Vasicek Single Factor Model

(1)

Alexandra Kochend¨ orfer

7. Februar 2011

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Problem Setting

I Consider portfolio with N different credits of equal size 1.

I Each obligor has an individual default probability.

I In case of default of the n’th obligor we lose the whole n’th position in portfolio.

I What can we say about the loss distribution?

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U.S. Corporate Default Rates Since 1920

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Why is default correlation important

Why is default correlation important?

Consider, for two default events A and B

I default probabilities p _A and p _B

I joint default probability p _AB

I conditional default probability p _A|B

I correlation ρ _AB between default events These quantities are connected by

p _A|B = p _AB p _B

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Why is default correlation important

ρ AB = Cov (A, B)

pVar(A)pVar(B) = p AB − p _A p B

pp _A (1 − p _A )p _B (1 − p _B ) The default probabilities are usually very small p A = p B = p 1

p _AB = p _A p _B + ρ _AB p

p _A (1 − p _A )p _B (1 − p _B ) ≈ p ² + ρ _AB · p ≈ ρ _AB · p p _A|B = p _A + ρ _AB

p B

p p _A (1 − p _A )p _B (1 − p _B ) ≈ ρ _AB

The joint default probability and conditional default probability are

dominated by the correlation coefficient.

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Independent/perfectly dependent defaults

Independent Defaults

Consider N independent default events D ₁ , . . . , D _N with p _D

₁

= · · · = p _D

_N

= p ⇒ Number of defaults ∼ B(p, N).For N = 100, p = 0.05

p (%) 1 2 3 4 5 6 7 8 9 10

99.9(%) VaR 5 7 9 11 13 14 16 17 19 20

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Perfectly dependent defaults

Consinder default correlation ρ _ij = 1 for all pairs ij . 1 = p D

1

D

2

− p _D

₁

p D

2

pp _D

₁

(1 − p _D

₁

)p _D

₂

(1 − p _D

₂

) = p D

1

D

2

− p ² p(1 − p)

⇒ p _D

₁

_D

₂

= p D

1

= p D

2

= p i.e. D 1 ∩ D ₂ = D 1 = D 2 a.s.

⇒ p _D

₁

_D

₂

_D

₃

= p D

2

D

3

= p ⇒ D 1 ∩ D ₂ ∩ D ₃ = D 1 a.s. . . .

⇒ D ₁ ∩ · · · ∩ D _N = D ₁ a.s. ⇒ p _D

₁

_...D

_N

= p

All loans in the portfolio defaults with probability p, none with

probability 1 − p.

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Perfectly dependent defaults

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Vasicek Single Factor Model Modelling Default Correlation

Data sources

Data Sources

I Actual Rating and Default Events.

+ Objective and direct.

– Joint defaults are rare events, sparse data sets.

I Credit Spread.

+ Incorporate information on markets, observable.

– No theoretical link between credit spread correlation and default correlation.

I Equity correlation.

+ Data easily available, good quality.

– Connection to credit risk not obvious, needs a lot of

assumptions.

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Default triggered by firm’s value

Default triggered by Firm’s Value

The firm value (A n,t ) 0≤t≤1 of each obligor n ∈ {1, . . . , N} is modelled as in Black-Scholes model, hence at terminal time t = 1 with A _n,1 = A _n we have

A _n = A _n,0 exp

µ _n − σ ² _n 2

+ σ _n B _n

with some standard normal variable B n .

The r.v. (B ₁ , . . . , B _N ) are jointly normally distributed with covariance matrix Σ = (ρ _ij ) _ij , where ρ _ij denotes the asset correlation between assets i and j .

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The obligor n defaults if the asset value falls below a perspecified barrier C _n (debts)

D _n = 11 _{A

_n

_<C

_n

_} The default probability of the n’s debtor is

p D

n

= P(D n = 1) = P(A n < C n ) = P(B n < c n ) = Φ(c n ) with default barrier

c n = log _A ^C

ⁿ

n,0

− µ _n σ _n

We can assume the individual default probabilities p D

n

as given

and compute c _n = Φ ⁻¹ (p _D

_n

) and vice versa.

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The joint distribution of B _i determines the dependency structure of default variables uniquely

P(D 1 = 1, . . . , D N = 1) = P(B 1 < c 1 , . . . , B N < c N )

= Φ _N (Φ ⁻¹ (p _D

₁

), . . . , Φ ⁻¹ (p _D

_n

); Σ) In case with two assets with correlation ρ _1,2 = ρ _2,1 , the default correlation can be computed via

ρ = P(D ₁ = 1, D ₂ = 1) − p _D

₁

p _D

₂

pp _D

₁

(1 − p _D

₁

)p _D

₂

(1 − p _D

₂

)

= Φ 2 (Φ ⁻¹ (p D

1

), Φ ⁻¹ (p D

2

); ρ 1,2 ) − p D

1

p D

2

pp _D

₁

(1 − p _D

₁

)p _D

₂

(1 − p _D

₂

)

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We need

I N(N − 1)/2 asset correlations of Σ

I N individual default probabilities

I Additional assumptions on the structure of B _i to reduce the

number of parameters.

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Vasicek Single Factor Model Vasicek Single Factor Model

Vasicek Single Factor Model

Assume, that the logarithmic return B n can be written as B _n = √

ρ · Y + p

1 − ρ · _n

with some constant ρ ∈ [0, 1] and N + 1 independent standard normally distributed r.v. Y , 1 , . . . , N .

Interpretation

I Y is a common systematic risk factor affecting all firms (state of economy)

I _n are idiosyncratic factors independent across firms (management, innovations)

I Corr (B i , B j ) = ρ controls the proportions between systematic and idiosyncratic factors, empirically around 10%.

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Conditional on the realisation of the systematic factor Y

I the logarithmic returns B n are independent ( for a constant y variables √

ρ · y + √

1 − ρ · _n are independent)

I default variables D n = 11 {B

_n

<c

n

} are independent as function of B _n

The only effect of Y is to move B _n closer or further away from

barrier c _n .

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Loss distribution in finite portfolio

Theorem

For ρ ∈ (0, 1) and same default probabilities p = p _D

₁

= · · · = p _D

_N

the conditional default probability is given by

p(y ) := P[B n < c | Y = y ] = Φ Φ ⁻¹ (p) − √ ρ · y

√ 1 − ρ

. The number of defaults L = P N

i =1 D _i has the following distribution

P(L ≤ m) =

m

X

k=0

N k

· Z ∞

−∞

p(y ) ^k · (1 − p(y )) ^N−k · φ(y )dy

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Proof

The probability of k defaults is P(L = k) = E(P({L = k} | Y )) =

Z ∞

−∞

P(L = k | Y = y )φ(y )dy , where φ is density of Y . The defaults D _n are independent

conditional on Y , hence

P(L = k | Y = y ) = N k

· p(y ) ^k · (1 − p(y )) ^N−k Thus, for m ∈ {1, . . . , N} we have

P(L ≤ m) =

m

X

k=0

N k

· Z ∞

−∞

p(y ) ^k · (1 − p(y )) ^N−k · φ(y )dy

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Loss Distibutions for different ρ

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VaR Levels for different ρ with N = 100 and p = 5%

ρ(%) 99.9(%)VaR 99.(%)VaR

0 13 11

1 14 12

10 27 19

30 55 35

50 80 53

Independent defaults

p (%) 1 2 3 4 5 6 7 8 9 10

99.9(%) VaR 5 7 9 11 13 14 16 17 19 20

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Large Homogeneous Portfolio Approximation

Definition (Large Homogeneous Portfolio LHP)

I p _D

₁

= · · · = p _D

_N

= p

I portfolio is weighted with ω ₁ ^(N) , . . . , ω _N ^(N) , P N

n=1 ω ^(N) n = 1, such that

N→∞ lim

N

X

n=1

(ω n ^(N) ) ² = 0

The portfolio is not dominated by few loans much larger then the rest.

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Definition (Loss Rate)

The portfolio loss rate is defined by

L ^(N) =

N

X

n=1

ω _n ^(N) D _n ∈ [0, 1]

Lemma

Following holds for the LHP

E(L ^(N) | Y ) = p(Y ) = Φ Φ ⁻¹ (p) − √ ρ · Y

√ 1 − ρ

Var (L ^(N) | Y ) =

N

X

n=1

(ω n ^(N) ) ² · p(Y ) · (1 − p(Y ))

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Proof

Linearity of conditional expectation yields

E(L ^(N) | Y ) =

N

X

n=1

ω ^(N) n E(D n | Y )

=

N

X

n=1

ω ^(N) _n P(D _n | Y ) = p(Y )

N

X

n=1

ω ^(N) _n = p(Y )

D n are independent conditional on Y , thus

Var (L ^(N) | Y ) =

N

X

n=1

(ω _n ^(N) ) ² Var (D _n | Y )

=

N

X

n=1

(ω n ^(N) ) ² · p(Y ) · (1 − p(Y ))

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Theorem

The portfolio loss rate in LHP converges in probability for N → ∞.

L ^{(N) P} → p(Y ) = Φ Φ ⁻¹ (p) − √ ρ · Y

√ 1 − ρ

Proof

For the large portfolio the variation of loss rate given Y tends to 0

Var (L ^(N) | Y ) =

N

X

n=1

(ω n ^(N) ) ² · p(Y ) · (1 − p(Y ))

≤ 1 4

N

X

n=1

(ω ^(N) _n ) ² −−−−→

N→∞ 0

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This provides convergence in L ² :

E((L ^(N) − p(Y )) ² ) = E((L ^(N) − E(L ^(N) | Y )) ² )

= E(E((L ^(N) − E(L ^(N) | Y )) ² | Y ))

= E(Var (L ^(N) | Y )) −−−−→

N→∞ 0

Convergence in L ² implies convergence in probability i.e. for all

> 0:

N→∞ lim P

L ^(N) − p(Y ) >

= 0

The law of L ^(N) converges weakly to the law of p(Y ), i.e.

P(L ^(N) ≤ x) −−−−→

N→∞ P(p(Y ) ≤ x )

for all x , where the distribution function of p(Y ) is continuous.

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Theorem (Approximative Distribution of Loss Rate in LHP)

P(p(Y ) ≤ x ) = Φ √1 − ρ · Φ ⁻¹ (x ) − Φ ⁻¹ (p)

√ ρ

, x ∈ [0, 1]

Proof

P(p(Y ) ≤ x ) = P

Φ Φ ⁻¹ (p) − √ ρ · Y

√ 1 − ρ

≤ x

= P

Y ≤

√ 1 − ρ · Φ ⁻¹ (x ) − Φ ⁻¹ (p)

√ ρ

= Φ √1 − ρ · Φ ⁻¹ (x ) − Φ ⁻¹ (p)

√ ρ

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Approximative density of loss rate with p = 2%, ρ = 10%

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Properties of Loss Rate Distribution

E(p(Y )) = lim

N→∞ E(L ^(N) ) = lim

N→∞

N

X

n=1

ω ^(N) _n p = p

Because of convergence we can easily compute α-Quantiles of loss rate distribution for large N

P(L ^(N) ≤ α) ≈ Φ √1 − ρ · Φ ⁻¹ (α) − Φ ⁻¹ (p)

√ ρ

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I When ρ → 1

P(L ^(∞) ≤ α) = 1 − p = P(L ^(∞) = 0) for all α ∈ (0, 1) P(L ^(∞) = 1) = p

All loans default with prob. p, none with 1 − p.

I When ρ → 0

P(L ^(∞) ≤ α) = 0 for α < p

P(L ^(∞) ≤ α) = 1 for α ≥ p ⇒ P(L ^(∞) = p) = 1 With the Law of Large Numbers the loss in Binomial model tends almost surly to

1 N

N

X

i =1

D _i → p

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Simulated Loss Distibution

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Vasicek Single Factor Model Conclusion

Conclusion

The Vasicek Single Factor Model provides a closed form Loss Rate Distribution

lim

N→∞ P(L ^(N) ≤ x) = Φ √1 − ρ · Φ ⁻¹ (x ) − Φ ⁻¹ (p)

√ ρ

for a Large Homogeneous Portfolio, which depends only on two parameters p and ρ and gives a good fit to market data.

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Vasicek Single Factor Model Conclusion

Vasicek Single Factor Model

Vasicek Single Factor Model

Alexandra Kochend¨ orfer

7. Februar 2011

Problem Setting

I Consider portfolio with N different credits of equal size 1.

I Each obligor has an individual default probability.

I In case of default of the n’th obligor we lose the whole n’th position in portfolio.

I What can we say about the loss distribution?

Contents

Default Correlation Definition

Why is default correlation important Independent/perfectly dependent defaults Modelling Default Correlation

Data sources

Default triggered by firm’s value Vasicek Single Factor Model

Loss distribution in finite portfolio

Large Homogeneous Portfolio Approximation

Conclusion

Definition Default correlation is the phenomenon that the likelihood of one obligor defaulting on its debt is affected by whether or not another obligor has defaulted on its debts.

I Positive correlation: one firm is the creditor of another

I Negative correlation: the firms are competitors Drivers of Default Correlation

I State of the general economy

I Industry-specific factors

Oil industry: 22 companies defaulted over 1982-1986.

Thrifts: 19 defaults over 1990-1992.

Casinos/hotel chains: 10 defaults over 1990-1992.

U.S. Corporate Default Rates Since 1920

Why is default correlation important?

Consider, for two default events A and B

I default probabilities p A and p B

I joint default probability p AB

I conditional default probability p A|B

I correlation ρ AB between default events These quantities are connected by

p A|B = p AB p B

ρ AB = Cov (A, B)

pVar(A)pVar(B) = p AB − p A p B

pp A (1 − p A )p B (1 − p B ) The default probabilities are usually very small p A = p B = p  1

p AB = p A p B + ρ AB p

p A (1 − p A )p B (1 − p B ) ≈ p 2 + ρ AB · p ≈ ρ AB · p p A|B = p A + ρ AB

p B

p p A (1 − p A )p B (1 − p B ) ≈ ρ AB

The joint default probability and conditional default probability are

dominated by the correlation coefficient.

Independent Defaults

Consider N independent default events D 1 , . . . , D N with p D

= · · · = p D

= p ⇒ Number of defaults ∼ B(p, N).For N = 100, p = 0.05

p (%) 1 2 3 4 5 6 7 8 9 10

99.9(%) VaR 5 7 9 11 13 14 16 17 19 20

Perfectly dependent defaults

Consinder default correlation ρ ij = 1 for all pairs ij . 1 = p D

D

− p D

p D

pp D

(1 − p D

)p D

(1 − p D

) = p D

D

− p 2 p(1 − p)

⇒ p D

D

= p D

= p D

= p i.e. D 1 ∩ D 2 = D 1 = D 2 a.s.

⇒ p D

D

D

= p D

D

= p ⇒ D 1 ∩ D 2 ∩ D 3 = D 1 a.s. . . .

⇒ D 1 ∩ · · · ∩ D N = D 1 a.s. ⇒ p D

...D

= p

All loans in the portfolio defaults with probability p, none with

probability 1 − p.

Perfectly dependent defaults

Data Sources

I Actual Rating and Default Events.

+ Objective and direct.

I default probabilities p _A and p _B

I joint default probability p _AB

I conditional default probability p _A|B

I correlation ρ _AB between default events These quantities are connected by

p _A|B = p _AB p _B

pVar(A)pVar(B) = p AB − p _A p B

pp _A (1 − p _A )p _B (1 − p _B ) The default probabilities are usually very small p A = p B = p 1

p _AB = p _A p _B + ρ _AB p

p _A (1 − p _A )p _B (1 − p _B ) ≈ p ² + ρ _AB · p ≈ ρ _AB · p p _A|B = p _A + ρ _AB

p p _A (1 − p _A )p _B (1 − p _B ) ≈ ρ _AB

Consider N independent default events D ₁ , . . . , D _N with p _D

= · · · = p _D

Consinder default correlation ρ _ij = 1 for all pairs ij . 1 = p D

− p _D

pp _D

(1 − p _D

)p _D

(1 − p _D

− p ² p(1 − p)

⇒ p _D

_D

= p i.e. D 1 ∩ D ₂ = D 1 = D 2 a.s.

⇒ p _D

_D

_D

= p ⇒ D 1 ∩ D ₂ ∩ D ₃ = D 1 a.s. . . .

⇒ D ₁ ∩ · · · ∩ D _N = D ₁ a.s. ⇒ p _D

_...D

The firm value (A n,t ) 0≤t≤1 of each obligor n ∈ {1, . . . , N} is modelled as in Black-Scholes model, hence at terminal time t = 1 with A _n,1 = A _n we have

A _n = A _n,0 exp

µ _n − σ ² _n 2

+ σ _n B _n

The r.v. (B ₁ , . . . , B _N ) are jointly normally distributed with covariance matrix Σ = (ρ _ij ) _ij , where ρ _ij denotes the asset correlation between assets i and j .

The obligor n defaults if the asset value falls below a perspecified barrier C _n (debts)

D _n = 11 _{A

_<C

_} The default probability of the n’s debtor is

c n = log _A ^C

− µ _n σ _n

and compute c _n = Φ ⁻¹ (p _D

The joint distribution of B _i determines the dependency structure of default variables uniquely

= Φ _N (Φ ⁻¹ (p _D

), . . . , Φ ⁻¹ (p _D

); Σ) In case with two assets with correlation ρ _1,2 = ρ _2,1 , the default correlation can be computed via

ρ = P(D ₁ = 1, D ₂ = 1) − p _D

p _D

pp _D

(1 − p _D

)p _D

(1 − p _D

= Φ 2 (Φ ⁻¹ (p D

), Φ ⁻¹ (p D

pp _D

(1 − p _D

)p _D

(1 − p _D

I Additional assumptions on the structure of B _i to reduce the

Assume, that the logarithmic return B n can be written as B _n = √

1 − ρ · _n

with some constant ρ ∈ [0, 1] and N + 1 independent standard normally distributed r.v. Y , 1 , . . . , N .

I _n are idiosyncratic factors independent across firms (management, innovations)

1 − ρ · _n are independent)