CHAPTER ELEVEN

Bond Yields and Prices

CHAPTER ELEVEN

Bond Yields and Prices

Cleary / Jones

Investments: Analysis and Management

Learning ObjectivesLearning ObjectivesLearning ObjectivesLearning Objectives

To calculate the price of a bondTo calculate the price of a bond To explain the bond valuation processTo explain the bond valuation process To calculate major bond yield To calculate major bond yield

measures, including yield to maturity, measures, including yield to maturity, yield to call, and horizon returnyield to call, and horizon return

To account for changes in bond pricesTo account for changes in bond prices To explain and apply the concept of To explain and apply the concept of

durationduration

Bond Valuation PrincipleBond Valuation PrincipleBond Valuation PrincipleBond Valuation Principle

Intrinsic valueIntrinsic value– Is an estimated value Is an estimated value – Present value of the expected future Present value of the expected future

cash flowscash flows– Required to compute intrinsic valueRequired to compute intrinsic value

Expected future cash flowsExpected future cash flows Timing of expected cash flowsTiming of expected cash flows Discount rate, or required rate of return Discount rate, or required rate of return

by investorsby investors

Bond ValuationBond ValuationBond ValuationBond Valuation Value of a coupon bond with semi-Value of a coupon bond with semi-

annual payments:annual payments:

2n

2n

1tt

t

/2)r(1

MV

r/2)(1

/2C V

Biggest problem is determining the discount rate or required yield Required yield is the current market rate earned on comparable bonds with same maturity and credit risk

Interest RatesInterest RatesInterest RatesInterest Rates

Rates and basis pointsRates and basis points– 100 basis points are equal to one 100 basis points are equal to one

percentage pointpercentage point Short-term riskless rateShort-term riskless rate

– Provides foundation for other ratesProvides foundation for other rates– Approximated by rate on Treasury BillsApproximated by rate on Treasury Bills– Other rates differ because ofOther rates differ because of

Maturity differentialsMaturity differentials Security risk premiumsSecurity risk premiums

Interest RatesInterest RatesInterest RatesInterest Rates

Maturity differentialsMaturity differentials– Term structure of interest ratesTerm structure of interest rates

Accounts for the relationship between time Accounts for the relationship between time and yield for bonds the same in every and yield for bonds the same in every other respectother respect

Risk premiumRisk premium– Yield spread or yield differentialYield spread or yield differential– Associated with issuer’s particular Associated with issuer’s particular

situationsituation

Determinants of Interest Determinants of Interest RatesRates

Determinants of Interest Determinants of Interest RatesRates

Real rate of interestReal rate of interest– Rate that must be offered to persuade Rate that must be offered to persuade

individuals to save rather than individuals to save rather than consumeconsume

– Rate at which real capital physically Rate at which real capital physically reproduces itselfreproduces itself

Nominal interest rate Nominal interest rate – Function of the real rate of interest and Function of the real rate of interest and

expected inflation premiumexpected inflation premium

Market interest rates on riskless Market interest rates on riskless debt debt real rate +expected real rate +expected inflationinflation– Fisher HypothesisFisher Hypothesis

Real rate estimates obtained by Real rate estimates obtained by subtracting the expected inflation subtracting the expected inflation rate from the observed nominal rate from the observed nominal raterate

Determinants of Interest Determinants of Interest RatesRates

Determinants of Interest Determinants of Interest RatesRates

Yield to MaturityYield to MaturityYield to MaturityYield to Maturity

Yield to maturityYield to maturity– Rate of return on bonds most often Rate of return on bonds most often

quoted for investorsquoted for investors– Promised compound rate of return Promised compound rate of return

received from a bond purchased at the received from a bond purchased at the current market price and held to maturitycurrent market price and held to maturity

– Equates the present value of the expected Equates the present value of the expected future cash flows to the initial investmentfuture cash flows to the initial investment

Similar to internal rate of returnSimilar to internal rate of return

Yield to MaturityYield to MaturityYield to MaturityYield to Maturity

Solve for YTM (semi-annual Solve for YTM (semi-annual coupons):coupons):

2t

2n

1tt

t

YTM/2)(1MV

YTM/2)(1

/2CP

Investors earn the YTM if the bond Investors earn the YTM if the bond is held to maturity and all coupons is held to maturity and all coupons are reinvested at YTMare reinvested at YTM

Yield to CallYield to CallYield to CallYield to Call

Yield to a specified call date and Yield to a specified call date and call pricecall price

Substitute number of periods until Substitute number of periods until first call date for and call price for first call date for and call price for face value (semi-annual coupons)face value (semi-annual coupons)

2c

2c

1tt

t

YTC/2)(1

CP

YTC/2)(1

/2CP

Realized YieldRealized YieldRealized YieldRealized Yield

Rate of return actually earned on a Rate of return actually earned on a bond given the reinvestment of the bond given the reinvestment of the coupons at varying ratescoupons at varying rates

Can only be calculated after Can only be calculated after investment period is overinvestment period is over

Horizon return analysisHorizon return analysis– Bond returns based on assumptions Bond returns based on assumptions

about reinvestment rates about reinvestment rates

Bond Price ChangesBond Price ChangesBond Price ChangesBond Price Changes

Over time, bond prices that differ Over time, bond prices that differ from face value must changefrom face value must change

Bond prices move inversely to Bond prices move inversely to market yieldsmarket yields

The change in bond prices due to The change in bond prices due to a yield change is directly related a yield change is directly related to time to maturityto time to maturity and inversely and inversely related to coupon raterelated to coupon rate

Bond Price ChangesBond Price ChangesBond Price ChangesBond Price Changes

Holding maturity Holding maturity constant, a rate constant, a rate decrease will raise decrease will raise prices a greater prices a greater percent than a percent than a corresponding corresponding increase in rates will increase in rates will lower priceslower prices

Pri

ce

Market yield

Measuring Bond Price Measuring Bond Price Volatility: DurationVolatility: Duration

Measuring Bond Price Measuring Bond Price Volatility: DurationVolatility: Duration

Important considerationsImportant considerations– Different effects of yield changes on Different effects of yield changes on

the prices and rates of return for the prices and rates of return for different bondsdifferent bonds

– Maturity inadequate measure of a Maturity inadequate measure of a bond’s economic lifetimebond’s economic lifetime

– A measure is needed that accounts A measure is needed that accounts for both size and timing of cash flowsfor both size and timing of cash flows

DurationDurationDurationDuration

A measure of a bond’s lifetime, A measure of a bond’s lifetime, stated in years, that accounts for the stated in years, that accounts for the entire pattern (both size and timing) entire pattern (both size and timing) of the cash flows over the life of the of the cash flows over the life of the bondbond

The weighted average maturity of a The weighted average maturity of a bond’s cash flowsbond’s cash flows– Weights determined by present value of Weights determined by present value of

cash flowscash flows

Calculating DurationCalculating DurationCalculating DurationCalculating Duration

Need to time-weight present value Need to time-weight present value of cash flows from bondof cash flows from bond

tPriceMarket

)PV(CFD

n

1t

t

Duration depends on three factorsDuration depends on three factors– Maturity of the bondMaturity of the bond– Coupon paymentsCoupon payments– Yield to maturityYield to maturity

Duration RelationshipsDuration RelationshipsDuration RelationshipsDuration Relationships

Duration increases with time to Duration increases with time to maturity, but at a decreasing ratematurity, but at a decreasing rate– For coupon paying bonds, duration is always For coupon paying bonds, duration is always

less than maturityless than maturity– For zero coupon-bonds, duration equals For zero coupon-bonds, duration equals

time to maturitytime to maturity Duration increases with lower couponsDuration increases with lower coupons Duration increases with lower yield to Duration increases with lower yield to

maturitymaturity

Why is Duration Why is Duration Important?Important?

Why is Duration Why is Duration Important?Important?

Allows comparison of effective lives of Allows comparison of effective lives of bonds that differ in maturity, couponbonds that differ in maturity, coupon

Used in bond management strategies, Used in bond management strategies, particularly immunizationparticularly immunization

Measures bond price sensitivity to Measures bond price sensitivity to interest rate movements, which is very interest rate movements, which is very important in any bond analysisimportant in any bond analysis

Estimating Price Changes Estimating Price Changes Using DurationUsing Duration

Estimating Price Changes Estimating Price Changes Using DurationUsing Duration

Modified duration =D*=D/(1+r)Modified duration =D*=D/(1+r) D* can be used to calculate the D* can be used to calculate the

bond’s percentage price change bond’s percentage price change for a given change in interest ratesfor a given change in interest rates

r r)(1

D- price bond in %

ConvexityConvexityConvexityConvexity

Refers to the degree to which duration Refers to the degree to which duration changes as the yield to maturity changeschanges as the yield to maturity changes– Price-yield relationship is convex Price-yield relationship is convex

Duration equation assumes a linear Duration equation assumes a linear relationship between price and yieldrelationship between price and yield

Convexity largest for low coupon, long-Convexity largest for low coupon, long-maturity bonds, and low yield to maturitymaturity bonds, and low yield to maturity

Duration ConclusionsDuration ConclusionsDuration ConclusionsDuration Conclusions

To obtain maximum price volatility, To obtain maximum price volatility, investors should choose bonds with investors should choose bonds with the longest durationthe longest duration

Duration is additiveDuration is additive– Portfolio duration is just a weighted Portfolio duration is just a weighted

averageaverage Duration measures volatility, which Duration measures volatility, which

is not the only aspect of risk in bondsis not the only aspect of risk in bonds