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Chapter 18 - The Analysis and Valuation of Bonds
The Fundamentals of Bond ValuationThe present-value model
n
tn
p
tt
m i
P
i
CP
2
12)21()21(
2
Where:Pm=the current market price of the bondn = the number of years to maturityCi = the annual coupon payment for bond ii = the prevailing yield to maturity for this bond issuePp=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 ModelThe expected yield on the bond may be
computed from the market price
Where:
i = the discount rate that will discount the cash flows to equal the current market price of the bond
n
tn
p
ti
m i
P
i
CP
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Computing Bond YieldsYield Measure PurposeNominal 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
Promised yield to call Measures expected rate of return for bond held to first call date
Realized (horizon) yield 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.
Nominal Yield
Measures the coupon rate that a bond investor receives as a percent of the bond’s par value
Current YieldSimilar to dividend yield for stocksImportant 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 maturitySolve for i that will equate the current price to all cash flows from the bond to maturity, similar to IRR
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p
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m i
P
i
CP
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12)21()21(
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Computing the Promised Yield to Maturity
Two methods
• Approximate promised yield– Easy, less accurate
• Present-value model– More involved, more accurate
Approximate Promised Yield
Coupon + Annual Straight-Line Amortization of Capital Gain or Loss
Average Investment
2
APYmp
mpi
PPn
PPC
=
Present-Value Model
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tn
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ti
m i
P
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Promised Yield to CallApproximation
• May be less than yield to maturity
• Reflects return to investor if bond is called and cannot be held to maturity
2mc
mct
PPnc
PPC
AYC
Where:
AYC = approximate yield to call (YTC)
Pc = call price of the bond
Pm = market price of the bond
Ct = annual coupon payment
nc = the number of years to first call date
Promised Yield to CallPresent-Value Method
Where:
Pm = market price of the bond
Ci = annual coupon payment
nc = number of years to first call
Pc = call price of the bond
ncc
nc
tt
im i
P
i
CP
2
2
1 )1()1(
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Realized Yield Approximation
2
PPhp
PPC
ARYf
fi
Where:
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 YieldPresent-Value Method
hp
fhp
tt
tm i
P
i
CP
2
2
1 )21()21(
2/
Calculating Future Bond Prices
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
hpn
phpn
tt
if i
P
i
CP
22
22
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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 long-maturity 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
1BPB
EPB
Where:
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 behaviors1. 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
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
price
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)1(
)1(
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1
1
1
n
tt
n
tt
t
n
tt
t CPVt
i
Ci
tC
D
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
m
YTM1
durationMacaulay duration modified
Where:
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
iDP
P
mod100
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-value-weighted 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 toMaturity 0.02 0.04 0.06 0.08
1 0.995 0.990 0.985 0.9815 4.756 4.558 4.393 4.254
10 8.891 8.169 7.662 7.28620 14.981 12.980 11.904 11.23250 19.452 17.129 16.273 15.829
100 17.567 17.232 17.120 17.064
8 17.167 17.167 17.167 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
YTM100 mod
DP
P
Bond Convexity
• Modified duration is a linear approximation of bond price change for small changes in market yields
• Price changes are not linear, but a curvilinear (convex) function
iDP
P
mod100
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
For small changes this will give a good estimate, but this is a linear estimate on the tangent line
Pdi
dP
D mod
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
Pdi
Pd2
2
Convexity
