Chapter 18 - The Analysis and
Valuation of Bonds
The Fundamentals of Bond Valuation
The present-value model
2n
t
m
t
t 1
Pp
C 2
P 

2n
(1  i 2) (1  i 2)
Where:
Pm=the current market price of the bond
n = the number of years to maturity
Ci = the annual coupon payment for bond i
i = the prevailing yield to maturity for this bond issue
Pp=the par value of the bond
The Fundamentals of Bond
Valuation
• If yield < coupon rate, bond will be priced
at a premium to its par value
• If yield > coupon rate, bond will be priced
at a discount to its par value
• Price-yield relationship is convex (not a
straight line)
The Yield Model
The expected yield on the bond may be
computed from the market price
Pp
Ci 2
Pm  

t
2n
(1  i 2)
t 1 (1  i 2)
2n
Where:
i = the discount rate that will discount the cash flows to
equal the current market price of the bond
Computing Bond Yields
Yield Measure
Purpose
Nominal Yield
Measures the coupon rate
Current yield
Measures current income rate
Promised yield to maturity
Measures expected rate of return for bond held
to maturity
Measures expected rate of return for bond held
to first call date
Measures expected rate of return for a bond
likely to be sold prior to maturity. It considers
specified reinvestment assumptions and an
estimated sales price. It can also measure the
actual rate of return on a bond during some past
period of time.
Promised yield to call
Realized (horizon) yield
Nominal Yield
Measures the coupon rate that a bond
investor receives as a percent of the bond’s
par value
Current Yield
Similar to dividend yield for stocks
Important to income oriented investors
CY = Ci/Pm
where:
CY = the current yield on a bond
Ci = the annual coupon payment of bond i
Pm = the current market price of the bond
Promised Yield to Maturity
• Widely used bond yield figure
• Assumes
– Investor holds bond to maturity
– All the bond’s cash flow is reinvested at the
computed yield to maturity
Pp
Ci 2
Pm  

t
2n
(1  i 2)
t 1 (1  i 2)
2n
Solve for i that will
equate the current price
to all cash flows from
the bond to maturity,
similar to IRR
Computing the
Promised Yield to Maturity
Two methods
• Approximate promised yield
– Easy, less accurate
• Present-value model
– More involved, more accurate
Approximate Promised Yield
APY 
Ci 
Pp  Pm
n
Pp  Pm
2
=
Coupon + Annual Straight-Line Amortization of Capital Gain or Loss
Average Investment
Present-Value Model
Pp
Ci 2
Pm  

t
2n
(1  i 2)
t 1 (1  i 2)
2n
Promised Yield to Call
Approximation
• May be less than yield to maturity
• Reflects return to investor if bond is called
and cannot be held to maturity
Pc  Pm
C

t
P = call price of the bond
nc
AYC

P = market price of the bond
Pc  Pm
C = annual coupon payment
nc = the number of years to first call date
2
Where:
AYC = approximate yield to call (YTC)
c
m
t
Promised Yield to Call
Present-Value Method
2 nc
Ci / 2
Pc
Pm  

t
2 nc
(1  i )
t 1 (1  i )
Where:
Pm = market price of the bond
Ci = annual coupon payment
nc = number of years to first call
Pc = call price of the bond
Realized Yield Approximation
ARY 
Ci 
Where:
Pf  P
hp
Pf  P
2
ARY = approximate realized yield to call (YTC)
Pf = estimated future selling price of the bond
Ci = annual coupon payment
hp = the number of years in holding period of the bond
Realized Yield
Present-Value Method
2 hp
Pf
Ct / 2
Pm  

t
2 hp
(1  i 2)
t 1 (1  i 2)
Calculating Future Bond Prices
Pf 
2 n  2 hp

t 1
Pp
Ci / 2

t
2 n  2 hp
(1  i 2) (1  i 2)
Where:
Pf = estimated future price of the bond
Ci = annual coupon payment
n = number of years to maturity
hp = holding period of the bond in years
i = expected semiannual rate at the end of the holding period
What Determines Interest Rates
• Inverse relationship with bond prices
• Forecasting interest rates
• Fundamental determinants of interest rates
i = RFR + I + RP
where:
– RFR = real risk-free rate of interest
–
I = expected rate of inflation
– RP = risk premium
What Determines Interest Rates
• Effect of economic factors
–
–
–
–
real growth rate
tightness or ease of capital market
expected inflation
or supply and demand of loanable funds
• Impact of bond characteristics
–
–
–
–
credit quality
term to maturity
indenture provisions
foreign bond risk including exchange rate risk and country
risk
What Determines Interest Rates
•
•
•
•
•
Term structure of interest rates
Expectations hypothesis
Liquidity preference hypothesis
Segmented market hypothesis
Trading implications of the term structure
Expectations Hypothesis
• Any long-term interest rate simply
represents the geometric mean of current
and future one-year interest rates expected
to prevail over the maturity of the issue
Liquidity Preference Theory
• Long-term securities should provide higher
returns than short-term obligations because
investors are willing to sacrifice some
yields to invest in short-maturity obligations
to avoid the higher price volatility of longmaturity bonds
Segmented-Market Hypothesis
• Different institutional investors have
different maturity needs that lead them to
confine their security selections to specific
maturity segments
Trading Implications of the Term
Structure
• Information on maturities can help you
formulate yield expectations by simply
observing the shape of the yield curve
What Determines the
Price Volatility for Bonds
Bond price change is measured as the
percentage change in the price of the bond
Where:
EPB
1
BPB
EPB = the ending price of the bond
BPB = the beginning price of the bond
What Determines the
Price Volatility for Bonds
Four Factors
1. Par value
2. Coupon
3. Years to maturity
4. Prevailing market interest rate
What Determines the
Price Volatility for Bonds
Five observed behaviors
1. Bond prices move inversely to bond yields (interest rates)
2. For a given change in yields, longer maturity bonds post larger
price changes, thus bond price volatility is directly related to
maturity
3. Price volatility increases at a diminishing rate as term to maturity
increases
4. Price movements resulting from equal absolute increases or
decreases in yield are not symmetrical
5. Higher coupon issues show smaller percentage price fluctuation for
a given change in yield, thus bond price volatility is inversely
related to coupon
The Duration Measure
• Since price volatility of a bond varies
inversely with its coupon and directly with
its term to maturity, it is necessary to
determine the best combination of these two
variables to achieve your objective
• A composite measure considering both
coupon and maturity would be beneficial
The Duration Measure
n
C t (t )

t
t 1 (1  i )
D n

Ct

t
t 1 (1  i )
n
 t  PV (C )
t
t 1
price
Developed by Frederick R. Macaulay, 1938
Where:
t = time period in which the coupon or principal payment occurs
Ct = interest or principal payment that occurs in period t
i = yield to maturity on the bond
Characteristics of Duration
• Duration of a bond with coupons is always less
than its term to maturity because duration gives
weight to these interim payments
– A zero-coupon bond’s duration equals its maturity
• There is an inverse relation between duration and
coupon
• There is a positive relation between term to
maturity and duration, but duration increases at a
decreasing rate with maturity
• There is an inverse relation between YTM and
duration
• Sinking funds and call provisions can have a
dramatic effect on a bond’s duration
Modified Duration and Bond Price
Volatility
An adjusted measure of duration can be used
to approximate the price volatility of a bond
Macaulay duration
modified duration 
YTM
1
Where:
m
m = number of payments a year
YTM = nominal YTM
Duration and Bond Price Volatility
• Bond price movements will vary proportionally with
modified duration for small changes in yields
• An estimate of the percentage change in bond prices
equals the change in yield time modified duration
P
 100   Dmod  i
P
Where:
P = change in price for the bond
P = beginning price for the bond
Dmod = the modified duration of the bond
i = yield change in basis points divided by 100
Trading Strategies Using Duration
• Longest-duration security provides the maximum price
variation
• If you expect a decline in interest rates, increase the average
duration of your bond portfolio to experience maximum
price volatility
• If you expect an increase in interest rates, reduce the
average duration to minimize your price decline
• Note that the duration of your portfolio is the market-valueweighted average of the duration of the individual bonds in
the portfolio
Bond Duration in Years for Bonds Yielding
6 Percent Under Different Terms
COUPON RATES
Years to
Maturity
8
1
5
10
20
50
100
0.02
0.04
0.06
0.08
0.995
4.756
8.891
14.981
19.452
17.567
17.167
0.990
4.558
8.169
12.980
17.129
17.232
17.167
0.985
4.393
7.662
11.904
16.273
17.120
17.167
0.981
4.254
7.286
11.232
15.829
17.064
17.167
Source: L. Fisher and R. L. Weil, "Coping with the Risk of Interest Rate Fluctuations:
Returns to Bondholders from Naïve and Optimal Strategies," Journal of Business 44, no. 4
(October 1971): 418. Copyright 1971, University of Chicago Press.
Bond Convexity
• Equation 19.6 is a linear approximation of
bond price change for small changes in
market yields
P
 100   D mod  YTM
P
Bond Convexity
• Modified duration is a linear approximation
of bond price change for small changes in
market yields
P
 100   Dmod  i
P
• Price changes are not linear, but a
curvilinear (convex) function
Price-Yield Relationship for Bonds
• The graph of prices relative to yields is not a
straight line, but a curvilinear relationship
• This can be applied to a single bond, a portfolio of
bonds, or any stream of future cash flows
• The convex price-yield relationship will differ
among bonds or other cash flow streams
depending on the coupon and maturity
• The convexity of the price-yield relationship
declines slower as the yield increases
• Modified duration is the percentage change in
price for a nominal change in yield
Modified Duration
Dmod
dP
di

P
For small changes this will give a good
estimate, but this is a linear estimate on the
tangent line
Determinants of Convexity
The convexity is the measure of the curvature
and is the second derivative of price with
resect to yield (d2P/di2) divided by price
Convexity is the percentage change in dP/di
for a given change in yield
2
d P
2
di
Convexity 
P