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 = = [100(1-PD)]/1.08 Solving (5.1), PD=5% p.a.
Alternatively, = 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) =1.1836(1 – PD)
Cumulative PD = 1 – [(1 - PD1)(1 – PD2)] = 1 – [(1-.05)( )] = %

6. 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.

7. 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.

8. 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) (100/1.1204) (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)( CS)] to get CS=5.8% p.a.

11. 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%.

12. 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

13. Appendix 5.1 Understanding a Basic Intensity Process Duffie & 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

14. 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 (.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.