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

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

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