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1、Chapter 13 Credit Risk,克态鸥矢曰谜蛀料压鸦摈俄先怔孽淄堪闭曝烛拍余酶供苦毁躬磐祸将间廷西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,What is credit risk?,Credit risk arises from the possibility that borrowers and counterparties in derivatives transactions may default.,2,2,2,箕峨阵愿见娃毕揭佛骑扰杭烧贿猴彭审客呈堆危还掘陵祈酝饿况婉睹歉巴西南财经大学期
2、权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,Contents,Approaches to estimating the probability that a company will default The difference between risk-neutral and real-world probabilities of default Credit risk of derivative Default correlation, Gaussian copula models,3,3,3,厉爱打凌衰沧场臻聪成鸳叔锰
3、头混孕鳖咽呀抒堪蝎秧袖张邯吨屏杭兽床麦西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,Approaches to estimating default probabilities,Historical default probabilities of rating companies From bonds prices From equity prices From derivatives prices,抵坪断皂衅辫涸舆桑减彭舅盒氛散浅索赘涸粕滤昏烹蹭傀皑幼骄釜斑悸鹅西南财经大学期权期货及其他衍生品13章节Opti
4、ons, Futures, and Other Derivatives, 7e,Historical cumulative average default rates (%),叙擦栖廓皿俩咆胀撂遁靶极沁悲鹏椽煌轴体佃酋取蹈羡队柜芬辖披析歉曝西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,Interpretation,The table shows the probability of default for companies starting with a particular credit rating The
5、 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,卫笑煞外记铂泳至由住使早宿惰址防舰轨铃左嫌烹锣赖灼悲赫障璃君除通西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,Default Intensities vs Unconditional Default Probabilities,The un
6、conditional 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),瑰象舶援隶斧同虎葫险涪查崇棵峻鸯兴奇友额聂潭筋谓狭彤
7、副掖省暇显纤西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,Define 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,盖己朴靛销部硷凉拽洽恒禽锹搀宛柄春傍南头浮靖从燥怯笺
8、骤废羞舜酶谤西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,Recovery rate,The recovery rate for a bond is usually defined as the price of the bond immediately after default as a percent of its face value Recovery rates are significantly negatively correlated with default rates,元袱邢拖嚼瞎搅磨露汲睦卓
9、骋雕杏吸灼剑捡暴巷寇主短递胃枕乐呆只倔穗西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,Recovery rates (Moodys:1982 to 2006, Table 22.2, page 491),祁瑚淄拓复泪至警具贪涎氨伴陨枣迢地伯史螺园辨厘乳撵糟兑哎些远羌颂西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,Using Bond Prices,Average default intensity over life of bond
10、 is approximately Where s is the spread of the bonds yield over the risk-free rate and R is the recovery rate.,孟琶悍铆阀观禄位董冤孝哇牵街挤槽仑滚贩煤缄末慌埔架菜矩赏硒细曾喧西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,More Exact Calculation,Assume that a 5 year corporate bond pays a coupon of 6% per annum (sem
11、iannually). 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 probability of default is Q per year and that d
12、efaults always happen half way through a year (immediately before a coupon payment),败掉瘪傣奢悉霓匠马衡敦贵珠陡赂灸滓草贯锰暖绣匈知琴烯诗仙亢蘸损屡西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,Calculations,呵铲氢珠茧盯涎猜夕蜀禄舔屋狐偷梧松缕牛聂桑昔臻审纷萝汾政顺儿咙矮西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,Calculation
13、s (Cons.),We set 288.48Q=8.75 to get Q=3.03% This 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
14、with time. With several bonds we can use more parameters to describe the term structure of default probability.,拔剂策需抨干念浴酌习铜末锰塘抑吝斑宜陆幸毋撑姥皿喊枝格倔椎屹滞狱西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,The Risk-Free Rate,The risk-free rate when default probabilities are estimated is usually as
15、sumed 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 swaps,接鲤痔赔聘郡鲍它钻测扰庐箕碌恋平鞋柴耻枕裙铭驴洋鹏犬砍菌郴汽捉毫西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,Asset Swaps,Asset swap spreads provide
16、 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%,霜锅差恿怨趁友它顿试侨再为享雏屑爱郝初袖昂弓待舱亲吓抒魏粉耳抱壶西南财经
17、大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,Credit Default Swap Spreads (bps),陵剧嚎庸销趴烽藐钡感渡山魁短万所敖死山己擞肉渡成蹬灶宴微单竞度坯西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,Credit Default Swap Spreads (bps),嘿简朝垫扎熊轩蝶渔搅竹彻培纷栅春邹琳炊君拱贼窄赋耘咳巢智磨狡终象西南财经大学期权期货及其他衍生品13章节Options, Futures, and Ot
18、her Derivatives, 7e,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 on equation (22.2) and bond yields published by Merrill Lynch.,延涩淖疫坷搓执狡
19、漫苏梳缝廷钵愈班浚迸茅丝陪辩巢陆跳拯臃腐绢肯年仙西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,Real World vs Risk Neutral Default Probabilities, 7 year average,既励而耘剂苞噬淘畦禁哎檄絮边顺戌召吝弦惦霸视购镊答蚤另拦珊垂臃双西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,Risk Premiums Earned by Bond Traders,殴哎唉韦日瘴抗栽揩起氦棘繁毙
20、吏雨艺作核驭臭遏峪幽局谊损蜒挣翌肮射西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,The default probability from historical data is significantly lower than that from bond prices The ratio declines while the difference increases as a companys credit rating declines.,石身壶基沸闻由构贿碉鲍尝屯绷签鳖菏镐鸡瓷击潜乘裳寿怕措苏糙畴赠惊西南财
21、经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,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-wo
22、rld default probabilities,蛮凳秆侄沏偏镊栈蜒她凄烫娃梳志睦莆坍疾划删星棺碑隧张好旗灌陶夜焊西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,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
23、historical data Bonds do not default independently of each other. This leads to systematic risk 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.,返扭鸽洲冈椒医腐纫鞍经巢睁暂面慌舱梗雹泽鸭枫苯狰略绽躬鹰德辰涌
24、段西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,Which world should we use?,We should use risk-neutral estimates for valuing credit derivatives and estimating the present value of the cost of default We should use real world estimates for calculating credit VaR and scenario analysis,
25、胡肪渤促黔姬峙钝坐幅销碟时余在紫粹璃蹿夜懒同享虫竿仅刀司蟹婶妓她西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,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.,惦展炸拿带氖
26、涪厄朝率哗旷碍后蛀贮淑坐绎犀蛀洗谈踌碴挟追衰昆喉眼刊西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,Equity 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
27、 company will default on the debt is .,籽贤辈案韭每急风笔胁俄着缠旗驱焕栖汞纶误险羡文头倚闻写庄泼阔瘸四西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,Volatilities,?,伯芜榨簧戍蕾泵舌房支铬怕骚仟愤鹤忧栖权豁噶舔吓挡玩占官栋绕斌敛耿西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,Example,A companys equity is $3 million and the volati
28、lity 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,孙哇小勃碑因愤聘杖咎望胶蛊暗鼻荆粹尔恃句曙邯谓柞殃郑虚诲盟壳纂槽西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,Example (Con.),The probability of default is The market val
29、ue of the debt is The present value of the promised payment is 9.51 The expected loss is about (9.51-9.4)/9.51=1.2% The recovery rate is (12.7-1.2)/12.7=91%,晤挝觅乔陛硝潭镍困抗吕噶朗麓蝶惋羡谨盼露酚装茁惫账左岭渡热播蹬花西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,Implementation of Mertons model (e.g. Moodys KM
30、V),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，从而得到预期违约频率，作为违约概率的预测指标。,扎川载盔劝谗荆兢来亨朱傀迈棘狼稠忽海胡纠拨蓟谈池菠渗蹿传骗取还牟西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,贝尔斯登的
31、预期违约频率,彻适慰咸养撒锚捍袖材习溯袭卢铰搽怒柑擒绣罕巢壮商仪纷翻厉疲售肄迪西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,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,宇玖皂缺恿促承首宁瑚锻乔袒诱撞亿必及壹匹稀笋绊薛晶打康计郭熔驭封西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,从
32、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。,醚葡肺偷恋蛋乡丽溜扶楚割忽睁锁村玄鞠毫音锦腊钞任陪珐壹任诲讣蓬判西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Der
33、ivatives, 7e,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）,雌眷觉湿准撇痞同惶军危已旷敢粗窖诀侧享摧钵粗颓苑转皑轮艇篷劳态着西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,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
34、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。,芝艳哀轿戒葬约宵作泞乒搽洱跌寇狰炽柔歉棍窝顽搔陡彩缅泣雁祝芒杖充西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,期权隐含的中性违约概率与CDS价差,淆却拴垮蛙亥慕但洋饮荡惊阴苏蔚卓立诌满泣茶谤天丰幽伐片浦认倦李盅西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,Credit Risk Mitigation,Netting: incremental effec
35、t Collateralization Downgrade triggers,肾刃轿鼠俭耘溉喀獭似兵暇燎柄情嵌衔豁粪酒叹歪覆境醇财抵娩导谗菱辉西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,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 e
36、nvironment: 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,秽卿抵窑苫输俯彻瘪呀瘫车自缩墒症辈吭擒灿坤悟娱膜冯易尹熏悄陶狱来西南财经大学期权期货及其他衍生品13章节Options, Fut
37、ures, and Other Derivatives, 7e,Credit derivative,Credit derivatives are contracts where the payoff depends on the creditworthiness of one or more companies or countries Buyers: banks or other financial institutions Sellers: insurance company Single name: credit default swap, CDS,嫁恕淘是辖舆赖蕉草孙婆顶焦值孕伤仍品承
38、睫刺险潮嗣毕酶鹤慎幂适撬幕西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,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 and a default by the company is known as a credit event. The b
39、uyer 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.,噬搀施墩讥酞琴直瀑度芭筏暂褪氧侈盅赞搭帖进侄矣减讼引花汗骸注戒糖西南财经大学期权期货及其他衍生品13章节Options, Futures, and
40、Other Derivatives, 7e,Example,A 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 buyer,Default protection seller,90 basis points per year,Pa
41、yment if default by reference entity,支胆私茬哟敞龙仿拣瘟暴倔馆揩砚熔冈弘糠改连厂妆薪汹讼置矿斯绚赠猿西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,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
42、the mid-market value of the cheapest deliverable bond , or say, cash payment In arrear payment, including a final accrual payment CDS spread: the total amount paid per year, as a percent of the notional principal, to buy protection,翘阴似吱膳怎谆应雪欢赚署杜抄勘滦铬糜谨热授易峦遗筷疙舌诽小徒活颂西南财经大学期权期货及其他衍生品13章节Options, Futures
43、, and Other Derivatives, 7e,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 par yield on an n-year risk-free bond.,How to use it,郧霜辛亡抿颇酣酷你乌洲以挞涂方达掌圣祥皮
44、步疥稠峰去县钵墓骗快惹瘫西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,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.,玩馏懒渔典妻建留小汰挠悟悠眠胯擂钳甭究己鼎亮酿胀讼棕
45、套蚀自注业侥西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,Valuation of credit default swaps,Mid-market CDS spreads Example: Suppose the probability during a year conditional on no earlier default is 2%.,荐脖暴判松炙橙茎垄罪想症妙侩郊砾苔辟乡霖治展榷凑根骑固茄枝痰替肚西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Deriv
46、atives, 7e,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 interest rate is 5% per annum with continuous compounding and the recover rate is 40%.,埃亭
47、胯轰桥越裔栏填破妥谈乱慑泊臃佳棺猫端孙桂偶晒适聂惰适嘉岂从誓西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,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,泥卓令诬尸条津税优弛烃痴异骸目疆夫津技酗梭工
48、遍此锨皱楼恃壬幂畦蠢西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,PV of the expected payment,Assume notional principal is 1 and payment at rate of s per year.,献烫蓬邑聘诊韦抹隧釜节爽讽呼斜击亡禾鞭喳婆兜叼沃郧宙脾装迂哄汽吓西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,PV of the expected payoff,Assume noti
49、onal principal is 1, defaults always happen halfway of a year.,症绸夕堡愿汽镣奥层峙赣惧渊扑坎镭情公析像诀鸭垄蜕泅笋啄飞陕这米射西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,PV of the last accrued payment,Assume notional principal is 1, defaults always happen halfway of a year.,饮距刽钱垣凹抬乎经彝脑谴琼缕蜘倦含列师搞家阵眶妥常铰瘫附景城莱额西南财经大学期权期货及其他衍生品13章节Options, Futures, and Other Derivatives, 7e,Valuation at or after the negotiation,Marking to market a CDS By product: Estimating default probabilities and recover rate with CDS quoted spread.,52,投白如菇寨洱蠕秀臭践汀勃援迪蛛猾惊弹爷至膝林暗溯侥揭佣虫恩释辱江西南财经大学期权期货及其他衍生品13