Tải bản đầy đủ

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.
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.

Published by:

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

Confidential

ii

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

Confidential

1

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.

Confidential

2

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

Confidential

3

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




Confidential

1
2

4

Release Date: 12-February-1987



Tài liệu bạn tìm kiếm đã sẵn sàng tải về

Tải bản đầy đủ ngay

×