CHAPTER 33 VALUING BONDS

1 1 CHAPTER 33 VALUING BONDSThe value of a bond is the present value of the expected cash flows on the bond ,discounted at an interest rate that is appropriate to the riskiness of that bond . Since thecash flows on a straight bond are fixed at issue, the value of a bond is inversely related tothe interest rate that investors demand for that bond . The interest rate charged on a bondis determined by both the general level of interest rates, which applies to all BONDS andfinancial investments, and the default premium specific to the entity issuing the CHAPTER examines the determinants of both the general level of interest rates and themagnitude of the default premia on specific BONDS . The general level of interest ratesincorporates expected inflation and a measure of real return and reflects the termstructure, with BONDS of different maturities carrying different interest rates.

2 The defaultpremia varies across time, depending in large part on the health of the economy andinvestors' risk often have special features embedded in them that have to be factored intothe value. Some of these features are options - to convert into stock (convertible BONDS ),to call the bond back if interest rates go down (callable BONDS ) and to put the bond back tothe issuer at a fixed price under specific circumstances (putable BONDS ). Other bondcharacteristics, such as interest rate caps and floors, have option features. Some of theseoptions reside with the issuer of the bond , some with the buyer of the bond , but they allhave to be priced. Option pricing models can be used to value these special features andprice complex fixed income securities. Some special features in BONDS such as sinkingfunds, subordination of further debt and the type of collateral may affect the prices ofbonds, as Prices and Interest RatesThe value of a straight bond is determined by the level of and changes in interestrates.

3 As interest rates rise, the price of a bond will decrease and vice versa. This inverserelationship between bond prices and interest rates arises directly from the present valuerelationship that governs bond The Present Value RelationshipThe value of a bond , like all financial investments, is derived from the presentvalue of the expected cash flows on that bond , discounted at an interest rate that reflectsthe default risk associated with the cash flows. There are two features that set bondsapart from equity investments. First, the cash flows on a bond , , the coupon paymentsand the face value of the bond , are usually set at issue and do not change during the life ofthe bond . Even when they do change, as in floating rate BONDS , the changes are generallylinked to changes in interest rates.

4 Second, BONDS usually have fixed lifetimes, unlikestocks, since most bonds1 specify a maturity date. As a consequence, the present valueof a 'straight bond ' with fixed coupons and specified maturity is determined entirely bychanges in the discount rate, which incorporates both the general level of interest rates andthe specific default risk of the bond being present value of a bond , expected to mature in N time periods, with couponsevery period can be calculated. PV of bond = Coupont(1+r)tt=1t=N +Face Value(1+r)Nwhere,Coupont = Coupon expected in period tFace Value = Face value of the bondr = Discount rate for the cash flowsThe discount rate used to calculate the present value of the bond will vary from bond tobond depending upon default risk, with higher rates used for riskier BONDS and lower ratesfor safer the bond is traded, and a market price is therefore available for it, the internalrate of return can be computed for the bond , , the discount rate at which the presentvalue of the coupons and the face value is equal to the market price.

5 This internal rate ofreturn is called the yield to maturity on the bond . 1 Console BONDS are the exception to this rule, since they are are several details, relating to both the magnitude and timing of cash flows,that can affect the value of a bond and its yield to maturity. First, the coupon payment ona bond may be semi-annual, in which case the discounting has to allow for the semi-annualcash flows. (The first coupon will be discounted back half a year, the second one year, thethird a year and a half and so on.) Second, once a bond has been issued, it accrues couponinterest between coupon payments and this accrued interest has to be added on to theprice of the bond , when VALUING the : VALUING a straight bond at issueThe following is a valuation of a thirty-year Government bond at the time ofissue.

6 The coupon rate on the bond is , and the market interest rate is Theprice of the bond can be calculated. PV of bond = ( )tt=1t=30 +1,000( )30=$ is based upon annual coupons. If the calculation is based upon semi-annual coupons,the value of the bond is:PV of bond = ( )tt= +1,000( )30=$ : VALUING a seasoned straight bondThe following is a valuation of a seasoned Government bond , with twenty yearsleft to expiration and a coupon rate of The next coupon is due in two current twenty-year bond rate is The value of the bond can be of bond = ( )tt= + ( )2/12+1,000( ) $ bond trades at well above face value, because of its high coupon rate. Note that thesecond term of the equation is the present value of the next A Measure of Interest Rate Risk in BondsWhen the fact that the cash flows on a bond are fixed at issue is combined with thepresent value relationship governing bond prices, there is a clear rationale for why interest4changes affect bond prices so directly.

7 Any increase in interest rates, either at theeconomy wide level or because of an increase in the default risk of the company issuingthe bond , will lower the present value of the stream of expected cash flows and hence thevalue of the bond . Any decrease in interest rates will have the opposite effect of interest rate changes on bond prices will vary from bond to bond andwill depend upon a number of characteristics of the bond .(a) the maturity of the bond - Holding coupon rates and default risk constant, increasingthe maturity of a straight bond will increase its sensitivity to interest rate changes. Thepresent value of cash flows changes much more for cash flows further in the future, asinterest rates change, than for cash flows which are nearer in time.

8 Figure illustratesthe present values of six BONDS - a 5-year, a 10-year, a 15-year, a 20-year, a 30-year and a50-year BONDS , all with 8% coupons for a range of interest longer-term BONDS are much more sensitive to interest rate changes than the shorterterm BONDS . For instance, an increase in interest rates from 8% to 10% results in a declinein value of for the five-year bond and of for the fifty-year BONDS .(b) the coupon rate of the bond - Holding maturity and default risk constant, increasing thecoupon rate of a straight bond will decrease its sensitivity to interest rate changes. Sincehigher coupons result in more cash flows earlier in the bond 's life, the present value willFigure : bond Values and Interest Rates$ $ $ $ $ $1, $1, $1, year10 year15 years20 years30 years50 yearsBond Maturitiesr=6%r=7%r=8%r=9%r=10%5change less as interest rates change.

9 At the extreme, if the bond is a 'zero-coupon' bond ,the only cash flow is the face value at maturity, and the present value is likely to varymuch more as a function of interest rates. Figure illustrates the percentage changes inbond prices for six thirty-year BONDS with coupon rates ranging from 0% to 10% for arange of interest BONDS with the lower coupons are much more sensitive, in percentage terms, tointerest rate changes than those with higher the maturity and the coupon rate are the key determinants of how sensitivethe price of a bond is to interest rate changes, a number of other factors impinge on thissensitivity. Any special features that the bond has, including convertibility and callability,make the maturity of the bond less definite and can therefore affect the bond price'ssensitivity to interest rate changes.

10 If there is any relationship between the level ofinterest rates and the default premia on BONDS , the default risk of a bond can affect itsprice A More Formal Measure of Interest Rate Risk - DurationFigure : Percent Change in bond Price - Interest rate changes from 8% RatePercent Change in bond PriceInterest rate drops 2%Interest rate drops 1%Interest rate rises 1%Interest rate rises 2%6 Since the interest rate risk of a bond is a significant component of its total risk, amore formal measure of interest risk is needed, which consolidates the effects of maturity,coupon rates and the bond 's special features. To arrive at this measure, consider thepresent value relationship developed earlier in this CHAPTER . PV of bond = Coupont(1+r)tt=1t=N +Face Value(1+r)NDifferentiating the bond price with respect to interest rate should provide a formalmeasure of bond price sensitivity to interest rate changes.