1、FIXED INCOME SECURITIESLECTURE 6BOND RISK债券风险（1）利率风险；（2）违约风险（信用风险）；（3）通货膨胀风险；（4）提前偿还风险；（5）汇率风险；（6）流动性风险。Default Risk(1)Default Risk:Uncertainty that the realized return will deviate from the expected return because the issuer will fail to meet the contractual obligations specified in the indenture.T
2、he major concern is failure to meet interest and principal payments.Default Risk:Points (2)Most investors do not directly access a bonds default risk,but instead use the quality ratings provided by Moodys,S&P,and Fetch to evaluate the degree of risk.AAA,AA,A,BBB,C,Default Rates:.12%per year for all
3、bonds since WWII.1980s:3.27%per year on junk bonds.1990-1991:9%on junk bonds Default Risk:Points (3)Salomon Brothers and Hutch Studies:Empirical studies that looked at the relation between default risk premium(RP)and the state of the economy.RPYTMYTMCGeneric low quality bondAGeneric high quality bon
4、drelationRP widens in recession or theecationRP narrows inansion or theectationCA.:(exp).exp(exp).Default Risk:Points (4)Johnston Study:Study looked at the yield curves for different quality bonds.Found that the YC for lower quality bonds tended on average to be negatively sloped.Reason:Greater conc
5、ern over the repayment of principal on low quality bonds.MCYTMDefault Risk:Points (5)McEnnally-Boardman Study:Applied the Evans and Archer methodology to bond portfolios grouped in terms of their quality ratings.Found that lower quality bond portfolios had less risk because of their lower correlatio
6、ns.nAAAnBCall Risk(1)Call Risk:Uncertainty that the realized return will deviate from the expected return because the issuer calls the bond,forcing the investor to reinvest in a market with lower rates.Note:When a bond is called the holder receives the call price(CP).Since the CP usually exceeds the
7、 principal,the return the investor receives over the call period is often greater than the initial YTM.The investor,though,usually has to reinvest in a market with lower rates which often causes his return for the investment period to be less than the initial YTM.Call Risk(2)Example:Compare the ARR
8、for the call period with the ARR for the investment period for a bond that is called.Buy:10-year,10%annual coupon bond at par($1000);callable at 110:CP=$1100.Assume:HD=10 years.Flat YC at 10%.YC stays at 10%until the end of year 3.Year 3,the YC shifts down to a flat 8%and the bond is called.Investor
9、 reinvest at 8%for the next 7 years.Call Risk(3)ARRCall Date ValuePARRCall Date ValueCBCC032111268100 110100 11010011001431$1431$1000.(.)(.)ARRHD ValuePARRARRHDBHDHDHD071037101108109381126810810938$1431(.)$1000.(.)(.).ARRYTMHD938%10%.Call Risk(4)Points:Call Risk Premium:Price Compression:Call featur
10、es put limitations on the price-yield curve.At the rate where the bond could be called(YTM*),the YC flattens,with the price equal to the CP.RPYTMYTMCallNCRP greater in highererest rate periodsRRPint Call Risk(5)Points:Need for a valuation model for callable bonds different from the PV model.A 10-yea
11、r callable bond may be more like a 3-year bond.Binomial Tree Model or option pricing model.YTMPB0AAYTM*ACPAA NonCallable BondAA Callable Bond债券利率风险包括:价格(市场）风险;再投资风险 所有债券价格都受利率变化的影响对于高等级债券利率风险更重要债券价值和收益率的关系 假定未来现金流的数额和时间都是确定的，其风险通过贴现率来调整 例子:ATT 6s 09 面值=$1,000;到期日:Oct.15,2009 年利息=(6%of$1,000)/2=$30 支
12、付日:April 15,October 15 YTM Bond Price PV of Coupons +PV of Par5.14%1,050.03349.03701.005.64%1,020.58343.08677.516.14%992.13337.27654.866.64%964.63331.61633.027.14%938.04326.08611.967.64%912.34320.69591.658.14%887.50315.43572.068.64%863.47310.30553.169.14%840.23305.29534.939.64%817.75300.41517.35该债券在
13、2002年10月15日的价值债券价格和收益率关系 债券价格和收益率反方向变化$0$50$100$150$2000%2%4%6%8%10%12%14%16%利率价格债券价格和到期日之间的关系 随着到期日的临近，债券的价格趋向于面值10%Return6%Return758595105115125109876543210距到期日的年数Price债券价格波动性的特点债券价格波动性的特点 1、价格的利率敏感性与债券的票面利率具有反向关系。其他因素相同时，低票面利率债券比高票面利率债券价格的利率敏感性更强。2、价格的利率敏感性与债券的到期时间长短具有正向关系。其他因素相同时，长期债券比短期债券价格的利率敏
14、感性更强。3、随着到期时间的增长，价格的利率敏感性增加，但是增加得越来越慢。债券价格波动性的特点（续）债券价格波动性的特点（续）4、收益率上升导致价格下跌的幅度比等规模的收益率降低带来的价格上涨的幅度小，这被称为价格波动的不对称性。5、价格的利率敏感性与债券的初始收益率水平具有反向关系。其他因素相同时，债券的初始收益率较低时，价格的利率敏感性更强。Bond Pricing Relationships Inverse relationship between price and yield An increase in a bonds yield to maturity results in a
15、 smaller price decline than the gain associated with a decrease in yield Long-term bonds tend to be more price sensitive than short-term bondsBond Pricing Relationships(cont.)As maturity increases,price sensitivity increases at a decreasing rate Price sensitivity is inversely related to a bonds coup
16、on rate Price sensitivity is inversely related to the yield to maturity at which the bond is selling价格价格收益率收益率价格和收益率关系ExamplesBondCoupon Maturity Initial YTMA12%5 years10%B12%30 years10%C3%30 years10%D3%30 years6%ABCDChange in yield to maturity(%)Percentage change in bond price0期限越长的债券价格的利率敏感性越大ex.A
17、BC票面利息($)909090面值 1,000 1,000 1,000Moodys RatingAaAaAa期限 5 yrs.10 yrs.15 yrs.YTM9%10%11%价格1,000939856Let yields decrease by 10%(8.1%,9%,and 9.9%respectively).新价格:1,0361,000931%Price change:3.6%6.6%8.8%债券期限长度和利率风险 债券期限越长，利率风险越大$0$50$100$150$200$2500%2%4%6%8%10%12%14%16%RatePrice10 Year20 Year5 YearIn
18、terest Rate Risk Bond Pricing TheoremsAs maturity increases,price sensitivity increases at a decreasing rateex.See previous slide.The price change for B is 3%higher than A,but the price change for C is only 2.2%higher than B.oBond prices are more sensitive to a decline in i-rates than a rise in i-ra
19、tesLet yields increase by 10%(9.9%,11%,12.1%respectively).New prices are:966882790%Price changes:-3.4%-6.1%-7.7%Compare these price changes with the ones resulting from a decline in i-rates provided above.Interest Rate Risk Bond Pricing TheoremsoLow coupon bond prices are more sensitive to i-rate ch
20、anges than high coupon bond pricesex.ABCoupon($)60100Face Value1,000 1,000Moodys RatingAaAaTerm-to-maturity10 yrs.10 yrs.YTM12%12%Price661887Let yields decrease to 11%.New prices are:706942%price changes:6.7%6.2%利息额的大小与利率风险 票面利率越低，利率风险越大.$0$50$100$150$2001%3%5%7%9%11%13%15%RatePriceCouponZero While
21、term to maturity is a major determinant of interest rate risk,it is not the only factor Duration:proper measure of interest rate sensitivity An elasticity concept percentage change in bond price as a result of a one percentage change in yield Can be derived from the PV formula(note“minus”sign)Durati
22、on 1)y1/()y1(P/PDDuration 2 A measure of the effective maturity of a bond The weighted average of the times until each payment is received,with the weights proportional to the present value of the paymentWhy is duration important?Measures the interest rate risk of a bond Allows comparison across bon
23、ds that differ in coupon rate,yield and maturity Also a measure of the effective maturity of a bond(weighted average maturity of a bonds cash flows)An essential concept in bond portfolio management,particularly in immunization strategies(protecting bond portfolios from interest rate risk)久期 到期期限不能很好
24、地衡量债券的实际期限 需要考虑利息额的大小 计算平均值:“有效期限”(Macaulay)久期久期:现金流发生时间的加权平均值,权重为各现金流的现值占所有现金流现值总和的比重:TtttTtttYTMCFYTMtCFD11)1()1(Duration:CalculationtttwCFyice()1PrDt wtTt1CFCashFlowforperiodttDuration Calculation8%BondTimeyearsPayment PV of CF(10%)Weight C1 XC418072.727.0765.076528066.116.0690.1392Sum31080811.42
25、0950.263.85391.00002.56172.77748%BondTimeyearsPaymentPV of CF(10%)WeightC1 XC4.54038.095.0395.019714036.281.0376.03761.52.0401040sum34.553855.611964.540.0358.88711.000.05371.77421.8852例子:久期 计算开发银行债券的久期 发行人发行人 开发银行 面值面值 100 票面利率票面利率 8%期限期限 10 年 假设该债券市场价格为114.72元，要求的利率为 6%。存续期可计算如下:年存续期445.772.114)06.
26、01(10010)06.01(810110ttt零息债券的久期计算 -面值面值 100 到期日支付的利息额到期日支付的利息额 105.45 期限期限10 年 要求的利率为 6%，给定价格为P0=114.72，存续期计算如下:1072.114)06.01()45.105100(1010存续期久期通常,久期比债券的到期期限要短零息债券的久期正好等于其到期期限 Duration can be used to estimate the percentage change in price for a given change in yield:Duration/Price Relationshipy1
27、)y1(DPPy1DD*yDPP*wOr,if we denote D*=modified duration Example:y=+100 basis points,D*=5 P/P=-5 x 0.01=-0.05 or 5%Duration can be used to estimate the percentage change in price for a given change in yield:Duration/Price Relationshipy1)y1(DPPy1DD*yDPP*wOr,if we denote D*=modified duration Example:y=+
28、100 basis points,D*=5 P/P=-5 x 0.01=-0.05 or 5%Rules for DurationRule 1 The duration of a zero-coupon bond equals its time to maturityRule 2 Holding maturity constant,a bonds duration is higher when the coupon rate is lowerRule 3 Holding the coupon rate constant,a bonds duration generally increases
29、with its time to maturityCAPPP 1(1)TCrPACr(1)TAPAr 11()CAdPdPr dPrDPdrPdrdr (1)CCAdPPTCPdrrrr1AAdPP Tdrr Zero coupon bonds Level perpetuity Par bonds(C=r,P=A)1(1)(1)CAPPCDTPrPr1(1(1)(1)TDrrDuration of Annuity(Mortgage)1(1)TrPMr1(1)TdPPMTdrrrr(1)1(1)1Tr dPrTDPdrrrRules for Duration(contd)Rule 4 Holdi
30、ng other factors constant,the duration of a coupon bond is higher when the bonds yield to maturity is lowerRule 5 The duration of a level perpetuity is equal to:(1)rrRules for Duration(contd)Rule 6 The duration of an annuity(mortgage)is equal to:1(1)1TrTrrDuration of portfolio The duration of portfo
31、lio is a weighted sum of the duration of the individual bondswith weighting coefficients proportional to individual bond prices.Duration of portfolio1nAkkAkAt PVDP1nBkkBkBt PVDP Duration of portfolio1()NAABBABkkkkP DP DtPVPVAABBP DP DPAABBP DP DDPP久期缺口分析 DURgap=DURa-（L/A*DURl）%NW-DURgapi/（1+i）一家商业银行
32、的资产价值为100，负债价值为95，当利率从10%上升到15%时，这家银行的净资产价值有何变化？DURgap=2.7-（95/100*1.03）=1.72%NW-1.720.05/（1+0.10）=-7.8%久期和凸性PriceDurationPricing Error from convexityTaylor expansion221d PConvexitydyP2221()()()2dPd PP yyP ydydyerrordydyCorrection for ConvexityModify the pricing equation:Convexity is Equal to:Nttttt
33、yCFP122)1(y)(11Where:CFt is the cashflow(interest and/or principal)at time t.2)y(Convexity21y*DPPConvexity(1)Duration is a measure of the slope of the price-yield curve at a given point-first-order derivative.Convexity is a measure of the change in the slope of the price-yield curve-second-order der
34、ivative.Convexity measures how bowed-shaped the price-yield curve is.Convexity(2)Property:The greater a bonds convexity,the greater its capital gains and the smaller its capital losses for given absolute changes in yields.YTMPB0y0PB0YTMy1y2K GainK Lossy0y1y2K GainK Lossyyyy1020Convexity(3)Measure:An
35、nualized ConvexityConvexity for n lengthn2Convexity(4)The convexity measure for a bond which pays coupons each period and its principal at maturity:ConvexityCyyCMyyM MFC yyPMMMB2111211132120()()()(/)()Convexity(5)Example:The convexity in half-years for a 10-year,9%coupon bond selling at par(F=100)an
36、d paying coupons semiannually is 225.43 and its annualized convexity is 56.36:ConvexityAnnualized Convexity2 450451110452 45 20045104520 21 10045 0451045100225432254325636320221222(.).(.)(.)()(.)(.)()()(./.)(.).Convexity(6)Uses:Descriptive Parameter:Greater k-gains and smaller k-losses the greater a
37、 bonds convexity.Estimation of :Using Taylor Expansion,a better estimate of price changes to discrete changes in yield than the duration measure can be obtained by combining duration and convexity measures.%/%PyBConvexity(7)Taylor Expansion:For 10-year,9%bond,an increase in the annualized yield by 2
38、00 BP(9%to 11%)would lead to an estimated 11.87%decrease in price using Taylor Expansion(the actual is 12%):%()PModified DurationyConvexityyB122%.(.).(.).PB 65021256 360211872Convexity(8)Note:Using Taylor Expansion the percentage increases in price are not symmetrical with the percentage decreases f
39、or given absolute changes in yields.yfromtoPB9%11%65021256 360211872:%.(.).(.).yfromtoPB9%7%65021256 360214132:%.(.).(.).Convexity example8%BondTime(half year)PaymentPV of CF(10%)Weight(t2+t)xC414038.095.0395.079024036.281.0376.225734401040sum34.553855.611964.540.0358.88711.000.429917.741318.4759Con
40、vexity example Like duration,convexity is computed as a weighted average of the terms(t2+t)(rather than t),and divided by(1+y)2 In the above example,convexity is equal to:18.4759/1.052=16.7582Duration:OK for small changes in the yield Convexity-adjusted approximation provides a more accurate measure
41、 for larger changes in yield Investors like convexity why?Convexity of Two Bonds0Change in yield to maturity(%)Percentage change in bond priceBond ABond B久期和凸性假定 Derivation and application of duration and convexity assumes:Term structure is flat Shifts are parallel Bonds have no imbedded options债券久期
42、的近似计算 yPPP02近似久期例子 期限为25年票面利率为6%的债券到期收益率为 9%,最初的价格P0 为 70.3570 步骤 1:将收益率从 9%提高到 9.1%.y 为0.001.债券价格降为(P+)69.6164 步骤 2:将收益率从 9%降为 8.9%.债券价格则(P-)为 71.1105 近似久期可计算如下:例子62.10)001.0)(3570.70(26164.691105.71近似久期Derivation of Duration and Convexity(1)Duration:Take derivative with respect to y:PCFyCFyPCFyCFy
43、CFyBtttMtttMBMM0110112211111 ()()()()()dPdyCFyCFyM CFyBMM ()()()()()()()1121112231Derivation of Duration and Convexity(2)Factor out dPdyCFyCFyM CFyBMM ()()()()()()()1121112231()()11111yydPdyyCFyCFyM CFydPdyyCFyCFyMCFydPdyyPV CFPV CFMPV CFBMMBMMBM 1111211111121111121122112212()()()()()()()()()()()()(
44、)()()()()()()()()Derivation of Duration and Convexity(3)Divide through by P:dPdyPdPPdyyPV CFPPV CFPMPV CFPBBBMB1111210200 /()()()()()()()Modified DurationdPPdyytPV CFPtMtB/()()1110Derivation of Duration and Convexity(1)Convexity:Take the derivative of dPdyCFyCFyM CFyBMM ()()()()()()()1121112231 2100
45、1200220)2(423122)1()()1(1)1()1(/1:exp)1()1()1(6)1(2tMttBBMtttBBBBMMByCFPVttPPyCFttConvexitydyPdPdyPdsummationaasressandPbythroughDivideyCFMMyCFyCFdyPdDefinitionVaR is defined as the predicted worst-case loss at a specific confidence level(e.g.99%)over a certain period of time.VaR=expected profit/los
46、s-worst case loss at a specific confidence level c(e.g.99%)V=current marked-to-market value of the portfolio,R=Return over the horizon H,=Expected return,R=the return corresponding to the worst case loss at the c(e.g.99%)confidence level VaR(H;c)=E(V)-V=V(-R)If R is normally distributed with mean,and standard deviation,then Prob(R R)=Prob(R-)/(R-)/)=1-cThe cut-off return Rcan be expressed as:R=+VaR(H;c)=-V P=-PD y（P）=-PD （y）2.33-2.331%
