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Chapter 5

Chapter 5. Reduced Form Models: KPMG’s Loan Analysis System and Kamakura’s Risk Manager. Estimating PD: An Alternative Approach. Merton’s OPM took a structural approach to modeling default: default occurs when the market value of assets fall below debt value

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Chapter 5

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  1. Chapter 5 Reduced Form Models: KPMG’s Loan Analysis System and Kamakura’s Risk Manager

  2. Estimating PD: An Alternative Approach • Merton’s OPM took a structural approach to modeling default: default occurs when the market value of assets fall below debt value • Reduced form models: Decompose risky debt prices to estimate the stochastic default intensity function. No structural explanation of why default occurs.

  3. A Discrete Example:Deriving Risk-Neutral Probabilities of Default • B rated $100 face value, zero-coupon debt security with 1 year until maturity and fixed LGD=100%. Risk-free spot rate = 8% p.a. • Security P = 87.96 = [100(1-PD)]/1.08 Solving (5.1), PD=5% p.a. • Alternatively, 87.96 = 100/(1+y) where y is the risk-adjusted rate of return. Solving (5.2), y=13.69% p.a. • (1+r) = (1-PD)(1+y) or 1.08=(1-.05)(1.1369)

  4. Multiyear PD Using Forward Rates • Using the expectations hypothesis, the yield curves in Figure 5.1 can be decomposed: • (1+0y2)2 = (1+0y1)(1+1y1) or 1.162=1.1369(1+1y1) 1y1=18.36% p.a. • (1+0r2)2 = (1+0r1)(1+1r1) or 1.102=1.08(1+1r1) 1r1=12.04% p.a. • One year forward PD=5.34% p.a. from: (1+r) = (1- PD)(1+y) 1.1204=1.1836(1 – PD) • Cumulative PD = 1 – [(1 - PD1)(1 – PD2)] = 1 – [(1-.05)(1-.0534)] = 10.07%

  5. The Loss Intensity Process • Expected Losses (EL) = PD x LGD • If LGD is not fixed at 100% then: (1 + r) = [1 - (PDxLGD)](1 + y) Identification problem: cannot disentangle PD from LGD.

  6. Disentangling PD from LGD • Intensity-based models specify stochastic functional form for PD. • Jarrow & Turnbull (1995): Fixed LGD, exponentially distributed default process. • Das & Tufano (1995): LGD proportional to bond values. • Jarrow, Lando & Turnbull (1997): LGD proportional to debt obligations. • Duffie & Singleton (1999): LGD and PD functions of economic conditions • Unal, Madan & Guntay (2001): LGD a function of debt seniority. • Jarrow (2001): LGD determined using equity prices.

  7. KPMG’s Loan Analysis System • Uses risk-neutral pricing grid to mark-to-market • Backward recursive iterative solution – Figure 5.2. • Example: Consider a $100 2 year zero coupon loan with LGD=100% and yield curves from Figure 5.1. • Year 1 Node (Figure 5.3): • Valuation at B rating = $84.79 =.94(100/1.1204) + .01(100/1.1204) + .05(0) • Valuation at A rating = $88.95 = .94(100/1.1204) +.0566(100/1.1204) + .0034(0) • Year 0 Node = $74.62 = .94(84.79/1.08) + .01(88.95/1.08) • Calculating a credit spread: 74.62 = 100/[(1.08+CS)(1.1204+CS)] to get CS=5.8% p.a.

  8. Kamakura’s Risk Manager • Based on Jarrow (2001). • Decomposes risky debt and equity prices to estimate PD and LGD processes. • Fundamental explanatory variables: ROA, leverage, relative size, excess return over market index return, monthly equity volatility. • Type 1 error rate of 18.68%.

  9. Noisy Risky Debt Prices • US corporate bond market is much larger than equity market, but less transparent • Interdealer market not competitive – large spreads and infrequent trading: Saunders, Srinivasan & Walter (2002) • Noisy prices: Hancock & Kwast (2001) • More noise in senior than subordinated issues: Bohn (1999) • In addition to credit spreads, bond yields include: • Liquidity premium • Embedded options • Tax considerations and administrative costs of holding risky debt

  10. Appendix 5.1Understanding a Basic Intensity ProcessDuffie & Singleton (1998) • 1 – PD(t) = e-ht where h is the default intensity. Expected time to default is 1/h. • A rated firm: h=.001: expected to default once every 1,000 years. • B rated firm: h=.05: expected to default once every 20 years. • If have a portfolio with 1,000 A rated loans and 100 B rated loans, then there are 6 expected defaults per year = (1000*.001)+(100*.05)=6

  11. Systemic Default Intensities • hit = pitJt + Hit where Jt=intensity of arrival systemic events, Hit=firm-specific intensity of default arrival. • Cyclical increases default intensities. For example, if A (B) rated default intensity increases to .0012 (.055) then portfolio expects 6.7 defaults per year. • Figure 5.4 compares default intensity term structure for high credit risk (h=400 bp) vs. low credit risk (h=5 bp). The model can generate realistic default term structures.

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