固定收益证券Bond-Price-Volatility.ppt

Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,*,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,1,Chapter 4,Bond Price Volatility,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,2,Learning Objectives,After reading this chapter,you will understand,the price-yield relationship of an option-free bond,the factors that affect the price volatility of a bond when yields change,the price-volatility properties of an option-free bond,how to calculate the price value of a basis point,how to calculate and interpret the Macaulay duration,modified duration,and dollar,duration of a bond,why duration is a measure of a bonds price sensitivity to yield changes,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,3,Learning Objectives,(continued),After reading this chapter,you will understand,the spread duration measure for fixed-rate and floating-rate bonds,how to compute the duration of a portfolio and contribution to portfolio duration,limitations of using duration as a measure of price volatility,how price change estimated by duration can be adjusted for a bonds convexity,how to approximate the duration and convexity of a bond,the duration of an inverse floater,how to measure a 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 Hall,4-,4,Exhibit 4-2,Shape of Price-Yield Relationship for an Option-Free Bond,Price,Maximum Price,Yield,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,5,Price Volatility Characteristicsof Option-Free Bonds,There are,four properties,concerning the price volatility of an option-free bond:,Although the prices of all option-free bonds move in the opposite direction from the change in yield required,the percentage price change is not the same for all bonds.,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.,For large changes in the required yield,the percentage price change is not the same for an increase in the required yield as it is for a decrease in the required yield.,For a given large change in basis points,the percentage price increase is greater than the percentage price decrease.,An explanation for these four properties of bond price volatility lies in the,convex shape,of the price-yield relationship.,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,6,Price Volatility Characteristicsof Option-Free Bonds,(continued),Characteristics of a Bond that Affect its Price Volatility,There are,two characteristics,of an option-free bond that determine its price volatility:,coupon,and,term to maturity,.,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.,Second,for a given 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 Hall,4-,7,EXHIBIT 4-3,Instantaneous Percentage Price Change for Six Hypothetical Bonds,Six hypothetical bonds,priced initially to yield 9%:,9%coupon,5 years to maturity,price=100.0000,9%coupon,25 years to maturity,price=100.000,6%coupon,5 years to maturity,price=88.1309,6%coupon,25 years to maturity,price=70.3570,0%coupon,5 years to maturity,price=64.3928,0%coupon,25 years to maturity,price=11.0710,Yield(%)Change to:,Change in Basis Points,Percentage Price Change(coupon/maturity in years),9%/5,9%/25,6%/5,6%/25,0%/5,0%/25,6.00,-300,12.80,38.59,13.47,42.13,15.56,106.04,7.00,-200,8.32,23.46,8.75,25.46,10.09,61.73,8.00,-100,4.06,10.74,4.26,11.60,4.91,27.10,8.50,-50,2.00,5.15,2.11,5.55,2.42,12.72,8.90,-10,0.40,1.00,0.42,1.07,0.48,2.42,8.99,-1,0.04,0.10,0.04,0.11,0.05,0.24,9.01,1,-0.04,-0.10,-0.04,-0.11,-0.05,-0.24,9.10,10,-0.39,-0.98,-0.41,-1.05,-0.48,-2.36,9.50,50,-1.95,-4.75,-2.05,-5.09,-2.36,-11.26,10.00,100,-3.86,-9.13,-4.06,-9.76,-4.66,-21.23,11.00,200,-7.54,-16.93,-7.91,-18.03,-9.08,-37.89,12.00,300,-11.04,-23.64,-11.59,-25.08,-13.28,-50.96,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,10,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,10,Measures of Bond Price Volatility,(continued),Price Value of a Basis Point,The price value of a basis point,also referred to as the dollar value 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 Hall,4-,11,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,11,Measures of Bond Price Volatility,(continued),Yield Value of 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 Hall,4-,12,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,12,Measures of Bond Price Volatility,(continued),Duration,The Macaulay duration is one measure of the approximate change in price for a small change in yield:,where,P,=,price of the bond,C,=,semiannual coupon interest,(in dollars),y,=,one-half the yield to maturity or required yield,n,=,number of semiannual periods,(number of years times 2),M,=,maturity value,(in dollars),Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,13,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,13,Measures of Bond Price Volatility,(continued),Duration,Investors 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,dy,=,change in yield,P,=,price of the bond,.,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,14,EXHIBIT,4-5,Calculation of Macaulay Duration and Modified Duration for 5-Year,9%,Bond Selling to Yield 9%,Coupon rate:,9.00%,Term(years):5,Initial yield:9.00%,Period,t,Cash Flow,PV of$1 at 4.5%,PV of CF,t PVCF,1,$4.50,0.956937,4.306220,4.30622,2,4.50,0.915729,4.120785,8.24156,3,4.50,0.876296,3.943335,11.83000,4,4.50,0.838561,3.773526,15.09410,5,4.50,0.802451,3.611030,18.05514,6,4.50,0.767895,3.455531,20.73318,7,4.50,0.734828,3.306728,23.14709,8,4.50,0.703185,3.164333,25.31466,9,4.50,0.672904,3.028070,27.25262,10,$104.50,0.643927,67.290443,672.90442,100.00000,826.87899,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,15,EXHIBIT 4-6,Calculation of Macaulay Duration and Modified Duration for 5-Year,6%,Bond Selling to Yield 9%,Coupon rate:,6.00%,Term(years):5,Initial yield:9.00%,Period,t,Cash Flow,PV of$1 at 4.5%,PV of CF,t PVCF,1,$3.00,0.956937,2.870813,2.87081,2,3.00,0.915729,2.747190,5.49437,3,3.00,0.876296,2.628890,7.88666,4,3.00,0.838561,2.515684,10.06273,5,3.00,0.802451,2.407353,12.03676,6,3.00,0.767895,2.303687,13.82212,7,3.00,0.734828,2.204485,15.43139,8,3.00,0.703185,2.109555,16.87644,9,3.00,0.672904,2.018713,18.16841,10,103.00,0.643927,66.324551,663.24551,Total,88.130923,765.89520,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,16,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,16,Measures of Bond Price Volatility,(continued),Properties of Duration,The 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 bond durations.,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,17,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,17,Measures of Bond Price Volatility,(continued),Approximating the Percentage Price Change,The below equation can be used to approximate the percentage 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 can be interpreted as the approximate percentage change in price for a 100-basis-point change in yield.,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,18,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,18,Measures of Bond Price Volatility,(continued),Portfolio Duration,The 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 portfolio 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 Hall,4-,19,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,19,Measures of Bond Price Volatility,(continued),Portfolio Duration,Portfolio managers look at portfolio duration for sectors of the bond market.,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,20,EXHIBIT 4-7,Calculation of Duration and Contribution to Portfolio Duration for a Asset Allocation to Sectors of the Lehman Brothers U.S.Aggregate Index:October 26,2007,Sector,Portfolio Weight,Sector Duration,Contribution to Portfolio Duration,Treasury,0.000,4.95,0.00,Agency,0.121,3.44,0.42,Mortgages,0.449,3.58,1.61,Commercial Mortgage-Backed Securities,0.139,5.04,0.70,Asset-Backed Securities,0.017,3.16,0.05,Credit,0.274,6.35,1.74,1.000,4.52,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,21,EXHIBIT 4-8,Calculation of Duration and Contribution to the Lehman Brothers Aggregate Index Duration:October 26,2007,Sector,Weight in Index,Sector Duration,Contribution to Index Duration,Treasury,0.230,4.95,1.14,Agency,0.105,3.44,0.36,Mortgages,0.381,3.58,1.36,Commercial Mortgage-Backed Securities,0.056,5.04,0.28,Asset-Backed Securities,0.010,3.16,0.03,Credit,0.219,6.35,1.39,Total,1.000,4.56,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,22,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,22,Measures of Bond Price Volatility,(continued),Portfolio Duration,While 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 for the recommended portfolio is expected given the greater allocation to non-Treasury sectors.,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,23,EXHIBIT 4-9,Calculation of Spread Duration and Contribution to Portfolio Spread Duration for an Asset Allocation to Sectors of the Lehman Brothers U.S.Aggregate Index:October 26,2007,Sector,Portfolio Weight,Sector Spread Duration,Contribution to Portfolio Spread Duration,Treasury,0.000,0.00,0.00,Agency,0.121,3.53,0.43,Mortgages,0.449,3.62,1.63,Commercial Mortgage-Backed Securities,0.139,5.04,0.70,Asset-Backed Securities,0.017,3.16,0.05,Credit,0.274,6.35,1.79,Total,1.000,4.60,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,24,EXHIBIT 4-10,Calculation of Spread Duration and Contribution to the Lehman Brothers Aggregate Index Spread Duration:October 26,2007,Sector,Weight in Index,Sector Spread Duration,Contribution to Index Spread Duration,Treasury,0.230,0.00,0.00,Agency,0.105,3.53,0.37,Mortgages,0.381,3.62,1.38,Commercial Mortgage-Backed Securities,0.056,5.04,0.28,Asset-Backed Securities,0.010,3.16,0.03,Credit,0.219,6.53,1.43,Total,1.000,3.49,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,25,Convexity,Because all the duration measures are only 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 Hall,4-,26,Exhibit 4-12,Line Tangent to the Price-Yield Relationship,Price,Yield,y,*,p,*,Actual Price,Tangent Line at,y*,(estimated price),Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,27,Convexity,(continued),The approximation will always understate the actual price.,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,28,Exhibit 4-13.,Price Approximation Using Duration,Price,Yield,y,3,p,*,Actual Price,Tangent Line at,y*,(estimated price),y,*,y,1,y,2,y,4,Error in Estimating Price Based only on Duration,Error in Estimating Price Based only on Duration,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,29,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,29,Convexity,(continued),Measuring Convexity,The 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:,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,30,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,Convexity,(continued),Measuring Convexity,In general,if the cash flows occur,m times per year,convexity,is adjusted to an annual figure as follows:,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,31,EXHIBIT 4-14,Calculation 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:,9.00%,Price:100,Period,t,Cash Flow,1/(1.045),t+2,t(t+1)CF,t(t+1)CF,(1.045),t+2,1,4.50,0.876296,9,7.886,2,4.50,0.838561,27,22.641,3,4.50,0.802451,54,43.332,4,4.50,0.767895,90,69.110,5,4.50,0.734828,135,99.201,6,4.50,0.703185,189,132.901,7,4.50,0.672904,252,169.571,8,4.50,0.643927,324,208.632,9,4.50,0.616198,405,249.560,10,104.50,0.589663,11,495,6,778.186,12,980,7,781.020,Copyright 2010 Pearson Education,Inc.Publishing as Prentice Hall,4-,32,EXHIBIT 4-15,Calculation of Convexity Measure and Dollar Convexity Measure for Five-Year,6%,Bond Selling to Yield 9%,Coupon rate:,6.00%,Term(years):5 Initial yield:,9.00%,Price:88.1309,Period,t,Cash Flow,1/(1.045),t+2,t(t+1)CF,t(t+1)CF,(1.045),t+2,1,3.00,0.876296,6,5.257,2,3.00,0.838561,18,15.094,3,3.00,0.802451,36,28.888,4,3.00,0.767895,60,46.073,5,3.00,0.734828,90,66.134,6,3.00,0.703185,126,88.601,7,3.00,0.672904,168,113.047,8,3.00,0.643927,216,139.088,9,3.00,0.616198,270,166.373,10,103.00,0.589663,11,330,6,680.891,12,32