1、資管所 陳竑廷22.1 Credit Ratings22.2 Historical Data22.3 Recovery Rate22.4 Estimating Default Probabilities from bond price22.5 Comparison of Default Probability estimates22.6 Using equity price to estimate Default Probabilities Credit Risk Arise from the probability that borrowers and counterparties in d
2、erivatives transactions may default.S&P AAA,AA,A,BBB,BB,B,CCC,CC,C Moody Aaa,Aa,A,Baa,Ba,B,Caa,Ca,C Investment grade Bonds with ratings of BBB(or Baa)and abovebestworstFor a company that starts with a good credit rating default probabilities tend to increase with timeFor a company that starts with a
3、 poor credit rating default probabilities tend to decrease with time The unconditional default probability the probability of default for a certain time period as seen at time zero39.717-30.494=9.223%The default intensity(hazard rate)the probability of default for a certain time period conditional o
4、n no earlier default100 30.494=69.506%0.09223/0.69506=13.27%ttimetosurvivingcompanytheofyprobabilitcumulativethetVttimeatsintensitiedefaultthet:)(:)(defaultearliernooncondtionalttandttimebetweendefaultofyprobabilitthett:)(etdtVtVtdttdVtttVtttVttVttVttVttVtttVttVtV0)()()()()()()()()()()()()()()()()(Q
5、(t):the probability of default by time t(22.1)eettdttVtQ)()(11)(1)(0 Defined as the price of the bond immediately after default as a percent of its face value Moody found the following relationship fitting the data:Recovery rate=59.1%8.356 x Default rate Significantly negatively correlated with defa
6、ult ratesSource:Corporate Default and Recovery Rates,1920-2006 Assumption The only reason that a corporate bond sells for less than a similar risk-free bond is the possibility of default In practice the price of a corporate bond is affected by its liquidity.raterecoveryexpectedtheRyieldbondcorporate
7、theofspreadthesyearperintentisydefaultaveragethe:Rs1(22.1)%33.34.0102.0200%40bpsR)1(11)1()1(*1*1*)1()(RssReReRessrrff11R11-1-fre1*1*)1(freR*1*)1(Taylor expansion Suppose that Face value=$100,Coupon=6%per annum,Last for 5 years Corporate bond Yield:7%per annum$95.34 Risk-free bond Yield:5%per annum$1
8、04.094 The expected loss=104.094 95.34=$8.75Q:the probability of default per year288.48Q=8.75Q=3.03%0 1 2 3 4 5e-0.05*3.5 The default probabilities estimated from historical data are much less than those derived from bond pricesThe probability of the bond surviving for T years is(22.1)(1ln(1)(1)()(t
9、QtttQett%11.000759.01ln71)7(1ln71)7(Q A-rated bonds,Merrill Lynch 1996/12 2007/10The average yield was 5.993%The average risk-free rate was 5.289%The recovery rate is 40%16.14.0105298.005993.01Rs(22.2)0.11*(1-0.4)=0.066 Risk-neutral default probabilities implied from bond yields Value credit derivat
10、ives or estimate the impact of default risk on the pricing of instruments Real-world default probabilities implied from historical data Calculate credit VaR and scenario analysis Unfortunately,credit ratings are revised relatively infrequently.The equity prices can provide more up-to-date informatio
11、nIf VT D,ET=VT-D)0,max(DVETTV0 And 0 cant be directly observable.But if the company is publicly traded,we can observe E0.Mertons model gives the value firms equity at time T asSo we regard ET as a function of VTWe write)0,max(DVETT(*)()(*)()(2211tdwVdtVdVtdwdtVdVtdwEdtEdEtdwdtEdEVVEEdVVEdELemma sIto
12、By V offunction a is EOther term without dW(t),so ignore itReplace dE,dV by(*)(*)respectivelyWe compare the left hand side of the equation above with that of the right hand side)()()(221tdwVVEdtVVEtdwVdtVVEtdwEdtEVVEVEVEVVEEtdWVVEtdWEdtVVEdtE)()(and21(22.4)Suppose thatE0=3(million)r=0.05 D=10 E=0.80
13、 T=1Solvingthen getV0=12.400=0.2123N(-d2)=12.7%2020010021000),(),()N(:),()N()N(:),(VGVFminimizeVdEVGdDedVEVFVVVEVrTVSolvingF(x,y)2+G(x,y)2=(D2)2+(E2)2F(x,y)=A2*NORMSDIST(LN(A2/10)+(0.05+B2*B2/2)/B2)-10*EXP(-0.05)*NORMSDIST(LN(A2/10)+(0.05+B2*B2/2)/B2-B2)G(x,y)=NORMSDIST(LN(A7/10)+(0.05+B7*B7/2)/B7)*A7*B7 Initial V0=12.40,0=0.2123 Initial V0=10,0=0.1%.)/.-.(.e.E V*.-917.12/)2.17.12(raterecovery The2151949519 loss expected The51910payment promised theof luepresent va The4934012debt theof mark value The%7.12)N(-d1050002
