The Fundamentals of Bond ValuationThe present-value modelWhere: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 ValuationIf yield < coupon rate, bond will be priced at a premium to its par valueIf yield > coupon rate, bond will be priced at a discount to its par valuePrice-yield relationship is convex (not a straight line)

The Present Value ModelThe value of the bond equals the present value of its expected cash flowswhere: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 Yield ModelThe expected yield on the bond may be computed from the market pricewhere:i = the discount rate that will discount the cash flows to equal the current market price of the bond

Computing Bond YieldsYield Measure PurposeNominal YieldMeasures the coupon rateCurrent yieldMeasures current income ratePromised yield to maturityMeasures expected rate of return for bond held to maturityPromised yield to callMeasures expected rate of return for bond held to first call dateRealized (horizon) yieldMeasures 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 YieldMeasures the coupon rate that a bond investor receives as a percent of the bonds par value

Current YieldSimilar to dividend yield for stocksImportant to income oriented investorsCY = Ci/Pm where: CY = the current yield on a bondCi = the annual coupon payment of bond iPm = the current market price of the bond

Promised Yield to MaturityWidely used bond yield figureAssumesInvestor holds bond to maturityAll the bonds cash flow is reinvested at the computed yield to maturity

Computing the Promised Yield to MaturitySolve for i that will equate the current price to all cash flows from the bond to maturity, similar to IRR

Computing Promised Yield to Call

where:Pm = market price of the bondCi = annual coupon paymentnc = number of years to first callPc = call price of the bond

Realized (Horizon) YieldPresent-Value Method

Calculating Future Bond Priceswhere:Pf = estimated future price of the bondCi = annual coupon paymentn = number of years to maturityhp = holding period of the bond in yearsi = expected semiannual rate at the end of the holding period

Yield Adjustments for Tax-Exempt BondsWhere:FTEY = fully taxable yield equivalenti = the promised yield on the tax exempt bondT = the amount and type of tax exemption (i.e., the investors marginal tax rate)

Bond Valuation Using Spot Rates

where: Pm = the market price of the bond Ct = the cash flow at time tn = the number of yearsit = the spot rate for Treasury securities at maturity t

What Determines Interest RatesInverse relationship with bond pricesForecasting interest ratesFundamental determinants of interest ratesi = RFR + I + RP where:RFR = real risk-free rate of interest I = expected rate of inflation RP = risk premium

What Determines Interest RatesEffect of economic factorsreal growth ratetightness or ease of capital marketexpected inflationor supply and demand of loanable fundsImpact of bond characteristicscredit qualityterm to maturityindenture provisionsforeign bond risk including exchange rate risk and country risk

Term Structure of Interest RatesIt is a static function that relates the term to maturity to the yield to maturity for a sample of bonds at a given point in time.Term Structure TheoriesExpectations hypothesisLiquidity preference hypothesisSegmented market hypothesisTrading implications of the term structure

Spot Rates and Forward RatesCreating the Theoretical Spot Rate CurveCalculating Forward Rates from the Spot Rate Curve

Expectations HypothesisAny 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 TheoryLong-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 HypothesisDifferent institutional investors have different maturity needs that lead them to confine their security selections to specific maturity segments

Trading Implications of the Term StructureInformation on maturities can help you formulate yield expectations by simply observing the shape of the yield curve

Yield SpreadsSegments: government bonds, agency bonds, and corporate bondsSectors: prime-grade municipal bonds versus good-grade municipal bonds, AA utilities versus BBB utilitiesCoupons or seasoning within a segment or sectorMaturities within a given market segment or sector

Yield SpreadsMagnitudes and direction of yield spreads can change over time

What Determines the Price Volatility for BondsBond price change is measured as the percentage change in the price of the bondWhere:EPB = the ending price of the bondBPB = the beginning price of the bond

What Determines the Price Volatility for BondsFour Factors1. Par value2. Coupon3. Years to maturity4. Prevailing market interest rate

What Determines the Price Volatility for BondsFive 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 maturity3. Price volatility increases at a diminishing rate as term to maturity increases4. Price movements resulting from equal absolute increases or decreases in yield are not symmetrical5. Higher coupon issues show smaller percentage price fluctuation for a given change in yield, thus bond price volatility is inversely related to coupon

What Determines the Price Volatility for BondsThe maturity effectThe coupon effectThe yield level effectSome trading strategies

The Duration MeasureSince 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 objectiveA composite measure considering both coupon and maturity would be beneficial

The Duration MeasureDeveloped 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

Characteristics of Macaulay DurationDuration of a bond with coupons is always less than its term to maturity because duration gives weight to these interim paymentsA zero-coupon bonds duration equals its maturityThere is an inverse relationship between duration and couponThere is a positive relationship between term to maturity and duration, but duration increases at a decreasing rate with maturityThere is an inverse relationship between YTM and durationSinking funds and call provisions can have a dramatic effect on a bonds duration

Modified Duration and Bond Price VolatilityAn adjusted measure of duration can be used to approximate the price volatility of an option-free (straight) bondWhere:m = number of payments a yearYTM = nominal YTM

Modified Duration and Bond Price VolatilityBond price movements will vary proportionally with modified duration for small changes in yieldsAn estimate of the percentage change in bond prices equals the change in yield time modified durationWhere: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

Trading Strategies Using Modified DurationLongest-duration security provides the maximum price variationIf you expect a decline in interest rates, increase the average modified duration of your bond portfolio to experience maximum price volatilityIf you expect an increase in interest rates, reduce the average modified duration to minimize your price declineNote that the modified duration of your portfolio is the market-value-weighted average of the modified durations of the individual bonds in the portfolio

Bond Duration in Years for Bonds Yielding 6 Percent Under Different Terms

Sheet: Table 10.3

Table 16.6

Bond Duration for Bond Yielding 6 Percent under Different Terms

COUPON RATES

Years to

Maturity

0.02

0.08

1.0

0.995

0.981

5.0

4.756

4.558

4.393

4.254

10.0

8.891

8.169

7.662

7.286

20.0

14.981

12.98

11.904

11.232

50.0

19.452

17.129

16.273

15.829

100.0

17.567

17.232

17.12

17.064

8.0

17.167

17.167

17.167

17.167

Source: L. Fisher and R. L. Weil, "Coping with the Risk of Interest Rate Fluctuations:

(October 1971): 418. Copyright 1971, University of Chicago Press.

Bond ConvexityModified duration is a linear approximation of bond price change for small changes in market yields

However, price changes are not linear, but a curvilinear (convex) function

Price-Yield Relationship for BondsThe graph of prices relative to yields is not a straight line, but a curvilinear relationshipThis can be applied to a single bond, a portfolio of bonds, or any stream of future cash flowsThe convex price-yield relationship will differ among bonds or other cash flow streams depending on the coupon and maturityThe convexity of the price-yield relationship declines slower as the yield increasesModified duration is the percentage change in price for a nominal change in yield

Modified DurationFor small changes this will give a good estimate, but this is a linear estimate on the tangent line

Determinants of ConvexityThe convexity is the measure of the curvature and is the second derivative of price with resect to yield (d2P/di2) divided by priceConvexity is the percentage change in dP/di for a given change in yield

Determinants of ConvexityInverse relationship between coupon and convexityDirect relationship between maturity and convexityInverse relationship between yield and convexity

Modified Duration-Convexity EffectsChanges in a bonds price resulting from a change in yield are due to:Bonds modified durationBonds convexityRelative effect of these two factors depends on the characteristics of the bond (its convexity) and the size of the yield changeConvexity is desirable

Duration and Convexity for Callable BondsIssuer has option to call bond and pay off with proceeds from a new issue sold at a lower yieldEmbedded optionDifference in duration to maturity and duration to first callCombination of a noncallable bond plus a call option that was sold to the issuerAny increase in value of the call option reduces the value of the callable bond

Option Adjusted DurationBased on the probability that the issuing firm will exercise its call optionDuration of the non-callable bondDuration of the call option

Convexity of Callable BondsNoncallable bond has positive convexityCallable bond has negative convexity

Limitations of Macaulay and Modified DurationPercentage change estimates using modified duration only are good for small-yield changesDifficult 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 shiftInitial assumption that cash flows from the bond are not affected by yield changes

Effective DurationMeasure of the interest rate sensitivity of an assetUse a pricing model to estimate the market prices surrounding a change in interest ratesEffective DurationEffective ConvexityP- = the estimated price after a downward shift in interest ratesP+ = the estimated price after a upward shift in interest ratesP = the current priceS = the assumed shift in the term structure

Effective DurationEffective duration greater than maturityNegative effective durationEmpirical duration

Empirical DurationActual percent change for an asset in response to a change in yield during a specified time period

Yield Spreads With Embedded OptionsStatic Yield SpreadsConsider the total term structureOption-Adjusted SpreadsConsider changes in the term structure and alternative estimates of the volatility of interest rates