Ch-05 - Fixed Income Securities and Markets Chapter 5...

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Unformatted text preview: Fixed Income Securities and Markets Chapter 5 One-Factor Measures of Price OneSensitivity How much price changes as interest rates change Initial + shifted rate curve Initial computing prices is computing straightforward Defining changes in interest rates Defining Many different possible changes in interest rates Parallel shift in curve Parallel Each spot rate moves in some Each proportion to its maturity Inverse proportion to maturity Inverse 1 Interest Rate Factor An interest rate factor is a random An variable that impacts interest rates in some way. One factor driving all interest rates One that factor itself is an interest rate. that Two or more factors. Two Factors are not interest rates. Factors Measures of Price Sensitivity Ch. 5: very generally, one-factor Ch. onemeasures of price sensitivity Ch. 6: the special case of parallel yield Ch. shifts Ch. 7: multi-factor formulations. Ch. multiCh. 8: to model interest rate changes Ch. empirically. DV01 P(y): price-rate function of a fixedP(y): pricefixedincome security, where y is an interest rate factor. For example: the term structure of For interest rates is flat at 5% and the rates move up and down in parallel. 2 In this 5% flat environment As of Feb 15, 2001 As US Treasury bond: 5% coupon, US maturity Feb. 15, 2011 One-year European call option struck Oneat par on the 5s of Feb 15, 2011 This option gives its owner the right to This purchase some face amount of the bond after exactly one year at par. Price expressed as a percent of face value Price expressed as a percent of face value 3 DV01: dollar value of an '01 (i.e., .01%) The change in the value (price) of a The fixed income security for a one-basis onepoint decline in rates. DV 01 ≡ − ΔP 10, 000 × Δy (ΔP)/(Δy) ==> Slope of the line ==> Use points relatively close ==> In the limit, the slope of the line tangent to the price-rate curve at the desired rate pricelevel ==> derivative DV 01 ≡ − 1 dP ( y ) 10, 000 dy 4 DV01 at 4% DV01 − ΔP 8.0866 − 8.2148 =− = .0641 10, 000 × Δy 10, 000 × (4.01% − 3.99%) This chapter's definition: very general. This A hedging example, part I: Hedging a call option If a market maker sells $100 million If face value of the call option and rates are at 5%, how might the market maker hedge interest rate exposure by trading in the underlying bond? Buy or sell the bond? Buy 5 .0779 .0369 = 100, 000, 000 × 100 100 .0369 = $47,370, 000 F = 100, 000, 000 × .0779 .0369 = $36,900 100, 000, 000 × 100 .0779 = $36,901 $47,370, 000 × 100 − FA × DV 01A FB = DV 01B F× Duration Duration measures the percentage Duration change in the value of a security for a unit change in rates (10,000 basis points). D≡− 1 ΔP P Δy 1 dP P dy (108.0901 − 108.2615) /108.1757 D=− = 7.92 4.01% − 3.99% ΔP = − DΔy P ΔP = (−7.92 × .0001) × 108.1757 = −0.0857 D≡− 6 Duration Terminology Effective duration: Effective duration computed for any assumed change in the term structure of interest rates. Macaulay duration or modified Macaulay duration: interest rate sensitivity with respect to a change in yield-to-maturity. yield- to- Convexity 7 Convexity Convexity: how interest rate sensitivity Convexity: changes with rates 1 d 2P C= P dy 2 100 − 100.0780 = −779.8264 5% − 4.99% Δ 2 P −779.0901 + 779.8264 = = 7363 Δy 2 5.005% − 4.995% 1 Δ 2 P 7363 C= = = 73.63 P Δy 2 100 8 Measuring the price sensitivity of portfolios P = ∑ Pi dP dP =∑ i dy dy 1 dP 1 dPi =∑ 10, 000 dy 10, 000 dy DV 01 = ∑ DV 01i 1 P 1 − P − 1 dPi dP = ∑− dy P dy P 1 dPi dP = ∑− i dy P Pi dy Pi Di P P C = ∑ i Ci P D=∑ A hedging example: the negative convexity of callable bonds A callable bond is a bond that the callable issuer may repurchase or call at some fixed set of prices on some fixed set of dates Price = (price of bond) - (value of Price embedded option) 9 At 5%, callable bond price = At 100-3.0501=96.9499 100DV01=0.0779-0.0369=0.0410 DV01=0.0779Convexity: -223= Convexity: 103.15%X73.63-3.15%X9503.33 103.15%X73.63- First-order approximation First- ΔP ≈ − DΔy P Estimating price changes and returns with DV01, duration, and convexity 1 d 2P 2 dP P ( y + Δy ) ≈ P( y ) + Δy + Δy 2 dy 2 dy 1 1 d 2P 2 ΔP 1 dP ≈ Δy + × Δy 2 P dy 2 P P dy 1 ΔP ≈ − DΔy + C Δy 2 2 P %ΔP = −120.82 × 0.0025 + (1/ 2)9503.3302 × 0.00252 = −0.30205 + 0.02970 = −27.235% 2.2194 = 3.0501× (1 − 27.235%) 10 11 ...
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