vasicek’s model of distribution of losses in a large...

Vasicek’s Model of Distribution of Losses in a Large, Homogeneous Portfolio

Stephen M SchaeferLondon Business School

Credit Risk ElectiveSummer 2012

Vasicek’s Model

• Important method for calculating distribution of loan losses:� widely used in banking

� used in Basel II regulations to set bank capital requirements

Merton-model Approach to Distribution of Portfolio Losses 2

• Motivation linked to distance-to-defaultanalysis

• But, model of dependence is Gaussian Copulaagain

• Key assumptions(apart from Gaussian dependence)� homogeneous portfolio(equal investment in each credit)

� very large numberof credits

Motivation: Merton’s Model

• In Merton model value of risky debt depends on firm value and default risk is correlated because firm values are correlated(e.g., via common dependence on market factor).

• Value of firm i at time T:

2exp( ( (1/ 2) ) ) where ~ (0,1)V V T NTµ σ εσ ε= − + %%


• We will assume that correlation between firm valuesarises because of correlation between surprise in individual firm value (ει) and market factor (m)

,

surprise in

2exp( ( (1/ 2) ) ) where ~ (0,1), ,

expected value of

V i i

CR

V V T NiT i i iV i

CR

Tµ σ εσ ε= − + %

144424443

%

14243

Correlation structure: Gaussian Copula

• Suppose correlationbetween each firm’s value and the market factor is the sameand equal to sqrt(ρ)ρ)ρ)ρ).

• This means that we may model correlation between the ε’s as

1 , 1,

andi im v i Nε ρ ρ= + − = K


and

( , )i jcorr ε ε ρ=• Where m and vi are independent N(0,1) random variables

and ρ is common to all firms

• Notice that if vi ~ N(0,1) and m ~ N(0,1) then εi ~ N(0,1)

Structural Approach, contd.

• From our analysis of distance-to-default, we know that under the Merton Model a firm defaults when:

,

2, , ,

1( ) / where ln( / )

2V ii D i i V i D i i iR T T R B Vε µ σ σ ≤ − − =

• The unconditional (natural) probability of default, p, is therefore:


therefore:

1 1

2 2

2 2, , , ,

,

Prob( ) ( )

i

D i i V i D i i V i

V i i

R Rp

T TN

T Tε

σ σ

µ σ µ σ≡ <

− − − −=

• In this model we assume that the default probability, p, is constant across firms

Idea: Single Common Factor and Large Homogeneous Portfolio

• Working out the distribution of portfolio losses directly when the ε’s are correlated is not easy

• But, if we work out the distribution conditional


• But, if we work out the distribution conditional on the market shock, m, then we can exploit the fact that the remaining shocks are independent and work out the portfolio loss distribution

Structural Approach, contd.

• Default occurs when:1 2( )R µ σ− −

• The shock to the return, εi, is related to the common and idiosyncratic shocks by:

1i im vε ρ ρ= + −


1

2

2, , 1

,

1

( )1 ( )

or

( )

1

i i

i

D i i V i

V i

Rv

v

m N pT

N p m

εσ

µ σρ ρ

ρρ

−

−

<

<

− −= + − =

−−

Vasicek and the Intensity Model

• We’ll see later that the Vasicek model is essentially the same as the intensity model when:� the intensity is the samefor all the names; and

� the numberof names becomes large

� equal investment in each name� equal investment in each name

� we use theGaussian copula


The Default Condition in Vasicek

• A large value of mmeans a “good” shock to the market (high asset values)

1( )

1iv

N p mρρ

−

<−

−


(high asset values)

• The larger the value of m– the market shock –the more negative the idiosyncratic shock, vi, has to be to trigger default

• The higher the correlation, ρ, between the firm shocks, the larger the impact of mon the critical value of vi.

Conditional Default Probability

• Conditional onthe realisation of the common shock, m, the probability of default is therefore:

1

Prob(default Prob( )

)= 1

i

N p mm v

ρρ

−

−

< − −


1

11

( )( ), say

1

( )and therfore = ( ( ))

1

N p mN m

N p mN m

ρ θρ

ρ θρ

−

−−

−= =

−

−=

−

The relation between θ(m) and m• For a given market shock, m, θ(m) gives the conditional

probability of default on an individual loan

Hi corr


Lo corr

Implications of Conditional Independence

• For a given value of m, as the number of loans in the portfolio → ∞, the proportionof loans in the portfolio that actually default converges to the probability θ (m) <<<< KEY IDEA

• In the chart, if the market shock is


m* then the ACTUALproportion of defaults in the portfolio converges to 15% as # loans → ∞

The critical value of m

• For a given actualfrequency of loss, θ, we can calculate the corresponding value of the market shock, m(θ) that will produce exactly that level of loss:

1( ) ( )

1

N p mN

ρ θθρ

− −=

−


( )

( )

11

1 1

1

( ) ( )

1

( ) 1( )

N p mN

N p Nm

ρ

ρ θθρ

ρ θθ

ρ

−−

− −

−

−=

−

− −=

The distribution of portfolio loss • Since the proportion ofportfolio losses decreaseswith m, the

probability that the proportion of loans that default(L) is less than θis:

( ) ( ) ( )

( )

1 1

1 1

( ) 1Prob = prob ( ) prob

( ) 1

N p NL m m m

N p N

ρ θθ θ

ρ

ρ θ

− −

− −

− −< > = >

− −


( )

( ) ( )

1 1

1 1

( ) 1

1 ( )Prob

N p NN

N N pL N

ρ θρ

ρ θθ

ρ

− −

− −

− −= −

− −< =

Loan Loss Distribution

• This result gives the cumulative distribution of the fraction of loans that defaultin a well diversified homogeneous portfoliowhere the correlation in default comes from dependence on a common factor

• Homogeneitymeans that each loan has:

( ) ( )1 11 ( )Prob

N N pL N

ρ θθ

ρ

− − − −< =


• Homogeneitymeans that each loan has:

� the same default probability, p� (implicitly) the same loss-given-default� the same correlation, ρ, across different loans

• The distribution has two parameters� default probability, p� correlation, ρ

Loan Loss Distribution with p = 1% and ρ = 12% and 0.6%

0.04

0.05

0.06

0.07

0.08

p = 1.5% rho = 12.0%

p = 1.5% rho = 0.6%


0

0.01

0.02

0.03

0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00%

Portfolio Loan Loss (%(

Example of Vasicek formula Applied to Bank Portfolio


Source: Vasicek

Relationship between the Vasicek model and the intensity model with the

Gaussian CopulaGaussian Copula


Fundamentally, Vasicek model gives same results Intensity model and Gaussian copula (!)

• Default condition in Vasicek model:

1

2

2, , 1

,

( )1 ( )i i

D i i V i

V i

Rvm N p

Tε

σ

µ σρ ρ −<

− −= + − =


• In other words, whether a normally distributed N(0,1) variable is larger or smaller than a given fixed number, 1( )N p−

and .. in intensity model (with the Gaussian copula) .. the same (!)

• In the intensity model default occurs when

1ln(1 ) * where ( )i i i iU U Nτ τ ε

λ= − − ≤ =

i.e., when the default time τi is smaller than the “maturity”τ*


i

• Define 1

* ln(1 *) and * ( *)U U Nτ ελ

= − − =

• Then default occurs when

*iε ε≤

Or.. in pictures ..


1ln(1 ) * where ( )i i i iU U Nτ τ ε

λ= − − ≤ =*iε ε≤

• If the value of e that we draw is smaller than the critical value

• Then τi is less than τ* and we have a default

Example

• Intensity model with 1000 names and equal intensity and Vasicek model with model with equal default probability and correlation


The bottom line ..

• The Vasicek model is the same as the intensity model with a Gaussian copula, identical default probabilities and a large number of names.


Applications

• Vasicek’s obtains a formula for the distribution of losses with:� singlecommon factor

� homogeneousportfolio

� large numberof credits


• But the approach can be generalisedto a much more realistic (multi-factor) correlation structure and granularity in the portfolio holdings

� quite widely usedin banking for management of risk of loan portfolio

Takeaways

• Vasicek’s formulagives useful quick method for generating distribution of lossesin large portfolio� in one-factor versionfundamentally the same as

Gaussian copula� exploitation of conditional independenceis useful

idea


idea

• Applicationstend to be in risk managementof actual loan losses (natural distribution) rather than pricing (risk-neutral distribution)� less evidenceof poor performancein natural

distribution (same story about structural model again)

vasicek’s model of distribution of losses in a large...

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