##### Chapter Nine

1. The concept of ‘economic capital’ is discussed in the chapter titled ‘Capital—risk, regulation and adequacy’.

2. This approach is taken by Stephanou, Constantinos and Mendoza, Juan Carlos, 2005, ‘Credit risk measurement under Basel II: An overview and implementation issues for developing countries’, World Bank Policy Research Working Paper 3556, 11.

4. A detailed description of these methods is provided in Scheurmann, Til, 2004, ‘What do we know about Loss given Default?’, Wharton Financial Institutions Center Working Paper Series 04–01.

5. Altman, Edward I, Brooks Brady, Andrea Resti and Andrea Sironi, ‘The Link Between Default and Recovery Rates: Theory, Empirical Evidence and Implications’, *Journal of Business* 78, no. 6 (2005): 2203–2227.

6. Basel Committee on Banking Supervision (2006), ‘International Convergence of Capital Measurement and Capital Standards: A Revised Framework, Comprehensive Version’, www.bis.org

7. Department of the Treasury, Federal Reserve System and Federal Insurance Corporation (2006), ‘Basel 2 Capital Accord—Notice of Proposed Rulemaking’, 341–342.

8. Market risk is discussed in detail in chapter on bank investments.

9. Banks’ capital requirements and the concept of ‘economic capital’ are discussed in detail in chapter titled ‘Capital—Risk, regulation and adequacy’.

10. Discussed in Chapter as above.

11. The technical document of Credit Metrics by J P Morgan can be accessed at www.riskmetrics.com.

12. Notable in this context are the published research work of Edward Altman (1991) of New York University, and Lucas and Lonski of Moody's Investor Service.

13. VaR is a commonly used abbreviation of Value at Risk, used prevalently in market risk determination. In simple terms, VaR measures the maximum expected loss for a given holding period (of a security or a portfolio of securities) for a given confidence level (as chosen by the portfolio manager). For example, a portfolio with daily VaR of Rs. 1 crore with 99 per cent confidence signifies that there is a 1 per cent chance that the portfolio will lose more than Rs. 1 crore over the next 24 hours. VaR is discussed in detail in the chapter on ‘Investments’.

14. You can access the transition matrix on the Web site of CRISIL at http://www.crisil.com/credit-ratings-risk-assessment/crisil-rating-default-study-2007.pdf, accessed on 4 March 2009.

15. The other probability used in credit analysis is the ‘risk neutral probability’ of default, where the expected return required by all investors is the risk-free rate. See Box 9.1 for the distinction between risk neutral and real world probabilities.

16. CRISIL also measures ‘accuracy ratios’ reflecting the model's robustness and predictive ability.

17. Accessed at http://www.crisil.com/credit-ratings-risk-assessment/crisil-rating-default-study-2007.pdf on 19 March 2009.

18. It is also important to distinguish between the ‘promised’ return on a loan (which is the loan price), and the ‘expected’ return, which factors in expected losses.

19. The choice of the optimal time horizon is discussed in detail in the Technical document on CreditMetrics™, by J P Morgan, accessed at www.riskmetrics.com.

20. Several studies have been carried out for the construction of the ‘transition matrix’. The first was a series of articles by Altman and Kao in 1991 (a&b) and 1992 (a&b), followed by special and periodical studies by Moody's and Standard and Poor's.

21. CreditMetrics™—Technical document.

22. Altman *et al*., in their December 2001 report titled ‘Analysing and explaining default recovery rates’, submitted to International Swaps and Derivatives Association (ISDA), ascribe the following reasons for the relatively little attention to recovery rates. One, that credit pricing models and risk management applications tend to focus on the systematic risk component of credit risk, and two, credit risk models traditionally assume RR to be independent of PD.

23. Detailed explanations on forward zero curves can be found in the annexure to the chapter on ‘Risk Management in Banks’.

24. CreditMetrics technical document, page 27, accessed at www.riskmetrics.com.

25. CreditMetrics Technical document, 2007 update, Appendix D, page 156.

26. The first percentile of a normal distribution M(m, *σ*^{2}) is (mean – 2.33*σ*) which equals – 18.46.

27. The primary motivation to develop quantitative portfolio models to manage credit risk is the need to address ‘concentration risk’, which has been discussed in an earlier section of this chapter, and in the previous chapter.

28. The normal distribution is the classic ‘bell-shaped’ curve described in statistics. The bivariate normal distribution has two variables which are possibly correlated. An extension of this is the multivariate normal distribution, features of which are covered in standard texts on statistics.

29. T. Wilson, ‘Portfolio credit risk I’, *Risk* 10, no. 9 (September 1987).

T. Wilson. ‘Portfolio credit risk II’, *Risk* 10, no. 10 (October 1997).

30. Thomas C Wilson., ‘Portfolio Credit Risk,’ *FRBNY Economic Policy review* (October 1998): 71–82.

31. The illustration is based on the numerical example given in Wilson's paper quoted above.

32. In a large portfolio of borrowers, as more and more borrowers are added to the portfolio, the standard deviation of the portfolio losses (the variability in portfolio loss) would fall, up to a point. The residual risk that cannot be eliminated by diversifying the portfolio is called the ‘market risk’ or ‘systematic risk’. This risk is considered non-diversifiable and arises due to market/industry or economy-related factors. The risk that can be diversified is called ‘non-systematic’ or ‘idiosyncratic’ or ‘firm-specific’ or ‘non-diversifiable’ risk. The following diagram clarifies:

33. ‘Calibration’ is simply the process of assigning a default probability to a rating or a homogeneous segment of borrowers.

34. The ‘gamma distribution’ is a skew distribution that approximates to the normal distribution when the mean is large. This continuous probability distribution that has two parameters—a scale and a shape parameter—called *α* and *β*. It is fully described by its mean *μ* and standard deviation *σ*, such that *μ* = *αβ* and *σ ^{2} = αβ^{2}*. It is prevalently used in engineering, science and business to model continuous variables that are always positive and have a skewed distribution. Examples of its use include ‘queuing’ models—such as estimating the flow of material in manufacturing and distribution processes, assessing the load on web servers and applications in telecom exchanges. Its skewed profile makes it an appropriate model in climatology, say for rainfall pattern assessment, or in financial services, such as for modelling magnitude of loan defaults or insurance claims.

35. Stephen Kealhofer, John McQuown and Oldrich Vasicek founded KMV Corporation in 1989. Their credit risk models are based on a modification of Merton's asset value model. The firm was subsequently acquired by Moodys corporation, and details of various versions of the model can be accessed at www.kmvmoodys.com.

36. Please refer to any standard corporate finance textbook for an understanding of ‘present value’, ‘free cash flows’ and ‘discount rate’.

37. For a discussion on the Equity Multiplier, a measure of ‘leverage’ and the impact of debt on a firm's balance sheet, please refer to chapters on ‘Bank financial statements’ and the annexure containing ‘financial ratios’ in the chapter on ‘Uses of funds—the lending function’.

38. ‘Options’ are discussed in detail in the chapter on ‘risk management’.

39. Fischer Black and Myron Scholes ‘The pricing of options and corporate liabilities’, *Journal of Political Economy* 81, no. 3 (May–June 1973): 637–659.

40. Robert C Merton, ‘On the Pricing of Corporate Debt: The Risk Structure of Interest Rates.’, *Journal of Finance*. 29, no. 2 (May 1974): 449–470.

41. Fischer Black and John Cox, ‘Valuing Corporate Securities: Some Effects of Bond Indenture Provisions’, *Journal of Finance* 31, no. 2 (May 1976):351–36

42. There are two basic types of options, ‘call’ and ‘put’. A ‘call’ gives the option holder the right to buy the underlying asset for the strike price, whereas a ‘put’ gives the option holder the right to sell the underlying asset for the exercise price.

43. A random variable *X* is ‘lognormal’ if its natural logarithm *Y =* log(*X*) follows a normal distribution. This means that *Y* is a normal distribution with mean *μ* and variance *σ ^{2}*. It is prevalently used in modelling stock or volatile asset prices where we assume that asset returns are normally distributed.

44. The Modigliani–Miller theorem is a Nobel Prize winning proposition that states that under perfect market conditions, the value of the firm is independent of its capital structure. Refer M. Miller and F. Modigliani, ‘The cost of capital, corporation finance and the theory of investment,’ *American Economic Review* (June 1958).

45. The term structure of interest rates is discussed in the chapter on ‘Risk management’.

46. A ‘stochastic process’ is simply a collection of random variables—variables whose values change in an uncertain manner—taking values in a common time space. A diffusion type stochastic process is one modelled in continuous time. Many statistical inferential problems can be described in terms of diffusions, through forming stochastic differential equations.

47. Merton in his 1974 paper (page 450) refers to the stochastic differential equation *dV* = (*αv* – *C*)*dt* + *σVdz*, where *α* is the instantaneous expected rate of return on the firm per unit time, *C* is the total dollar payouts by the firm per unit time to either its shareholders or liabilities-holders (e.g., dividends or interest payments) if positive, and it is the net dollars received by the firm from new financing if negative; *σ ^{2}* is the instantaneous variance of the return on the firm per unit time and

*dz*is a standard Gauss—Wiener process. For more on the Gauss—Wiener process, please refer to a standard book on statistics.

48. European call options can be exercised only on the maturity date of the option. If the option can be exercised at any time before maturity, it is an ‘American option’. Options basics are discussed in the chapter on ‘risk management’.

49. Also called the KMV–Merton model, the KMV–Merton model was developed by the KMV Corporation in the late 1980s. It was successfully marketed by KMV until KMV was acquired by Moodys in April 2002. The model is now sold to subscribers by Moodys KMV.

50. Peter J Crosbie and Jeffrey R.Bohn ‘Modelling default risk’, KMV LLC, 2002.

51. Ibid, 3.

52. Moodys KMV company, 2006, ‘Structural Modelling in Practice’, presentation by Dr. Jeffrey R Bohn, Head of Research, Moodys KMV company.

53. Crosbie, Peter J, and Bohn, Jeffrey R, 2003, ‘Modelling default risk—Modelling methodology’, Moodys KMV company (Credit Monitor®, EDFCalc®, Private Firm Model®, KMV®, CreditEdge, Portfolio Manager, Portfolio Preprocessor, GCorr, DealAnalyser, CreditMark, the KMV logo, Moodys RiskCalc, Moodys Financial Analyst, Moody's Risk Advisor, LossCalc, Expected Default Frequency and EDF are trademarks of MIS Quality Management Corp.)

54. Ibid, 13

55. Ibid, 13, Figure 8.

56. Assume that firms with DD = 4 number 6000 at a particular point in time. Also assume that 20 firms out of these defaulted after one year. Then EDF 1st year = 20/6000 = 0.0033 = 0.33 per cent or 33 bps.

57. Amnon Levy, ‘An overview of modelling credit portfolios’, Moody's KMV, Figure 7, Moodys KMV company (2008): 15.

58. Moodys KMV offers a suite of products and services on credit risk modelling and management, information on which can be accessed at www.moodyskmv.com.

59. The technical document of CreditRisk+™ can be accessed at http://wwwcsfb.com/institutional/research/assets/creditrisk.pdf, accessed on 21 April 2009.

60. In the context of two state models, a typical approach is by means of Bernoulli random variables.

61. In its simplest form, a Poisson distribution assumes that every borrower can have multiple defaults, as contrasted with the Bernoulli distribution given earlier, which assumes that there are only two outcomes—the counterparty either defaults or survives. One of the essential assumptions of the CreditRisk+ model is that the individual probabilities of default are sufficiently small for the compound Bernoulli distribution of default events to be approximated by a Poisson distribution.

62. Derived from the PGF for a portfolio of independent borrowers/counterparties, explained in the CreditRisk+™ document, Appendix A (1997): 34–35.

63. Based on the information in CreditRisk+™ document, Appendix A (1997): 33.

64. We can illustrate the PGF through a numerical example. If there is a portfolio *X* with only one borrower, who has a probability of default of, say, 5 per cent over the next 1 year, the number of defaults over the next 1 year will have a probability distribution as follows: *P* (*X* defaults, i.e. *X =* 1) = 0.05 and *P* (*X* does not default, i.e. *X =* 0) = 0.95. The PGF would be P(z) = 0.95z^{0} + 0.05z^{1}. This is the PGF defined in CreditRisk+ technical document, Appendix A, equation (3), page 34. The same numerical example can be presented in a different way. If the portfolio value is defined as the expected exposure loss, then *X* will have the probability distribution at the end of one year as *P*(*X* defaults, recovery = 0) = 0.05 and *P*(*X* does not default, recovery = 5, say by way of interest + principal) = 0.95. Now the PGF would be *P*(*z*) = 0.05z^{0} + 0.95z^{5}.

65. The numerical example in the previous footnote will help understanding this notation.

66. Philippe Artznerand and Freddy Delbaen ‘Default Risk Insurance and In-Complete Markets’, *Mathematical Finance* 5 (1995): 187–195.

67. Robert A. Jarrow and Stuart M. Turnbull, ‘Pricing Derivatives on Financial Securities Subject to Credit Risk’, *Journal of Finance* 50, no. 1 (1995): 53–86.

68. Darrell Duffie and Kenneth J. Singleton, ‘Modelling Term Structures of Defaultable Bonds’, *Review of Financial Studies* 12 (1999): 687–720.

69. The interested reader can access the mathematical derivations from the papers quoted above, as well as subsequent work and books by the above researchers and others.

70. If the default process *N _{T}* had a zero trend, that is, the expected future default given current information is exactly the current value of the process, it would be a ‘fair game’ or a ‘martingale’.

71. D. Madan and H. Unal, ‘Pricing the Risks of Default’. *Review of Derivatives Research 2* (1998), 121–160.

72. Robert A. Jarrow, David Lando and Stuart M Turnbull, ‘A Markov Model for the Term Structure of Credit Risk Spreads’, *The Review of Financial Studies* 10, no. 2 (Summer 1997): 481–523.

73. These documents are available at http://www.kamakuraco.com/, accessed on 2 May 2009.

74. The basic form of this model can be found in Chava, Sudheer and Robert A Jarrow (2004), ‘Banruptcy prediction with industry effects’, downloaded from http://ssrn.com, accessed on 2 May 2009.

75. Hybrid models typically combine two or more credit risk modelling approaches. The Jarrow–Merton Hybrid model incorporates features of the Jarrow–Chava model and the Merton structural model.

76. Robert A. Jarrow and Philip Protter, ‘Structural Versus Reduced Form Models: A New Information Based Perspective’, *Journal of investment management* 2, no. 2 (2004): 1–10.

77. Ibid, 8

78. According to the British Bankers’ Association (BBA) 2006 credit derivatives survey, the largest contributor to CDS growth has been CDS Index trading. The survey points out that about two-thirds of banks’ derivatives volume is from trading activities. Unlike credit insurance contracts, credit derivatives are negotiable and attract large secondary trading. The Chicago Board of Trade has introduced CDR Liquid 50 NAIG Index futures contract, which enables the CBOT to facilitate growth of the CDS market. More information on CDS trading can be accessed at www.cbot.com. In addition, Creditex has online trading platforms for credit derivatives (www.creditex.com).

79. Wai-Yan Cheng, ‘Recent Advances in Default Swap Valuation’, *Journal of derivatives* (Fall 2001): 18–27.

80. Credit events that trigger a default swap may include one or more of the following: failure to meet payment obligations when due, bankruptcy, repudiation, material adverse restructuring of debt, obligation acceleration or obligation default. These are as defined by ISDA.

81. One basis point is (1/100) per cent.

82. Default swap spreads, though economically comparable to bond yield spreads, do not require specification of a benchmark risk-free yield curve. Second, changes in credit quality get reflected faster in CDS spreads than in bond yield spreads. Hence, CDS spreads are able to reflect credit risk with more speed and accuracy.

83. A loan (or bond) is a deliverable obligation if it has a maturity less than 30 years, is a G6 currency, and not subordinated to the reference asset (for more details, see the ISDA Web site).

84. The final price is the market value of the reference obligation on the default date as computed by a specified calculation agent (by a specified valuation method). In practice, one or more dealers are asked for quotes on the reference obligation. Of the quotes obtained, the highest and the lowest quotes are ignored. The average of the remaining quotes is considered the value of the reference obligation.

85. D. Duffie, ‘Credit Swap Valuation’, *Financial Analysts Journal* (January/February 1999): 73–87.

86. Please refer to Box 9.1 in Section II of this chapter for the distinction between risk neutral and real world probabilities.

87. From here on, we follow the approach of Hull and White (2000) in their paper Hull J. and A. White, ‘Valuing Credit Default Swaps I: No Counterparty Default Risk’, *Journal of Derivatives* 8, no. 1 (Fall 2000): 29–40.

88. Arbitrage is the simultaneous purchase and sale of an asset, in different markets or in different forms, in order to profit from a difference in the price. Arbitrage exists as a result of market inefficiencies; it provides a mechanism to ensure prices do not deviate substantially from fair value for long periods of time. The ‘no arbitrage’ argument in valuing CDS is explained in detail in Hull J. and A. White, ‘Valuing Credit Default Swaps I: No Counterparty Default Risk’, *Journal of Derivatives*, (Fall 2000): 29–40.

89. In accordance with the practice in the interest rate swap market, the protection buyer is the ‘fixed rate’ payer, while the protection seller is the ‘floating rate’ payer. The protection buyer is also indicated by terminologies, such as (a) long CDS, (b) CDS buyer, (c) credit risk seller and, (d) fixed rate payer; while the protection seller is also called (a) short CDS, (b) CDS seller, (c) credit risk buyer and (d) floating rate payer. Another thing to remember is that with CDS, buying protection is ‘short’ the reference asset and selling protection is ‘long’ the reference asset. This is because buying protection implies ‘selling’ the reference asset. The market value of the premium leg acts like a short in that as the price of the CDS falls, the market value of the trade increases. The opposite is applicable to the protection leg.

90. J. C. Hull and Alan White, ‘Valuing Credit Default Swaps II: Modelling Default Correlations’, *Journal of derivatives* 8 (2001): 12–21.

91. R. A. Jarrow and S. M. Turnbull, ‘Pricing Derivatives on Financial Securities Subject to Credit Risk’, *Journal of Finance* 50 (1995): 53–85.

92. Dominic O'Kane and Stuart Turnbull, ‘Valuation of Credit Default Swaps’, Fixed Income quantitative research, Lehman Bros, QCR Quarterly, vol 2003-Q1/Q2 (2003): 1–19. The model is illustrated through easy to understand numerical examples.

93. Binomial tree is used in pricing options.

94. D. Duffie and K. J. Singleton, ‘Modelling Term Structure of Defaultable Bonds’, Review of Financial Studies. 12 (1999): 687–720.

95. Note that ‘conditional probabilities’ are also called ‘default intensities’, which form an important part of the reduced-form models discussed in the previous section.

96. There are two ways to price risky (defaultable) debt—one is to discount the risky (default-adjusted) cash flows by the risk-free rate, and the other, to discount risk-free cash flows by risk-adjusted (default spread) rates. Since we do not know the risky rates, it is easier to price the debt by discounting risky cash flows with the risk-free rate, which is easier to obtain or infer. On the other hand, calculating of risky cash flows is also simple—this is the expected cash flow based on probability of default. This is the reason for wide usage of the risk-free rate in the pricing methodology.

97. ‘Swaps’ in general is discussed in the chapter on ‘Risk management in banks’.

98. A good resource to understand cash flow and market value CDOs is the ‘CDO Handbook’ published by J P Morgan Securities Inc (2002).

99. This sample CDO structure can be found in ‘Markit Credit Indices: A Primer’, Markit group Ltd., May 2009, accessed at www.markit.com on 16 May 2009.

100. Based on the example given in ‘Markit Credit Indices: A Primer’, published in May 2009, accessed at www.markit.com.

101. After the financial market collapse in 2007, this method too has drawn flak, and has been named as one of the reasons for the miscalculation by investors. However, 100. for those interested in an analysis of this approach, please refer Christopher Finger , ‘Issues in the Pricing of Synthetic CDOs’, Riskmetrics Group, working paper 04–01 (2004), at www.riskmetrics.com.

102. Summarized from Padmalatha Suresh ‘Credit Derivatives—Still Relevant?’, Treasury Management, The ICFAI University Press, (March 2009): 36–42.

103. 2008, Survey of credit underwriting practices, Office of the comptroller of currency, USA.

104. The term ‘underwriting standards’, as used in the above report, ‘refers to the terms and conditions under which banks extend or renew credit, such as financial and collateral requirements, repayment programs, maturities, pricing and covenants. Conclusions about ‘easing’ or ‘tightening’ represent OCC examiners’ observations during the survey period. A conclusion that the underwriting standards for a particular loan category have eased or tightened does not necessarily indicate that all the standards for that particular category have been adjusted.'

105. ‘Survey of credit underwriting practices, 2008’, Office of the comptroller of the currency, USA (June 2008).

106. Treasury Secretary Geithner's testimony can be accessed at www.ustreas.gov/press/releases/tg71.htm. A description of the four components of the Treasury Framework and a more detailed outline of the systemic risk component are available at www.ustreas.gov/press/releases/tg72.htm.

107. In the United States, the industry has been working closely with regulators, such as the Federal Reserve Board (FRB). The FRB and fourteen dealer banks started working in 2005 to address operational inefficiencies and backlogs in the processing of OTC CDS trades. This group was known as the ‘Fed 14’. Since then other banks have joined these efforts.

108. ISDA's big bang can be accessed at http://www.isda.org/bigbangprot/docs/Big-Bang-Protocol.pdf, accessed on 26 May 2009.

109. For more on credit event auctions, access www.markit.com/cds. The site also contains a useful document titled ‘Credit event auction primer’.

110. Freddie Mac, the acronym for the ‘Federal Home Loan Mortgage Corporation’ (FHLMC), is a government-sponsored enterprise (GSE) of the US federal government. Its primary objective is to expand the secondary market for mortgages in the United States. It buys mortgages on the secondary market, pools and packages them, and sells them as mortgage-backed securities (MBS). In September 2008, consequent to its role in the credit crisis, the federal government took over Freddie Mac and Fannie Mae (Federal National Mortgage Association), the other mortgage loan purchaser in the private sector. In 2008, both institutions together owned or guaranteed about half of the US’ $12 trillion mortgage market. Hence, both were at high risk from the subprime crisis.