Vasicheck prezentation

Vasicek Single Factor Model


Alexandra Kochendorfer

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’thposition in portfolio.

I What can we say about the loss distribution?

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Contents

Default CorrelationDefinitionWhy is default correlation importantIndependent/perfectly dependent defaults

Modelling Default CorrelationData sourcesDefault triggered by firm’s value

Vasicek Single Factor ModelLoss distribution in finite portfolioLarge Homogeneous Portfolio Approximation

Conclusion

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Default Correlation

Definition

Definition Default correlation is the phenomenon that thelikelihood of one obligor defaulting on its debt is affected bywhether 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 economyI Industry-specific factors

I Oil industry: 22 companies defaulted over 1982-1986.I Thrifts: 19 defaults over 1990-1992.I Casinos/hotel chains: 10 defaults over 1990-1992.

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Default Correlation

Definition

U.S. Corporate Default Rates Since 1920

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Default Correlation

Why is default correlation important

Why is default correlation important?

Consider, for two default events A and B

I default probabilities pA and pB

I joint default probability pAB

I conditional default probability pA|BI correlation ρAB between default events

These quantities are connected by

pA|B =pAB

pB

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Default Correlation

Why is default correlation important

ρAB =Cov(A,B)√

Var(A)√

Var(B)=

pAB − pApB√pA(1− pA)pB(1− pB)

The default probabilities are usually very small pA = pB = p � 1

pAB = pApB + ρAB

√pA(1− pA)pB(1− pB) ≈ p2 + ρAB · p ≈ ρAB · p

pA|B = pA +ρAB

pB

√pA(1− pA)pB(1− pB) ≈ ρAB

The joint default probability and conditional default probability aredominated by the correlation coefficient.

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Default Correlation

Independent/perfectly dependent defaults

Independent DefaultsConsider N independent default events D1, . . . ,DN withpD1 = · · · = pDN

= p ⇒ Number of defaults ∼ B(p,N).ForN = 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 19 208 / 33


Default Correlation


Perfectly dependent defaults

Consinder default correlation ρij = 1 for all pairs ij .

1 =pD1D2 − pD1pD2√

pD1(1− pD1)pD2(1− pD2)=

pD1D2 − p2

p(1− p)

⇒ pD1D2 = pD1 = pD2 = p i.e. D1 ∩ D2 = D1 = D2 a.s.

⇒ pD1D2D3 = pD2D3 = p ⇒ D1 ∩ D2 ∩ D3 = D1 a.s. . . .

⇒ D1 ∩ · · · ∩ DN = D1 a.s.⇒ pD1...DN= p

All loans in the portfolio defaults with probability p, none withprobability 1− p.

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Default Correlation


Perfectly dependent defaults

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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 (An,t)0≤t≤1 of each obligor n ∈ {1, . . . ,N} ismodelled as in Black-Scholes model, hence at terminal time t = 1with An,1 = An we have

An = An,0 exp

{(µn −

σ2n

2

)+ σnBn

}with some standard normal variable Bn.

The r.v. (B1, . . . ,BN) are jointly normally distributed withcovariance matrix Σ = (ρij)ij , where ρij denotes the assetcorrelation between assets i and j .

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The obligor n defaults if the asset value falls below aperspecified barrier Cn (debts)

Dn = 11{An<Cn}

The default probability of the n’s debtor is

pDn = P(Dn = 1) = P(An < Cn) = P(Bn < cn) = Φ(cn)

with default barrier

cn =log Cn

An,0− µn

σn

We can assume the individual default probabilities pDn as givenand compute cn = Φ−1(pDn) and vice versa.

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The joint distribution of Bi determines the dependencystructure of default variables uniquely

P(D1 = 1, . . . ,DN = 1) = P(B1 < c1, . . . ,BN < cN)

= ΦN(Φ−1(pD1), . . . ,Φ−1(pDn); Σ)

In case with two assets with correlation ρ1,2 = ρ2,1, the defaultcorrelation can be computed via

ρ =P(D1 = 1,D2 = 1)− pD1pD2√

pD1(1− pD1)pD2(1− pD2)

=Φ2(Φ−1(pD1),Φ−1(pD2); ρ1,2)− pD1pD2√

pD1(1− pD1)pD2(1− pD2)

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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 Bi to reduce thenumber of parameters.

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Vasicek Single Factor ModelAssume, that the logarithmic return Bn can be written as

Bn =√ρ · Y +

√1− ρ · εn

with some constant ρ ∈ [0, 1] and N + 1 independent standardnormally distributed r.v. Y , ε1, . . . , εN .

Interpretation

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

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

I Corr(Bi ,Bj) = ρ controls the proportions between systematicand idiosyncratic factors, empirically around 10%.

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Conditional on the realisation of the systematic factor Y

I the logarithmic returns Bn are independent ( for a constant yvariables

√ρ · y +

√1− ρ · εn are independent)

I default variables Dn = 11{Bn<cn} are independent as functionof Bn

The only effect of Y is to move Bn closer or further away frombarrier cn.

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Loss distribution in finite portfolio

TheoremFor ρ ∈ (0, 1) and same default probabilities p = pD1 = · · · = pDN

the conditional default probability is given by

p(y) := P[Bn < c | Y = y ] = Φ

(Φ−1(p)−√ρ · y√

1− ρ

).

The number of defaults L =∑N

i=1 Di has the followingdistribution

P(L ≤ m) =m∑

k=0

(N

k

)·∫ ∞−∞

p(y)k · (1− p(y))N−k · φ(y)dy

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ProofThe probability of k defaults is

P(L = k) = E(P({L = k} | Y )) =

∫ ∞−∞

P(L = k | Y = y)φ(y)dy ,

where φ is density of Y . The defaults Dn are independentconditional 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∑

k=0

(N

k

)·∫ ∞−∞

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 13 111 14 12

10 27 1930 55 3550 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 pD1 = · · · = pDN= p

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

(N)N ,

∑Nn=1 ω

(N)n = 1,

such that

limN→∞

N∑n=1

(ω(N)n )2 = 0

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

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Definition (Loss Rate)

The portfolio loss rate is defined by

L(N) =N∑

n=1

ω(N)n Dn ∈ [0, 1]

LemmaFollowing holds for the LHP

E(L(N) | Y ) = p(Y ) = Φ

(Φ−1(p)−√ρ · Y

√1− ρ

)Var(L(N) | Y ) =

N∑n=1

(ω(N)n )2 · p(Y ) · (1− p(Y ))

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ProofLinearity of conditional expectation yields

E(L(N) | Y ) =N∑

n=1

ω(N)n E(Dn | Y )

=N∑

n=1

ω(N)n P(Dn | Y ) = p(Y )

N∑n=1

ω(N)n = p(Y )

Dn are independent conditional on Y , thus

Var(L(N) | Y ) =N∑

n=1

(ω(N)n )2Var(Dn | Y )

=N∑

n=1

(ω(N)n )2 · p(Y ) · (1− p(Y ))

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

L(N) P→ p(Y ) = Φ

(Φ−1(p)−√ρ · Y

√1− ρ

)

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

Var(L(N) | Y ) =N∑

n=1

(ω(N)n )2 · p(Y ) · (1− p(Y ))

≤ 1

4

N∑n=1

(ω(N)n )2 −−−−→

N→∞0

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This provides convergence in L2:

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 L2 implies convergence in probability i.e. for allε > 0:

limN→∞

P(∣∣∣L(N) − p(Y )

∣∣∣ > ε)

= 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.26 / 33




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

)27 / 33




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

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Properties of Loss Rate Distribution

E(p(Y )) = limN→∞

E(L(N)) = limN→∞

N∑n=1

ω(N)n p = p

Because of convergence we can easily compute α-Quantiles of lossrate 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 tendsalmost surly to

1

N

N∑i=1

Di → p

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Simulated Loss Distibution

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Conclusion

Conclusion

The Vasicek Single Factor Model provides a closed form Loss RateDistribution

limN→∞

P(L(N) ≤ x) = Φ

(√1− ρ · Φ−1(x)− Φ−1(p)

√ρ

)

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

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Conclusion

Bibliography

Vasicek : The Distribution of Loan Portfolio Value, Risk(2002).

Martin, Reitz, Wehn : Kredit und Kreditrisikomkodelle,Vieweg, (2006).

Schonbucher : Faktor Models: Portfolio Credit Risks WhenDefaults are Correlated, Journal of Risk Finance (2001).

Elizalde : Credit Risk Models IV: Understanding and pricingCDOs, discussion paper (2005).

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Economy & Finance

Vasicheck prezentation