The Valuation of Bonds
Minggu 9
Bond Values
• Bond values are discussed in one of two
ways:
– The dollar price
– The yield to maturity
• These two methods are equivalent since a
price implies a yield, and vice-versa
Bond Yields
• There are several ways that we can
describe the rate of return on a bond:
– Coupon rate
– Current yield
– Yield to maturity
– Modified yield to maturity
– Yield to call
– Realized Yield
The Coupon Rate
• The coupon rate of a bond is the stated
rate of interest that the bond will pay
• The coupon rate does not normally
change during the life of the bond, instead
the price of the bond changes as the
coupon rate becomes more or less
attractive relative to other interest rates
• The coupon rate determines the dollar
amount of the annual interest payment:
Annual Pmt  Coupon Rate  Face Value
The Current Yield
• The current yield is a measure of the
current income from owning the bond
• It is calculated as:
Annual Pmt
CY 
Face Value
The Yield to Maturity
• The yield to maturity is the average annual
rate of return that a bondholder will earn
under the following assumptions:
– The bond is held to maturity
– The interest payments are reinvested at the
YTM
• The yield to maturity is the same as the
bond’s internal rate of return (IRR)
The Modified Yield to Maturity
• The assumptions behind the calculation of the
YTM are often not met in practice
• This is particularly true of the reinvestment
assumption
• To more accurately calculate the yield, we can
change the assumed reinvestment rate to the
actual rate at which we expect to reinvest
• The resulting yield measure is referred to as the
modified YTM, and is the same as the MIRR for
the bond
The Yield to Call
• Most corporate bonds, and many older
government bonds, have provisions which allow
them to be called if interest rates should drop
during the life of the bond
• Normally, if a bond is called, the bondholder is
paid a premium over the face value (known as
the call premium)
• The YTC is calculated exactly the same as YTM,
except:
– The call premium is added to the face value, and
– The first call date is used instead of the maturity date
The Realized Yield
• The realized yield is an ex-post measure
of the bond’s returns
• The realized yield is simply the average
annual rate of return that was actually
earned on the investment
• If you know the future selling price,
reinvestment rate, and the holding period,
you can calculate an ex-ante realized yield
which can be used in place of the YTM
(this might be called the expected yield)
Calculating Bond Yield
Measures
• As an example of the calculation of the bond
return measures, consider the following:
– You are considering the purchase of a 2-year bond
(semiannual interest payments) with a coupon rate of
8% and a current price of $964.54. The bond is
callable in one year at a premium of 3% over the face
value. Assume that interest payments will be
reinvested at 9% per year, and that the most recent
interest payment occurred immediately before you
purchase the bond. Calculate the various return
measures.
– Now, assume that the bond has matured (it was not
called). You purchased the bond for $964.54 and
reinvested your interest payments at 9%. What was
your realized yield?
Calculating Bond Yield Measures
(cont.)
Timeline
if not called
Timeline
if called
-964.54
40
40
40
1,000
40
0
1
2
3
4
-964.54
40
1,030
40
0
1
2
Calculating Bond Yield Measures
(cont.)
• The yields for the example bond are:
– Current yield = 8.294%
– YTM = 5% per period, or 10% per year
– Modified YTM = 4.971% per period, or
9.943% per year
– YTC = 7.42% per period, or 14.84% per year
– Realized Yield:
• if called = 7.363% per period, or 14.725% per year
• if not called = 4.971% per period, or 9.943% per
year
Bond Valuation in Practice
• The preceding examples ignore a couple
of important details that are important in
the real world:
– Those equations only work on a payment
date. In reality, most bonds are purchased in
between coupon payment dates. Therefore,
the purchaser must pay the seller the accrued
interest on the bond in addition to the quoted
price.
– Various types of bonds use different
assumptions regarding the number of days in
a month and year.
Valuing Bonds Between Coupon Dates
(cont.)
• Imagine that we are halfway between coupon
dates. We know how to value the bond as of the
previous (or next even) coupon date, but what
about accrued interest?
• Accrued interest is assumed to be earned
equally throughout the period, so that if we
bought the bond today, we’d have to pay the
seller one-half of the period’s interest.
• Bonds are generally quoted “flat,” that is, without
the accrued interest. So, the total price you’ll
pay is the quoted price plus the accrued interest
(unless the bond is in default, in which case you
do not pay accrued interest, but you will receive
the interest if it is ever paid).
Valuing Bonds Between Coupon Dates
(cont.)
• The procedure for determining the quoted
price of the bonds is:
– Value the bond as of the last payment date.
– Take that value forward to the current point in
time. This is the total price that you will
actually pay.
– To get the quoted price, subtract the accrued
interest.
• We can also start by valuing the bond as
of the next coupon date, and then discount
that value for the fraction of the period
remaining.
Valuing Bonds Between Coupon Dates
(cont.)
• Let’s return to our original example (3 years,
semiannual payments of $50, and a required
return of 7% per year).
• As of period 0 (today), the bond is worth
$1,079.93. As of next period (with only 5
remaining payments) the bond will be worth
$1,067.73. Note that:
1
1067.73  1079.931.035  50
P1
P0
Interest earned
• So, if we take the period zero value forward one
period, you will get the value of the bond at the
next period including the interest earned over
the period.
Valuing Bonds Between Coupon Dates
(cont.)
• Now, suppose that only half of the period has
gone by. If we use the same logic, the total price
of the bond (including accrued interest) is:
0.5
1079.931.035  1098.66
• Now, to get the quoted price we merely subtract
the accrued interest:
QP  1098.66  25  1073.66
• If you bought the bond, you’d get quoted
$1,073.66 but you’d also have to pay $25 in
accrued interest for a total of $1,098.66.
Day Count Conventions
• Historically, there are several different assumptions that
have been made regarding the number of days in a
month and year. Not all fixed-income markets use the
same convention:
– 30/360 – 30 days in a month, 360 days in a year. This is used in
the corporate, agency, and municipal markets.
– Actual/Actual – Uses the actual number of days in a month and
year. This convention is used in the U.S. Treasury markets.
• Two other possible day count conventions are:
– Actual/360
– Actual/365
• Obviously, when valuing bonds between coupon dates
the day count convention will affect the amount of
accrued interest.
The Term Structure of
Interest Rates
• Interest rates for bonds vary by term to
maturity, among other factors
• The yield curve provides describes the
yield differential among treasury issues of
differing maturities
• Thus, the yield curve can be useful in
determining the required rates of return for
loans of varying maturity
Types of Yield Curves
Rising
Declining
Flat
Humped
Today’s Actual Yield Curve
U.S. Treasury Yield Curve
24 April 2002
Data Source: http://www.ratecurve.com/yc2.html
Term to Maturity
Y
R
30
Y
R
20
Y
R
15
Y
R
10
Y
R
7
Y
R
5
Y
R
4
Y
R
3
Y
R
2
Y
EA
R
Y
D
A
0
18
D
A
Y
6.00%
5.00%
4.00%
3.00%
2.00%
1.00%
90
YLD
4.75%
1.25%
1.75%
1.71%
1.88%
2.19%
3.23%
3.74%
4.18%
4.43%
4.91%
5.10%
5.64%
5.76%
5.61%
Yield
Maturity
PRIME
DISC
FUNDS
90 DAY
180 DAY
YEAR
2 YR
3 YR
4 YR
5 YR
7 YR
10 YR
15YR
20 YR
30 YR
Explanations of the Term
Structure
• There are three popular explanations of
the term structure of interest rates (i.e.,
why the yield curve is shaped the way it
is):
– The expectations hypothesis
– The liquidity preference hypothesis
– The market segmentation hypothesis
(preferred habitats)
• Note that there is probably some truth in
each of these hypotheses, but the
expectations hypothesis is probably the
most accepted
The Expectations Hypothesis
• The expectations hypothesis says that
long-term interest rates are geometric
means of the shorter-term interest rates
• For example, a ten-year rate can be
considered to be the average of two
consecutive five-year rates (the current
five-year rate, and the five-year rate five
years hence)
• Therefore, the current ten-year rate must
be:
5
5
1  R10   10 1  R5  1t 5 R5 
The Expectations Hypothesis
(cont.)
• For example, if the current five-year rate is 8%
and the expected five-year rate five years from
now is 10%, then the current ten-year rate must
be:
1t R10   10 1.085 1.105
• In an efficient market, if the ten-year rate is
anything other than 8.995%, then arbitrage will
bring it back into line
• If the ten-year rate was 9.5%, then people would
buy ten-year bonds and sell five-year bonds until
the rates came back into line
The Expectations Hypothesis
(cont.)
• The ten-year rate can also be thought of a
series of five two-year rates, ten one-year
rates, etc.
• Note that since the ten-year rate is
observable, we normally would solve for
an expected future rate
• In the previous example, we would usually
solve for the expected five-year rate five
years from now:
1 t 5 R 5   5
1 t R10 
5
1 t R 5 
10
1.10 
1.08995
5
1.08
10
5
The Liquidity Preference
Hypothesis
• The liquidity preference hypothesis contends
that investors require a premium for the
increased volatility of long-term investments
• Thus, it suggests that, all other things being
equal, long-term rates should be higher than
short-term rates
• Note that long-term rates may contain a
premium, even if they are lower than short-term
rates
• There is good evidence that such premiums
exist
The Market Segmentation
Hypothesis
• This theory is also known as the preferred
habitat hypothesis because it contends
that interest rates are determined by
supply and demand and that different
investors have preferred maturities from
which they do no stray
• There is not much support for this
hypothesis
S
D
S
D
Banks
Insurance
Companies
Bond Price Volatility
• Bond prices change as any of the
variables change:
– Prices vary inversely with yields
– The longer the term to maturity, the larger the
change in price for a given change in yield
– The lower the coupon, the larger the
percentage change in price for a given
change in yield
– Price changes are greater (in absolute value)
when rates fall than when rates rise
Measuring Term to Maturity
• It is difficult to compare bonds with
different maturities and different coupons,
since bond price changes are related in
opposite ways to these variables
• Macaulay developed a way to measure
the average term to maturity that also
takes the coupon rate into account
• This measure is known as duration, and is
a better indicator of volatility than term to
maturity alone
Duration
• Duration is calculated as:
N
D

t 1
Pmt t  t 
1  i
t
Bond Pr ice
• So, Macaulay’s duration is a weighted
average of the time to receive the present
value of the cash flows
• The weights are the present values of the
bond’s cash flows as a proportion of the
bond price
Calculating Duration
• Recall our earlier example bond with a
YTM of 5% per six-months:
D
-964.54
40
40
40
1,000
40
0
1
2
3
4
40
40
40
1040
1 
2

3






2
3
4 4
. 
105
. 
. 
. 
105
105
105
964.54

3636.76
 3.77
964.54
• Note that this is 3.77 six-month periods,
which is about 1.89 years
Notes About Duration
• Duration is less than term to maturity,
except for zero coupon bonds where
duration and maturity are equal
• Higher coupons lead to lower durations
• Longer terms to maturity usually lead to
longer durations
• Higher yields lead to lower durations
• As a practical matter, duration is generally
no longer than about 20 years even for
perpetuities
Modified Duration
• A measure of the volatility of bond prices is
the modified duration (higher DMod =
higher volatility)
• Modified duration is equal to Macaulay’s
duration divided by 1 + per period YTM
D Mod
D

1  i
• Note that this is the first partial derivative
of the bond valuation equation wrt the
yield
Why is Duration Better than
Term?
• Earlier, it was noted that duration is a
better measure than term to maturity. To
see why, look at the following example:
• Suppose that you are comparing two fiveyear bonds, and are expecting a drop in
yields of 1% almost immediately. Bond 1
has a 6% coupon and bond 2 has a 14%
coupon. Which would provide you with the
highest potential gain if your outlook for
rates actually occurs? Assume that both
bonds are currently yielding 8%.
Why is Duration Better than Term?
(cont.)
• Both bonds have equal maturity, so a
superficial investigation would suggest that
they will both have the same gain.
However, as we’ll see bond 2 would
actually gain more.
5
60
1000


5
t


t
5
1
.
08
1
.
08
D1  t 1
D Mod ,1  4.30
1.08
920.15
 4.44
 3.98
5
D2 
120
1000
 1.08 t   1.08 5
t 1
t
5
D Mod , 2  4.11
 3.81
1.08
1159.71
 4.11
Why is Duration Better than Term?
(cont.)
• Note that the modified duration of bond 1 is
longer than that of bond two, so you would
expect bond 1 to gain more if rates actually drop.
– Pbond 1, 8%= 920.15; Pbond 1, 7%= 959.00; gain = 38.85
– Pbond 2, 8%= 1159.71; Pbond 2, 7%= 1205.01; gain =
45.30
• Bond 1 has actually changed by less than bond
2. What happened? Well, if we figure the
percentage change, we find that bond 1 actually
gained by more than bond 2.
• %Dbond 1 = 4.22%; %Dbond 2 = 3.91% so your
gain is actually 31 basis points higher with bond
1.
Why is Duration Better than Term?
(cont.)
• Bond price volatility is proportionally related to
the modified duration, as shown previously.
Another way to look at this is by looking at how
many of each bond you can purchase.
• For example, if we assume that you have
$100,000 to invest, you could buy about 108.68
units of bond 1 and only 86.23 units of bond 2.
• Therefore, your dollar gain on bond 1 is
$4,222.14 vs. $3,906.15 on bond 2. The net
advantage to buying bond 1 is $315.99.
Obviously, bond 1 is the way to go.
Convexity
• Convexity is a measure of the curvature of
the price/yield relationship
DMod = Slope of Tangent Line
Convexity
Yield
• Note that this is the second partial
derivative of the bond valuation equation
wrt the yield
Calculating Convexity
• Convexity can be calculated with the
following formula:
1
C
1  i
2



N
 1  i 
CFt
t
t 1

t t 


2
VB
• For the example bond, the convexity (per
period) is:

  402
40 12  1
. 
105
1
2
2
. 
105
2
  403
2
. 
105
. 
105
2
C
964.54
  10404
3
3
2
. 
105
4

4
17,820.73
16,163.93
.
 11025

 16.758
964.54
964.54
Calculating Convexity (cont.)
• To make the convexity of a semi-annual
bond comparable to that of an annual
bond, we can divide the convexity by 4
• In general, to convert convexity to an
annual figure, divide by m2, where m is the
number of payments per year
Calculating Bond Price
Changes
• We can approximate the change in a
bond’s price for a given change in yield by
using duration and convexity:
DV    D  Di  V    0.5  C  V   Di 
2
B
Mod
B
B
• If yields rise by 1% per period, then the
price of the example bond will fall by
33.84, but the approximation is:

DVB   359
.  0.01  964.54  0.5  16.75  964.54  0.01
2
.
  34.63  0.81  3382
Solved Examples (on a
payment date)
Bond 1
Term (years)
Yield
Coupon
Face Value
Value
Duration
Mod. Duration
Convexity
2
3%
100
1000
1133.94
1.91
1.86
5.33
Bond 2
3
5%
80
1000
1081.70
2.79
2.66
9.88
Bond 3
4
7%
60
1000
966.13
3.67
3.43
15.54
Bond 4
5
9%
40
1000
805.52
4.58
4.20
22.46
Bond 5
6
11%
20
1000
619.25
5.62
5.06
31.24
Bond 6
7
13%
0
1000
425.06
7.00
6.19
43.86