固定收益证-券Bond-Price-Volati课件.ppt

1、Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-1Chapter 4 Bond Price Volatility Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-2Learning ObjectivesAfter reading this chapter, you will understandv the price-yield relationship of an option-free bondv the facto

2、rs that affect the price volatility of a bond when yields changev the price-volatility properties of an option-free bondv how to calculate the price value of a basis pointv how to calculate and interpret the Macaulay duration, modified duration, and dollarv duration of a bondv why duration is a meas

3、ure of a bonds price sensitivity to yield changesCopyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-3Learning Objectives (continued)After reading this chapter, you will understandv the spread duration measure for fixed-rate and floating-rate bondsv how to compute the duration of a p

4、ortfolio and contribution to portfolio durationv limitations of using duration as a measure of price volatilityv how price change estimated by duration can be adjusted for a bonds convexityv how to approximate the duration and convexity of a bondv the duration of an inverse floaterv how to measure a

5、 portfolios sensitivity to a nonparallel shift in interest rates (key rate duration and yield curve reshaping duration)Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-4Exhibit 4-2Shape of Price-Yield Relationship for an Option-Free BondPriceMaximum PriceYieldCopyright 2010 Pearso

6、n Education, Inc. Publishing as Prentice Hall4-5Price Volatility Characteristicsof Option-Free BondsvThere are four properties concerning the price volatility of an option-free bond:(i)Although the prices of all option-free bonds move in the opposite direction from the change in yield required, the

7、percentage price change is not the same for all bonds.(ii)For very small changes in the yield required, the percentage price change for a given bond is roughly the same, whether the yield required increases or decreases.(iii)For large changes in the required yield, the percentage price change is not

8、 the same for an increase in the required yield as it is for a decrease in the required yield.(iv)For a given large change in basis points, the percentage price increase is greater than the percentage price decrease.v An explanation for these four properties of bond price volatility lies in the conv

9、ex shape of the price-yield relationship.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-6Price Volatility Characteristicsof Option-Free Bonds (continued)v Characteristics of a Bond that Affect its Price VolatilityThere are two characteristics of an option-free bond that determin

10、e its price volatility: coupon and term to maturity.1)First, for a given term to maturity and initial yield, the price volatility of a bond is greater, the lower the coupon rate.This characteristic can be seen by comparing the 9%, 6%, and zero-coupon bonds with the same maturity.2)Second, for a give

11、n coupon rate and initial yield, the longer the term to maturity, the greater the price volatility. This can be seen in Exhibit 4-3 (See Overhead 4-9) by comparing the five-year bonds with the 25-year bonds with the same coupon.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-7EXH

12、IBIT 4-3 Instantaneous Percentage Price Change for Six Hypothetical BondsSix hypothetical bonds, priced initially to yield 9%:9% coupon, 5 years to maturity, price = 100.00009% coupon, 25 years to maturity, price = 100.0006% coupon, 5 years to maturity, price = 88.13096% coupon, 25 years to maturity

13、, price = 70.35700% coupon, 5 years to maturity, price = 64.39280% coupon, 25 years to maturity, price = 11.0710Yield (%) Change to:Change in Basis PointsPercentage Price Change (coupon/maturity in years)9% / 59% / 256% / 56% / 250% / 50% / 256.00-30012.8038.5913.4742.1315.56106.047.00-2008.3223.468

14、.7525.4610.0961.738.00-1004.0610.744.2611.604.9127.108.50-502.005.152.115.552.4212.728.90-100.401.000.421.070.482.428.99-10.040.100.040.110.050.249.011-0.04-0.10-0.04-0.11-0.05-0.249.1010-0.39-0.98-0.41-1.05-0.48-2.369.5050-1.95-4.75-2.05-5.09-2.36-11.2610.00100-3.86-9.13-4.06-9.76-4.66-21.2311.0020

15、0-7.54-16.93-7.91-18.03-9.08-37.8912.00300-11.04-23.64-11.59-25.08-13.28-50.96Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-8EXHIBIT 4-4 Price Change for a 100-Basis-Point Change in Yield for a 9% 25-Year Bond Trading at Different Yield LevelsYield Level (%)Initial PriceNew Pri

16、ce aPrice DeclinePercent Decline7$123.46$110.74$12.7210.308110.74100.0010.749.709100.0090.879.139.131090.8783.077.808.581183.0776.366.718.081276.3670.555.817.611370.5565.505.057.161465.5061.084.426.75 a As a result of a 100-basis-point increase in yield.Copyright 2010 Pearson Education, Inc. Publish

17、ing as Prentice Hall4-9Measures of Bond Price Volatilityv Money managers, arbitrageurs, and traders need to have a way to measure a bonds price volatility to implement hedging and trading strategies.v Three measures that are commonly employed:1) price value of a basis point2) yield value of a price

18、change3) durationCopyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-10Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-10Measures of Bond Price Volatility (continued)v Price Value of a Basis PointThe price value of a basis point, also referred to as the dollar val

19、ue of an 01, is the change in the price of the bond if the required yield changes by 1 basis point.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-11Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-11Measures of Bond Price Volatility (continued)v Yield Value of

20、 a Price Change Another measure of the price volatility of a bond used by investors is the change in the yield for a specified price change.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-12Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-12Measures of Bond Pri

21、ce Volatility (continued)v Duration The Macaulay duration is one measure of the approximate change in price for a small change in yield:where P = price of the bondC = semiannual coupon interest (in dollars)y = one-half the yield to maturity or required yieldn = number of semiannual periods (number o

22、f years times 2)M = maturity value (in dollars)12121111Macaulay durationnnCCnCnM + +. . .+ + yyyyPCopyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-13Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-13Measures of Bond Price Volatility (continued)v Duration Invest

23、ors refer to the ratio of Macaulay duration to 1 + y as the modified duration. The equation is:where y = one-half the yield to maturity or required yield. The modified duration is related to the approximate percentage change in price for a given change in yield as given by:where dP = change in price

24、, dy = change in yield, P = price of the bond.1Macaulay durationymodified duration1dP dy Pmodified durationCopyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-14EXHIBIT 4-5 Calculation of Macaulay Duration and Modified Duration for 5-Year 9% Bond Selling to Yield 9%Coupon rate: 9.00%

25、Term (years): 5Initial yield: 9.00%Period, tCash FlowPV of $1 at 4.5%PV of CFt PVCF1$ 4.500.9569374.3062204.306222 4.500.9157294.1207858.241563 4.500.8762963.94333511.830004 4.500.8385613.77352615.094105 4.500.8024513.61103018.055146 4.500.7678953.45553120.733187 4.500.7348283.30672823.147098 4.500.

26、7031853.16433325.314669 4.500.6729043.02807027.2526210 $104.500.64392767.290443672.90442100.00000826.87899Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-15EXHIBIT 4-6 Calculation of Macaulay Duration and Modified Duration for 5-Year 6% Bond Selling to Yield 9%Coupon rate: 6.00%T

27、erm (years): 5Initial yield: 9.00%Period, tCash FlowPV of $1 at 4.5%PV of CFt PVCF1$ 3.000.9569372.8708132.870812 3.000.9157292.7471905.494373 3.000.8762962.6288907.886664 3.000.8385612.51568410.062735 3.000.8024512.40735312.036766 3.000.7678952.30368713.822127 3.000.7348282.20448515.431398 3.000.70

28、31852.10955516.876449 3.000.6729042.01871318.1684110103.000.64392766.324551663.24551Total88.130923765.89520Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-16Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-16Measures of Bond Price Volatility (continued)v Proper

29、ties of DurationThe modified duration and Macaulay duration of a coupon bond are less than the maturity.The Macaulay duration of a zero-coupon bond equals its maturity; but a zero-coupon bonds modified duration is less than its maturity.Lower coupon rates generally have greater Macaulay and modified

30、 bond durations.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-17Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-17Measures of Bond Price Volatility (continued)v Approximating the Percentage Price ChangeThe below equation can be used to approximate the percen

31、tage price change for a given change in required yield:where dP = change in price, P = price of the bond and dy = change in yield.Suppose that the yield on any bond changes by 100 basis points. Then, substituting 100 basis points (0.01) for dy into the above equation, we get:Thus, modified duration

32、can be interpreted as the approximate percentage change in price for a 100-basis-point change in yield.()dP dyPmodified duration() 0.01() 1%dP Pmodified durationmodified durationCopyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-18Copyright 2010 Pearson Education, Inc. Publishing as

33、 Prentice Hall4-18Measures of Bond Price Volatility (continued)v Portfolio DurationThe duration of a portfolio is simply the weighted average duration of the bonds in the portfolios.Portfolio managers look at their interest rate exposure to a particular issue in terms of its contribution to portfoli

34、o duration. This measure is found by multiplying the weight of the issue in the portfolio by the duration of the individual issue given as:contribution to portfolio duration = weight of issue in portfolio duration of issue.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-19Copyrig

35、ht 2010 Pearson Education, Inc. Publishing as Prentice Hall4-19Measures of Bond Price Volatility (continued)v Portfolio Duration Portfolio managers look at portfolio duration for sectors of the bond market.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-20EXHIBIT 4-7 Calculation

36、of Duration and Contribution to Portfolio Duration for a Asset Allocation to Sectors of the Lehman Brothers U.S. Aggregate Index: October 26, 2007SectorPortfolio WeightSector DurationContribution to Portfolio DurationTreasury0.0004.950.00Agency0.1213.440.42Mortgages0.4493.581.61Commercial Mortgage-B

37、acked Securities0.1395.040.70Asset-Backed Securities0.0173.160.05Credit0.2746.351.741.0004.52Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-21EXHIBIT 4-8Calculation of Duration and Contribution to the Lehman Brothers Aggregate Index Duration: October 26, 2007SectorWeight in Inde

38、xSector DurationContribution to Index DurationTreasury0.2304.951.14Agency0.1053.440.36Mortgages0.3813.581.36Commercial Mortgage-Backed Securities0.0565.040.28Asset-Backed Securities0.0103.160.03Credit0.2196.351.39Total1.0004.56Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-22Cop

39、yright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-22Measures of Bond Price Volatility (continued)v Portfolio DurationWhile the portfolio and the index have the same duration, the spread duration for the recommended portfolio is 4.60 vs. 3.49 for the index.The larger spread duration fo

40、r the recommended portfolio is expected given the greater allocation to non-Treasury sectors.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-23EXHIBIT 4-9Calculation of Spread Duration and Contribution to Portfolio Spread Duration for an Asset Allocation to Sectors of the Lehman

41、Brothers U.S. Aggregate Index: October 26, 2007SectorPortfolio WeightSector Spread DurationContribution to Portfolio Spread DurationTreasury0.0000.000.00Agency0.1213.530.43Mortgages0.4493.621.63Commercial Mortgage-Backed Securities0.1395.040.70Asset-Backed Securities0.0173.160.05Credit0.2746.351.79T

42、otal1.0004.60Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-24EXHIBIT 4-10Calculation of Spread Duration and Contribution to the Lehman Brothers Aggregate Index Spread Duration: October 26, 2007SectorWeight in IndexSector Spread DurationContribution to Index Spread DurationTreas

43、ury0.2300.000.00Agency0.1053.530.37Mortgages0.3813.621.38Commercial Mortgage-Backed Securities0.0565.040.28Asset-Backed Securities0.0103.160.03Credit0.2196.531.43Total1.0003.49Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-25Convexityv Because all the duration measures are only

44、approximations for small changes in yield, they do not capture the effect of the convexity of a bond on its price performance when yields change by more than a small amount.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-26Exhibit 4-12 Line Tangent to the Price-Yield Relationship

45、PriceYieldy*p*Actual PriceTangent Line at y*(estimated price)Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-27Convexity (continued)vThe approximation will always understate the actual price.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-28Exhibit 4-13. Price

46、 Approximation Using DurationPriceYieldy3p*Actual PriceTangent Line at y*(estimated price)y*y1y2y4Error in Estimating Price Based only on DurationError in Estimating Price Based only on DurationCopyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-29Copyright 2010 Pearson Education, In

47、c. Publishing as Prentice Hall4-29Convexity (continued)v Measuring ConvexityThe percentage change in the price of the bond due to convexity or the convexity measure is:Where The percentage price change due to convexity is:221 PdPdyconvexity measure 212dPdyPconvexity measure22221(1)(1) (1)(1)=ntntPt

48、tCn nMdyydyCopyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-30Copyright 2010 Pearson Education, Inc. Publishing as Prentice HallConvexity (continued)v Measuring ConvexityIn general, if the cash flows occur m times per year, convexity is adjusted to an annual figure as follows:2 co

49、nvexity measure in yearconvexity measure in m period per year mCopyright 2010 Pearson Education, Inc. Publishing as Prentice Hall4-31EXHIBIT 4-14Calculation of Convexity Measure and Dollar Convexity Measure for Five-Year 9% Bond Selling to Yield 9%Coupon rate: 9.00% Term (years): 5 Initial yield: 9.

50、00% Price: 100Period, tCash Flow1/(1.045)t+2t(t + 1)CFt(t + 1)CF(1.045) t+214.50 0.87629697.88624.500.8385612722.64134.500.8024515443.33244.500.7678959069.11054.500.73482813599.20164.500.703185189132.90174.500.672904252169.57184.500.643927324208.63294.500.616198405249.56010104.500.58966311,4956,778.