1、 Credit risk arises from the possibility that borrowers and counterparties in derivatives transactions may default.22Approaches to estimating the probability that a company will defaultThe difference between risk-neutral and real-world probabilities of defaultCredit risk of derivativeDefault correla
2、tion,Gaussian copula models33Approaches to estimating default probabilitiesHistorical default probabilities of rating companiesFrom bonds pricesFrom equity pricesFrom derivatives pricesHistorical cumulative average default rates(%)InterpretationThe table shows the probability of default for companie
3、s starting with a particular credit ratingThe probability that a bond initially rated Baa will default during the second year is 0.506-0.181=0.325Default probability change with timeDefault Intensities vs Unconditional Default ProbabilitiesThe unconditional default probability is the probability of
4、default for a certain time period as seen at time zeroThe conditional default probability is the probability of default for a certain time period conditional on no earlier default(say,default intensity or hazard rate)Define V(t)as cumulative probability of the company surviving to time t.Taking limi
5、ts,we getDefine Q(t)as the probability of default by time t.Where is the average default intensity between 0 and t(1()(1()()V ttV ttV t ()()()V ttV tV t t 0()()()()()()1()1tdt tdVtVtdtVteQtVte ()tRecovery rateThe recovery rate for a bond is usually defined as the price of the bond immediately after
6、default as a percent of its face valueRecovery rates are significantly negatively correlated with default ratesRecovery rates(Moodys:1982 to 2006,Table 22.2,page 491)Using Bond Prices Average default intensity over life of bond is approximately Where s is the spread of the bonds yield over the risk-
7、free rate and R is the recovery rate.()1stR()tMore Exact CalculationAssume that a 5 year corporate bond pays a coupon of 6%per annum(semiannually).The yield is 7%with continuous compounding and the yield on a similar risk-free bond is 5%(continuous compounding).Price of risk-free bond is 104.09;pric
8、e of corporate bond is 95.34;expected loss from defaults is 8.75.Suppose that the probability of default is Q per year and that defaults always happen half way through a year(immediately before a coupon payment)CalculationsCalculations(Cons.)We set 288.48Q=8.75 to get Q=3.03%This analysis can be ext
9、ended to allow defaults to take pace more frequentlyInstead of assuming a constant unconditional probability of default we can assume a constant default intensity or a particular pattern for the variation of default probabilities with time.With several bonds we can use more parameters to describe th
10、e term structure of default probability.The Risk-Free RateThe risk-free rate when default probabilities are estimated is usually assumed to be the LIBOR/swap zero rate(or sometimes 10 bps below them)To get direct estimates of the spread of bond yields over swap rates we can look at asset swapsAsset
11、SwapsAsset swap spreads provide a direct estimate of the spread of bond yields over the LIBOR/swap curve.If the asset swap spread is 150 bps and the LIBOR/swap zero curve is flat at 5%.The expected loss from default over the 5-year life of the bond is therefore$6.55.6.55=288.48*Q,Q=2.27%Credit Defau
12、lt Swap Spreads(bps)Credit Default Swap Spreads(bps)Comparison historical vs bondCalculation of default intensities using historical data are based on equation(22.1)and table(22.1);From equation(22.1),we haveThe calculations using bond prices are based on equation(22.2)and bond yields published by M
13、errill Lynch.(7)1/7ln1(7)0.0011Q0.05993 0.052980.01161 0.4Real World vs Risk Neutral Default Probabilities,7 year averageRisk Premiums Earned by Bond TradersThe default probability from historical data is significantly lower than that from bond pricesThe ratio declines while the difference increases
14、 as a companys credit rating declines.Real World vs.Risk-Neutral Default ProbabilitiesThe default probabilities backed out of bond prices or credit default swap spreads are risk-neutral default probabilitiesThe default probabilities backed out of historical data are real-world default probabilitiesP
15、ossible reasons for these resultsCorporate bonds are relatively illiquidThe subjective default probabilities of bond traders may be much higher than the estimates from Moodys historical dataBonds do not default independently of each other.This leads to systematic risk that cannot be diversified away
16、.Bond returns are highly skewed with limited upside.The non-systematic risk is difficult to diversify away and may be priced by the market.Which world should we use?We should use risk-neutral estimates for valuing credit derivatives and estimating the present value of the cost of defaultWe should us
17、e real world estimates for calculating credit VaR and scenario analysisMertons modelMertons model regards the equity as an option on the assets of the firm.In a simple situation the equation value iswhere is the value of the firm and is the debt repayment required.max(,0)TVDTVDEquity vs.Assets An op
18、tion pricing model enables the value of the firms equity today,to be related to the value of its assets today,and the volatility of its assets,The risk-neutral probability that the company will default on the debt is .0012()()rTEVN dDeN d0E0VV2()NdVolatilities0010()EVVEVEEVN dVVwhere?ExampleA compan
19、ys equity is$3 million and the volatility of the equity is 80%The risk-free rate is 5%,the debt is$10 million and time to debt maturity is 1 yearSolving the two equations yields012.4 and 21.23%VVExample(Con.)The probability of default is The market value of the debt is The present value of the promi
20、sed payment is 9.51The expected loss is about(9.51-9.4)/9.51=1.2%The recovery rate is(12.7-1.2)/12.7=91%2()0.127Nd009.4VEImplementation of Mertons model(e.g.Moodys KMV)Mertons model produces a good ranking of default probabilities(risk-neutral or real-world)Moody 公司把股票当于公司资产期权的思想计算出风险中性世界的违约距离，再利用拥有
21、的海量历史违约数据库，建立起风险中性违约距离与现实世界违约率之间的对应关系，从而得到预期违约频率，作为违约概率的预测指标。贝尔斯登的预期违约频率从期权价格中引出风险中性违约概率 由于股票是公司资产的期权，这样股票期权就是期权的期权，其价格可以表达为：运用最大熵的办法（Capuano,2008)就可以从公司同期限的所有期权价格中估计出 和D0max(,0)()=(,0)()TTiirTTiTTVrTTiTTVD KCeVDKf VdVeVDKf VdV()Tf V从期权价格中可以推导出风险中性违约概率运用上述方法，我们就可根据2008年3月14日贝尔斯登将于2008年3月22日到期的期权价格，计
22、算出贝尔斯登的风险中性违约概率和公司价值的概率分布。贝尔斯登于2008年3月14日被摩根大通接管。下图显示，市场对贝尔斯登一周后的命运产生巨大分歧，公司价值大涨大跌的概率远远大于小幅变动的概率，这样的分布与正常情况的分布有天壤之别。可见期权价格可以让我们清楚地看出市场在非常时期对未来的特殊看法。贝尔斯登风险中性违约概率和公司价值概率分布（2008年3月14日）风险中性违约概率风险中性违约概率虽然不同于现实概率，但其变化可以反映现实世界违约概率的变化。在金融危机时期，它可能比CDS价差能更敏感地反映出违约概率的变化。在贝尔斯登于2008年3月14日被接管前后，根据上述方法计算出来的风险中性概率每
23、天的变化比CDS的价差更敏感。这是因为在金融危机期间，金融机构自身的信用度大幅降低，造成在OTC市场交易的CDS交易量急剧萎缩，价差大幅扩大，信号失真。期权隐含的中性违约概率与CDS价差Credit Risk MitigationNetting:incremental effectCollateralizationDowngrade triggersDefault correlationThe credit default correlation between two companies is a measure of their tendency to default at about t
24、he same timeFactors (1)macroeconomic environment:good economy =low number of defaults (2)Same industry or geographic area:companies can be similarly or inversely affected by an external event (3)credit contagion:connections between companies can cause a ripple effect Credit derivativeCredit derivati
25、ves are contracts where the payoff depends on the creditworthiness of one or more companies or countriesBuyers:banks or other financial institutionsSellers:insurance companySingle name:credit default swap,CDSHow does CDS works?This is a contract that provides insurance against the risk of a default
26、by particular company.The company is known as the reference entity and a default by the company is known as a credit event.The buyer of the insurance obtains the right to sell bonds issued by the company for their face value when a credit event occurs.The sellers of the insurance agrees to buy the b
27、onds for their face value when a credit event occur.ExampleA 5-year credit default swap on March 1,2009.The notional principal is$100 million.The buyer agrees to pay 90 basis points annually for protection against default by the reference entity.Default protection buyerDefault protection seller90 ba
28、sis points per yearPayment if default by reference entityMechanismIf not default,reference entity pays$900,000 on March 1 of each 2010-2014If default,e.g.June 1,2012;(1)specifies physical settlement;(2)determine the mid-market value of the cheapest deliverable bond,or say,cash paymentIn arrear payme
29、nt,including a final accrual paymentCDS spread:the total amount paid per year,as a percent of the notional principal,to buy protectionCDS and Bond yieldsA CDS can be used to hedge a position in a corporate bond.The n-year CDS spread should be approximately equal to the excess of the par yield on an
30、n-year corporate bond over the par yield on an n-year risk-free bond.How to use itCDS and Cheapest-to-deliver bondBonds typically have the same seniority,but they may not sell for the same percentage of face value immediately after a default.Search a cheapest-to-deliver bond.(1)LRMid-market CDS spre
31、adsExample:(1)Suppose the probability during a year conditional on no earlier default is 2%.Time(year)Time(year)default probabilitydefault probabilitysurvival probabilitysurvival probability1 10.020.020.980.982 20.01960.01960.96040.96043 30.01920.01920.94120.94124 40.01880.01880.92240.92245 50.01840
32、.01840.90390.9039 (2)Default always happen halfway through a year and that payments on the credit default swap are made once a year at the end of each year.(3)The risk-free interest rate is 5%per annum with continuous compounding and the recover rate is 40%.1Default 123450Default 2Default 3Default 4
33、Default 5PayoffAccrual payment.Payment 1Payment 2Payment 3 Payment 4 Payment 5Survival probabilityDefault probabilityAssume notional principal is 1 and payment at rate of s per year.Time(year)survival probabilityexpected paymentdiscount factorpv of expected payment10.980.98s0.95120.9322s20.96040.960
34、4s0.90480.8690s30.94120.9412s0.86070.8101s40.92240.9224s0.81870.7552s50.90390.9039s0.77880.7040stotal4.0704sAssume notional principal is 1,defaults always happen halfway of a year.Time(year)default probabilityrecovery rateexpected paymentdiscount factorpv of expected payment10.020.40.0120.95120.0117
35、20.01960.40.01180.90480.010930.01920.40.01150.86070.010240.01880.40.01130.81870.009550.01840.40.01110.77880.0088total0.0511Assume notional principal is 1,defaults always happen halfway of a year.Time(year)default probabilityaccrual paymentdiscount factorpv of expected payoff10.020.5s0.97530.0098s20.01960.5s0.92770.0091s30.01920.5s0.88250.0085s40.01880.5s0.83950.0079s50.01840.5s0.79850.0073stotal0.0426sMarking to market a CDSBy product:Estimating default probabilities and recover rate with CDS quoted spread.524.07040.04260.05114.07040.04260.0511ssss
