Transcription
Pricing Fixed-Income Securities
The Relationship Between Interest Rates and OptionFree Bond Prices Bond Prices A bond’s price is the present value of the future couponpayments (CPN) plus the present value of the face (par)value (FV)Price CPN1CPN 2CPN 3CPNn FV . (1 r )1 (1 r )2 (1 r )3(1 r )nnCPNtFV t(1 i )nt 1 (1 i ) Bond Prices and Interest Rates are Inversely RelatedPrice Par Bond Yield to maturity coupon rateDiscount Bond Yield to maturity coupon ratePremium Bond Yield to maturity coupon rate
Relationship between price and interest rate ona 3-year, 10,000 option-free par value bond thatpays 470 in semiannual interestFor a given absolute changein interest rates, thepercentage increase in abond’s price will exceed thepercentage decrease. ’sThis asymmetric pricerelationship is due to theconvex shape of the curve-plotting the price interestrate relationship.10,155.24D 155.2410,000.00D - 152.279,847.73Bond Prices ChangeAsymmetrically toRising and FallingRates8.89.4 10.0Interest Rate %
The Relationship Between Interest Ratesand Option-Free Bond Prices Maturity Influences Bond PriceSensitivity Forbonds that pay the same couponrate, long-term bonds changeproportionally more in price than doshort-term bonds for a given ratechange.
The effect of maturity on the relationshipbetween price and interest rate on fixedincome, option free bonds ’sFor a given coupon rate, the prices oflong-term bonds changeproportionately more than do theprices of short-term bonds for a givenrate 9.4%, 3-year bond9.4%, 6-year bond8.89.4 10.0Interest Rate %
The effect of coupon on the relationshipbetween price and interest rate on fixedincome, option free bonds% change in priceFor a given change in market rate, thebond with the lower coupon willchange more in price than will thebond with the higher coupon.MarketPrice ofPrice of ZeroRate 9.4% BondsCoupon8.8% 10,155.24 7,723.209.4%10,000.007,591.3710.0%9.847.737,462.15 1.74 1.550- 1.52- 1.709.4%, 3-year bondZero Coupon, 3-year bond8.89.4 10.0Interest Rate %
Duration and Price Volatility Duration as an Elasticity Measure Maturity simply identifies how much timeelapses until final payment. It ignores all information about the timingand magnitude of interim payments. Duration is a measure of the effectivematurity of a security. Duration incorporates the timing and size of asecurity’s cash flows. Duration measures how price sensitive asecurity is to changes in interest rates. The greater (shorter) the duration, the greater(lesser) the price sensitivity.
Duration and Price Volatility Duration as an Elasticity Measure Durationis an approximate measure ofthe price elasticity of demand% Change in Quantity DemandedPrice Elasticity of Demand % Change in Price
Duration and Price Volatility Duration as an Elasticity Measure Thelonger the duration, the larger thechange in price for a given change ininterest rates.DPDuration - PDi(1 i) Di DP - Duration P (1 i)
Duration and Price Volatility Measuring Duration Durationis a weighted average of thetime until the expected cash flows froma security will be received, relative tothe security’s price Macaulay’s DurationknCFt (t)CFt (t) tt(1 r)(1 r)t 1D t k1 CFtPrice of the Security t(1 r)t 1
Duration and Price Volatility Measuring Duration Example What is the duration of a bond with a 1,000 face value, 10% coupon, 3 yearsto maturity and a 12% YTM?100 1 100 2 100 3 1,000 3 1232,597.6(1.12)(1.12)(1.12)(1.12) 3D 2.73 years31001000951.96 t(1.12) 3t 1 (1.12)
Duration and Price Volatility Measuring Duration Example What is the duration of a bond with a 1,000 face value, 10% coupon, 3 yearsto maturity but the YTM is 5%?100 * 1 100 * 2 100 * 3 1,000 * 3 1233,127.31(1.05)(1.05)(1.05)(1.05) 3D 2.75 years1136.161,136.16
Duration and Price Volatility Measuring Duration Example What is the duration of a bond with a 1,000 face value, 10% coupon, 3 yearsto maturity but the YTM is 20%?100 * 1 100 * 2 100 * 3 1,000 * 3 1232,131.95(1.20)(1.20)(1.20)(1.20) 3D 2.68 years789.35789.35
Duration and Price Volatility Measuring Duration Example What is the duration of a zero couponbond with a 1,000 face value, 3 yearsto maturity but the YTM is 12%?1,000 * 32,135.34(1.12) 3D 3 years1,000711.78(1.12) 3 By definition, the duration of a zerocoupon bond is equal to its maturity
Duration and Price Volatility Comparing Price Sensitivity Thegreater the duration, the greaterthe price sensitivity Macaulay' s Duration DP - Di P(1 i) Macaulay' s DurationModified Duration (1 i)
Duration and Price Volatility Comparing Price Sensitivity WithModified Duration, we have anestimate of price volatility:DP% Change in Price - Modified Duration * DiP
Comparative price sensitivityindicated by durationType of Bond3-Yr. ZeroInitial market rate (annual)9.40%Initial market rate (semiannual)4.70%Maturity value 10,000Initial price 7,591.37Duration: semiannual periods6.00Modified duration5.73Rate Increases to 10% (5% Semiannually)- 130.51Estimated DP-1.72%Estimated DP / PInitial elasticity0.26936-Yr. Zero9.40%4.70% 10,000 5,762.8812.0011.463-Yr. Coupon9.40%4.70% 10,000 10,0005.375.126-Yr. Coupon9.40%4.70% 10,000 10,0009.449.02- 198.15-3.44%0.5387- 153.74-1.54%0.2406- 270.45-2.70%0.4242 DP - Duration [Di / (1 i)] P DP / P - [Duration / (1 i)] Diwhere Duration equals Macaulay's duration.
a 3-year, 10,000 option-free par value bond that pays 470 in semiannual interest 8.8 9.4 10.0 Interest Rate % 's D 155.24 D - 152.27 Bond Prices Change Asymmetrically to Rising and Falling Rates For a given absolute change in interest rates, the percentage increase in a bond's price will exceed the percentage decrease.
