Limiting Loan Loss Probability Distribution
KMV Corporation COPYRIGHT 1991, KMV CORPORATION, SAN FRANCISCO, CALIFORNIA, USA. All rights reserved. Document Number: 999-0000-046. Revision 1.0.0. This is a highly confidential document that contains information that is the property of KMV Corporation or Kealhofer, McQuown, Vasicek Development, L.P. (collectively, “KMV”). This document is being provided to you under the confidentiality agreement that exists between your company and KMV. This document should only be shared on a need to know basis with other employees of your business, excluding independent contractors, consultants or other agents. By accepting this document, you agree to abide by these restrictions; otherwise you should immediately return the document to KMV. Any other actions are a violation of the owner’s trade secret, copyright and other proprietary rights. Any other actions are also a violation of the previously mentioned confidentiality agreement. KMV retains all trade secret, copyright and other proprietary rights in this document. KMV Corporation and the KMV Logo are registered trademarks of KMV Corporation. Portfolio Manager™, Credit Monitor™, Global Correlation Model™, GCorr™, Private Firm Model™, EDF Calculator™, EDFCalc™, Expected Default Frequency™ and EDF™ are trademarks of KMV Corporation. All other trademarks are the property of their respective owners.
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Oldrich Alfons Vasicek
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Limiting Loan Loss Probability Distribution LIMITING LOAN LOSS PROBABILITY DISTRIBUTION Oldrich Vasicek, 8/9/91
The cumulative probability that the percentage loss on a portfolio of n loans does not exceed θ is [ nθ ]
Fn ( θ ) = ∑ Pk k =0
where Pk are given by an integral expression in Oldrich Vasicek’s memo, Probability of Loss on Loan Portfolio, 2/12/87 (attached at the end of this note). The substitution
1 s= N N −1 ( p ) − ρu 1− ρ
(
)
in the integral gives Fn (θ ) as [ nθ ]
Fn ( θ ) = ∑ ( k =0
1
n k
) ∫ s (1 − s ) k
n− k
dW ( s )
0
where
1 W (s) = N ρ
(
1 − ρ N −1 ( s ) − N −1 ( p )
)
By the law of large numbers, [ nθ ]
lim ∑ ( nk )s k (1 − s ) n →∞
n−k
=0
if θ < s
=1
if θ > s
k =0
and therefore the cumulative distribution function of loan losses on a very large portfolio is
F∞ ( θ ) = W ( θ )
This is a highly skewed distribution. Its density is
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f∞ ( θ) =
1 1− ρ exp − ρ 2ρ
(
)
1 − ρ N −1 ( θ ) − N −1 ( p ) + 2
2 1 −1 N ( θ)) ( 2
Its mean, median and mode are given by
θ= p 1 N −1 ( p ) θmed = N 1− ρ 1 − ρ −1 θmode = N N ( p ) for ρ < 1 − 2ρ
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Limiting Loan Loss Probability Distribution PROBABILITY OF LOSS ON LOAN PORTFOLIO Oldrich Vasicek, 2/12/87
Consider a portfolio consisting of n loans in equal dollar amounts. Let the probability of default on any one loan be p, and assume that the values of the borrowing companies’ assets are correlated with a coefficient ρ for any two companies. We wish to calculate the probability distribution of the percentage gross loss L on the portfolio, that is,
Pk = P L =
k , k = 0,1,… , n n
Let Ait be the value of the i-th company’s assets, described by a logarithmic Wiener process
dAi = rAi dt + σ i Ai dzi where zit , i =1, 2, …, n are Wiener processes with
E ( dzi ) = dt 2
E ( dzi ) ( dz j ) = ρ dt , i ≠ j The company defaults on its loan if the value of its assets drops below the contractual value of its obligations Di payable at time T. We thus have
p = P [ AiT < Di ] = N ( −ci )
where
ci =
1 log Ai 0 − log Di + rT − 12 σ2T ) ( σ T
and N is the cumulative normal distribution function. Because of the joint normality and the equal correlations, the processes zi can be represented as
zi = bx + aεi , i = 1, 2,… , n where
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b = ρ , a = 1− ρ and
E ( dx ) = dt 2
E ( d εi ) = dt 2
E ( dx )( d εi ) = 0
E ( d εi ) ( d ε j ) = 0 , i ≠ j The term bx can be interpreted as the i-th company exposure to a common factor x (such as the state of the economy) and the term aεi represents the company’s specific risks. Then
k Pk = P L = n
= ( nk ) P [ A1T < D1 ,… , AkT < Dk , Ak +1T ≥ Dk +1 ,… , AnT ≥ Dn ] = ( nk ) = ( nk )
∞
∫ P[ A
1T
< D1 ,… , AkT < Dk , Ak +1T ≥ Dk +1 ,… , AnT ≥ Dn | xT = u ] d P [ xT < u ]
−∞ ∞
∫ P c
1
T + bxT + aε1T < 0,..., ck T + bxT + aε kT < 0, ck +1 T + bxT + aε k +1T ≥ 0,
−∞
… , cn T + bxT + aε nT ≥ 0| xT = u ] d P [ xT < u ] =(
∞
n k
) ∫ N − c +abu −∞
k
c + bu 1 − N − a
n−k
dN ( u )
In terms of the original parameters p and ρ, we have
Pk = (
) ∫ N 11− ρ N −1 ( p ) − ρu −∞ ∞
n k
(
)
k
1 N −1 ( p ) − ρu 1 − N 1− ρ
(
)
n−k
dN ( u ) , k = 0,1,..., n
Note that the integrand is the conditional probability distribution of the portfolio loss given the state of the economy, as measured by the market increase or decline in terms of its standard deviations.
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