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
• Reduced form models: Decompose risky
debt prices to estimate the stochastic default
intensity function. No structural
explanation of why default occurs.
2
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)
3
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%
4
Figure 5.1 Yield curves.
Spot
Yield
16%
13.69%
14%
B Rated ZeroCoupon Bond
A Rated ZeroCoupon Bond
Zero-Coupon
Treasury Bond
11.5%
10%
8%
1 Yr.
2 Yr.
Time to Maturity
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
Figure 5.2 The multiperiod loan migrates over
many periods.
A
B Risk
Grade
B
C
D
0
1
2
3
4
Time
9
Figure 5.3 Risky debt pricing.
Period 0
Period 1
$85.43
Period 2
94%
5.66%
1%
$67.14
94%
$80.28
$100 A Rating
1%
94%
$100 B Rating
0.34%
5%
5%
$0 Default
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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%.
Bond Prices:
B = B[t, T, i, (t, X(t)), (t, X(t)), (t,T,X(t)), , S(t,X(t))]
Equity Prices:  = [ t, T, i, (t, X(t)), , S(t,X(t))]
where t is the current period; T is the bond’s time to maturity; i is the stochastic defaultfree interest rate process; (t, X(t)) is the default intensity process, i.e., the risk neutral
PD; (t, X(t)) is the recovery rate (1 – LGD); (t,T,X(t)) is the liquidity premium;  is a
stock market bubble factor; and S(t,X(t)) is the liquidating dividend on equity in the event
of bond default.
11
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
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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
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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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Figure 5.4 Term structure of coupon-strip (zero-recovery) yield spreads.
Source:
Duf f ie and Singleton (1998), p. 20.
400
h(0) 5 bps
h(0) 400 bps
350
300
250
200
150
100
50
0
0
1
2
3
4
5
6
Maturity (Years)
7
8
9
10
15