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1、Chapter 13 Credit Risk1西南财经大学期权期货及其他衍生品1What is credit risk?Credit risk arises from the possibility that borrowers and counterparties in derivatives transactions may default.222西南财经大学期权期货及其他衍生品1ContentsApproaches to estimating the probability that a company will defaultThe difference between risk-ne
2、utral and real-world probabilities of defaultCredit risk of derivativeDefault correlation, Gaussian copula models3333西南财经大学期权期货及其他衍生品1Approaches to estimating default probabilitiesHistorical default probabilities of rating companiesFrom bonds pricesFrom equity pricesFrom derivatives prices4西南财经大学期权期
3、货及其他衍生品1Historical cumulative average default rates (%)5西南财经大学期权期货及其他衍生品1InterpretationThe table shows the probability of default for companies 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 prob
4、ability change with time6西南财经大学期权期货及其他衍生品1Default Intensities vs Unconditional Default ProbabilitiesThe unconditional default probability is the probability of default for a certain time period as seen at time zeroThe conditional default probability is the probability of default for a certain time p
5、eriod conditional on no earlier default(say, default intensity or hazard rate)7西南财经大学期权期货及其他衍生品1Define V(t) as cumulative probability of the company surviving to time t.Taking limits, we getDefine Q(t) as the probability of default by time t.Where is the average default intensity between 0 and t8西南财
6、经大学期权期货及其他衍生品1Recovery rateThe recovery rate for a bond is usually defined as the price of the bond immediately after default as a percent of its face valueRecovery rates are significantly negatively correlated with default rates9西南财经大学期权期货及其他衍生品1Recovery rates (Moodys:1982 to 2006, Table 22.2, page
7、 491)(Moodys:1982 to 2006, Table 22.2, page 491)10西南财经大学期权期货及其他衍生品1Using Bond Prices Average default intensity over life of bond is approximately Where s is the spread of the bonds yield over the risk-free rate and R is the recovery rate.11西南财经大学期权期货及其他衍生品1More Exact CalculationAssume that a 5 year
8、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; price of corporate bond is 95.34; expected loss from defaults is 8.75.Suppose that the
9、probability of default is Q per year and that defaults always happen half way through a year (immediately before a coupon payment)12西南财经大学期权期货及其他衍生品1Calculations13西南财经大学期权期货及其他衍生品1Calculations (Cons.)We set 288.48Q=8.75 to get Q=3.03%This analysis can be extended to allow defaults to take pace more
10、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 the term structure of default probability.14
11、西南财经大学期权期货及其他衍生品1The Risk-Free RateThe risk-free rate when default probabilities are estimated is usually assumed to be the LIBOR/s rate( or sometimes 10 bps below them)To get direct estimates of the spread of bond yields over s we can look at asset swaps15西南财经大学期权期货及其他衍生品1Asset SwapsAsset s provide
12、 a direct estimate of the spread of bond yields over the LIBOR /s.If the asset s is 150 bps and the LIBOR /s 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%16西南财经大学期权期货及其他衍生品1Credit Default S (bps)17西南财经大学期权期货及其他衍生品1Credi
13、t Default S (bps)18西南财经大学期权期货及其他衍生品1Comparison 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 Merrill L
14、ynch.19西南财经大学期权期货及其他衍生品1Real World vs Risk Neutral Default Probabilities, 7 year average20西南财经大学期权期货及其他衍生品1Risk Premiums Earned by Bond Traders21西南财经大学期权期货及其他衍生品1The default probability from historical data is significantly lower than that from bond pricesThe ratio declines while the difference incr
15、eases as a companys credit rating declines.22西南财经大学期权期货及其他衍生品1Real World vs. Risk-Neutral Default ProbabilitiesThe default probabilities backed out of bond prices or credit default s are risk-neutral default probabilitiesThe default probabilities backed out of historical data are real-world default
16、probabilities23西南财经大学期权期货及其他衍生品1Possible 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 ri
17、sk that cannot be diversified away.Bond returns are highly skewed with limited upside. The non-systematic risk is difficult to diversify away and may be priced by the market.24西南财经大学期权期货及其他衍生品1Which world should we use?We should use risk-neutral estimates for valuing credit derivatives and estimatin
18、g the present value of the cost of defaultWe should use real world estimates for calculating credit VaR and scenario analysis25西南财经大学期权期货及其他衍生品1Mertons 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 fi
19、rm and is the debt repayment required.26西南财经大学期权期货及其他衍生品1Equity vs. Assets An option 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
20、 is .27西南财经大学期权期货及其他衍生品1Volatilities?28西南财经大学期权期货及其他衍生品1ExampleA companys 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 yields29西南财经大学期权期货及其他衍生品1Example (Con.)The probability
21、of default is The market value of the debt is The present value of the promised 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%30西南财经大学期权期货及其他衍生品1Implementation of Mertons model (e.g. Moodys KMV)Mertons model produces a good ranking of default p
22、robabilities (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。31西南财经大学期权期货及其他衍生品1贝尔斯登的预期违约频率32西南财经大学期权期货及其他衍生品1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D33西南财经大学
23、期权期货及其他衍生品1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。34西南财经大学期权期货及其他衍生品1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）35西南财经大学期权期货及其他衍生品1风险中
24、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。36西南财经大学期权期货及其他衍生品1期权隐含的中性违约概率与CDS价差37西南财经大学期权期货及其他衍生品1Credit Risk MitigationNetting: incremental effectColl
25、ateralizationDowngrade triggers38西南财经大学期权期货及其他衍生品1Default correlationThe credit default correlation between two companies is a measure of their tendency to default at about the same timeFactors (1) macroeconomic environment: good economy = low number of defaults (2) Same industry or geographic area:
26、 companies can be similarly or inversely affected by an external event (3) credit contagion: connections between companies can cause a ripple effect 39西南财经大学期权期货及其他衍生品1Credit derivativeCredit derivatives are contracts where the payoff depends on the creditworthiness of one or more companies or count
27、riesBuyers: banks or other financial institutionsSellers: insurance companySingle name: credit default swap, CDS40西南财经大学期权期货及其他衍生品1How does CDS works?This is a contract that provides insurance against the risk of a default by particular company. The company is known as the reference entity and a def
28、ault 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 bonds for their face value when a credit event occur.41西南财经大学期权期货及其他衍生品1Exam
29、pleA 5-year credit default s 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 basis points per yearPayment if default by reference entity4
30、2西南财经大学期权期货及其他衍生品1MechanismIf 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 payment, including a final accr
31、ual paymentCDS spread: the total amount paid per year, as a percent of the notional principal, to buy protection43西南财经大学期权期货及其他衍生品1CDS 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 n-y
32、ear corporate bond over the par yield on an n-year risk-free bond.How to use it44西南财经大学期权期货及其他衍生品1CDS 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.45西南财经大
33、学期权期货及其他衍生品1Valuation of credit default swapsMid-market CDS spreadsExample:(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 20.01960.01960.96040.9604
34、3 30.01920.01920.94120.94124 40.01880.01880.92240.92245 50.01840.01840.90390.903946西南财经大学期权期货及其他衍生品1Valuation of credit default swaps (cons.) (2) Default always happen halfway through a year and that payments on the credit default s made once a year at the end of each year. (3) The risk-free interes
35、t rate is 5% per annum with continuous compounding and the recover rate is 40%.47西南财经大学期权期货及其他衍生品11Default 123450Default 2Default 3Default 4Default 5PayoffAccrual payment.Payment 1Payment 2Payment 3 Payment 4 Payment 5Survival probabilityDefault probability48西南财经大学期权期货及其他衍生品1PV of the expected payme
36、ntAssume notional principal is 1 and payment at rate of s per year.Time(year)survival probabilityexpected paymentdiscount factorpv of expected payment10.980.98s0.95120.9322s20.96040.9604s0.90480.8690s30.94120.9412s0.86070.8101s40.92240.9224s0.81870.7552s50.90390.9039s0.77880.7040stotal4.0704s49西南财经大
37、学期权期货及其他衍生品1PV of the expected payoffAssume notional principal is 1, defaults always happen halfway of a year.Time(year)default probabilityrecovery rateexpected paymentdiscount factorpv of expected payment10.020.40.0120.95120.011720.01960.40.01180.90480.010930.01920.40.01150.86070.010240.01880.40.01
38、130.81870.009550.01840.40.01110.77880.0088total0.051150西南财经大学期权期货及其他衍生品1PV of the last accrued paymentAssume notional principal is 1, defaults always happen halfway of a year.Time(year)default probabilityaccrual paymentdiscount factorpv of expected payoff10.020.5s0.97530.0098s20.01960.5s0.92770.0091s30.01920.5s0.88250.0085s40.01880.5s0.83950.0079s50.01840.5s0.79850.0073stotal0.0426s51西南财经大学期权期货及其他衍生品1Valuation at or after the negotiationMarking to market a CDSBy product: Estimating default probabilities and recover rate with CDS quoted spread.52西南财经大学期权期货及其他衍生品1
