# Probability of loss on loan portfolio

Probability of Loss on Loan
Portfolio

KMV Corporation
COPYRIGHT  1987, KMV CORPORATION, SAN FRANCISCO, CALIFORNIA, USA. All rights
reserved. Document Number: 999-0000-056. Revision 1.0.0.
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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
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Authors:

KMV Corporation
1620 Montgomery Street, Suite 140
San Francisco, CA 94111 U.S.A.
Phone: +1 415-296-9669
FAX: +1 415-296-9458
email: support@kmv.com
website: http: // www.kmv.com

Oldrich Alfons Vasicek

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Release Date: 12-February-1987

Probability of Loss on Loan Portfolio
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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Release Date: 12-February-1987

KMV Corporation

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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Release Date: 12-February-1987

Probability of Loss on Loan Portfolio
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. 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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Release Date: 12-February-1987

KMV Corporation

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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Release Date: 12-February-1987

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