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1、Chapter 13 Credit Risk,What is credit risk?,Credit risk arises from the possibility that borrowers and counterparties in derivatives transactions may default.,2,2,2,Contents,Approaches to estimating the probability that a company will default The difference between risk-neutral and real-world probab
2、ilities of default Credit risk of derivative Default correlation, Gaussian copula models,3,3,3,Approaches to estimating default probabilities,Historical default probabilities of rating companies From bonds prices From equity prices From derivatives prices,Historical cumulative average default rates
3、(%),Interpretation,The table shows the probability of default for companies starting with a particular credit rating The probability that a bond initially rated Baa will default during the second year is 0.506-0.181=0.325 Default probability change with time,Default Intensities vs Unconditional Defa
4、ult Probabilities,The unconditional default probability is the probability of default for a certain time period as seen at time zero The conditional default probability is the probability of default for a certain time period conditional on no earlier default(say, default intensity or hazard rate),De
5、fine V(t) as cumulative probability of the company surviving to time t. Taking limits, we get Define Q(t) as the probability of default by time t. Where is the average default intensity between 0 and t,Recovery rate,The recovery rate for a bond is usually defined as the price of the bond immediately
6、 after default as a percent of its face value Recovery rates are significantly negatively correlated with default rates,Recovery 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 o
7、ver the risk-free rate and R is the recovery rate.,More Exact Calculation,Assume 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
8、 104.09; price 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),Calculations,Calculations (Cons.),We set 288.48Q=8.75 to get Q=3.03% Thi
9、s analysis can be extended to allow defaults to take pace more frequently Instead 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 pa
10、rameters to describe the term structure of default probability.,The Risk-Free Rate,The 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 ca
11、n look at asset swaps,Asset Swaps,Asset 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
12、.6.55=288.48*Q, Q=2.27%,Credit Default Swap Spreads (bps),Credit Default Swap Spreads (bps),Comparison historical vs bond,Calculation of default intensities using historical data are based on equation (22.1) and table (22.1); From equation (22.1), we have The calculations using bond prices are based
13、 on equation (22.2) and bond yields published by Merrill Lynch.,Real World vs Risk Neutral Default Probabilities, 7 year average,Risk Premiums Earned by Bond Traders,The default probability from historical data is significantly lower than that from bond prices The ratio declines while the difference
14、 increases as a companys credit rating declines.,Real World vs. Risk-Neutral Default Probabilities,The default probabilities backed out of bond prices or credit default swap spreads are risk-neutral default probabilities The default probabilities backed out of historical data are real-world default
15、probabilities,Possible reasons for these results,Corporate bonds are relatively illiquid The subjective default probabilities of bond traders may be much higher than the estimates from Moodys historical data Bonds do not default independently of each other. This leads to systematic risk that cannot
16、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.,Which world should we use?,We should use risk-neutral estimates for valuing credit derivatives and estimating the present value of the cost
17、 of default We should use real world estimates for calculating credit VaR and scenario analysis,Mertons model,Mertons model regards the equity as an option on the assets of the firm. In a simple situation the equation value is where is the value of the firm and is the debt repayment required.,Equity
18、 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 is .,Volatilities,?,Example,A companys equity is $3 million an
19、d 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 year Solving the two equations yields,Example (Con.),The probability of default is The market value of the debt is The present value of the promised payment is 9.51 The expected los
20、s is about (9.51-9.4)/9.51=1.2% The recovery rate is (12.7-1.2)/12.7=91%,Implementation 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
21、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D,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
22、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,C
23、redit Risk Mitigation,Netting: incremental effect Collateralization Downgrade triggers,Default correlation,The credit default correlation between two companies is a measure of their tendency to default at about the same time Factors (1) macroeconomic environment: good economy = low number of default
24、s (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 derivative,Credit derivatives are contracts where the payoff depends on the creditworthiness of one or mo
25、re companies or countries Buyers: banks or other financial institutions Sellers: insurance company Single name: credit default swap, CDS,How 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 an
26、d 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 bonds for their face value when a credit event occur.,Example,A 5-ye
27、ar 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 buyer,Default protection seller,90 basis points per year,Payment if default by reference entity
28、,Mechanism,If not default, reference entity pays $900,000 on March 1 of each 2010-2014 If default, e.g. June 1, 2012 ; (1) specifies physical settlement; (2) determine the mid-market value of the cheapest deliverable bond , or say, cash payment In arrear payment, including a final accrual payment CD
29、S spread: the total amount paid per year, as a percent of the notional principal, to buy protection,CDS and Bond yields,A 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-year corporate bond over the p
30、ar yield on an n-year risk-free bond.,How to use it,CDS and Cheapest-to-deliver bond,Bonds 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.,Valuation of credit default swaps,Mid-market CDS s
31、preads Example: Suppose the probability during a year conditional on no earlier default is 2%.,Valuation of credit default swaps (cons.),(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 inter
32、est rate is 5% per annum with continuous compounding and the recover rate is 40%.,1,Default 1,2,3,4,5,0,Default 2,Default 3,Default 4,Default 5,Payoff,Accrual payment,Payment 1,Payment 2,Payment 3,Payment 4,Payment 5,Survival probability,Default probability,PV of the expected payment,Assume notional
33、 principal is 1 and payment at rate of s per year.,PV of the expected payoff,Assume notional principal is 1, defaults always happen halfway of a year.,PV of the last accrued payment,Assume notional principal is 1, defaults always happen halfway of a year.,Valuation at or after the negotiation,Marking to market a CDS By product: Estimating default probabilities and recover rate with CDS quoted spread.,52,