畢博上海銀行—CreditRiskMgmtSysAnalyticsEffectofsyscreditriskonloanporfolio

1、精品文檔，值得擁有The Effect of Systematic Credit Risk on Loan Portfolio Value at Risk and on Loan PricingSubmitted to the JP Morgan CreditMetrics MonitorDr. Barry Belkin, Daniel H. Wagner AssociatesDr. Lawrence R. Forest, Jr.,KPMGUsing a contingent claims model framework, CreditMetricsTM derives correlation

2、s between ratings migrations of different borrowers from the observed correlations between equity values of the different industries and countries of those borrowers. As a variation of that analysis, we treat correlations here as arising from a single systematicrisk factor. Applying this single-fact

3、or model, in Section I we evaluate the effect of systematic credit risk on loan portfolio value at risk. In SectionII we focus on individual loan transactions and assess the effect of systematic risk on credit spreads and therefore loan pricing and valuation decisions. We attach great importance to

4、this effect, because arbitrage theory provides the basis for the Loan Analysis SystemSM (LAS) developed by KPMG Peat Marwick to support commercialloan valuation and pricing. Our results show that one must properly account for systematic factors separately from specific factors if one is to assess ri

5、sk accurately at both the loan portfolio and loan transaction level1.I. Systematic Credit Risk at the Loan Portfolio LevelIn this Section we evaluate the effect of systematic credit risk on loan portfolio value at risk. We compare the approach that considers only two credit states (default and no-de

6、fault) with the one that distinguishes among multiple non-default states (the full-state model). In addition, we compare the portfolio payoff distribution under the full-state model with the corresponding Gaussian distribution that would obtain if all borrowers were uncorrelated. Not surprisingly, w

7、e find that the Gaussian distribution poorly approximates actual value at risk. The two-state model provides a better estimate, though we still observe important discrepancies.To model credit-risk correlation, we follow the CreditMetricsTM approach described in 1 and assume that transitions in discr

8、ete ratings grades occur as a result of migration in an underlying process that measures“ distancefrom” default. To simplify the presentation, we work with default distance transformed into a normalized risk score whose one-period changes have a unit normal distribution.We index borrowers by i and l

9、et U i denote the one-period change in the normalized risk score of borrower i . To represent correlation in a straightforward way, we assume that we can write eachU i as:1/16精品文檔，值得擁有U ii Z1ii(1)Here Z represents a unit normal random variable measuring the composite effect of economic factors influ

10、encing the rating-score migration of all borrowers. We referto this systematic component of credit risk as Z-risk.”Thei“represent mutuallyindependent unitnormal variables (also independent of Z), each specific toanindividual borrower.We refer to the credit risk induced by eachias “ -risk. ”Theparame

11、teridetermines thefraction ofthe varianceofU ithatisattributable to Z-risk. The correlation between U i andU j forany ijisijSince an investor can eliminate-risk through diversification, it commandsno risk premium. Z-risk, on the other hand, appears in every portfolio of loans and in broad-based port

12、folios of other assets, no matter how varied. Since Z-risk cannot be diversified away, it accounts for the risk premium or “ unexpectedcredit” loss charged on loans.In what follows, we analyze the typical way in which Z-risk and credit correlation affect portfolio-wide risk. To simplify the analysis

13、,we consider a loan portfolio in which each borrower has the same exposure toZ-risk. We thus replace the borrower specific i by a single value common to all borrowers.2 The model in Equation (1) then reduces to:(2)U iZ1i .The correlation between any pair of borrowers is now.The two extreme casesare:

14、 (i) -risk only () and (ii) Z-risk only (1).0We define ratings by partitioning theU iintodisjoint bins x k(i ) ,x k(i )1 )defined by boundary values x k(i ) . We fix thex k(i )so that the one-period ratingstransitionprobabilitiescalculated from(2) agree withobserved historical ratingmigrationprobabi

15、lities.This methodof calibrating themodel (2) results inaseparate set of ratings grade breakpoints for each starting grade3.This conflicts with the view that U i represents a simple transform of default distance. Also, by tying default to the position of U i at the end of the analysis period only, w

16、e depart from the view that default represents a trapping state in continuous time. Despite these limitations, the model has proved useful as a starting point in evaluating credit correlation.2/16精品文檔，值得擁有Consider a two-period simple (option-free) term loan L. Let X denote the payoff to the lender a

17、t the end of the first period. 4 We apply the standard variance decomposition formula to X :(3)V arX E Z V arX| Z VaZr EX | Z .The termEZ Var X | Z represents the average amount of payoff variability inducedby i Itmeasures(diversifiable)-risk. The term VarZ E X | Z measuresthesystematic variability

18、in loan payoff induced byZ -risk.Consider a loan portfolioNthat includes a large numberN of L-type loansmade to different borrowers, each with the same initial rating. 5Let SN denote thepayoff ofthe portfolioN .The payoffs ofthe loans inNare conditionallyindependent givenZ.It follows, therefore, tha

19、tE Z V arSN| Z NE Z Var X | Z (4)VarZ E SN| Z N 2Var Z E X | Z .The systematic term in the variance decomposition ofSNscales with N 2 , while thediversifiable term scales with N . Thus, given positive correlation among the ratings migrations of different borrowers, for large enough N the systematic

20、risk will dominate in the variance decompositionV arSN EZ V arSN | Z V aZrE SN | Z .(5)For independent identically distributed variables, the central limittheoremapplies to the partial sums SN , with a normalization that scales withN .In thepresent circumstance, we see that if any limit law holds, t

21、he normalization must scalewithN .We now ask whether any counterpart to the central limit theorem holds.Toanswer thisquestion, wecreated aspreadsheet thatcalculates theprobability distribution for the payofffrom a portfolio of two-period loans. We setN10,000 and determined the distribution for the p

22、ortfolio net present value.6We also looked at the“ normalized net present value,” meaning the deviation fromportfolio s meanNPV divided by NVar SN .This represents the unexpectedportfoliogain or loss measured inunits of standard deviation.Weuse thisnormalized variable to quantify portfolio value at

23、risk.3/16精品文檔，值得擁有We analyzed a two-year loan under the following assumptions:? A rating system with seven non-default rating grades (Aaa, Aa , A , Baa, Ba, B, and Caa)?One-year rating grade transition matrix taken from Moody s Rep(see 4)?Constant risk-free interest rate = 4.5%?No loan origination o

24、r holding cost?Fixed loss in the event of default LIED( ) = 40%? Credit risk premiums (unexpected loss) with a flat forward termstructure: Aaa: 3.0 bps, Aa: 4.8 bps, A: 10.0 bps,Baa: 16.0bps, Ba: 120.0 bps, B: 150.0 bps, Caa: 300.0 bps? One of two values for credit migration correlation:.7.15,. 25We

25、 obtain a highly skewed value-at-risk distribution with long lower tail (seeFigure 1).We illustrate this by examining the limit densityf0forthe normalizedportfoliovalue at riskin the case where (i) each borrowerisrated Ba at loanorigination, (ii) the correlation parameterhas the value .25, and (iii)

26、 each loan has acouponofLIBOR+171.9 bps(which represents paratorigination). 8Indeterminingf0 ,wetranslatedtheportfoliopayoffbyitsmeanvalueESN $10,11610.and then scaled the result bythe portfoliopayoff standarddeviationN$96.00.For comparison, we have also displayed the limit densityf 1that results if

27、 oneassigns a two-period loan the value $1 (par value) at the end of period 1 if that loan does not default during period 1 and the value 1 LIED if the loan defaults (see again Figure 1)9. This collapses the migration process to two states: default and non-default. Under this default/no default mode

28、l, the mean portfolio payoff is $10,116.25, quite close to the value of $10,116.10 obtained under the full-state model. However, the portfolio payoff standard deviation is $83.26, substantially below the $96.00 value for the full-state model10.Finally, we show the density f2, the unit Gaussian densi

29、ty that would obtain for normalized value at risk if the migration processes of the borrowers were statistically independent. In that case, the central limit theorem would apply unconditionally and the density for the normalized portfolio payoff would closely resemble the unit normal.4/16精品文檔，值得擁有Bo

30、thf0 andf1 are markedly asymmetric and have modes that are skewed tothe right.The mode off1is more narrowly peaked than that off0 .The uppertail off0falls off smoothly, while that off1is sharply truncated.The lower tailsof the twodensities differ in ways notdiscernible in the diagram. Consider, fore

31、xample, theprobabilityofa portfolioloss.The expected portfolio profit of$116.10 corresponds to121.N .Thus, portfoliolosses occur ifthe payoff fallsshort of the mean by more than121. N .Underf0 such an outcome occurs withprobability .080. The corresponding probability underf1 is .065.So if we wereto

32、usef1toapproximate f 0, we would understate the probability ofa portfolioloss.11We of course observe a much more striking disparity betweenf 0and the unitnormal density f2 .The unit normal density ignores systematic risk, so it comes asno surprise that it poorly approximates f0 .The densityf0 is asy

33、mmetric, whereasthe Gaussian density is symmetric. The density f 0has a heavier lower tail. Underf2 , a portfolio loss occurs withprobability .113.Thus, if we were to use theGaussian as a proxy for f0, we would overestimate the probability of a loss.To summarize, our results show that if one doesn d

34、istinguish among different non-default grades or especially if one doesn t account for systematic credit risk, one will make important errors in estimating portfolio value-at-risk.Figure 1Densities for Loan Portfolio Value at Risk7.0Initial Borrower Rating Grade Ba6.0f0: Full Statef1: Default/Non-De

35、fault5.0f2: Gaussian4.03.02.01.00.0-3.0-2.0-1.00.01.0Portfolio Value (Units of Standard Deviation)5/16精品文檔，值得擁有II. Systematic Credit Risk at the Loan Transaction LevelIn modeling credit risk, most analysts work with an ordered set of several non-default states as well as a single trapping default st

36、ate. This multinomial framework creates a dilemma. One would like to apply arbitrage-free methods in pricing for credit risk.12 However, in a finite (discrete time and discrete risk rating) model, these methods have found successin uniquely identifying prices only for binomial credit risks. “ Binomi

37、al ” refers to a model with only two credit states-default and non-default.13In 3 the authors resolve this dilemma by introducing one-period binomial reference loans with payoffs that approximate the one-period payoffs of the actual multinomial loan. Specifically, they construct binomial loans with

38、payoff means and variances that match those of the one-period payoffs of the multinomial loan. For reasons that will be made clear below, we refer to this calibration as the total risk method. Each reference loan has only two possible payoff values. Thus, each one can be priced uniquely by arbitrage

39、. The authors then outline a recursive procedure for computing the value of the actual multiperiod multinomial loan from the associated values of the binomial reference loans14.The authors in 3 consider a portfolioNofNstatistically identicalmultinomial loans (each with the same multinomial payoff di

40、stribution) and aportfolio N of N statistically identical binomial loans whose payoffs are those of the associated reference loans. They then argue that the value of portfolio N andthe value of portfolioNmust approach equality as N.The argumentrelies on the central limit theorem15.However, the neede

41、d assumption of statistical independence is problematic. If one could construct arbitrarily large portfolios of independent but statistically identical loans, the credit risks being priced would be fully diversifiable and would command no market premium. This conflicts with the observation that loan

42、 spreads in the market include a component for unexpected as well as expected credit loss. The market therefore indicates that the ratings migration processes of borrowers reflect systematic credit risk16.This raises the question of how one should modify the calibration of the reference loan so as t

43、o account for systematic credit risk. The variance decomposition in Equation (2) above provides the answer. We observe that only systematic risk commands a risk premium. Therefore, for two credit risks to be priced the same by the market, they should have the same amount of systematic risk. Thus the

44、 calibration in 3 more properly would involve matching the systematic variance of the reference loan payoffs to the systematic variance of the multinomial loan payoffs. We refer to this scheme as thesystematic risk method.6/16精品文檔，值得擁有Alternatively, we could structure the binomial loan payoffs so th

45、at they created the least amount of systematic basis risk relative to the multinomial loan. In other worlds, we would structure the binomial loan to minimize the following expression:ZB2(6)E,EX |Z EX|Zin which X B denotes the year-1 payoff to the binomial loan. This basis risk minimizationmethodturn

46、s out to be mathematically equivalent to the systematic risk method if the conditional expectation EX B| Z of the binomial loan payoff and the conditional expectation E X| Z of the multinomial loan payoff have a correlation coefficient of unity. For the relevant range of values of the problem parame

47、ters, we have observed this correlation to be .99 or higher. Furthermore, we have separately applied the systematic risk method and the basis risk minimization method and observed that the two schemes produce nearly identical loan values and par spreads.We have created a spreadsheet that applies bot

48、h the total risk method and the alternative systematic risk method. In this spreadsheet,we have represented the systematic risk variableZ using 1,000 equiprobability bins. For each discrete value of Z representing a bin, we determine the conditional moments for the payoff distribution both of an ind

49、ividual loan and a portfolio of 10,000 loans statistically identical to the given loan. In these calculations, we make strong use of the conditional independence of the portfolio loan payoffs for each value ofZ.We applied the total variance decomposition to the previously described two-period loan w

50、ith the borrower initially in rating grade Ba, a (par) credit spread of 171.9 bps, and .25. The results indicate that, for an individual loan, most of the payoff variance (about 95%) reflects diversifiable risk (see Table 1). If we use the total risk calibration scheme in fashioning the binomial ref

51、erence loan, we find that its payoff distribution understates the systematic variance of actual loan payoffs17.At the portfolio level, virtually all of the payoff variance (99.7%) derives from systematic risk (see Table 2). Furthermore, we now see a substantial gap between the payoff variances for t

52、he binomial- and multinomial-loan portfolios.Consequently, the second-order match enforced for individual transactions does not carry over to the portfolio.Table 1Variance Decomposition:Individual LoanMultinomial LoanBinomial Loan7/16精品文檔，值得擁有EX E Z Var X| Z VarZ E X | Z Var X X ESN E Z Var SN | Z V

53、arZ E SN| Z Var SN SN $1.011610$1.011610.001750.001772.000092.000069.001842.001841$.042913$.042912Table 2Variance Decomposition:Loan PortfolioMultinomial LoanBinomial Loan$10,116.10$10,116.1017.5017.729,197.686,913.579,215.186,931.29$96.00$83.25These results show that the small systematic component of risk for individual loans almost entirely determines credit risk for large