
ç1 +
÷
è
2 ø
6%
30
= 119 .600%
404
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 2 Exercise Solutions (Continued)
Note that n=29; x=0
To get a quick estimate of the price, we could note that the duration of a
15-year is about 9, and since rates fell 2% from the rate that prices the
bond at par (8%), the price rose about 9×2% = 18%. But, since the bond
has positive convexity, we know the bond’s price rises more in a
declining interest rate environment, so the price is at least 118%.
4. Many bonds pay interest twice per year, but their coupons are
quoted on an annual basis. That is, an 8% 2-year U.S. Treasury
note pays a 4% coupon twice per year. What is the bond’s annual
yield-to-maturity if it is priced at par on a coupon date?
Since the bond is priced at par on a coupon date, its yield-to-maturity is
8.00% BEY. Its annual yield is
ö
(1 + y Annual )= æçè 1 + 8%
÷
2 ø
y Annual
=
2
8.160%
405
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 2 Exercise Solutions (Continued)
5. If a 10-year Treasury bond with a 7% coupon is issued today at
a price of 99-24 (99.750%), what is its bond-equivalent yield-tomaturity? Its annual yield-to-maturity?
Bond-equivalent:
7%
100%
24
2
PV = Price = å
%
i +
20 = 99
32
yö
i =1 æ
y
æ
ö
ç1 + ÷
ç1 + ÷
è
è
2ø
2ø
20
y BEY
= 7.035%
You can calculate the bond-equivalent yield with a calculator, but a good
estimate can be made by comparing the current price to par (when the
bond’s yield equals its coupon). Since the modified duration of a
10-year coupon bond is about 7, the yield change (from the coupon rate)
required to cause the bond to trade at a quarter-point discount can be
calculated as:
DurationModified = -
Dy @ -
DP
1 dP
1
´
@´
P dy
100% Dy
DP
1
1
- 8 / 32% 8 / 32%
´
@´
@
= 0.036%
P DurationModified
100%
7
7
Then y = Dy + c = 0.036% + 7 .000% = 7 .036%
Note that, although the actual duration of this bond is 7.1, our
assumption of a duration of 7 was still able to estimate the yield very
accurately.
2
y ö
æ
Annual yield: (1 + y Annual ) = ç 1 + BEY ÷ Þ y Annual = 7 .159%
è
2 ø
406
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 2 Exercise Solutions (Continued)
6. For settlement on June 26, 1996, the price on the February 15,
1997 STRIPS was 96.444%. The yield is quoted as the yield to
the stated maturity date, but that day is a Saturday and the cash
is not delivered until the following Monday. What is the
difference between the quoted yield and the yield actually earned
by the investor?
The price of the STRIPS is set by the market; the corresponding quoted
yield is dependent upon that price, but also by the conventions for that
yield. STRIPS are settled on price; the corresponding yield is not a
settlement quantity.
100%
PV = 96 .444% =
yö
æ
ç1 + ÷
è
2ø
1+ 50
182
1
é
ù
æ 100% ö 1+ 50 / 182
ê
ú = 5.762%
y = 2´ ç
1
êè 96 .444% ÷ø
ú
úû
ëê
For bonds that mature on “bad” days, the yield may be quoted to that
day, but the yield to receipt of cash is different. To calculate the yield to
the cash flow receipt strictly according to the methodology for
computing partial periods, the term of the zero is one plus the
actual/actual period between June 26, 1996 and August 15, 1996 plus
actual/actual period between February 15, 1997 and February 17, 1997.
100%
PV = 96 .444% =
yö
æ
ç1 + ÷
è
2ø
1+ 50 + 2
182 181
1
é
ù
1+ 50 / 182 + 2 / 181
100%
æ
ö
ê
y = 2´ ç
- 1ú = 5.712%
êè 96 .444% ÷ø
ú
úû
ëê
407
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 2 Exercise Solutions (Continued)
The yield difference is –5 bp, i.e., the effective earnings on the bond are
5 bp less than the stated yield. Note that the period from February 15,
1996 to August 15, 1996 has 182 days because of leap year, while the
period from February 15, 1997 to August 15, 1997 has 181 days.
7. Is the price of a bond above or below par if its yield is less than
its coupon?
Since the bond’s yield is less than its coupon, the present value is greater
than par, but that does not necessarily mean the price is above par
(although it usually is). For example, consider a 1¼-year semi-annual
bond that pays an 8.000% coupon and has a 7.990% yield. For this bond,
n=2 and x=0.5.
PV =
æ 7 .990% ö 100% ´ 7 .990% - 8.000%
8.000% ´ ç 1 +
÷+
2
è
2 ø
æ 7 .990% ö
ç1 +
÷
è
2 ø
æ 7 .990% ö
7 .990% ´ ç 1 +
÷
è
2 ø
Accrued = x ´
0. 5
= 101.992%
8%
c
= 0.5 ´
= 2.000%
2
f
Price = PV - Accrued = 99.992%
So, the price is less than par, even though the yield is less than the
coupon.
408
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 2 Exercise Solutions (Continued)
8. Which has a longer duration, a 7-year zero-coupon bond yielding
7.20% (BEY) or a 10-year 7.25% coupon bond yielding 7.20%
(BEY)?
For the zero: D =
14 / 2
= 6 .757
7 .20%
1+
2
For the coupon bond, estimate the duration by calculating prices at
7.19% and 7.21% (using the two-sided estimate):
PVy =7 .19% = 100.423%
PVy =7 .20% = 100.352%
PVy =7 .21% = 100.282%
D=-
(100.282% - 100.423%) = 7 .033
100.352% ´ 0.02%
The coupon bond has a slightly longer duration.
409
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 2 Exercise Solutions (Continued)
9. As long as you can safely stuff cash under your mattress (nonnegative interest rates), what is the most you would ever pay for
a bond that matures in eight years and has a 7% coupon paid
annually? What if the bond paid a semi-annual coupon? Could
interest rates ever become negative?
8
P=å
i =1
7%
i
+
100%
(1 + y) (1 + y)8
PriceMax = Price y =0% = 8 ´ 7% + 100% = 156%
Note that the present value of a bond at a yield of 0% is the sum of its
nominal cash flows.
For a semi-annual coupon bond:
7%
100%
2
P(y) = å
i +
16
yö
i =1 æ
yö
æ
ç1 + ÷
ç1 + ÷
è
è
2ø
2ø
16
PriceMax = Price y =0% = 16 ´ 3.5% + 100% = 156%
The difference between the present value of two bonds is a function of the
discounting of future cash flows. Since the maximum value is when
yields are 0%, compounding will have no effect. There would be a
difference in values if there were seven and a half years until maturity
instead of eight years. The semi-annual bond would only have 15 coupon
periods left, while the annual bond would still have eight coupons, each
one twice as big as the semi-annual bond’s coupons. The annual bond
would thus have an extra half coupon and be worth 3½% more.
Interest rates can only be negative when there is a cost of holding cash.
This could occur if there was a high safety risk in holding cash, so that
one would pay for another party to take that risk.
410
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 2 Exercise Solutions (Continued)
10. A bond issued by company A has a 6% coupon and matures
February 15, 2026. The U.S. Treasury bond that matures the
same date also has a coupon of 6% and is priced at 86-18+
(86.578125%). Is the price of company A’s bond greater or less
than 86-18+?
Since the corporate bond is the same as the Treasury bond except for an
additional layer of credit risk, its yield must be higher. So, the PV of the
corporate is less than the PV of the Treasury. Since each has the same
accrued interest (except for some very small differences from differing
day-count conventions), the price of the corporate must be less than
86-18+.
11. If three bonds promise the following cash flows, which is worth
the most? Estimate the duration of each at a 7% semi-annual
yield.
Years from Now
1
Cash Flow A ($)
Cash Flow B ($)
1,000
400
2
Cash Flow C ($)
500
3
1,000
600
1,000
4
1,000
700
1,000
800
1,000
5
411
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 2 Exercise Solutions (Continued)
As long as yields are greater than 0%, bond A is worth the most. The
proof:
Years from Now
A Cumulative
Cash Flow ($)
B Cumulative
Cash Flow ($)
C Cumulative
Cash Flow ($)
1
1,000
400
0
2
1,000
900
0
3
2,000
1,500
1,000
4
3,000
2,200
2,000
5
3,000
3,000
3,000
Bond A always has cumulative cash receipts that are greater than or
equal to the cumulative cash receipts on bond B or C. Bond A is,
therefore, receiving cash earlier and will have higher present value at any
positive discount rate.
To estimate the duration at 7% yield, calculate the PV of each at 7%, then
the PVs at some different yield, say 7.01%. This is a one-sided duration
estimate, which is not as accurate as a two-sided estimate, which
considers the effect of an increase and decrease in rates.
Present
Value at
7.00% ($)
Present
Value at
7.01% ($)
Change
in Present
Value ($)
Change in
Yield (%)
Duration
DP / PV
Dy
Bond A
2,506.42
2,505.80
– 0.62
0.01
2.47
Bond B
2,395.95
2,395.21
– 0.74
0.01
3.09
Bond C
2,281.83
2,280.96
– 0.87
0.01
3.82
412
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 2 Exercise Solutions (Continued)
12. A perpetual bond pays coupons forever, but never matures. If a
perpetual bond pays a 7% coupon annually and is priced at
95%, what is its yield? What is its duration? What is its
convexity? How does its convexity compare to a zero-coupon
bond with the same duration?
4
P=å
i =1
æ 7%
7%
7% ö
ç
÷
L
=
+
+
+
lim
(1 + y )n ÷ø
(1 + y )i n®4 çè (1 + y ) (1 + y )2
7%
æ 7%
7%
7% ö
P
÷
= lim çç
+L+
+
2
n
(1 + y ) n®4 è (1 + y )
(1 + y ) (1 + y ) n+1÷ø
Taking the difference, and factoring for P,
P-
æ (1 + y )
æ
1 ö
1 ö
P
÷
= Pçç 1 ÷÷ = Pçç
(1 + y ) è (1 + y )ø è (1 + y ) 1 + y ÷ø
æ 7%
æ y ö
7% ö
÷
= Pçç
÷÷ = lim çç
n+1 ÷
n® 4 (1 + y )
(
)
+
1
y
(
)
+
1
y
è
ø
è
ø
Solving for P and removing terms not involving n from the limit gives
P=
æ 7% öù
1é
ç
÷ú
ê7% - nlim
® 4 ç (1 + y )n ÷
y ëê
è
øûú
Since the denominator of the limit grows to infinity as n increases,
P=
c 7%
=
y
y
And so when P =95%, y = 7.368%
413
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 2 Exercise Solutions (Continued)
D=C=
y æ 7% ö 1
1 dP
1
´
=´ ç- 2 ÷ = =
= 13.57
P dy
7% è y ø y 7 .368%
y
1 d 2P
2 ´ 7% 2
´ 2 =
´
= 2 = 368.37
P dy
y3
y
7%
It is interesting to note that here C = 2 ´ D 2. Compare this to
C = D 2 æç 1 +
è
1ö
÷
nø
for a zero-coupon bond. The perpetual has higher convexity than a zerocoupon bond with the same duration, because its cash flows are more
dispersed. Furthermore, the duration and convexity of the perpetual do
not depend on its coupon. The dispersion of the cash flows for the
perpetual is as great as possible for a bond with a normal structure.
Therefore, most noncallable securities have a convexity that lies between
D 2 and 2×D 2.
414
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 2 Exercise Solutions (Continued)
13. One year ago, a bank loaned you enough to purchase a home
with a 30-year fixed-rate mortgage requiring a payment of
$1000 per month. Mortgage payments are level across the life of
the note, so each payment comprises both interest and principal.
The monthly interest rate on the mortgage is 8%. What was its
original face value? What is the balance today? What is the
BEY? Who is the issuer?
Original Face (valued at par at origination):
360
P=å
i =1
$1,000
8% ö
æ
ç1 +
÷
è
12 ø
i
=
$1,000
8% ö
æ
ç1 +
÷
è
12 ø
1
+
$1,000
8% ö
æ
ç1 +
÷
è
12 ø
2
+L+
$1,000
8% ö
æ
ç1 +
÷
è
12 ø
360
Using the annuity valuation methodology:
P
$1,000
$1,000
$1,000
=
+L+
+
2
360
361
8% ö æ
æ
8%
8%
8%
ö
ö
æ
ö
æ
ç1 +
÷
ç1 +
÷
ç1 +
÷
è
12 ø çè 1 + 12 ÷ø
è
è
12 ø
12 ø
Taking the difference, and then factoring P from the right hand side,
P-
$1,000
$1,000
P
=
361
8% ö æ
8% ö æ
æ
8%
ö
+
+
1
1
ç
÷ ç
÷
è
12 ø è
12 ø çè 1 + 12 ÷ø
é
ù
éæ
ê
ú
ê çè 1 +
1
ú = Pê
= P ê1 8%
æ
ö
ê
ú
êæ
+
1
ç
÷
ê è
ú
ê çè 1 +
ø
12
ë
û
ë
ù
é 8% ù
8% ö
÷
ú
ê
ú
1
12 ø
12
ú
ê
ú
=P
8% ö æ
8% ö ú
8%
æ
ö
ê
ú
÷ ç1 +
÷ú
ç1 +
÷ú
ê
12 ø è
12 ø û
12 ø û
ëè
415
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 2 Exercise Solutions (Continued)
Solving for P,
$1,000 360
P=å
i =1
$1,000
8% ö
æ
ç1 +
÷
è
12 ø
i
=
$1,000
8% ö
æ
ç1 +
÷
è
12 ø
8%
12
360
= $136 ,283
Current Face (valued at par in one year):
The balance today can be computed using the same formula, except with
348 payments instead of 360 payments. Note that the “balance” today is
the principal outstanding, not the PV of the mortgage today, since rates
(and so the discounting of the cash flows) have changed. However, the
balance is the present value when the yield equals the coupon, because
then the security is priced at par.
$1,000 348
P=å
i =1
$1,000
8% ö
æ
ç1 +
÷
è
12 ø
i
=
$1,000
8% ö
æ
ç1 +
÷
è
12 ø
8%
12
348
= $135,145
416
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 2 Exercise Solutions (Continued)
There is an alternative methodology for figuring out the balance, using
an amortization table:
Beginning
Principal ($)
Payment ($)
Interest
Paid ($)
Principal
Paid ($)
Ending
Balance ($)
1
136,283
1,000
909
91
136,192
2
136,192
1,000
908
92
136,100
3
136,100
1,000
907
93
136,007
4
136,007
1,000
907
93
135,914
5
135,914
1,000
906
94
135,820
6
135,820
1,000
905
95
135,726
7
135,726
1,000
905
95
135,630
8
135,630
1,000
904
96
135,535
9
135,535
1,000
904
96
135,438
10
135,438
1,000
903
97
135,341
11
135,341
1,000
902
98
135,243
12
135,243
1,000
902
98
135,145
Period
The bond-equivalent yield can be calculated (at the origination yield and
8% compounded monthly) using the following formula:
2
yMonthly ö
æ
yBEY ö
æ
÷
ç1 +
÷ = ç1 +
è
2 ø
12 ø
è
12
Þ yBEY
6
ù
éæ
yMonthly ö
ê
= 2 ´ ç1 +
÷ - 1ú = 8.135%
12 ø
ú
êè
û
ë
The issuer is you, the homeowner! The bank holds the mortgage.
417
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
159
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 3 Exercise Solutions
U.S. Treasury Prices from Tuesday, June 25, 1996 Pack
1. Under what conditions will the STRIPS curve lie above the
coupon curve on a plot of maturity vs. yield (maturity on the
x-axis)? When will it lie below the coupon curve?
As long as the coupon curve is strictly upward sloping, then the STRIPS
curve will lie above it. Similarly, if the coupon curve is strictly
downward sloping, then the STRIPS curve will lie below it.
It is probably easier to think of the problem in the other direction.
Consider a STRIPS curve that is upward sloping for all maturities. Now,
consider a coupon bond and a zero that mature on the same date. The
PV of the coupon bond can be estimated by discounting each of its cash
flows at the STRIPS rate corresponding to the payment date of the cash
flow. The final coupon payment and principal redemption are discounted
at the same yield at which the STRIPS is discounted (they are paid on
the same date). Since all shorter STRIPS yields are lower than the yield
for the payment made on the maturity date, the coupon payments of the
coupon bond are discounted at a lower rate than the maturity-date
payment. So, the all-in discount rate for the coupon bond is lower,
because it is a weighted average of all the discount rates of the
cash flows — the coupon payments and the principal redemption.
For each maturity, the discount rate for the coupon bond will be less than
the discount rate for the STRIPS, so the STRIPS curve lies above the
coupon curve.
419
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 3 Exercise Solutions (Continued)
U.S. Treasury Prices from Tuesday, June 25, 1996 Pack
2. Estimate the closing price and accrued interest for the UST
6.875% of July 31, 1999 if its yield-to-maturity falls 10 bp (from
6.548%).
On June 25, 1996 (for settlement June 26, 1996), the 6.875% note due
July 31, 1999 was priced at 100-285 to yield 6.548%. If the yield falls
10 bp today, the yield will be 6.448%. To calculate the price and accrued
interest, start with the PV at the new yield:
n = Number of Full Semi-Annual Payment Periods Between Settlement and Maturity =
6
x=
Days between January 31, 1996 and June 26, 1996 147
=
= 0 .807692
Days between January 31, 1996 and July 31, 1996 182
6.875%
100%
2
PV = å
i +1- x +
6 +1- x =
i =0 æ
yö
yö
æ
ç1 + ÷
ç1 + ÷
2ø
2ø
è
è
6
y ö 100% ´ y - 6.875%
æ
6.875% ´ ç 1 + ÷ +
6
2ø
è
yö
æ
ç1 + ÷
2ø
è
yö
æ
y ´ ç1 + ÷
2ø
è
1- x
PVy=6.448% = 103.949294%
Accrued = x ´
c
6.875%
= 0.807692 ´
= 2.776442%
2
2
Price y =6 .448%= PVy =6 .448% - Accrued = 103.949294% - 2.776442%
= 101.172851% @ 101-05+
420
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 3 Exercise Solutions (Continued)
U.S. Treasury Prices from Tuesday, June 25, 1996 Pack
To estimate the price, use the definition:
Duration =
DurationQuote Sheet
æ
ç1 +
è
yö
÷
2ø
@-
1
DPV
1
DP
´
=´
PV
PV Dy
Dy
PV = Price + Accrued = 100.894531% + 2.776442% = 103.670973%
DP @
DurationQuote Sheet
2.77
´ PV ´ Dy = ´ 103.670973% ´ -0.1%
yö
6.548% ö
æ
æ
ç1 + ÷
ç1 +
÷
è
è
2ø
2 ø
= 0.278065%
Py=6 .448% @ 100.894531% + 0.278065% = 101.172596% @ 101-05 +
3. Two separate Treasury issues mature on August 15, 1997. Why
do their durations differ?
They have different coupons, so the present-value-weighted average
times to receive cash flows are different. The 8.625% of August 15,
1997 has the highest coupon, so a higher percentage of its PV comes
from its coupons. A dollar of coupon income contributes less to duration
than a dollar of principal because it is received sooner. Macaulay
duration is defined as the present-value-weighted average time until cash
flow receipt, and duration happens (by coincidence) to be Macaulay
duration divided by a single discount factor. Consequently, the duration
of the higher coupon bond with the same maturity is shorter.
421
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 3 Exercise Solutions (Continued)
U.S. Treasury Prices from Tuesday, June 25, 1996 Pack
4. Given the quote sheet price for the UST 6.875% of July 31, 1999
(100-285), calculate the bond’s yield, modified duration, price
duration, Macaulay duration, accrued interest, and the value of
an 01 and a 32nd.
n = Number of Full Semi-Annual Payment Periods Between Settlement and Maturity = 6
x=
Days between January 31, 1996 and June 26, 1996 147
=
= 0.807692
Days between January 31, 1996 and July 31, 1996 182
Accrued = x ´
c
6.875%
= 0.807692 ´
= 2.776442%
2
2
PV = 100.894531% + 2.776442% = 103.670974%
6.875%
100%
2
PV = 103.670974% = å
i +1- x +
6 +1- x =
i =0 æ
yö
yö
æ
ç1 + ÷
ç1 + ÷
è
è
2ø
2ø
6
æ
6.875% ´ ç 1 +
è
y ö 100% ´ y - 6.875%
÷+
6
2ø
yö
æ
ç1 + ÷
è
2ø
yö
æ
y ´ ç1 + ÷
è
2ø
1-x
Use a calculator to get y = 6.548%
DurationModified PV @
[
]
PVy =6 .549% - PV6 .547% / PVy =6 .548%
DPV / PV
== 2.68
Dy
0.002%
422
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 3 Exercise Solutions (Continued)
U.S. Treasury Prices from Tuesday, June 25, 1996 Pack
The duration on the quote sheet is Macaulay duration:
æ
DurationQuote Sheet = DurationModified PV ´ ç 1 +
è
DurationModified Price @
yö
6 .548% ö
æ
÷ = 2.68 ´ ç 1 +
÷ = 2.77
è
2ø
2 ø
[
]
- P6.547% / Py =6.548%
P
- DP / P
= - y =6.549%
= 2.75
Dy
0.002%
The value of an 01 is the dollar price change of the bond, in hundredths
of a percent, for a 1-bp movement in interest rates, i.e., dollar duration.
If the dollar duration is 277.850% and yields fall 1 bp, the price changes
by 277.850% × 0.01%=0.0277850%.
The value of a 1/32 in bp is calculated from:
D@-
DPV / PV
, so
Dy
Value32nd = Dy @ -
DPV
DP
-1/32 ´ 1%
=== 0.011% = 1.1 bp
D ´ PV
DurationDollar
277.850%
5. Is the price of a 2-year fixed-rate bond more or less sensitive to
movements in interest rates than the price of a 2-year
floating-rate bond? Why?
The 2-year fixed-rate bond is more sensitive to interest rates than the
floater.
The coupon on a floating-rate bond resets periodically. So, if the coupon
resets to some market-based rate every quarter, every month, etc., then
the price will be near par on each reset date. For a bond that resets
423
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 3 Exercise Solutions (Continued)
U.S. Treasury Prices from Tuesday, June 25, 1996 Pack
its rate quarterly, market rate movements will only affect the value of the
bond for that quarter. If rates rise, the coupon will be below market for
only the remainder of that quarter. So, the price sensitivity is similar to
a bond with a maturity of three months or less.
6. If the 113/4% of November 15, 2014 falls in price to 135-00, what
is its yield-to-call for settlement on June 26, 1996?
Callable Treasuries are callable five years prior to maturity and are
redeemed at par.
n = Number of Full Semi-Annual Payment Periods Between Settlement and First Call = 26
x=
Days between May 15, 1996 and June 26, 1996
42
=
= 0.228261
Days between May 15, 1996 and November 15, 1996 184
26
P = 135% = å
i =0
5.875%
yö
æ
ç1 + ÷
è
2ø
i +1- x
+
100%
yö
æ
ç1 + ÷
è
2ø
26 +1- x
- x ´ 5.875%
Using a calculator, the yield is 7.548%
We can verify this yield by recomputing the price
æ
11.75% ´ ç 1 +
è
P=
y ö 100% ´ y - 11.75%
÷+
26
2ø
yö
æ
ç1 + ÷
è
2ø
yö
æ
y ´ ç1 + ÷
è
2ø
1-x
- x ´ 5.875% = 135.000%
424
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 3 Exercise Solutions (Continued)
U.S. Treasury Prices from Tuesday, June 25, 1996 Pack
7. A trader has given you the 5¾% of August 15, 2003 as a
benchmark for a corporate bond. On your Telerate screen, the
5-year is now trading at 100, the 10-year is now trading at
101, and the trader looks very busy. How would you estimate the
current price of your benchmark?
Our objective is to replicate the interest rate sensitivity of a 7-year
Treasury using the more liquid 5- and 10-year Treasuries.
Closing
Price
Accrued
(%)
(%)
Current
Price
(%)
Closing
Yield
(%)
Current
Yield
(%)
Modified
PV
Duration
Par
5.750% 8/15/03
93-282
2.085
95-002
6.846
6.638
5.59
100.00
6.500% 5/31/01
99-03
0.462
100-00
6.717
6.498
4.14
48.03
6.875% 5/15/06
99-18
0.785
101-00
6.935
6.733
7.03
47.99
Bond
The butterfly portfolio must have the same market value and duration on
each leg. The yield changes of the hedge side of the portfolio are then
weighted to estimate the yield change of the benchmark. The equations
that must be satisfied are:
Proceeds:
ParBarbell - Short ´ PVBarbell - Short + ParBarbell - Long ´ PVBarbell - Long
= ParBullet ´ PVBullet
Dollar duration:
ParBarbell - S ´ PVBarbell - S ´ DBarbell - S + ParBarbell - L ´ PVBarbell - L ´ DBarbell - L
= ParBullet ´ PVBullet ´ DBullet
425
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 3 Exercise Solutions (Continued)
U.S. Treasury Prices from Tuesday, June 25, 1996 Pack
Solving these equations gives:
ParBarbell - Short =
ParBarbell - Long =
(
ParBullet ´ PVBullet ´ DBarbell-Long - DBullet
(
PVBarbell - Short ´ DBarbell -Long - DBarbell - Short
)
)
ParBullet ´ PVBullet ´ (DBullet - DBarbell - Short )
(
PVBarbell - Long ´ DBarbell - Long - DBarbell - Short
)
The following intermediate table has the values needed to compute the
par weights:
Price
(%)
Accrued
(%)
PV (%)
5.750% 8/15/03
93-282
2.085
95.968
5.78
5.59
6.500% 5/31/01
99-03
0.462
99.555
4.28
4.14
6.875% 5/15/06
99-18
0.785
100.347
7.27
7.03
Bond
Macaulay
Modified
Duration PV Duration
Plugging these values into the above formula gives ParShort = 48.03 and
ParLong = 47.99 when ParBullet = 100.00. Since the 5- and 10-year
position hedges the 5.750% position, the price change on the 5.750%
must be the par-weighted price change on the hedge portfolio:
DP5.750% = 48.03 ´
36
46
29
´ 1% + 47.99 ´
´ 1% =
32
32
32
426
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 3 Exercise Solutions (Continued)
U.S. Treasury Prices from Tuesday, June 25, 1996 Pack
8. You sell the 5¾% of August 15, 2003 (at the closing price) and
hedge with the 5-year and the 10-year. The Fed tightens, and the
curve flattens. Do you hang your head in shame or do a
victory lap?
You own a barbell portfolio. The best hedge would use the weights from
the prior problem. When the yield curve flattens (assuming a “linear”
flattening: The yield increase on the bullet is the same as the durationinterpolated yield increase on the barbell bonds), the 10-year position
will rally more than the 5-year position will suffer, because it has longer
duration. Therefore, the barbell portfolio will outperform the bullet
portfolio. A victory lap is in order here. However, there is no guarantee
that the flattening will be linear; with a non-linear flattening, both a gain
and a loss are possible.
Linear Flattening
Yield Change (bp)
Bullet (Short)
Barbell-Short (Long)
Barbell-Long (Long)
Net
New Price (%)
6.49
10.00
3.00
93.535518
98.683157
99.351215
Profit & Loss ($)
0.347
– 0.197
– 0.101
0.049
Sample Non-Linear Flattening
Bullet (Short)
Barbell-Short (Long)
Barbell-Long (Long)
Net
Yield Change (bp)
New Price (%)
Profit & Loss ($)
5.00
10.00
3.00
93.615020
98.683157
99.351215
0.268
– 0.197
– 0.101
– 0.031
427
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
159
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 4 Exercise Solutions
1. What is the price for trade on June 25, 1996 for settlement on
May 15, 1997 of the 5.875% UST of March 31, 1999 if the term
repo rate is 5% and the bond’s yield for regular settlement is
6.475%? Assume coupons are reinvested at the term repo rate.
The arbitrage-free condition maintains that investors should be
indifferent between investing money in a short-term repo agreement and
effectively locking in a short-term rate by buying the security and
simultaneously arranging a forward sale:
(Price
Spot
rd ö
+ AccruedSpot ´ æç 1 +
÷ = PriceForward + Accrued Forward + FVCoupons
è
360 ø
AccruedSpot =
)
5.875%
Days Between March 31, 1996 and June 26, 1996
´
= 1.396516%
2
Days Between March 31, 1996 and September 30, 1996
Accrued Forward =
5.875%
Days Between March 31, 1997 and May 15, 1997
´
= 0.722336%
2
Days Between March 31, 1997 and September 30, 1997
n = Number of Full Periods Between June 26, 1996 and March 31, 1999 = 5
x=
Days Between March 31, 1996 and June 26, 1996
87
=
= 0.475410
Days Between March 31, 1996 and September 30, 1996 183
PVSpot
5.875%
100%
2
=å
i +1- x +
n+1- x =
i =0 æ
yö
yö
æ
ç1 + ÷
ç1 + ÷
è
è
2ø
2ø
n
6.475% ö 100% ´ 6.475% - 5.875%
æ
5.875% ´ ç 1 +
÷+
5
è
2 ø
6.475% ö
æ
ç1 +
÷
è
2 ø
6.475% ö
æ
6.475% ´ ç 1 +
÷
è
2 ø
1-x
429
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 4 Exercise Solutions (Continued)
PriceSpot = PVSpot - AccruedSpot = 99.889228% - 1.396516% = 98.492711% @ 98 -156
Two coupons are paid between trade date and settlement: one on
September 30, 1996, and the other on March 31, 1997. We assume that
the coupons are reinvested at the term repo rate of 5% (simple interest
actual/360).
The future values of the coupons are
FVCoupon 1 =
5.875% æ
Days Between September 30, 1996 and May 15, 1997 ö
´ ç 1 + 5% ´
÷ = 3.030113%
è
ø
2
360
FVCoupon 2 =
5.875% æ
Days Between March 31, 1997 and May 15, 1997 ö
´ ç 1 + 5% ´
÷ = 2.955859%
è
ø
2
360
Applying the original formulas,
(Price
Spot
Days Between June 26, 1996 and May 15, 1997 ö
æ
+ AccruedSpot ´ ç 1 + 5% ´
÷
è
ø
360
)
= PriceForward + Accrued Forward + FVCoupons
PriceForward = 97.662061% @ 97- 211
430
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 4 Exercise Solutions (Continued)
2. You are offered the 2-year for settlement on September 30, 1996
at 99-20. If the yield today (June 25, 1996) is 6.30%, is that price
fair?
The current 2-year is the 6.000% due May 31, 1998.
AccruedSpot =
6.000%
Days Between May 31, 1996 and June 26, 1996
´
= 0.426230%
2
Days Between May 31, 1996 and November 30, 1996
Accrued Forward =
6.000% Days Between May 31, 1996 and September 30, 1996
´
= 2.000000%
2
Days Between May 31, 1996 and November 30, 1996
The yield of 6.30% corresponds to a price of 99.457356% for settlement
June 26, 1996. No coupons are paid during the holding period.
Therefore:
(Price
Spot
Days Between June 26, 1996 and September 30, 1996 ö
æ
+ AccruedSpot ´ ç 1 + r ´
÷ = PriceForward + Accrued Forward
è
ø
360
r=
æ PriceForward + Accrued Forward
ö
360
´ç
- 1÷÷ = 6 .538%
Days Between June 26, 1996 and September 30, 1996 çè PriceSpot + AccruedSpot
ø
)
So, the party providing financing is earning an excessive return
(prevailing repo rates are approximately 5.50%; the break-even repo rate
is even higher than the yield on the note). Unless you absolutely cannot
hold the security before September 30, you should reconsider.
431
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 4 Exercise Solutions (Continued)
3. If the yield on the 30-year is 7.50% and three-month repo is
5.5%, estimate the expected change in yield and price for the
bond over that period.
The definition of modified duration:
DurationModified @
- DPV PV (y - r ) ´ t
@
Dy
Dy
From Chapter 2, the duration of a 30-year bond is approximately 12.
Therefore,
1
(7.5% - 5.5%) ´
(y - r ) ´ t
Dy @
DurationModified
=
12
4 = 0.042%
The actual forward price and yield on the 30-year, the 6% of February
15, 2026, is defined by:
and September 26, 1996 ö
(PriceSpot + AccruedSpot )´æç 1 + r ´ Days Between June 26, 1996
÷
360
è
ø
= Price Forward + Accrued Forward + FVCoupons
PriceSpot = 82.244969%
AccruedSpot = 2.175824%
AccruedForward = 0.684783%
FVCoupons = 3.019250%
PriceForward = 81.903341%
YieldForward = 7.538%
Dy Actual = 0.038%
So our estimate was pretty good.
432
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 4 Exercise Solutions (Continued)
4. Under market-expectations theory, forward rates equal expected
future rates. So, money invested at today’s spot rate and then
reinvested at a forward rate has the same future value as money
invested today until the end of the forward reinvestment period.
If simple-interest three-month rates are 4.50% and six-month
rates are 5%, what is the implied three-month rate three months
forward?
Investing $100 today for six months should provide the same future
value as investing $100 for three months and then reinvesting the
proceeds for three months at the implied three-month forward rate. This
forward rate cannot be locked in. However, if the market’s view of the
forward rate were to differ from the implied forward rate, that would
imply an imbalance in the three-month and six-month rates that would
modify investor willingness to take positions in the securities.
D[7 / 1 / 96,1 / 1 / 97 ]ö æ
D[7 / 1 / 96,10 / 1 / 96 ]ö æ
D[10 / 1 / 96,1 / 1 / 97 ]ö
æ
÷
ç 1 + 5% ´
÷ = ç 1 + 4.5% ´
÷ ´ ç 1 + rf ´
è
ø è
ø è
ø
360
360
360
æ 1 + 5% ´ 184 ö = æ 1 + 4.5% ´ 92 ö ´ æ 1 + r ´ 92 ö
ç
÷ ç
÷ ç
÷
f
è
360 ø è
360 ø è
360 ø
184 ö
é æ
ù
ç 1 + 5% ´
÷
ê
ú
è
ø
360
360
rf =
´ê
- 1ú = 5.437%
92 ê æ 1 + 4.5% ´ 92 ö
ú
ç
÷
êë è
úû
360 ø
433
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 4 Exercise Solutions (Continued)
5. Consider the August 15, 2023 bond for trade on June 25, 1996
and settlement on June 15, 1997. How much difference is there
in the yield-to-maturity if the term repo rate changes from 5% to
3%?
From the quote sheet, the bond has a 6.250% coupon and a price of
88-14+.
(Price
Spot
AccruedSpot =
rd ö
+ AccruedSpot ´ æç 1 +
÷ = PriceForward + Accrued Forward + FVCoupons
è
360 ø
)
6.250%
Days Between February 15, 1996 and June 26, 1996
´
= 2.266484%
2
Days Between February 15, 1996 and August 15, 1996
Accrued Forward =
6.250%
Days Between February 15, 1997 and June 15, 1997
´
= 2.071823%
2
Days Between February 15, 1997 and August 15, 1997
FVCoupon 1 =
Days Between August 15, 1996 and June 15, 1997 ö 6 .250% æ
304 ö
6 .250% æ
´ ç1 + r ´
´ ç1 + r ´
÷=
÷
è
ø
è
2
360
2
360 ø
FVCoupon 2 =
6 .250% æ
Days Between Feb 15, 1997 and June 15, 1997 ö 6 .250% æ
120 ö
´ ç1 + r ´
´ ç1 + r ´
÷=
÷
è
ø
è
2
360
2
360 ø
Price (%)
Yield (%)
5% Repo Rate
86.674138
7.409
3% Repo Rate
84.963597
7.578
Change
–1.710541
0.169
434
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 4 Exercise Solutions (Continued)
Rough estimate:
11.5
5%
3%
´
(
)
r
r
´
t
( 1 2)
12 = 0.159722% = 16 bp
Dy @
=
DurationModified
12
So, again, the estimate is pretty good.
435
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
159
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 5 Exercise Solutions
1. What is the yield-to-maturity or internal rate of return of a
portfolio of $2,000 face amount of 2-year STRIPS priced at
88.60% and $1,000 face amount of 3-year STRIPS priced at
82.30%? What is the dollar-duration-weighted yield? What is
the market-value-weighted yield?
PV = $2,000 ´ 88.60% + $1,000 ´ 82.30% = $2, 595 =
$2,000
æç 1 +
è
yö
÷
2ø
4
+
$1,000
æç 1 +
è
yö
÷
2ø
6
Solving for yield by trial and error: y=6.332%
Note that the portfolio looks like two cash flows, $2,000 two years from
now and $1,000 three years from now. We know the portfolio present
value: it is just the sum of the present values of the two bonds. So,
calculating the portfolio yield-to-maturity is the same as calculating the
yield of a bond with those two cash flows. The yield-to-maturity is only
an estimate of three-year rate of return assuming that, in two years, the
maturing STRIPS can be reinvested at 6.33%. This arbitrary assumption
highlights the deficiency of yield-to-maturity and internal rate of return
as measures of potential return.
437
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 5 Exercise Solutions (Continued)
Term
Par
(Years) ($)
Price
(%)
Yield
(%)
Modified Market
Duration Value ($)
MV×Y
($)
MV×D
($)
MV×D×Y
($)
2
2,000
88.60
6.144
1.94
1,772
109
3,438
211
3
1,000
82.30
6.660
2.90
823
54
2,390
158
2,595
163
5,828
369
From the above table:
Yield Dollar-Duration-Weighted =
Yield Market-Value-Weighted =
369
= 6 .331%
5 ,828
163
= 6 .289 %
2 ,595
438
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 5 Exercise Solutions (Continued)
2a. What is the total bond-equivalent rate of return of a 3-year
7% annual coupon bond selling at par and held by the investor
until maturity? Do three cases: 1) reinvest all cash flows at 5%,
2) reinvest all cash flows at 7%, and 3) reinvest all cash flows at
9%.
æ
7% ´ ç 1 +
è
rö
÷
2ø
4
Future Value
of Year 1
Cash Flow
r (%)
(%)
æ
7% ´ ç 1 +
è
rö
÷
2ø
2
107%
1
é
ù
6
Total
æ
ö
ê
2´ ç
÷ - 1ú
êè 100% ø
ú
úû
ëê
Future Value
of Year 2
Cash Flow
(%)
Future Value
of Year 3
Cash Flow
(%)
Total Future
Value
(%)
BondEquivalent
Rate of
Return (%)
5
7.727
7.354
107.000
122.081
6.762
7
8.033
7.499
107.000
122.531
6.889
9
8.348
7.644
107.000
122.992
7.019
439
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 5 Exercise Solutions (Continued)
2b. What is the expected rate of return if all three scenarios are
equally likely?
The correct way to compute expected rate of return is to calculate the
rate of return of the expected total future value:
Present Value = 100%
Future ValueExpected =
Rate of ReturnExpected
1 3
´ å PVScenario = 122.535%
3 i =1
1
é
ù
6
122
.
535
%
æ
ö
ê
= 2´ ç
÷ - 1ú = 6.890%
êè 100% ø
ú
ë
û
440
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 5 Exercise Solutions (Continued)
Assume June 26, 1996 settlement.
3a. Which has a higher yield-to-maturity (bond-equivalent internal
rate of return):
• The 5-year Treasury (6.500% due May 31, 2001, priced at
99-03), or
• The same-duration portfolio comprising the
– 2-year Treasury (6.000% due May 31, 1998, priced at
99-14+), and
– 10-year Treasury (6.875% due May 15, 2006, priced at
99-18)?
In an upward-sloping yield curve environment, a barbell portfolio
usually has a higher internal rate of return because the higher yield of the
longer bond is weighted more heavily.
Instead of calculating the internal rate of return precisely, we can
estimate it using the dollar-duration-weighted yield. A portfolio with the
same cost and duration as the 6.500% bond can be constructed using the
butterfly weights.
Present
Modified
Value (%) Yield/IRR (%) Duration
Coupon (%)
Maturity
Par ($)
Bullet
6.500
5/31/01
100.00
99.555
6.717
4.13
Barbell
6.000
5/31/98
55.02
99.879
6.302
1.79
6.875
5/15/06
44.44
100.347
6.935
7.03
6.784
4.13
441
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 5 Exercise Solutions (Continued)
3b. Which portfolio has a higher one-year rate of return if cash flow
is reinvested at 5½% and horizon yields equal spot yields?
The bullet portfolio (the 5-year Treasury) has a higher one-year rate of
return under this scenario, even though it has a lower internal rate of
return. Since yields did not change and no securities matured, both
portfolios effectively earned their market-value-weighted yield. The
yield of the bullet portfolio is 7 bp higher than the market-valueweighted yield of the barbell portfolio.
Note that the rate of return for the barbell portfolio is calculated using
the total horizon value.
Coupon
(%)
Maturity
Par ($)
Present Horizon Horizon Horizon
Value
Value of
Yield
Value
($)
Coupons ($) (%)
($)
BE Rate
of Return
(%)
Bullet
6.500
5/31/01 100.00
99.555
6.616
6.717
106.331
6.694
Barbell
6.000
5/31/98
55.02
54.958
3.360
6.302
58.468
6.289
6.875
5/15/06
44.44
44.597
3.117
6.935
47.729
6.902
106.197
6.564
99.555
442
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 5 Exercise Solutions (Continued)
3c. Which portfolio has a higher one-year rate of return if cash flow
is reinvested at 5½% and horizon yields equal forward yields
(assuming a 5½% repo rate)?
Both portfolios have the same one-year rate of return under this scenario:
the repo rate! The forward price enforces a rate of return that equals the
repo rate. The slight deviation is due to the almost-offsetting effect of
1) a 365-day year for simple interest, which results in a higher annual
rate of return, and 2) compounding, which results in a lower bondequivalent rate of return.
Coupon
(%)
Maturity
Par ($)
Present Horizon Horizon Horizon
Value
Value of
Yield
Value
($)
Coupons ($) (%)
($)
BE Rate
of Return
(%)
Bullet
6.500
5/31/01 100.00
99.555
6.616
7.081
105.107
5.501
Barbell
6.000
5/31/98
55.02
54.958
3.360
7.221
58.023
5.501
6.875
5/15/06
44.44
44.597
3.117
7.159
47.084
5.501
105.107
5.501
99.555
443
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 5 Exercise Solutions (Continued)
3d. Which portfolio has a higher one-year rate of return if each of
the two preceding scenarios are equally likely? What is the
expected rate of return for each portfolio?
The bullet portfolio has the higher average rate of return.
Coupon
(%)
Maturity
Par ($)
Present Spot Yield Horizon Average BE Rate
Value
Horizon
Yield
Horizon of Return
($)
Value ($) Value ($) Value ($)
(%)
Bullet
6.500
5/31/01 100.00
99.555
Barbell
6.000
5/31/98
55.02
54.958
58.468
58.023
58.246
5.895
6.875
5/15/06
44.44
44.597
47.729
47.084
47.407
6.203
105.652
6.033
106.331 105.107 105.719
99.555
6.098
444
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 6 Exercise Solutions
1. Use the Black–Derman–Toy tree to price a 4-year floor struck at
6.50%.
0.73% +
1 æ 2.33%
0.16% ö
´ç
+
÷
2 è 1 + 5.17% 1 + 6.98% ø
The floor is worth 1.28%, less than the 6.50% cap from the text, which
was worth 2.26%. The yield curve is upward-sloping, so the tree is
biased upwards, which adds value to the cap relative to the floor.
445
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 6 Exercise Solutions (Continued)
2. Provide an example of a risk that is hedged by each of the option
types or strategies in this section.
Option Type
Hedging Use
Call
An investor that is concerned that the price of an asset may rise sometime
in the future
Put
An investor that is concerned that the price of an asset may fall sometime
in the future
Straddle
An investor that needs to hedge the sensitivity of a portfolio to volatility
Cap
A bank that is hedging against the increased cost of deposits that would
accompany an increase in short rates
Floor
The risk that future floating-rate payments may decline
Spread
An oil refiner that is concerned about the risk that the price of crude oil
may increase while the cost of gasoline may decline
Binary
A corporation that is involved in a merger negotiation using a contract
that offers the other company an out if interest rates rise more than a target
amount may hedge the economic risk of the deal’s falling through using a
binary option
Look-Back
An investor that needs to prove execution at the best possible price
Knock-Out
A corporation that is acquiring another company may want to hedge
the risk inherent in the target’s business; the corporation may want the
hedge to expire if certain conditions occur that would correspond to the
deal’s breaking down
Knock-In
The target company may want to hedge the risk inherent in its business
if the conditions occur that would correspond to the deal’s breaking down
Asian
A copper refiner that is required to sell at the average price over some
period may want to hedge the cost of acquiring raw material based on
the average price over the same period
Bermudan
An investor that only periodically has the information required to
determine if an adverse event has occurred would only want to exercise
concurrently with that information and would prefer the lower premium of
the Bermudan option relative to the American option
446
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 6 Exercise Solutions (Continued)
3. Use the Black–Derman–Toy tree to price a 9% coupon 4-year bond
callable at par starting in one year. What is the option worth?
How does that compare to the value of the option on the 71/2%
bond?
The option on the 9% bond is worth 4.81%, while the option on the
71/2% bond in Chapter 6 was worth 1.18%. The higher the coupon, the
higher the value of an embedded call option. Note that the 9% coupon
bond is worth more than the 71/2% bond, even though the embedded
option in the 9% coupon bond is worth more.
447
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 6 Exercise Solutions (Continued)
4. What combination of options could you use to provide a fixed
profit as long as the price of the underlier stayed between 90%
and 110%?
A strategy that combined selling a put struck at 90% and selling a call
struck at 110% would provide a constant profit as long as the price of the
underlier stayed between 90% and 110%. The option writer would
receive premium from selling the put and from selling the call. The put
would not be exercised unless the price of the underlier fell below 90%.
Likewise, the call would not be exercised unless the price of the
underlier rose above 110%. Between 90% and 110%, neither option
would be exercised, and the writer would receive the total premiums,
which are fixed.
448
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 6 Exercise Solutions (Continued)
5. The daily closing yields on a security fluctuate as follows: 7.00%,
7.10%, 7.20%, 7.05%, 6.90%, 6.90%. What are the two measures
of volatility for this period? What are the annualized volatilities?
Hint: Are there any non-business days during this period?
The two measures of volatility are
Sample Standard Deviation (of percentage yield changes between days):
n
åx
i =1
2
i
- n ´ x2
n-1
0.129% - 5 ´ ( -0.275%)
=
= 177
. %
5-1
2
Range Volatility Estimate (only valid for log-normal distribution):
1
´ ln( High) - ln( Low)
4 ´ ln(2)
[
]=
2
1
´ ln(7 .20%) - ln(6 .90%)
4 ´ ln(2)
[
] = 2.56%
2
These estimates are very different because the actual daily yield changes
are far from log-normally distributed.
There are two methods for annualizing these volatility estimates
(illustrated for the sample standard deviation measure of volatility).
Business Day Annualization:
VAnnual = V5-day ´
252
= 12.56%
5-1
Calendar Day Annualization:
VAnnual = V7-day ´
365
= 12.77%
7 -1
Note that the six observations reflect five yield changes.
449
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
159
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 7 Exercise Solutions
Prices as of June 25, 1996
1. The June 25, 1996 September 1996 futures price was 107-05.
Assume that there were three deliverable bonds: the 11¼% of
February 15, 2015 (priced at 141-09+), the 8% of November 15,
2021 (priced at 108-19+) and the 6% of February 15, 2026
(priced at 86-18+). Assume the short-term rate is 5%. Which
bond would be the cheapest-to-deliver if interest rates increased
by 70 bp?
The strategy for solving this problem is to find some reference point for
the bond and futures prices under the new yield-curve environment. If
there was no option value or arbitrage, the basis of the cheapest-todeliver would be the carry value. We can calculate the basis and the
basis net of carry (BNOC) for each bond using the following formulas:
Basis = Price Bond - Price Futures ´ Factor Bond
= ValueCarry + ValueOptions + Value Arbitrage
= ValueCoupon Accrual plus Reinvestment of Coupons Paid - Value Financing on Price+Accrued + BNOC
BNOC = Basis - (ValueCoupon Accrual plus Reinvestment of Coupons Paid - Value Financing on Price+Accrued )
é
Date Delivery - DateCouponi ö ù
Coupon k æ
´ å ç 1 + Repo Actual ´
÷ú
ê Accrued Delivery - Accrued Spot +
2
360
øú
i =1 è
= Basis - ê
ê
ú
Date Delivery - Date Spot
ê-(Price Spot + Accrued Spot )´ Repo Actual ´
ú
360
ë
û
Present
Coupon
Price
Basis
Value
Yield
Basis
Net of
(%)
Maturity
(%)
Factor
(%)
(%)
(%)
Carry (%)
11.250
2/15/2015
141-09+
1.3089
145.377
7.191
1.040
–0.009
8.000
11/15/2021
108-19+
1.0000
109.522
7.252
1.453
0.826
6.000
2/15/2026
86-18+
0.7751
88.754
7.089
3.521
3.111
451
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 7 Exercise Solutions (Continued)
Prices as of June 25, 1996
Note that the basis net of carry for the 11¼% bond is almost zero. This
is because the repo rate of 5% is nearly the implied repo rate (given in
the text) of 5.02%. In a different rate environment, if we hold the
assumption that the cheapest-to-deliver has a basis net of carry of
–0.009, we can determine what the price of the futures would be if each
bond were the cheapest-to-deliver. The bond that implies the lowest
futures price is the actual cheapest-to-deliver.
Basis = ValueCarry + BNOC = ValueCoupon Accrual plus Reinvestment of Coupons Paid - ValueFinancing on Price+ Accrued + BNOC
DateDelivery - DateCouponi ö ù
é
Coupon k æ
´ å ç 1 + Repo Actual ´
÷ú
ê Accrued Delivery - Accrued Spot +
2
360
øú
i =1 è
ê
=ê
ú
DateDelivery - DateSpot
ê
ú
BNOC
+
´
´
+
Repo
Actual
êë (PriceSpot Accrued Spot )
úû
360
Price Futures (Implied ) =
Price Bond - Basis
Factor Bond
Keep in mind that the repo rate also rose by 70 bp.
Present
Coupon
Price
(%)
(%)
11.250
Basis
Implied
Value
Yield
Net of
Basis
Futures
Factor
(%)
(%)
Carry (%)
(%)
Price (%)
132-15+
1.3089
136.564
7.891
–0.009
0.908
100-17
8.000
100-161
1.0000
101.417
7.952
–0.009
0.536
99-31
6.000
79-13
0.7751
81.582
7.789
–0.009
0.347
102-00
452
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 7 Exercise Solutions (Continued)
Prices as of June 25, 1996
The futures price would then be 99-31, and the 8% bond would be the
cheapest-to-deliver. At this futures price, the basis table would look like:
Present
Coupon
Price
Basis
Value
Yield
Basis
Net of
(%)
Maturity
(%)
Factor
(%)
(%)
(%)
Carry (%)
11.250
2/15/2015
132-15+
1.3089
136.564
7.891
1.635
0.719
8.000
11/15/2021
100-161
1.0000
101.417
7.952
0.535
–0.010
6.000
2/15/2026
79-13
0.7751
81.582
7.789
1.920
1.564
2. The September 1996 futures price on June 25, 1996 was 107-05.
How much would the cheapest-to-deliver, the 11¼% of February
15, 2015 (priced at 141-09+), have to richen before the 8% of
November 15, 2021 (priced at 108-19+) becomes the cheapest-todeliver?
For a fixed futures price, the cheapest-to-deliver security has the highest
implied repo rate. The implied repo rate satisfies the following equation:
é
ù
DateDelivery - DateSpot ö
æ
÷ - Accrued Delivery ú
ê PriceSpot + AccruedSpot ´ ç 1 + RepoImplied ´
360
è
ø
ú
PriceFutures ´ Factor = ê
ê
ú
DateDelivery - DateCouponi ö
Coupon k æ
´ å ç 1+ RepoImplied ´
ê
ú
÷
2
360
ø
i =1 è
êë
úû
(
)
Solving for the implied repo rate:
RepoImplied =
Coupon
´k
2
DateDelivery - DateCouponi
PriceFutures ´ Factor + Accrued Delivery - PriceSpot - AccruedSpot +
(Price
Spot
)
+ AccruedSpot ´
DateDelivery - DateSpot
360
-
Coupon k
´å
2
l =1
360
453
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 7 Exercise Solutions (Continued)
Prices as of June 25, 1996
The implied repo rate for the 11¼% bond is 5.02%, while the implied
repo for the 8% bond is 2.17%. The 11¼% bond, with the higher implied
repo rate, is the cheapest-to-deliver. The question is, what price for the
11¼% bond will give it a repo rate below 2.17%? In order to find that
price, we can solve for the break-even spot price:
Price Break-Even Spot
é
Date Delivery - DateCouponi ö ù
Coupon k æ
´ å ç 1+ Repo Implied ´
÷ú
ê Price Futures ´ Factor + Accrued Delivery +
2
360
øú
i=1 è
ê
ê
ú
Date Delivery - Date Spot ö
æ
- Accrued Spot ´ ç 1+ Repo Implied ´
ê
ú
÷
360
è
ø
ê
úû
=ë
Date Delivery - Date Spot
1+ Repo Implied ´
360
The break-even spot price is 142-12 (7.11% yield). Therefore, if the
11¼% bond richens by 341/2/32 (8 bp), it will have a lower implied repo
rate, and the 8% bond will be the cheapest-to-deliver.
There is another methodology for arriving at this answer. The current
implied repo rate of the cheapest-to-deliver, the 11¼% bond, is 5.02%.
This is near market repo rates. Any differences between this implied
repo rate and the actual repo rate owe to the value of options or arbitrage
embedded in the futures. The implied repo rate for the 8% bond, 2.17%,
is far from market repo rates. If the price of the 8% bond did not change,
but the price of the futures did, the bond’s implied repo rate would
change. So the question becomes, “What futures price causes the
implied repo rate on the 8% bond to become 5.02%, and what does that
imply about the price of the 11¼% bond?” At an implied repo rate of
5.02% and a price of 108-19+ on the 8% bond, the futures price would
be 108-00. At a 5.02% implied repo rate, that price implies a price of
142-123 for the 11¼% bond, almost exactly the same as above.
454
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 7 Exercise Solutions (Continued)
Prices as of June 25, 1996
3a. The June 25, 1996 price of the September 1996 futures was
107-05. The cheapest-to-deliver was the 11¼% of February 15,
2015 (priced at 141-09+). Assume an outstanding liability of
$100,000,000 with a duration of 10. How many futures would
you buy to hedge it?
If the basis of the futures were zero, then
PriceCTD = Price Futures ´ Factor
DurationDollar = -
- Price Futures )´ Factor
¢
PriceCTD
(Price Futures
¢ - PriceCTD
DP
==Dy
Dy
Dy
The (Macaulay) duration from the quote sheet is 9.41. Price duration can
be calculated from Macaulay duration as follows:
DurationPrice =
DurationMacaulay
PV
´
= 9.35
y
Price
1+
2
Equating the dollar duration of the liability and the futures gives:
$100,000,000 ´ 10 =
107- 05
´ $100,000 ´ 9.35 ´ Contracts
100
and so the number of contracts to use as a hedge is 998. A more
sophisticated hedging strategy would account for the hedging bias
caused by ignoring the basis. Furthermore, there is some correlation
between long-term and short-term rates, and therefore, there is
correlation between the basis and bond prices. This phenomenon also
affects hedging with futures.
455
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 7 Exercise Solutions (Continued)
Prices as of June 25, 1996
3b.How big an interest rate move would it take to exhaust the initial
margin?
The initial margin on bond futures is currently 1.75%. Since the hedge
is long futures, it will lose money when interest rates rise. From the
definition of duration:
DPriceFutures
DurationPrice = 9.35 = -
PriceFutures
=Dy
-1.75%
107.156% Þ Dy = 0.17%
Dy
So the initial margin will be exhausted after a 17-bp rise in interest rates.
On any day that interest rates change by more than that, the clearing
corporation would be exposed to short-term counterparty risk because
the initial margin would not cover the mark-to-market on the futures.
3c. Assume a short-term rate of 5% and unchanged option values.
What would be the one-month bond-equivalent rate of return on
the futures if prices do not change? If yields do not change?
Given a constant basis net of carry, a constant repo rate, and horizon
prices, it is straightforward to calculate implied futures prices:
Basis = PriceBond - PriceFutures ´ FactorBond
PriceFutures (Implied) =
PriceBond - Basis
FactorBond
456
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 7 Exercise Solutions (Continued)
Prices as of June 25, 1996
Basis
Implied
7/25/96
Present
Yield
Net of
Basis
Futures
Scenario
Price (%)
Value (%)
(%)
Carry (%)
(%)
Price (%)
Constant Price
141-09+
146.304
7.185
–0.009
0.710
107-13
Constant Yield
141-07
146.226
7.191
–0.009
0.711
107-11
Forward Price
140-312
145.982
7.209
–0.009
0.713
107-05
Note that the basis varies slightly between scenarios. As the market
value of the cheapest-to-deliver changes, the total financing cost
changes; financing cost is a component of basis. These futures prices
imply a return on the initial investment of initial margin:
Futures
Futures
6/25/96
7/25/96
Bond-Equivalent
Scenario
Value (%)
Value (%)
Rate of Return (%)
Constant Price
1.75
2.00
251.63
Constant Yield
1.75
1.94
172.11
Forward Price
1.75
1.75
0.00
457
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
159
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 8 Exercise Solutions
1. Price a 25-year 10% semi-annual-pay bond callable in five years
at a price of 102.500% that sinks 5% each of the last 10 years (to
maturity, call, and average life). Use a spread of 100 bp over the
Treasury curve. What is the risk of misinterpretation for a
$10 million block?
The average life, assuming 5% of principal is repaid at the end of each
of years 15 to 24 and the remaining 50% of principal repaid at maturity
at the end of year 25, is 22.25 years. This can be calculated using a
spreadsheet or using the useful trick:
n
åi =
i =1
n ´ (n + 1)
2
The average principal repayment date is then
æ 24 ´ 25 14 ´ 15 ö
5% ´ ç
÷ + 50% ´ 25 = 22.25
2 ø
è 2
The Treasury benchmark yields in the table below are tabulated in the
Treasury pack at the end of Chapter 3:
Yield-to-
Term (Years)
Benchmark
Benchmark
Security
Yield (%)
Yield (%)
Price (%)
Maturity
25
81/8% due 5/15/21
7.257
8.257
118.317
Call
5
61/2% due 5/31/01
6.717
7.717
111.036
Average Life
22.25
9% due 11/15/18
7.252
8.252
117.653
Misinterpreting a yield quotation can cause an error of over 7% of par.
459
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 8 Exercise Solutions (Continued)
Pricing the bond as a series of “bondlets,” one for each sinking-fund
payment, at the yield-to-average-life gives a more accurate price
computation of 117.532%, which is close to, but not exactly, the price of
a single bond using the average life in place of maturity.
2. What is the price of a 25-year 10% bond callable at par in five
years if the spread is 100 bp over the Treasury curve? What
about if the bond is putable in five years? What about if the bond
is both callable and putable?
Investor’s
Yield-to-
Price-to-
Price-to-
Embedded
Option Date
Maturity
Option Date
Option
YTM (%)
(%)
(%)
(%)
Price (%)
Short Call
8.257
7.717
118.317
109.324
109.324
Long Put
8.257
7.717
118.317
109.324
118.317
8.257
7.717
118.317
109.324
109.324
Short Call & Long Put
460
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 8 Exercise Solutions (Continued)
3. A zero slash bond pays a coupon that starts in the future, so it has
a zero-coupon component and a normal bond component. If an
issuer’s yield is 8.500%, what is the coupon of a semi-annual-pay
par-priced 20-year bond with a 5-year zero-coupon period?
What is the duration?
To handle the zero-coupon period, redefine x, add the following
definition for m, then discount the value of the bond back an additional
m periods:
x is the length of the accrual period using the appropriate
calendar for the partial period (30/360 in this case), 0#x<1, and
m is the number of full coupon periods between the next date on which
a coupon would have been paid, but for the deferral period, and the
first actual coupon date.
Price =
æ
yö
vy - c
cç 1+ ÷ +
n
fø æ
è
yö
ç1 + ÷
fø
è
æ
yö
yç1 + ÷
fø
è
m+1- x
-x´
c
f
461
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 8 Exercise Solutions (Continued)
Luckily, this formula can be solved for the coupon c:
Price -
v
æ
yö
ç1 + ÷
fø
è
n + m+1- x
=
æ
yö
c
cç 1+ ÷ n
fø æ
è
yö
ç1 + ÷
fø
è
æ
yö
yç1 + ÷
fø
è
m+1- x
-x´
c
f
éæ
ù
yö
1
ê ç 1+ ÷ ú
n
fø æ
yö
êè
ú
ç1 + ÷
ê
fø
è
xú
ú
=c´ê
m+1- x
fú
ê
æ
yö
ê yç1 + ÷
ú
fø
è
ê
ú
êë
úû
Price -
v
æ
yö
ç1 + ÷
fø
è
100% -
n+ m+1- x
100%
29 +10 +1
8.500% ö
æ
ç1 +
÷
è
2 ø
= 14.653%
=
c=
8.500% ö
1
æ
æ
yö
1
ç 1+
÷ç 1+ ÷ 29
n
è
2 ø æ
fø æ
è
8.500% ö
yö
+
1
ç
÷
ç1 + ÷
è
2 ø
fø
è
x
10+1
m+1- x
8.500% ö
f
æ
æ
yö
8.500% ´ ç 1 +
÷
yç 1 + ÷
è
2 ø
fø
è
In this case, n=29, m=10, and x=0.
To estimate the duration,
DurationModified @ -
DP
P = - Py = 8 .51% - Py = 8 .49% = 12.36
Dy
0.02%
The precise duration is 12.39. For comparison, the duration of a regular,
20-year, par-priced 8.500% coupon bond is 9.54.
462
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 8 Exercise Solutions (Continued)
4. A step-up bond pays a coupon that steps up after a period of time.
The step-up date often coincides with a call date. What is the
yield-to-call and yield-to-maturity of a 20-year 8% semi-annualpay coupon bond priced at 102%, with a coupon step-up after
10 years to 10% and a 10-year par call date? How would you
hedge this bond?
The yield-to-call for this bond can be computed using the standard bond
price formula (x=0, n=19, c=8%):
Price =
æ
y ö
vyc - c
cç 1+ c ÷ +
n
f ø æ
è
yc ö
ç1 + ÷
f ø
è
æ
y ö
yc ç 1 + c ÷
f ø
è
8% +
1-x
-x´
c
=
f
y ö 100% ´ yc - 8%
æ
8% ´ ç 1+ c ÷ +
19
è
2ø
yc ö
æ
ç1 + ÷
è
2ø
y ö
æ
yc ç 1 + c ÷
è
2ø
1
100% ´ yc - 8%
=
yc ö
æ
ç1 + ÷
è
2ø
yc
20
yc = 7.709%
463
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 8 Exercise Solutions (Continued)
To handle the step-up coupon, redefine n, add the following definitions
for m and d, value the bond as of the step-up date, and value that bond,
plus the annuity, back an additional m periods:
n
is the number of full coupon periods between the step-up date and
the maturity of the bond,
d
is the stepped-up coupon rate, and
m is the number of full coupon periods between the next coupon date
and the step-up date.
x=0, n=20, m=19
d+
æ
y ö
cç 1+ m ÷ +
f ø
è
Price =
vy m - d
æ
y ö
ç1 + m ÷
f ø
è
æ
y ö
ç1 + m ÷
f ø
è
æ
y ö
ym ç 1 + m ÷
f ø
è
1- x
n
-c
10% +
8% +
m
-x´
c
=
f
100% ´ y m - 10%
y ö
æ
ç1 + m ÷
è
2ø
y ö
æ
ç1 + m ÷
è
2ø
ym
20
- 8%
20
y m = 8.402%
The bond trades strongly to the call, since yc is 70 bp lower than ym.
Therefore, the bond could be hedged as a 10-year. The duration of the
hedge should increase as yields rise, to reflect a reduced probability of
call.
464
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 8 Exercise Solutions (Continued)
5. Estimate the duration of a 10-year LIBOR-flat semi-annual-pay,
semi-annual-reset floater priced at 90%. Assume 6-month
LIBOR semi-annually swaps to 8% fixed semi-annually (i.e., an
investor would be indifferent between receiving LIBOR for
10 years and receiving 8% for 10 years).
To estimate the duration:
The discount floater is equivalent to a par floater (with a higher spread
over LIBOR) and a negative annuity. The duration of the annuity can be
estimated as four, slightly less than the date to average payment of five
years, or it can be calculated exactly by determining the sensitivity of the
annuity stream to a small change in rates:
kPV Annuity = 10% =
k
8% + k ö
æ
ç1 +
÷
è
2 ø
8% + k
20
k = 1.576%
Duration @ -
PV y = 9 .577% - PV y = 9 .575%
DPV
1
1
´
=´
= 4.28
PV
Dy
10%
0.002%
465
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 8 Exercise Solutions (Continued)
The parameters for the discount floater can then be estimated as:
Proceeds Discount Floater = ProceedsPar Floater - Proceeds Annuity
DurationDollar, Discount Floater = DurationDollar, Par Floater - DurationDollar, Annuity
DurationDiscount Floater =
=
PVPar Floater ´ DurationPar Floater - PVAnnuity ´ DurationAnnuity
PVDiscount Floater
100% ´ 0 - 10% ´ 4.28
= - 0.475
90%
6. What is the price of the ABC 81/8% of April 15, 2016 for
settlement June 26, 1996 if the trader quotes the spread as 80 off
the old bond? Eighty off the curve?
Benchmark
Benchmark
Yield (%)
Yield (%)
Price (%)
71/4% due 5/15/16
7.235
8.035
100.866%
67/8% due 8/15/25
7.152
7.952
101.692%
A difference of almost 1% of par!
466
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 9 Exercise Solutions
1. Given the market data below, construct a swaps curve out to
30 years. Assume trade date is 7/22/96 and settlement date is
7/24/96.
Convexity
Maturity
Futures
Adjustment
Forward
Forward
Zero Price
Zero Rate
Price (%)
(bp)
Rate (%)
Price (%)
(%)
(%)
Settlement
7/24/1996
9/18/1996
12/18/1996
94.21
0.00
3/19/1997
6/18/1997
9/17/1997
93.89
93.72
93.57
– 0.30
– 0.60
– 1.00
12/17/1997
3/18/1998
6/17/1998
93.43
93.29
93.26
– 1.50
– 2.00
– 3.00
9/16/1998
12/16/1998
3/17/1999
93.19
93.13
93.04
– 3.70
– 4.30
– 4.90
6/16/1999
9/15/1999
12/15/1999
93.02
92.96
92.90
– 5.90
– 6.90
– 7.90
3/15/2000
6/21/2000
9/20/2000
12/20/2000
3/21/2001
6/20/2001
9/19/2001
7/24/2002
7/24/2003
7/24/2006
7/24/2008
7/24/2011
7/24/2016
7/24/2026
92.81
92.81
92.76
92.70
92.61
92.61
92.56
– 9.20
– 9.40
– 11.60
– 12.90
– 14.10
– 15.60
– 17.00
5.505
61.341
48.790
41.891
33.121
22.666
11.030
467
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 9 Exercise Solutions (Continued)
Step 1: The forward rate for each date with a futures contract is given
by the formula below.
Forward Rate =
100 - Eurodollar Futures Price + (Convexity Adjustment 100)
100
For 12/18/96:
Forward Rate =
100 - 94.21 + (0 100)
= 5.79%
100
Step 2: The forward price is the present value, at the beginning of the
period, of $1 at the end of the period discounted at the forward rate. The
forward price is related to the forward rate by the formula below.
Forward Price =
1
1 + Forward Rate ´
Days in Period
360 Days
There are 91 days in the period from 9/18/96 to 12/18/96, and the
forward rate for that period is 5.79%.
Forward Price =
=
1
1 + Forward Rate ´
1
1 + 5.79% ´
Days in Period
360 Days
91 Days
360 Days
= 98.558%
468
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 9 Exercise Solutions (Continued)
Step 3: The zero price for a future date is the present value, on 7/24/96,
of $1 on that future date. This present value is given by the product of
all the forward prices from the future date back to 7/24/96. For 12/18/96
the zero price is
Forward Price12 /18 / 96®9 /18 / 96 ´ Forward Price9 /18 / 96®7 / 24 / 96 = 98.558% ´ 99.151% = 97.721%
Step 4: The zero rate for any date is given by the formula below.
Zero Price =
1
y
ö
æ
ç 1 + Zero ÷
2 ø
è
n+1- x
According to the 30/360 day-count convention, there are 144 days from
7/24/96 to 12/18/96, so the equation becomes:
97 .721% =
1
y
æ
ö
ç 1 + Zero ÷
è
2 ø
144 /180
yZero=5.848%
Repeating these four steps allows us to generate the swaps curve out to
9/19/01.
469
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 9 Exercise Solutions (Continued)
Interpolated
Mid-market
Mid-market
Quoted
Mid-market
Maturity
Treasury (%)
Spread (bp)
Fixed Rate (%)
5-Years
6.609
6 -Years
6.654
30.5
6.959
7-Years
6.700
34.0
7.040
10-Years
6.836
37.0
7.206
12-Years
6.853
41.5
7.268
15-Years
6.880
47.5
7.355
20-Years
6.923
48.0
7.403
30-Years
7.010
38.5
7.395
Step 5: The Treasury yields are calculated by linear interpolation. For
maturities of T years where T is between five years and 10 years, the
formula is
æ YieldT - yr - Yield 5- yr ö æ Yield10- yr - Yield 5- yr ö
ç
÷ =ç
÷
T- yr - 5- yr
è
ø è 10- yr - 5- yr ø
For points between 10 years and 30 years, the formula is
æ YieldT - yr - Yield10- yr ö æ Yield 30- yr - Yield10- yr ö
ç
÷ =ç
÷
T - yr - 10- yr ø è 30- yr - 10- yr ø
è
The 6-year Treasury yield is given by
æ Yield6- yr - 6.609% ö æ 6.836% - 6.609% ö
ç
÷ =ç
÷ Þ Yield6- yr = 6.654%
è 6- yr - 5- yr ø è 10- yr - 5- yr ø
470
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 9 Exercise Solutions (Continued)
The 6-year swap spread is 30.5 bp so the current fixed rate for a 6-year
swap is
6.654% + 30.5 bp = 6.959%
Step 6: The present value today (7/24/96) of all the fixed-leg cash flows
on a market swap, including hypothetical principal repayment, is par.
The fixed-leg cash flows for a 6-year market swap are mapped out
below. Using the curve we have built so far, we can present value any
cash flow on or before 9/19/01. Zero rates for dates before 9/19/01 are
interpolated from the Eurodollar section of the swaps curve.
Days from
Today
Zero Rate
Zero Price
Cash Flow
(30/360)
(%)
(%)
($MM)
PV ($MM)
Settlement
7/24/1996
1/24/1997
180
5.896
97.136
3.480
3.380
7/24/1997
360
6.147
94.125
3.480
3.275
1/24/1998
540
6.327
91.080
3.480
3.169
7/24/1998
720
6.464
88.052
3.480
3.064
1/24/1999
900
6.567
85.084
3.480
2.961
7/24/1999
1,080
6.655
82.168
3.480
2.859
1/24/2000
1,260
6.732
79.316
3.480
2.760
7/24/2000
1,440
6.802
76.526
3.480
2.663
1/24/2001
1,620
6.861
73.819
3.480
2.569
7/24/2001
1,800
6.916
71.179
3.480
2.477
1/24/2002
1,980
3.480
7/24/2002
2,160
103.480
Total
29.165
The first 10 cash flows have a present value of $29.165MM.
471
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 9 Exercise Solutions (Continued)
The remaining cash flows must have a present value of
$100.00MM – $29.165MM = $70.835MM.
We make the assumption that zero rates increase linearly from the end of
the Eurodollar section (9/19/01) to the 6-year point (7/24/02). We
already know from the Eurodollar section that the zero rate on 9/19/01 is
6.9348%. Suppose the zero rate on 7/24/02 is z% and the zero rate on
1/24/02 is y%.
6.9328%
y%
9/19/01
127 days
z%
1/24/02
The relationship between y% and z% is
181 days
7/24/02
y% - 6.934%
z% - 6.934%
=
127 Days
127 Days + 181 Days
From the previous page, we know that there is a cash flow of $3.48MM
on 1/24/02 and a cash flow of $103.48MM on 7/24/02. Using y% and
z% to calculate the total present value of both cash flows results in the
following equation:
$3.48MM
y% ö
æ
ç1 +
÷
2 ø
è
11
+
$103.48MM
Z% ö
æ
ç1 +
÷
2 ø
è
12
= $70.835MM
This gives us two equations and two unknowns, which we can solve to
get y% = 6.966% and z% = 7.011%. It turns out that it is easiest to solve
this portion using a spreadsheet.
472
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 9 Exercise Solutions (Continued)
The table below summarizes the results for the 6-year point.
Days from
Today
Zero Rate
Zero Price
Cash Flow
(30/360)
(%)
(%)
($MM)
PV ($MM)
Settlement
7/24/1996
0
1/24/1997
180
5.896
97.136
3.480
3.380
7/24/1997
360
6.147
94.125
3.480
3.275
1/24/1998
540
6.327
91.080
3.480
3.169
7/24/1998
720
6.464
88.052
3.480
3.064
1/24/1999
900
6.567
85.084
3.480
2.961
7/24/1999
1,080
6.655
82.168
3.480
2.859
1/24/2000
1,260
6.732
79.316
3.480
2.760
7/24/2000
1,440
6.802
76.526
3.480
2.663
1/24/2001
1,620
6.861
73.819
3.480
2.569
7/24/2001
1,800
6.916
71.179
3.480
2.477
1/24/2002
1,980
6.966
68.618
3.480
2.388
7/24/2002
2,160
7.011
66.134
103.480
68.436
Total
100.000
The process is similar for the 7- , 10- , 12- , 15- , 20- , and 30-year points.
In this problem, we are given the zero price for these points, so the
equation
Price Zero =
1
y Zero ö
æ
ç1 +
÷
è
2 ø
n+1- x
allows us to calculate the zero rates directly.
473
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 9 Exercise Solutions (Continued)
The Entire Swaps Curve
As of July 22, 1996
Convexity
Maturity
Futures
Adjustment
Forward
Forward
Price (%)
(bp)
Rate (%)
Price (%)
Zero Price Zero Rate
(%)
(%)
Settlement
7/24/1996
5.505
99.151
99.151
5.766
12/18/1996
3/19/1997
9/18/1996
94.21
93.89
0.00
– 0.30
5.790
6.107
98.558
98.480
97.721
96.235
5.848
5.966
6/18/1997
9/17/1997
93.72
93.57
– 0.60
– 1.00
6.274
6.420
98.439
98.403
94.733
93.220
6.104
6.214
12/17/1997
3/18/1998
6/17/1998
9/16/1998
93.43
93.29
93.26
93.19
–
–
–
–
1.50
2.00
3.00
3.70
6.555
6.690
6.710
6.773
98.370
98.337
98.332
98.317
91.700
90.176
88.672
87.179
6.298
6.367
6.439
6.502
12/16/1998
93.13
– 4.30
6.827
98.304
85.700
6.550
3/17/1999
6/16/1999
93.04
93.02
– 4.90
– 5.90
6.911
6.921
98.283
98.281
84.229
82.780
6.590
6.637
9/15/1999
92.96
– 6.90
6.971
98.268
81.347
6.680
12/15/1999
3/15/2000
92.90
92.81
– 7.90
– 9.20
7.021
7.098
98.256
98.237
79.928
78.520
6.716
6.752
6/21/2000
9/20/2000
12/20/2000
92.81
92.76
92.70
– 9.40
– 11.60
– 12.90
7.096
7.124
7.171
98.105
98.231
98.220
77.032
75.669
74.322
6.790
6.823
6.851
3/21/2001
6/20/2001
9/19/2001
7/24/2002
7/24/2003
7/24/2006
7/24/2008
7/24/2011
7/24/2016
7/24/2026
92.61
92.61
92.56
– 14.10
– 15.60
– 17.00
7.249
7.234
7.270
98.201
98.204
98.195
72.984
71.674
70.380
66.134
61.341
48.790
41.891
33.121
22.666
11.030
6.876
6.906
6.934
7.011
7.105
7.307
7.384
7.504
7.561
7.485
474
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 9 Exercise Solutions (Continued)
2. Given the swaps curve from question 1, quote an unwind price
for the following swap. An investor is currently paying 7.40%
semi-annually 30/360 to receive 3-month LIBOR quarterly
actual/360 on a $400MM notional from 12/1/93 to 12/1/03. The
notional is amortizing according to the following schedule:
Notional Outstanding
Notional Maturing
Period Ending
($MM)
($MM)
12/1/96
400
0
12/1/97
400
0
12/1/98
400
0
12/1/99
400
0
12/1/00
300
100
12/1/01
200
100
12/1/02
100
100
12/1/03
0
100
Assume 3-month LIBOR was 5.50% on 6/1/96 and assume LIBOR from
7/24/96 to 9/1/96 is currently 5.4375%.
475
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 9 Exercise Solutions (Continued)
Step 1: Project all the cash flows on the fixed leg of the existing swap.
Hypothetical
Date
Interest Payments
Principal
($)
Repayments ($)
Total ($)
6/1/1996
12/1/1996
14,800,000
14,800,000
6/1/1997
14,800,000
14,800,000
12/1/1997
14,800,000
14,800,000
6/1/1998
14,800,000
14,800,000
12/1/1998
14,800,000
14,800,000
6/1/1999
14,800,000
14,800,000
12/1/1999
14,800,000
14,800,000
6/1/2000
14,800,000
14,800,000
12/1/2000
14,800,000
6/1/2001
11,100,000
12/1/2001
11,100,000
6/1/2002
7,400,000
12/1/2002
7,400,000
6/1/2003
3,700,000
12/1/2003
3,700,000
100,000,000
114,800,000
11,100,000
100,000,000
111,100,000
7,400,000
100,000,000
107,400,000
3,700,000
100,000,000
103,700,000
Step 2: We need to determine the current market rate for the fixed leg
of the swap by making sure all of the cash flows (including hypothetical
principal repayment) present value back to par. First, we interpolate zero
rates and calculate the corresponding zero prices. Then we guess a fixed
rate and project the cash flows. If the PV is greater than par, we reduce
the fixed rate until the PV is par. If the initial PV is less than par, we
increase the fixed rate until the PV is par.
476
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 9 Exercise Solutions (Continued)
In this case, the fixed rate is 6.954%. Note that the first period from
7/24/96 to 12/1/96 is a short period.
Date
Notional
Cash Flow
Zero Rate
Zero Price
Outstanding ($)
($)
(%)
(%)
PV ($)
Settlement
7/24/1996
12/1/1996
400,000,000
9,812,549
5.833
97.992
9,615,533
6/1/1997
400,000,000
13,907,550
6.078
95.022
13,215,268
12/1/1997
400,000,000
13,907,550
6.283
91.971
12,790,956
6/1/1998
400,000,000
13,907,550
6.426
88.942
12,369,635
12/1/1998
400,000,000
13,907,550
6.542
85.946
11,952,983
6/1/1999
400,000,000
13,907,550
6.629
83.024
11,546,553
12/1/1999
400,000,000
13,907,550
6.710
80.148
11,146,652
6/1/2000
400,000,000
13,907,550
6.782
77.340
10,756,106
12/1/2000
400,000,000
113,907,550
6.845
74.605
84,980,503
6/1/2001
300,000,000
10,430,662
6.899
71.952
7,505,068
12/1/2001
300,000,000
110,430,662
6.953
69.362
76,596,602
6/1/2002
200,000,000
6,953,775
6.998
66.859
4,649,212
12/1/2002
200,000,000
106,953,775
7.045
64.414
68,893,367
6/1/2003
100,000,000
3,476,887
7.091
62.031
2,156,744
12/1/2003
100,000,000
103,476,887
7.129
59.747
61,824,816
Total
400,000,000
477
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 9 Exercise Solutions (Continued)
Step 3: We combine the cash flows of the fixed legs of both swaps and
present value the result back to today. On the fixed legs, the dealer
receives $12,732,526 on 7/24/96.
Investor
Investor
Net to
Zero Price
PV
(%)
(%)
($)
Date
Pays ($)
12/1/1996
14,800,000
9,812,549
–4,987,451
5.833
97.992
–4,887,313
6/1/1997
14,800,000
13,907,550
–892,450
6.078
95.022
–848,026
12/1/1997
14,800,000
13,907,550
–892,450
6.283
91.971
–820,798
6/1/1998
14,800,000
13,907,550
–892,450
6.426
88.942
–793,762
12/1/1998
14,800,000
13,907,550
–892,450
6.542
85.946
–767,025
6/1/1999
14,800,000
13,907,550
–892,450
6.629
83.024
–740,945
12/1/1999
14,800,000
13,907,550
–892,450
6.710
80.148
–715,283
6/1/2000
14,800,000
13,907,550
–892,450
6.782
77.340
–690,221
12/1/2000 114,800,000 113,907,550
–892,450
6.845
74.605
–665,811
10,430,662
–669,338
6.899
71.952
–481,602
11,100,000 110,430,662
–669,338
6.953
69.362
–464,264
6,953,775
–446,225
6.998
66.859
–298,341
12/1/2002 107,400,000 106,953,775
–446,225
7.045
64.414
–287,432
3,476,887
–223,113
7.091
62.031
–138,399
12/1/2003 103,700,000 103,476,887
–223,113
7.129
59.747
–133,304
6/1/2001
12/1/2001
6/1/2002
6/1/2003
11,100,000
7,400,000
3,700,000
Receives ($) Investor ($)
Zero Rate
Total
–12,732,526
Step 4: If we combine the cash flows of the floating legs of the existing
swap and the hypothetical new swap, all of the future payments cancel
out. On the existing swap, the investor receives 3-month LIBOR
quarterly actual/360. On the hypothetical new swap, the investor pays
3-month LIBOR quarterly actual/360. The only exception is the
payments that occur at the end of the initial period.
478
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 9 Exercise Solutions (Continued)
On the existing swap, 3-month LIBOR was set at 5.500% for the 92-day
period from 6/1/96 to 9/1/96. At the end of the period, the investor
receives
æ 92 Days ö
$400,000 ,000 ´ 5.500% ´ ç
÷ = $5,622 ,222
è 360 Days ø
On the hypothetical new swap, LIBOR for the 39-day period from
7/24/96 to 9/1/96 is 5.4375%. At the end of the period, the investor pays
æ 39 Days ö
$400,000 ,000 ´ 5.4375% ´ ç
÷ = $2,356 ,250
è 360 Days ø
Netting the payments results in the investor receiving $3,265,972 on
9/1/96. The PV of this payment on 7/24/96 is
$3,265,972
= $3,246,715
æ
39 Days ö
ç 1 + 5.4375% ´
÷
360 Days ø
è
Step 5: On the fixed legs, the investor pays $12,732,526 on 7/24/96. On
the floating legs, the investor receives $3,246,715 on 7/24/96. The net
unwind price is a net payment of $9,485,811 by the investor on 7/24/96,
due to the decline in rates since the original swap was agreed.
479
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 9 Exercise Solutions (Continued)
3. Is a 6-month swap paying 6-month LIBOR (set at inception to
6%) and receiving 6% fixed semi-annually at-market? Why?
This swap is at-market only if the day-count basis on both legs is the
same. Typically, the LIBOR leg is actual/360 and the fixed leg is 30/360.
Six months is 181 to 184 days actual/360 and 180 days 30/360, so the
swap is not at-market. For example, if the swap is on the six months
from 6/1/96 to 12/1/96 on $100MM, the LIBOR leg pays
$100,000 ,000 ´ 6.00% ´
183 Days
= $3,050 ,000
360 Days
while the fixed leg pays
$100,000 ,000 ´ 6.00% ´
180 Days
= $3,000 ,000
360 Days
480
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 10 Exercise Solutions
1. Derive the formula for the monthly payment of a mortgage,
assuming a fixed-rate n-year mortgage with a c% rate and a B0
starting balance.
When the payments are computed, the mortgage is at par, so the coupon
equals the yield.
B0 =
PMT
PMT
PMT
+
+L+
2
(1 + c 12) (1 + c 12)
(1 + c 12)12´ n
B0
PMT
PMT
PMT
=
+L+
+
2
12
´
n
(1 + c 12) (1 + c 12)
(1 + c 12)
(1 + c 12)12´ n+1
B0 -
B0
PMT
PMT
=
(1 + c 12) (1 + c 12) (1 + c 12)12´ n+1
æ
ö
æ
1
1
1 ö
÷
B0 ´ ç 1 ÷ = PMT ´ ç
ç 1 + c 12 (1 + c 12)12 ´ n+1 ÷
1 + c 12 ø
è
è
ø
ö
æ
æ 1 + c 12
1
1
1 ö
÷
ç
B0 ´ ç
÷ = PMT ´
ç (1 + c 12) (1 + c 12)12 ´ n+1 ÷
è 1 + c 12 1 + c 12 ø
ø
è
æ
ö
æ c 12 ö
1
1
ç
÷
B0 ´ ç
÷ = PMT ´
ç (1 + c 12) (1 + c 12)12 ´ n+1 ÷
è 1 + c 12 ø
è
ø
æ
ö
1
ç
÷
B0 ´ c 12 = PMT ´ 1 12 ´ n ÷
ç
(1 + c 12) ø
è
PMT =
B0 ´ c 12
æ
ö
1
ç1 ÷
12 ´ n ÷
ç
(1 + c 12) ø
è
481
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 10 Exercise Solutions (Continued)
2. Fill in the following chart out to six months for a new-origination
30-year mortgage pool with an 8.00% rate, 0.50% servicing fee,
and 8% CPR, assuming that all prepayments are total payoffs.
Gross Coupon: 8.50%
Given the ending balance at time i–1 (which is the starting balance at
time i), the payment at time i–1, and the prepayment at time i–1,
æ
PPMTi -1 ö
PMTi = PMTi -1 ´ ç 1 ÷
Bi -1 + PPMTi -1 ø
è
Starting
Monthly
Net
Month Balance ($) Payment ($) Int ($)
Servicing
Sched
Prepay
Ending
Fee ($)
Prin ($)
Prin ($)
Balance ($)
1
1,000,000.00
7,689.13 6,666.67
416.67
605.80
6,920.19
992,474.01
2
992,474.01
7,635.89 6,616.49
413.53
605.87
6,868.07
985,000.07
3
985,000.07
7,583.02 6,566.67
410.42
605.93
6,816.32
977,577.81
4
977,577.81
7,530.51 6,517.19
407.32
606.00
6,764.93
970,206.88
5
970,206.88
7,478.37 6,468.05
404.25
606.07
6,713.89
962,886.93
6
962,886.93
7,426.58 6,419.25
401.20
606.13
6,663.20
955,617.59
482
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 10 Exercise Solutions (Continued)
3. Do the same exercise, but assume that 50% of the prepayments
are curtailments. How would this affect the life of the mortgage
pool?
Curtailments decrease the life of the mortgage. A 10% curtailment
means that 10% of the pool’s outstanding balance has been paid down
and some of the later cash flows have been eliminated.
Now,
æ
(1 - Curtailments) ´ PPMTi -1 ö
PMTi = PMTi -1 ´ ç 1 ÷
Bi -1 + PPMTi -1
è
ø
Starting
Monthly
Net
Servicing
Sched
Prepay
Ending
Fee ($)
Prin ($)
Prin ($)
Balance ($)
Month Balance ($) Payment ($) Int ($)
1
1,000,000.00
7,689.13 6,666.67
416.67
605.80
6,920.19
992,474.01
2
992,474.01
7,662.51 6,616.49
413.53
632.49
6,867.89
984,973.63
3
984,973.63
7,635.98 6,566.49
410.41
659.09
6,815.77
977,498.77
4
977,498.77
7,609.55 6,516.66
407.29
685.60
6,763.83
970,049.35
5
970,049.35
7,583.20 6,467.00
404.19
712.02
6,712.06
962,625.27
6
962,625.27
7,556.95 6,417.50
401.09
738.35
6,660.47
955,226.44
4. What is the bond-equivalent yield of a par mortgage that is
quoted at a 12% monthly yield?
A 12% mortgage yield pays 12% 12 on a monthly basis. Bondequivalent yield assumes semi-annual receipt of coupon interest.
Therefore,
2
12
y2 ö
y12 ö
æ
æ
ç1 + ÷ = ç1 +
÷
è
è
2ø
12 ø
Solving for y2, we see that
6
ù
éæ
y ö
y2 = 2 ´ êç 1 + 12 ÷ - 1ú . So, the BEY is 12.304%.
12 ø
úû
êëè
483
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 10 Exercise Solutions (Continued)
5. A homeowner borrows $100,000 at 10% with a 30-year fixed-rate
mortgage. How much principal is paid down in the fourth
month? How much interest is paid? If several mortgages
identical to this are pooled, under what circumstances will the
cash flow to the MBS investor be identical to the cash flow paid
by the homeowner?
There is a closed-form formula for the monthly balance of a mortgage.
Using it, the interest and principal payments for the given month can be
easily computed. The current balance of any given month must be equal
to the present value of the remaining cash flows. Assuming:
n = number of months passed
T = term of loan
r = monthly mortgage rate (or mortgage coupon divided by 12)
Bi = ending balance at time i
Bn =
PMT
PMT
PMT
Bn
PMT
PMT
+
+L+
=
+L+
2
T - n and
2
(1 + r ) (1 + r )
(1 + r ) (1 + r )
(1 + r )T - n +1
(1 + r )
PMT
æ r ö PMT
Bn ç
÷=
è 1 + r ø (1 + r ) (1 + r )T - n+1
From question 1, PMT =
1
ö
æ
ç1T -n ÷
(1 + r ) ÷
Þ Bn = PMT ´ çç
÷
r
÷
ç
ø
è
B0 ´ r
æ
1 ö÷
ç1 ç
(1 + r )T ÷ø
è
484
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 10 Exercise Solutions (Continued)
T -n
1 + r) - 1
(
n
´
1
+
r
Bn = B0 ´
(
)
(1 + r )T - 1
æ (1 + r )T - (1 + r )n ö
÷
Bn = B0 ç
T
ç
÷
è (1 + r ) - 1 ø
B3 = $99,866.18
B4 = $99,820.82
P4 = B3 – B4 = $45.36
I4 = B3 ×
10%
12
= $832.22
The only way the receipts by the investor can equal the payments by the
homeowner is if there is no servicing fee.
6. What is a 12% CPR in terms of SMM?
Recall that SMM = 1 – (1 – CPR)1/12. Therefore, a CPR of 12% is equal
to an SMM of 1.0596%.
485
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 10 Exercise Solutions (Continued)
7. Consider a 30-year FNMA 8.00% pool (55-day stated delay) with
348 months remaining until maturity with no historical
curtailments and no servicing expense. The original settlement
date is June 13, 1996, and the mortgage is trading at 102-14. The
drop is quoted at 14. Assume a money rate of 5.0% and
prepayments of 3.4% CPR. What is the implied financing rate to
the second FNMA settlement date of August 12, 1996?
•
The transaction will be in place from June 13, 1996 until August 12,
1996 (d = 60).
•
A 348-month, 8.00% coupon mortgage (with no curtailments and a
WAM of 348) has monthly payments (with respect to current
balance) of
PMT =
1-
8% / 12
1
= 0.740%
(1 + 8% / 12)348
•
The interest due in the first month is 0.667%, and so the scheduled
principal is 0.073%.
•
A CPR of 3.4% translates into
SMM = 1 - (1 - CPR)
1 12
= 0.288%
The principal prepayments (which apply to principal remaining after the
scheduled principal payment) is then
(
)
PrincipalPrepaid,1 = 100% - PrincipalScheduled ,1 ´ SMM = (1 - 0.073%) ´ 0.288% = 0.288%
PrincipalPaydown,1 = 0.361%
486
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
Chapter 10 Exercise Solutions (Continued)
•
The second month’s interest is (100% – 0.361%) × 8%/12 = 0.664%,
the total payment is the original payment scaled down for the amount
of principal prepayment 99.639% (99.639% + 0.288%)× 0.740% =
0.738%, and so the scheduled principal is 0.074%.
•
The second month’s principal prepayment is then
PrincipalPrepaid,2 = (99.639% - 0.074%) ´ SMM = 0.287%
PrincipalPaydown,2 = 0.360%
•
Since agency mortgages accrue from the beginning of the month,
AccruedSpot =
•
12 8%
´
= 0 .267%
30 12
Accrued Forward =
11 8%
´
= 0 .244%
30 12
The first month’s principal and interest payment is paid on July 25,
1996, and is reinvested for 18 days, and the second month’s principal
and interest payment is paid on August 25, 1996 (although August
25th is a Sunday) and is discounted for 13 days.
18 ö
1.024%
æ
FVCoupons + FVPrincipal = 1.028% ´ ç 1 + 5% ´
= 2.054%
÷+
è
13 ö
360 ø æ
ç 1 + 5% ´
÷
è
360 ø
•
The forward price is PriceForward = PriceSpot - Drop = 102-14 - 14 = 102-00
•
Plugging these variables into the formula,
rImplied =
360 é (100% - 0 .721%) ´ (102.000% + 0 .244%) + 2.053% ù
´ê
- 1ú = 4.999%
60 êë
102.438% + 0 .267%
úû
487
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159
159
This material has been prepared solely for information purposes and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or instrument or to
participate in any particular trading strategy. It is based on or derived from information generally available to the public from sources believed to be reliable. No
representation is made that information or formulae included herein are accurate or complete or that any results obtained therefrom are indicative of actual prices or returns.
All pricing and other data herein are derived from Morgan Stanley unless referenced otherwise. Past performance is not necessarily indicative of future results. Additional
information is available on request.
159