The Oxford Guide to Financial Modeling: Applications for Capital Markets, Corporate
Finance, Risk Management and Financial Institutions
Thomas S.Y. Ho and Sang Bin Lee
2004
Chapter 3: Bond Markets: The Bond Model
A bond is a debt instrument requiring the issuer (also called the debtor or borrower) to
repay to the lender/investor the amount borrowed plus interest over a specified period of
time. A typical (“plain vanilla”) bond issued in the United States specifies (1) a fixed
date when the amount borrowed (the principal) is due, and (2) the contractual amount of
interest, which typically is paid every six months. The date on which the principal is
required to be repaid is called the maturity date. Assuming that the issuer does not
default or redeem the issue prior to the maturity date, an investor holding this bond until
the maturity date is assured of a known cash flow.
The bond indenture is the contract specifying all contractual terms of the relationship
between the bond issuer and the bond owner. The bond trustee is typically a commercial
bank identified to monitor the bond issuer’s performance per the terms of the indenture.
Security Status – Some bond are secured by claims on specific assets of the issuer, so
that in the event of default (failure to make payment as scheduled) the bondholders have
a claim on specified assets or the proceeds from liquidation of the assets. Bonds of this
type are often referred to as mortgage bonds, collateralized bonds or collateralized
trust bonds.
Bonds which represent a general (unsecured) obligation of the issuer are often times
referred to as debentures. Claims represented by more junior bonds (subordinated
debentures) rank behind the claims of secured bondholders, general obligation
bondholders, and other unsecured creditors.
Time to maturity – bond pays principal value plus final coupon payment on specified
maturity date, T. Time to maturity (T – t)
Bond cash flows:
Maurity value = Face value = par value = (notional) principal = M
Coupon rate = c = annualized percentage of principal paid as interest
Bond price: as percentage of principal value, M=$100 for a semi-annual pay coupon bond
with n/2 years to maturity.
n
P 
C
t 1 (1  0.5 * y )
t
 M * (1  0.5 * y)  n
P = invoice price
C = 0.5*c*M
1
y = yield
M = maturity value or call price or put price
Taxation of bond income:
Coupon income subject to income tax
Price appreciation/depreciation (M – P(t)) subject to capital gains tax
 P(t) = Price (clean) = Invoice (dirty) price – accrued interest
 Invoice price = present value of remaining cashflows
Accrued Interest – In the United States the buyer of a coupon bond owes the seller
accrued interest through the settlement date of the transaction. Each marketplace i.e.
Treasury, corporate, municipal, has a specific set of conventions that are used to calculate
accrued interest.
clean price
accrued interest
invoice price
+
In an efficient market, the present value of the remaining bond payments is equivalent to
the invoice price.
Day counts include the settlement date and omit the cash flow date.
Treasury coupon markets (notes, bonds) use actual/actual day count convention.
Corporate, Municipal markets use 30/360 day count convention.
US – Treasury Bond 11 5/8 coupon, maturity 11/15/2004, issued 10/30/84
coupon 11.625
maturity
11/15/04
SD
bid
ask
ask
NCD
LCD
w
v accrued
ask
invoice
yield
8/9/99 124.08 124.14 6.12% 11/15/99 5/15/99 0.4674 0.5326 2.717 124.4375 127.15421
9/04/03
111.31 1.47% 11/15/03 5/15/03 0.6087
0.3913 3.538 111.96875 115.50675
8/31/04
102.03 1.422% 11/15/04 5/15/04 0.5870
0.4130 3.412 102.09375 105.50575
On any date except a coupon payment date bond transactions will involve accrued
interest. The buyer of the bond will pay the seller of the bond the “clean” price plus
accrued interest. The sum of the “clean” price and the accrued interest is often called the
invoice price; it is the cash outlay necessary to purchase the bond. The invoice price is
the present value of the bond’s remaining cash flows.
2
The quote information from Barron’s 8/9/99 reflects secondary market dealer quotes for
the 115/8% of 2004 Treasury bond.
The “clean” bid price is 124 8/32 = $124.25 per $100 face value. The dealer is willing to
pay $124.25 to purchase this bond.
The “clean” ask price is 124 14/32 = $124.4375 per $100 face value. The dealer is
willing to sell this bond for $124.4375.
The accrued interest for a transaction on the settlement date, 8/9/99, will depend on the
days since the last coupon payment and the day count conventions in the Treasury
market. In the Treasury market the actual/actual day count convention is used.
w
days between LCD and SD
 (1  v)
days between LCD and NCD
The fraction (w) is the portion of a six-month period since the last coupon payment date.
# days between 5/15/99 and 8/9/99 = 86
# days between 5/15/99 and 11/15/99 = 184
w = 86/184 = .467391
The calculation of accrued interest depends on; w, face value, and coupon rate. For this
example the face value is $100.
accrued interest = (1/2)*(.11625)*(0.46739)*(100) = 2.7167
To purchase this bond the invoice price = “clean” ask price + accrued interest.
invoice price = $124.4375 + 2.7167 = 127.1542.
Bond yield measures:
Yield to maturity – internal rate of return. – Measure of expected return if P(t) paid, bond
held to maturity, coupons received as promised, coupons reinvested at yield to maturity
through maturity date. Total return of investment i.e., (total future dollars)/P(t) scaled for
time and stated on annual basis = yield to maturity.
Yield to maturity = coupon rate ++ > invoice price = par value
Yield to maturity < coupon rate ++ > invoice price > par value
Yield to maturity > coupon rate ++ > invoice price < par value
Invoice price and yield to maturity are inversely related for straight bonds (non-callable,
non-convertible) the relationship is convex.
3
The yield to maturity of a portfolio is not the weighted average of the yield to maturities
of the individual bonds. To calculate the yield to maturity of a portfolio of bonds;
1. Determine the market value of the portfolio of bonds: (clean price and accrued
interest).
2. Determine the size and timing of the cash flows promised to accrue to the portfolio.
3. Find the required yield that makes the discounted value of the promised cash flows
equal to the market value of the bond portfolio.
BOND
A
B
C
Coupon
5%
6.50%
7%
Maturity
3
7
10
ytm
3%
4%
6%
M
Value
P/$
1.056971872
1.15132811
1.074387374
$1,000,000
$2,000,000
$3,000,000
$1,056,971.87 $2,302,656.22 $3,223,162.12 Current value
ytm
WA ytms
Cashflow map portfolio
Period
A
B
C
Portfolio
1
25000
65000
105000
195000
2
25000
65000
105000
195000
3
25000
65000
105000
195000
4
25000
65000
105000
195000
5
25000
65000
105000
195000
6
1025000
65000
105000
1195000
7
65000
105000
170000
8
65000
105000
170000
9
65000
105000
170000
10
65000
105000
170000
11
65000
105000
170000
12
65000
105000
170000
13
65000
105000
170000
14
2065000
105000
2170000
15
105000
105000
16
105000
105000
17
105000
105000
18
105000
105000
19
105000
105000
20
3105000
3105000
Portfolio
$6,582,790.21
5.1491%
4.8187%
190105.6096
185334.0657
180682.2849
176147.2611
171726.0639
1025958.586
142288.9476
138717.5751
135235.8419
131841.4983
128532.3508
125306.2611
122161.1444
1520212.243
71712.376
69912.43569
68157.67286
66446.95359
64779.17242
1867531.869
6582790.214
4
Call Provision (Option) – Issuer’s option to payoff bond before scheduled maturity date
as specified in call provision terms.

Callable for purpose of refinancing,

Callable for general business purposes (financial reorganizations, sinking fund
purposes, etc.)
Sinking Fund - Retirement of a portion of a bond issue at regularly scheduled intervals.
Conversion Option – Some bonds are convertible to equity or other assets at the option
of the bond holder.
Put Option – Some bonds can be sold back to the issuer on dates prior to the maturity
date.
Interest rate risk, bond characteristics, level of ytm, direction of change ytm:
n
P 
C
t 1 (1  0.5 * y )
t
 M * (1  0.5 * y)  n
P = invoice price
C = 0.5*c*M
y = yield
M = maturity value or call price or put price
Basis point = 1/100 of 1% , in decimal form 0.0001.
v=0;
ytm = 5%, coupon rate = 5%, maturity = 5 years, semi-annual, change in bps = 50 bps.
1. Increasing ytm will cause a smaller percentage price decline than decreasing ytm
the same number of basis points will cause a percentage price increase.
(Convexity)
5-year maturity, 5%-coupon
(Price ytm = 5%) = $100
(Price ytm = 5.50%) = $97.84, change = -2.16%
(Price ytm = 4.50%) = $102.22, change = +2.22%
2. The percentage change in price caused by a given change in ytm is greater the
smaller the initial ytm. (Convexity)
5-year maturity, 5%-coupon
(Price ytm = 5%) = $100
(Price ytm = 5.50%) = $97.84, change = -2.16%
5
5-year maturity, 5%-coupon
(Price ytm = 10%) = $80.70
(Price ytm = 10.50%) = $79.02, change = -2.08%
Price volatility and cash flows (timing and size):
3. The percentage change in price caused by a given change in ytm is greater the
longer the time to maturity. First bond has 5-year maturity. Second bond has
10-year maturity
5-year maturity, 5%-coupon
(Price ytm = 5%) = $100
(Price ytm = 5.50%) = $97.84, change = -2.16%
10-year maturity, 5%-coupon
(Price  ytm=5%) = $100
(Price ytm =5.5%)=$96.19, change = -3.81%
4. The percentage change in price caused by a given change in ytm is greater the
smaller the coupon rate. First bond has 5% coupon rate. Second bond is a zerocoupon 5-year bond.
5-year maturity, 5%-coupon
(Price ytm = 5%) = $100
(Price ytm = 5.50%) = $97.84, change = -2.16%
5-year maturity, 0%-coupon
(Price ytm=5%) = $78.12
(Price ytm=5.5%)=$76.24, change = -2.41%
Measures of interest rate risk:
Price Value of a Basis Point (PVBP) – a measure of localized price risk.
5-year maturity, 5%-coupon
(Price ytm = 5% )= $100
(Price ytm = 5.01% )= $99.95625, PVBP = $0.04375
10-year maturity, 5%-coupon
(Price ytm = 5% )= $100
(Price ytm = 5.01%) = $99.9221, PVBP = $0.07791
5-year maturity, 0%-coupon
(Price ytm=5%) = $78.12
(Price ytm=5.01%)=$78.08, PVBP = $0.03810
6
Yield value of a 32nd – A measure of localized price risk
1/32nd of $1 = $0.03125
5-year maturity, 5%-coupon
(ytmPrice=$100) = 5%
(ytmPrice = $99.96875) = 5.00714%, YV32 = 0.71423%
10-year maturity, 5%-coupon
(ytmPrice=$100) = 5%
(ytmPrice =$99.96875) = 5.00401%, YV32=0.0041%
5-year maturity, 0%-coupon
(ytmPrice =$78.12)=5%
(ytm Price=$78.0886) = 5.0082%, YV32=0.0082%
5-yr,c=5%,ytm=5%
P0
$100.00
PVBP
$0.0437
YV32
0.0071%
MaCaulay Modified
4.4854
4.3760
10-yr, c=5%,ytm=5%
5-yr,c=0%,ytm=5%
$100.00
$78.12
$0.0779
$0.0381
0.0041%
0.0082%
7.9894
5.0000
7.7946
4.8780
5-yr,c=5%,ytm=10%
$80.70
$0.0339
0.0092%
4.4138
4.2036
From Taylor series expansion of bond price-yield to maturity relationship, P=P(y):
dP  P( y1 )  P( y 0 ) 
P
1 2P
( y1  y 0 ) 
( y1  y 0 ) 2  Rm
2
y
2 y
Modified duration = D*
C
D* 
y
2
1  (1  y)  n(M  C y)  (1  y)
n
P
dP   D*  ( P)  (dy)
Macaulay’s duration = D
n

D
t C
t 1 (1  y )
t

n M
(1  y ) n
P
7
 ( n 1)

1 P
P y
Modified duration = D* = D/(1+y)
Duration is typically measured in years. It is necessary to divide duration measured in
six-month periods by two to restate on an annual basis.
Duration and market risk management:
A bond that has a Macaulay’s duration equal to the investor’s investment horizon is said
to immunize the investor against a change in the level of interest rates. Over the time
period equal to a bond’s Macaulay’s duration a change in value due to a parallel shift in
the yield curve (change in yield to maturity) is exactly balanced by the change in
reinvestment income and bond price change. When yield to maturity increases value falls
but reinvestment income increases. When yield to maturity decreases value increases but
reinvestment income decreases.
If the bond’s Macaulay’s duration is greater than an investor’s investment horizon the
bond will add interest rate risk exposure (reduce reinvestment risk exposure) to the
investor’s position. If the duration of a bond is shorter than the investor’s investment
horizon the bond will add reinvestment risk exposure (reduce interest rate risk exposure).
Duration does not contract on a year-to-year basis as bond approaches maturity.
5% Semi-annual coupon bond, yield-to-maturity 6%
30
25
20
15
10
5
4
3
Maturity
(years)
Macaulay’s 14.77 13.76 12.37 10.47
7.90
4.47
3.67
2.74
2
1
1.93
0.99
To maintain an immunized position an investor must rebalance the position as time
passes.
In addition to the time pattern of duration illustrated in the table, duration rises whenever
a bond pays a coupon. This regular fluctuation in duration is more pronounced for longer
maturity bonds.
Duration is a function of yield to maturity.
yield to
maturity
Macaulay’s
5% Semi-annual coupon bond, 30-year
5%
6%
7%
8%
15.84
14.77
13.73
12.75
9%
10%
11.82
10.96
To maintain an immunized position after substantial change in a bond’s yield to maturity
an investor must rebalance the position to protect against additional changes in yield to
maturity.
8
The duration (D or D*) of a portfolio is equal to the weighted average of the
component bond’s duration’s.
Convexity is determined by the second derivative of the bond price-yield to maturity
relationship :
2P
y
2

2C
y
3
1  (1  y)  2Cny
n
2
(1  y )  ( n 1)  n(n  1)( M  C )  (1  y )  ( n  2)
y
2P
 (4  P) years
y 2
From Taylor series expansion of bond price-yield to maturity relationship:
Convexity = CV =
dP  P( y1 )  P( y 0 ) 
P
1 2P
( y1  y 0 ) 
( y1  y 0 ) 2  Rm
2
y
2 y
1
 (CV )  ( P)  (dy ) 2
2
1
dP   D *  ( P)  (dy )   (CV )  ( P)  (dy ) 2  Rm
2
dP 
The convexity of a portfolio is the weighted average of the convexities of the
portfolio’s assets.
Yield spreads – differences in yield to maturity of financial assets.
Factors that affect yield spreads:
1.
2.
3.
4.
5.
6.
7.
Type of issuer – federal, agency, state, municipal, corporate
Perceived credit worthiness of issuer – credit rating
Collateral, sinking fund provisions, status in default
Term to maturity
Embedded options – call, conversion
Tax treatment of interest
Liquidity of issue.
On-the-run Treasury securities vs. off-the-run Treasury securities
http://www.publicdebt.treas.gov/
Monthly Statement of the Public Debt and Auction information
The term structure of interest rates – the relationship between spot rates and time to
maturity all other factors affecting yield held constant.
9
A spot rate is the yield to maturity on a zero-coupon bond. Each maturity has a
potentially unique spot rate.
The term structure of interest rates produced from the market quotes of Treasury
securities is referred to as the default risk free term structure.
The default risk free term structure is an important input in the pricing of other debt
market instruments such as bank loans, mortgages, corporate bonds, etc.
A U.S. Treasury bond can be thought of as a portfolio of zero coupon payments, each
occurring for a different holding period. The bond pricing equation can be expressed as a
sequence of lump sum payments discounted at spot rates corresponding to the holding
period of each payment.
n
(1) P  
Ct
t 1 (1  0.5  rt )
t

M
(1  0.5  rn ) n
Where r1 = annualized six-month spot rate, r2 = annualized one-year spot rate, …, rn is
the annualized rate applicable to a cash flow occurring at the last payment date of the
bond. (r1, r2, . . . rn) are the default risk free term structure of interest rates.
Value of 5%, 5-year,M=$100 and US Treasury spot rate curves
Feb-80 Feb-85 Feb-90 Feb-95 Feb-96 Feb-97 Sep-99
Value $75.07 $75.88
$85.99 $90.46 $97.90 $95.21 $96.43
ytm 11.73% 11.47% 8.49%
7.31%
5.49%
6.13%
5.83%
Spot Rates
6M
13.47%
8.93%
8.24% 6.34%
4.89%
5.35%
5.09%
12M
13.34%
9.32%
8.28% 6.66%
4.82%
5.57%
5.38%
18M
12.45%
8.96%
8.30% 6.98%
5.82%
5.75%
5.49%
24M
12.18%
9.18%
8.37% 7.05%
5.73%
5.88%
5.59%
30M
12.03%
9.82%
8.32% 7.14%
5.46%
5.96%
5.63%
36M
11.86% 11.12%
8.40% 7.16%
5.62%
6.02%
5.67%
42M
11.86% 10.42%
8.40% 7.26%
5.16%
6.10%
5.76%
48M
11.71% 10.76%
8.42% 7.30%
5.05%
6.16%
5.77%
54M
11.61% 11.03%
8.43% 7.33%
5.44%
6.21%
5.81%
60M
11.69% 11.64%
8.51% 7.33%
5.50%
6.14%
5.85%
10
Sep-03
$108.65
3.12%
1.03%
1.18%
1.44%
1.70%
1.96%
2.21%
2.46%
2.71%
2.97%
3.22%
n
(2) P  
Ct
t 1 (1  0.5  (rt
 RP ))
t

M
(1  0.5  (rn  RP )) n
(2) augments (1) to allow for risk premiums appropriate for the cash flows being priced.
The process for deriving the spot rate curve.
The spot rate curve (term structure) is extracted from prices of US Treasury securities (Tbills, notes, bonds or STRIPS). There are many methods used in practice. Here, I will
illustrate how to extract the spot rate curve from the set of on-the-run Treasury securities.
Steps to construct the spot rate curve from market information.
1. Identify set of Treasury securities to be utilized. This step depends on the
ultimate use of the spot rate curve. If the spot rate curve is produced to provide a
general picture of the term structure all available (longest time to maturity
possible) Treasury price (ytm) data, is required. If the spot rate curve is produced
to value a specific bond Treasury price (ytm) data spanning the cash flows of the
bond are necessary.
Data sources useful for identifying appropriate Treasury securities:
Yield Book: www.yieldbook.com
http://www.publicdebt.treas.gov/
2. Assemble the par coupon curve. First, find the yield to maturity of each of the
basic securities identified in step 1. The par coupon curve is the set of yields to
maturity and times to maturity from each of the Treasury securities identified in
step 1. Second, use linear interpolation (more advanced methods can be used) to
find the par coupon curve for six-month increments.
3. Bootstrap the spot rate curve from the par rate curve.
4. From the spot rate curve interpolate spot rates appropriate for the bond’s cash
flow dates.
Example:
On-the-run Treasury securities (time to maturity and yield to maturity)
settlement 8/1/2003
maturity coupon days years semi
ytm
T-bill 12/26/2003 NA
147 0.403 0.805479 0.94%
T-bill 3/18/2004 NA
230 0.630 1.260274 1.01%
11
T-note
T-note
T-note
T-note
8/31/2004
8/31/2005
8/15/2006
9/15/2008
2.1250%
2.0000%
2.3750%
3.1250%
396
761
1110
1872
1.085 2.169863 1.14%
2.085 4.169863 1.66%
3.041 6.082192 2.13%
5.129 10.25753 3.09%
The par coupon curve (par/coupon rate) is the ytm=coupon rate necessary to make a bond
with given maturity sell at par value. Par/coupon curve constructed using linear
interpolation between (times to maturity, ytm) of on-the-run Treasuries.
annual
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
semi
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
par/coupon
PV(spot)
rate
spot
0.9699%
0.995174 0.9699%
1.1157% 1.0056 0.9944 0.988931 1.1161%
1.3558% 1.0068 0.988931 0.982272 1.1960%
1.6158% 1.0081 0.983452 0.97557 1.2405%
1.8640% 1.0093 0.978009 0.968978 1.2645%
2.1098% 1.0105 0.972604 0.962451 1.2798%
2.3410% 1.0117 0.967235 0.956044 1.2885%
2.5709% 1.0129 0.961901
1.2946%
z1
z2
z3
z4
z5
z6
z7
z8
Constructing spot rate curve from par rate curve:
For semi-annual period = 2, using face value = $1
$1 
0.5 *1.1157%
1  0.5 *1.1157%

(1  0.5 * 0.9699%)
(1  0.5 * z 2 ) 2
For semi-annual period = 3, using face value = $1
0.5 * 1.3558%
0.5 *1.3558%
1  0.5 * 1.3558%


(1  0.5 * 0.9699%) (1  0.5 * 1.1161%) 2
(1  0.5 * z 3 ) 3
In general spot rates are extracted from the par coupon curve through an algebraic
process.
$1 
12





1 c
z n  2  



n

1
1


1

c


i 1  (1  z i ) i


2














1/ n





 1





where c = 0.5*coupon rate
Using spot rate curve to find the value of a 4% coupon, semi-annual pay bond with
payment dates 1/15, 7/15. Settlement date 8/01/03.
Bond
RP
Coupon
M
P
ytm
0.50%
4%
$100
106.56
1.7747%
Payment dates
1/15/2004
7/15/2004
1/15/2005
7/15/2005
1/15/2006
7/15/2006
days
167
349
533
714
898
1079
cash
years semi zero
z+RP
flow
0.458 0.915 0.9472% 1.4472% $2.0
0.956 1.912 1.1033% 1.6033% $2.0
1.460 2.921 1.1897% 1.6897% $2.0
1.956 3.912 1.2366% 1.7366% $2.0
2.460 4.921 1.2626% 1.7626% $2.0
2.956 5.912 1.2785% 1.7785% $102.0
sum
PV
1.986848
1.969694
1.951459
1.933483
1.915488
96.79841
106.5554
Forward rates
Forward Period:
Time Line in 6MO increments:
0-------------j-------------k
The forward period (j,k) begins j periods from today[0] and ends k periods from today.
The length of the forward period is k-j.
13
A forward rate is the rate earned during the forward period that makes the total return
from a sequential loan at zj rolled over at the forward rate, jfk, equivalent to a single loan
for k periods at zk.
Calculating a annualized forward rate from semi-annual spot rates:
Time Line in 6MO increments:
0-------------j-------------k
(1  0.5  z j ) j  (1  0.5 j f (k  j ) ) (k  j )  (1  0.5  z k ) k

 1  0.5  z k k
j f ( k  j )  2  
j
 1  0.5  z j



1

 (k  j ) 

 1




For these notes I will forego the algebraic adjustment accounting for semi-annual
compounding. The final result will be multiplied by 2.
zi = ½ x spot rate
ifj = ½ x forward rate
From the definition of a forward rate:
(1  z 2 ) 2  (1  z1 )  (11f1 )
and
(1  z3 ) 3  (1  z 2 ) 2  (1 2 f1 )
Substituting from the first expression into the second
(1  z3 ) 3  (1  z1 )  (11f1 )  (1 2 f1 )
Solving for the rate z3:


z3  2  3 (1  z1)  (11f1)  (1 2 f1) 1
Another way to write the same thing

z3  2  (1  z1 )  (11f1)  (1 2 f1)
1
3

1
Long rates (z3) are a geometric average of the short rate (z1) and forward rates (1f1 , 2f1).
14
Lending Strategies (a similar discussion concerning borrowing strategies is also
possible):
Consider the following setting:
Semi
1
Spot
z1
2
z2
Rate forward period
3%
1f1 = 5%
4%
In this circumstance there are two possible lending strategies for a two period loan.
Strategy #1: Buy two period zero coupon bond
Total return form Strategy #1 is 4%. This equivalent to earning z1 = 3% over the first
semi-annual period and 1f1 = 5% during the forward period. There is no interest rate risk
in Strategy #1.
Strategy #2 Sequential lending
i.e. purchase one-period zero coupon bond planning to roll over proceeds into a second
one-period bond at the end of the first period. The rate to be earned in the second period
is a random variable. Label this rate 1 ~
z1 . Interest rate risk is present in Strategy #2. The
realized rate during the second period could be less than 5%.
Suppose the decision maker expects that the one-period rate during the forward period
will be 6%, i.e. E0 (1~
z1 )  6% .
Which Strategy will the decision maker choose?
How will the decision maker’s choice be reflected in the price of zero coupon bonds?
If E(z)s equal forward rates then total return is equal for all investment strategies
spanning an investment horizon.
If E(z)s do not equal forward rates then total returns are not equal for all investment
strategies spanning an investment horizon.
Conditioned on E(z)s and decision maker’s risk tolerance, a decision maker may select a
strategy that produces an interest rate risk exposure if the expected return from the
investment strategy or expected cost from borrowing strategy is sufficient to compensate
for the additional interest rate risk exposure.
Term structure theories attempt to answer the question
15
“What determines the “shape” of the term structure?”
Ultimately the term structure is determined by the factors that affect the supply and
demand for fixed income securities of different maturities. The factors determine the
relative price of securities of different maturities and hence the relative yields.
Term structure theories are important in that they help market participants anticipate how
economic phenomena will affect the level and slope of the term structure.
Transacting in securities with time to maturity different than the participant’s investment
(borrowing) horizon exposes the market participant to either price or reinvestment risk.
The term structure theories discussed in the text (pages 111-116) are predicated on
different assumptions concerning market participant’s interest rate risk tolerance.
Pure expectations theory;
 Assumption – market participants are indifferent to the risk of trading in securities
whose maturity is different than the participant’s investment (borrowing) horizon.
 Outcome – Zero coupon bonds prices (spot rates) will adjust until forward rates are
equal to participant’s expectations of spot rates in future periods.
 Implication 1 – The shape of the term structure is determined by market participant’s
expectations of short rates in future periods.
jfk
= E0(jzk)
Implication 2 - The term structure will change level and/or shape when market
participant’s expectations of short rates change.
Implication 3 – Long-term spot rates are geometric averages of the shortest term spot
rate and market participants expectations of spot rates.


1
z3  2  (1  z1)  (1  E0 (1~
z2 ))  (1  E0 ( 2 ~
z3 ))  3 1
The Liquidity Theory, Preferred Habitat Theory and the Segmented Markets theory differ
in the assumed response of market participants to the risk of transacting in securities with
time to maturity different than the participant’s investment (borrowing) horizon.
These theories posit that market participants are averse to the risk of transacting in
securities with time to maturity that does not match their investment (borrowing) horizon.
The Liquidity Theory states that to get lenders to lend long term borrowers will have to
offer a higher expected return to overcome the lender’s loss of liquidity. Hence the price
of zero-coupon bonds reflects not only expected short rates as in the expectations
hypothesis, but also reflects the necessary premium required to entice lenders.
Liquidity theory;
16



Assumption – market participants are averse to the risk of trading in securities whose
maturity is different than the participant’s investment (borrowing) horizon.
Outcome – Zero coupon bond prices (spot rates) will adjust until forward rates reflect
participant’s expectations of short rates in future periods and the necessary liquidity
premiums.
Implication – The shape of the term structure is determined by market participant’s
expectations of short rates in future periods and the necessary liquidity premiums.
jfk
= E0(jzk) +jk
The term structure will change level and/or shape when market participant’s
expectations of short rates change and/or when liquidity premiums change.
Preferred Habitat theory;
 Assumption – market participants are averse to the risk of trading in securities whose
maturity is different than the participant’s investment (borrowing) horizon.
 Outcome – Zero coupon bond prices (spot rates) will adjust until forward rates reflect
participant’s expectations of short rates in future periods and the necessary premiums
to eliminate excess supply/excess demand for each segment of the term structure.
 Implication – The shape of the term structure is determined by market participant’s
expectations of short rates and the necessary premiums. Premiums can be either
positive or negative and need not be uniformly increasing.
jfk
= E0(jzk) +jk
The term structure will change level and/or shape when market participant’s
expectations of short rates change and/or when premiums change.
Segmented Markets theory;
 Assumption – market participants are extremely averse to the risk of trading in
securities whose maturity is different than the participant’s investment (borrowing)
horizon. In fact market participants will only transact in securities that have time to
maturity that matches their investment (borrowing) horizon.
 Outcome – Zero coupon bond prices (spot rates) reflect only the supply and demand
for securities. The borrowers who have a horizon equal to the time to maturity
determine the supply for a given time to maturity. The lenders who have a horizon
equal to the time to maturity determine the demand for a given time to maturity.
 Implication – The shape of the term structure is independent of expectations of future
interest rates and risk premiums. The term structure will change level and/or shape
when factors affecting either supply or demand in a maturity segment change.Yield
Spreads:
Money markets:
17
LIBOR – London interbank offer rate – rate at which US dollar deposits are offered on
loan. Deposits outside US banking system (not subject to reserve requirements and other
regulatory costs of US banking system)
LIBID – London interbank bid rate – rate bid for US dollars dollar deposits in the
interbank market.
LIBOR and LIBID reflect the credit rating (usually AA) of the money center banks
operating in this market.
LIBOR/LIBID interest cost/revenue computed using an actual/360 day count convention.
Treasury bills:
Treasury bills are pure discount securities issued by US Treasury. Treasury bills are sold
with an initial maturity of one-month, 3-months, 6-months and one-year. Cash
management bills are also sold very short original maturity.
In the secondary market, Treasury bills are quoted on a bank discount basis using a 360
day year. A quote of 3% indicates the price of the bill is calculated as a 3% discount
from face value.
Given, M=$1,000,000 and t = 82 a quote of Yd = 3% can be used to find the purchase
price for the bill:
D = dollar discount = $1,000,000 * 0.03 * 82/360 = $6,833.33
P = M – D = $1,000,000 – 6,833.33 = 993,166.67
To state the yield on a bill in a comparable format to the yield on Treasury notes and
bonds it is necessary to account for the discount interest of the bill and the 360 day
money-market convention.
For bills with less than 182 days till maturity
BEY =
D 365
$6,833.33 365

=
= 3.063%

P
t
$993,166.67 82
To calculate bond equivalent yield, for bills with maturity greater than 182 days it is also
necessary to account for the fact that bills pay no coupons but notes and bonds do.
Given, M=$1,000,000 and t = 200 a quote of Yd = 3% can be used to find the purchase
price for the bill:
D = dollar discount = $1,000,000 * 0.03 * 200/360 = $16,666.67
P = M – D = $1,000,000 – $16,666.67 = 983,333.33
18
BEY =
 2 t
2
365
2 t
365t 2  365
 11  M 
P
2 t
1
365
= 3.089%
To compare the yield of a Treasury bill with the yield offered on non-discount money
market instruments such as CDs the return on a Treasury bill must be adjusted to account
for the discount interest of the bill. The CD equivalent (money market equivalent) yield
on a Treasury bill can be found simply from the discount rate (all bill maturities)
CDEY =
360  Yd
360  t  Yd
For the 82 day bill examined above: CDEY =
360  0.03
 3.021%
360  82  0.03
Treasury auctions – Treasury bills notes and bonds are sold originally by the Treasury
through regularly scheduled auctions.
Details concerning the Treasury auctions (announcements, results, etc.) may be found at
http://www.publicdebt.treas.gov/
Bills are generally auctioned on a Monday. Buyers receive the bills the following
Thursday.
Bills auctioned in October 2003:
Recent Treasury Bill Auction Results
Term
Issue
Date
Maturity Discount Investment
Date
Rate % Rate %
Price
Per
$100
CUSIP
28-DAY 10-23-2003 11-20-2003
0.895
0.916
99.930 912795NX5
91-DAY 10-23-2003 01-22-2004
0.920
0.939
99.767 912795PG0
182-DAY 10-23-2003 04-22-2004
1.015
1.037
99.487 912795PV7
28-DAY 10-16-2003 11-13-2003
0.870
0.889
99.932 912795NW7
91-DAY 10-16-2003 01-15-2004
0.905
0.923
99.771 912795PF2
182-DAY 10-16-2003 04-15-2004
0.985
1.006
99.502 912795PU9
28-DAY 10-09-2003 11-06-2003
0.855
0.863
99.934 912795NV9
91-DAY 10-09-2003 01-08-2004
0.920
0.939
99.767 912795PE5
19
182-DAY 10-09-2003 04-08-2004
0.995
1.017
99.497 912795PT2
12-DAY 10-03-2003 10-15-2003
0.920
0.946
99.969 912795QH7
28-DAY 10-02-2003 10-30-2003
0.845
0.863
99.934 912795NU1
92-DAY 10-02-2003 01-02-2004
0.935
0.953
99.761 912795PD7
182-DAY 10-02-2003 04-01-2004
1.005
1.027
99.492 912795PS4
20