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Chapter 16The Analysis and

Valuation of Bonds

Innovative Financial InstrumentsDr. A. DeMaskey

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The Fundamentals of Bond ValuationThe present-value model

n

tnt

t

iM

iCV

2

120 )21()21(

2

Where:V0 = 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 issueM = the par value of the bond

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The Yield ModelThe expected yield on the bond may be computed from the current market price V0.

Where i is the discount rate that will discount the cash flows to equal the current market price of the bond.

n

tnt

t

iM

iCV

2

120 )21()21(

2

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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 (holding period) 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.

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Computing Bond YieldsYield Measure CalculationNominal Yield

Current yield

Promised yield to maturity

Promised yield to call

Realized (holding period) yield

Coupon Bond

Discount Bond

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Calculating Future Bond Prices

Where:Vf = estimated future price of the bond

Ci = annual coupon payment

n = number of years to maturityhp = holding period of the bond in yearsi = expected semiannual rate at the end of the holding period

hpn

hpn

tt

if i

Mi

CV 22

22

1 )21()21(2/

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Yield Adjustments for Tax-Exempt Bonds

Where:T = amount and type of tax exemption

T-1return annualFETY

The fully taxable equivalent yield (FTEY) takes into account the bond’s tax exemptions:

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Holding Period Yield With Differential Reinvestment Rates

• Estimate potential reinvestment rates• Calculate the ending wealth position

– Find the ending value of the coupons– Find the future price of the bond

• Compute the estimated holding period yield by equating the initial investment to the ending wealth position

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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

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Loanable Funds Theory• Interest rates are the price for loanable funds• Supply of loanable funds

– Federal Reserve– Domestic saving– Foreign saving

• Demand for loanable funds– Government– Corporations– Consumers

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Fundamental Determinants of Interest Rates

• i = f (Economic Forces + Issue Characteristics)• 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: collateral, call feature, sinking fund– foreign bond risk: exchange rate risk and country risk

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Term Structure of Interest Rates

• Maturity of a security

• Relationship between yields and maturity

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Types of Yield Curves

• Normal or upward sloping yield curve

• Inverted or downward sloping yield curve

• Horizontal or flat yield curve

• Humped yield curve

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Theories of Term Structure of Interest Rates

• Expectations hypothesis

• Liquidity preference hypothesis

• Segmented market hypothesis

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Expectations Hypothesis• Investor expectations of future interest rates• Long-term interest rates are the geometric

average of current and future 1-year interest rates

• Shape of yield curve– Rising– Declining– Horizontal– Humped

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Liquidity Preference Hypothesis

• Preference for short-term rather than long-term securities

• Shape of yield curve– Upward sloping– Downward sloping

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Segmented Markets Hypothesis

• Strong preference for securities of a particular maturity– Preferred habitat, or institutional, or hedging

pressure theory• Shape of yield curve

– Upward sloping– Downward sloping

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Trading Implications of the Term Structure

• The shape of the yield curve alone may contain information that is useful in predicting interest rates

• A downward sloping yield curve may indicate strong expectations of falling interest rates

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Yield Spreads• Difference in promised yields between two issues

or segments of the market• There are four major yield spreads:

– Segments: government bonds and corporate bonds– Sectors: high and low grade municipal bonds– Coupons or seasoning within a segment or sector– Maturities within a given market segment or sector

• Magnitudes and direction of yield spreads can change over time

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Bond Price Volatility• Interest rate sensitivity refers to the effect that yield

changes have on the price and rate of return for different bonds.

• A bond’s percentage price change is:

• The market price of a bond is a function of its par value (M), coupon (PMT), time to maturity (N), and prevailing market rate (i).

1bond of price Beginning

bond of price Ending%PB

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Malkiel Bond Theorems• Bond prices move inversely to bond yields (interest rates).• For a given change in yields, longer maturity bonds post

larger price changes. Thus, bond price volatility is directly related to maturity.

• Price volatility increases at a diminishing rate as term to maturity increases.

• Price movements resulting from equal absolute increases or decreases in yield are not symmetrical.

• Higher coupon bonds show smaller percentage price fluctuations for a given change in yield. Thus, bond price volatility is inversely related to coupon.

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Price-Yield Relationships

• The price volatility of a bond for a given change in yield is affected by:– The maturity effect– The coupon effect– The yield level effect

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Trading Strategies

• If a decline in interest rates is expected:– Long-maturity bond with low coupons– 30-year zero-coupon bond

• If an increase in interest rates is expected:– Short-maturity bond with high coupons– 1-year 8% coupon bond

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Duration

• Price volatility of a bond varies– inversely with its coupon and – directly with its term to maturity

• A composite measure, which considers both variables would be beneficial.

• Duration is a measure of the bond’s interest rate sensitivity.

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Duration Measures

• Macaulay Duration

• Modified Duration

• Effective Duration

• Empirical Duration

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The Macaulay Duration

price

)(

)1(

)1()(

1

1

1

n

tt

n

tt

t

n

tt

t CPVt

iC

itC

D

Developed by Frederick R. Macaulay, 1938Where: t = time period in which the coupon or principal payment occursCt = interest or principal payment that occurs in period t i = yield to maturity on the bond

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Characteristics of Duration• The duration of a coupon bond 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 relationship between term to maturity

and duration, but duration increases at a decreasing rate with maturity.

• There is an inverse relationship between YTM and duration.• Sinking funds and call provisions can have a dramatic effect

on a bond’s duration.

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Modified Duration

• An adjusted measure of duration can be used to approximate the price volatility of a bond.

• Modified duration is defined as:

mYTM1

DDmod

Where: m = number of payments a year YTM = nominal YTM

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Modified 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 times modified duration.

iDPP

mod100

Where:P = change in price for the bondP = beginning price for the bondDmod = the modified duration of the bondi = yield change in basis points divided by 100

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Trading Strategies Using Modified 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.

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Bond Convexity

• Modified duration is a linear approximation of bond price changes for small changes in market yields.

• Price changes are not linear, but a curvilinear (convex) function.

iDPP

mod100

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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.

• Modified duration is the percentage change in price for a nominal change in yield

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Desirability of Convexity

• The greater the convexity of a bond, the better its price performance.

• Based on the convexity of the price-yield relationship:– As yield increases, the rate at which the price of the

bond declines becomes slower– As yield declines, the rate at which the price of the

bond increases becomes faster

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Modified Duration

PdidP

D mod

• For small interest rate changes, this will give a good estimate.

• For larger changes, it will underestimate price increases and overestimate price decreases.

• This misestimate arises because the modified duration line is a linear estimate of a curvilinear relationship.

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Determinants of Convexity• Convexity is a measure of the curvature of the

price-yield relationship.• Since modified duration is the slope of the curve

at a given yield, convexity indicates changes in duration.

• Thus, convexity is the second derivative of price with respect to yield (d2P/di2) divided by price.

• Specifically, convexity is the percentage change in dP/di for a given change in yield.

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Determinants of Convexity

• The lower the coupon, the higher the convexity (-)

• The longer the maturity, the higher the convexity (+)

• The lower the yield to maturity, the higher the convexity (-)

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Modified Duration-Convexity Effects

• Changes in a bond’s price resulting from a change in yield are due to:– Bond’s modified duration– Bond’s convexity

• The relative effect of these two factors depends on the characteristics of the bond (its convexity) and the size of the yield change.

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Duration and Convexity for Callable Bonds

• The call provision is an example of an embedded option.• Option-adjusted duration is an estimate of duration based on

the probability that the bond will be called.– If interest rates > coupon rate, call is unlikely – If interest rates < coupon rate, call is likely

• A callable bond is a combination of a noncallable bond plus a call option that was sold to the issuer.

• The option has a negative value to the bond investor.• Thus, any increase in value of the call option reduces the

value of the callable bond.

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Option-Adjusted Duration

• Based on the probability that the issuing firm will exercise its call option– Duration of the non-callable bond

– Duration of the call option

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Convexity of Callable Bond

• Noncallable bond has positive convexity

• Callable bond has negative convexity

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Limitations of Macaulay and Modified Duration

• Percentage change estimates using modified duration are good only for small-yield changes.

• Difficult to determine the interest-rate sensitivity of a portfolio of bonds when there is a change in interest rates and the yield curve experiences a nonparallel shift.

• Callable bond duration depends on market conditions.

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Effective and Empirical Duration

• Effective Duration– A direct measure of the interest rate sensitivity

of a bond where it is possible to estimate price changes for an asset using a valuation model.

• Empirical Duration– Actual percent change for an asset in response

to a change in yield during a specified time period.