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Lecture 11Credit RiskOptions, Futures, Derivatives / March 3, 2008 1Credit RiskCredit Risk arises from the probability that borrowers and counterparts in derivatives transactionsmay default.We attempt to quantify the risk associated to credit risk.Credit Ratings• Moody’s and S&P provide ratings that describe the creditworthiness of corporate bonds.Moody’s Rating S&P RatingAaa AAAAa AAA ABaa BBBBa BBB BCaa CCCBonds with Aaa rating are considered to have little to no chance of default.• Moody’s subdivides categories such as Aa to Aa1, Aa2, Aa3, etc.• S&P subdivides categories such as AA to AA+, AA, AA−, etc.• Only Aaa or AAA are not subdivided.Options, Futures, Derivatives / March 3, 2008 2Historical Default ProbabilitiesWe can consider the historical default rates of the certain class of corporate bonds:For example:• A bond with an initial credit rating of A has a 0.23% chance of defaulting by the end of thethird year.• A bond with an initial credit rating of Caa has a 69.83% chance of defaulting by the end ofthe seventh year.Options, Futures, Derivatives / March 3, 2008 3We can compute the probability of default for a particular year from the table.• The probability of a Ba bond defaulting in the third year is6.00 − 3.48 = 2.52%• The probability of a Caa bond defaulting between the fifth and seventh year is69.36 − 60.83 = 8.53%The default probabilities are an increasing function of time, since ove r time a company with strongcredit may deteriorate. Companies with poor credit may default in a short period of time.We define two quantities:• Unconditional default probability: This is the probability of default during a particular yearas seen from the initial year.Example: The unconditional default probability of a Caa bond defaulting in the thirdyear is48.02 − 37.20 = 10.82%Options, Futures, Derivatives / March 3, 2008 4• Default intensities or Hazard rates: This is the probability of default during a particular yearas seen from the initial year conditioned on no default occurring earlier.Example: The probability that a Caa bond w ill last past year two is100 − 37.20 = 62.80%Therefore, the default intensity is0.10820.6280= 17.23%If we instead compute the default intensity λ(t) at time t over a shorter length of time ∆t. Thenλ(t)∆t is the probability of default between time t and time t + ∆t conditional on no earlierdefault.Options, Futures, Derivatives / March 3, 2008 5Default IntensitiesIf V (t) is the cumulative probability of the company surviving to time t, thenV (t + ∆t) − V (t) = −λ(t)V (t)∆tTaking limits we getdV (t)dt= e−Rt0λ(τ)dτDefine Q(t) as the probability of default by time t. It follows thatQ(t) = 1 − e−Rt0λ(τ)dτorQ(t) = 1 − e−λ(t)t(1)where λ(t) is the average default intensity between time 0 and time t.Options, Futures, Derivatives / March 3, 2008 6Recovery Rates• When a company goes bankrupt, those that are owed money by the company file claims againstthe assets of the company• Either the company reorganizes and creditors agree to partial payments or the assets areliquidated and the used to meet outstanding claims.• Recovery rates for a bond is normally defined as the bond’s market value immediately after adefault, as a percent of the face value.• Senior secured debt holders receive 51 cents to the dollar owed on average; whereas juniorsubordinated debt holders receive 25 cents to the dollar owed on average.Class Average recovery rate %Senior secured 51.6Senior unsecured 36.1Senior subordinated 32.5Subordinated 31.1Junior subordinated 24.5• Recovery rates are significantly negatively correlated with default rates.Options, Futures, Derivatives / March 3, 2008 7• It found that the following relationship provides a good fit to the data:Average recovery rate = 50.3 − 6.3 × Average default ratewhere both the average recovery rate and the average default rate are measured as percentages.Estimating Default Probabilities from Bond Prices• Probabilities of default can be estimate from the prices of bonds it has issued.• Assume that the reason a corporate bond sells for less than a risk-freebond is the possibility ofdefault.• Consider an approximate calculation. Suppose that a bond yields 200 basis points more than asimilar risk-free bond and that the expected re cove ry rate in the e vent of default is 40%.• The holder of a corporate bond must be expecting to lose 200 basis points (2% per year) fromdefaults. Given the recovery rate of 40%, this leads to an estimate of the probability of adefault per year conditional on no earlier default of0.021−0.4= 2.22%.• In generalh =s1 − R(2)where h is the default intensity per year, s is the spread of the corporate bond yield over therisk-free rate, and R is the expected recovery rate.Options, Futures, Derivatives / March 3, 2008 8More Exact CalculationA more exact calculation, suppose that the corporate bond we have been considering lasts for 5years, provides a coupon 6% per annum (paid semiannually) and that the yield on the corporatebond is 7% per annum (with continuous compounding).• The yield on a similar risk-free bond is 5% (with continuous compounding).• The yields imply that the price of the corporate bond is 95.34 and the price of the risk-freebond is 104.09.• The expected los s from default over the 5-year life of the bond is therefore 104.09-95.34 = 8.75.• Suppose that the probability of default per year (assumed in this simple example to be thesame each year) is Q.Consider for example the loss of default at 3.5 years. The expected value of the risk-free bond attime 3.5 years (using forward interest rates) is3 + 3e−0.05×0.5+ 3e−0.05×1.0+ 103e−0.05×1.5= 103.46Given the default of recovery rates in the previous section, the amount recovered if there is adefault is 40, so that the loss given default is 103.46-40 or $64.34.Options, Futures, Derivatives / March 3, 2008 9The present value of this loss is 54.01. The expected loss is therefore 54.01Q.Next consider for example the loss of default at 4.5 years. The expected value of the risk-free bondat time 3.5 years (using forward interest rates) is3 + 10 3e−0.05×0.5= 104.34Given the default of recovery rates in the previous section, the amount recovered if there is adefault is 40, so that the loss given default is 104.34-40 or $63.46. The present value of this loss is50.67. The expected loss is there fore 50.67.The total expected loss is 288.48Q. Setting this


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U of M MATH 5076 - Lecture notes

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