Vasicek Single Factor Model
Vasicek Single Factor Model
Alexandra Kochendörfer
7. Februar 2011
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Vasicek Single Factor Model
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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Vasicek Single Factor Model
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
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Vasicek Single Factor Model
Default Correlation
Definition
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
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Negative correlation: the firms are competitors
Drivers of Default Correlation
I
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State of the general economy
Industry-specific factors
I
I
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Oil industry: 22 companies defaulted over 1982-1986.
Thrifts: 19 defaults over 1990-1992.
Casinos/hotel chains: 10 defaults over 1990-1992.
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Vasicek Single Factor Model
Default Correlation
Definition
U.S. Corporate Default Rates Since 1920
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Vasicek Single Factor Model
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
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conditional default probability pA|B
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correlation ρAB between default events
These quantities are connected by
pA|B =
pAB
pB
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Vasicek Single Factor Model
Default Correlation
Why is default correlation important
Cov (A, B)
pAB − pA pB
p
ρAB = p
=p
Var (A) Var (B)
pA (1 − pA )pB (1 − pB )
The default probabilities are usually very small pA = pB = p 1
p
pAB = pA pB + ρAB pA (1 − pA )pB (1 − pB ) ≈ p 2 + ρAB · p ≈ ρAB · p
ρAB p
pA|B = pA +
pA (1 − pA )pB (1 − pB ) ≈ ρAB
pB
The joint default probability and conditional default probability are
dominated by the correlation coefficient.
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Vasicek Single Factor Model
Default Correlation
Independent/perfectly dependent defaults
Independent Defaults
Consider N independent default events D1 , . . . , DN with
pD1 = · · · = pDN = p ⇒ Number of defaults ∼ B(p, N).For
N = 100, p = 0.05
p (%)
99.9(%) VaR
1
5
2
7
3
9
4
11
5
13
6
14
7
16
8
17
9
19
10
20
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Vasicek Single Factor Model
Default Correlation
Independent/perfectly dependent defaults
Perfectly dependent defaults
Consinder default correlation ρij = 1 for all pairs ij.
pD D − p 2
pD1 D2 − pD1 pD2
= 1 2
1= p
p(1 − p)
pD1 (1 − pD1 )pD2 (1 − pD2 )
⇒ pD1 D2 = pD1 = pD2 = p i.e. D1 ∩ D2 = D1 = D2 a.s.
⇒ pD1 D2 D3 = pD2 D3 = 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 with
probability 1 − p.
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Vasicek Single Factor Model
Default Correlation
Independent/perfectly dependent defaults
Perfectly dependent defaults
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Vasicek Single Factor Model
Modelling Default Correlation
Data sources
Data Sources
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Actual Rating and Default Events.
+ Objective and direct.
– Joint defaults are rare events, sparse data sets.
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Credit Spread.
+ Incorporate information on markets, observable.
– No theoretical link between credit spread correlation and
default correlation.
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Equity correlation.
+ Data easily available, good quality.
– Connection to credit risk not obvious, needs a lot of
assumptions.
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Vasicek Single Factor Model
Modelling Default Correlation
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} is
modelled as in Black-Scholes model, hence at terminal time t = 1
with An,1 = An we have
σn2
+ σn Bn
An = An,0 exp
µn −
2
with some standard normal variable Bn .
The r.v. (B1 , . . . , BN ) are jointly normally distributed with
covariance matrix Σ = (ρij )ij , where ρij denotes the asset
correlation between assets i and j.
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Vasicek Single Factor Model
Modelling Default Correlation
Default triggered by firm’s value
The obligor n defaults if the asset value falls below a
perspecified 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 =
n
− µn
log ACn,0
σn
We can assume the individual default probabilities pDn as given
and compute cn = Φ−1 (pDn ) and vice versa.
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Vasicek Single Factor Model
Modelling Default Correlation
Default triggered by firm’s value
The joint distribution of Bi determines the dependency
structure 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 default
correlation can be computed via
P(D1 = 1, D2 = 1) − pD1 pD2
ρ= p
pD1 (1 − pD1 )pD2 (1 − pD2 )
=
Φ2 (Φ−1 (pD1 ), Φ−1 (pD2 ); ρ1,2 ) − pD1 pD2
p
pD1 (1 − pD1 )pD2 (1 − pD2 )
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Vasicek Single Factor Model
Modelling Default Correlation
Default triggered by firm’s value
We need
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N(N − 1)/2 asset correlations of Σ
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N individual default probabilities
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Additional assumptions on the structure of Bi 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 Bn can be written as
p
√
Bn = ρ · Y + 1 − ρ · n
with some constant ρ ∈ [0, 1] and N + 1 independent standard
normally distributed r.v. Y , 1 , . . . , N .
Interpretation
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Y is a common systematic risk factor affecting all firms (state
of economy)
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n are idiosyncratic factors independent across firms
(management, innovations)
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Corr (Bi , Bj ) = ρ controls the proportions between systematic
and idiosyncratic factors, empirically around 10%.
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Vasicek Single Factor Model
Vasicek Single Factor Model
Conditional on the realisation of the systematic factor Y
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the logarithmic returns Bn are independent ( for a constant y
√
√
variables ρ · y + 1 − ρ · n are independent)
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default variables Dn = 11{Bn <cn } are independent as function
of Bn
The only effect of Y is to move Bn closer or further away from
barrier cn .
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Vasicek Single Factor Model
Vasicek Single Factor Model
Loss distribution in finite portfolio
Theorem
For ρ ∈ (0, 1) and same default probabilities p = pD1 = · · · = pDN
the conditional default probability is given by
−1
√
Φ (p) − ρ · y
√
p(y ) := P[Bn < c | Y = y ] = Φ
.
1−ρ
P
The number of defaults L = N
i=1 Di has the following
distribution
m Z ∞
X
N
P(L ≤ m) =
·
p(y )k · (1 − p(y ))N−k · φ(y )dy
k
−∞
k=0
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Vasicek Single Factor Model
Vasicek Single Factor Model
Loss distribution in finite portfolio
Proof
The probability of k defaults is
Z
∞
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 independent
conditional on Y , hence
N
P(L = k | Y = y ) =
· p(y )k · (1 − p(y ))N−k
k
Thus, for m ∈ {1, . . . , N} we have
m Z ∞
X
N
P(L ≤ m) =
p(y )k · (1 − p(y ))N−k · φ(y )dy
·
k
−∞
k=0
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Vasicek Single Factor Model
Vasicek Single Factor Model
Loss distribution in finite portfolio
Loss Distibutions for different ρ
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Vasicek Single Factor Model
Vasicek Single Factor Model
Loss distribution in finite portfolio
VaR Levels for different ρ with N = 100 and p = 5%
ρ(%)
0
1
10
30
50
99.9(%)VaR
13
14
27
55
80
99.(%)VaR
11
12
19
35
53
Independent defaults
p (%)
99.9(%) VaR
1
5
2
7
3
9
4
11
5
13
6
14
7
16
8
17
9
19
10
20
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Vasicek Single Factor Model
Vasicek Single Factor Model
Large Homogeneous Portfolio Approximation
Large Homogeneous Portfolio Approximation
Definition (Large Homogeneous Portfolio LHP)
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pD1 = · · · = pDN = p
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portfolio is weighted with ω1 , . . . , ωN ,
such that
N
X
(N)
lim
(ωn )2 = 0
(N)
N→∞
(N)
(N)
n=1 ωn
PN
= 1,
n=1
The portfolio is not dominated by few loans much larger then the
rest.
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Vasicek Single Factor Model
Vasicek Single Factor Model
Large Homogeneous Portfolio Approximation
Definition (Loss Rate)
The portfolio loss rate is defined by
(N)
L
=
N
X
(N)
ωn Dn ∈ [0, 1]
n=1
Lemma
Following holds for the LHP
(N)
E(L
| Y ) = p(Y ) = Φ
Var (L(N) | Y ) =
√
Φ−1 (p) − ρ · Y
√
1−ρ
N
X
(N)
(ωn )2 · p(Y ) · (1 − p(Y ))
n=1
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Vasicek Single Factor Model
Vasicek Single Factor Model
Large Homogeneous Portfolio Approximation
Proof
Linearity of conditional expectation yields
(N)
E(L
| Y) =
=
N
X
n=1
N
X
(N)
ωn E(Dn | Y )
(N)
ωn P(Dn | Y ) = p(Y )
n=1
N
X
(N)
ωn
= p(Y )
n=1
Dn are independent conditional on Y , thus
Var (L(N) | Y ) =
=
N
X
(N)
(ωn )2 Var (Dn | Y )
n=1
N
X
(N)
(ωn )2 · p(Y ) · (1 − p(Y ))
n=1
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Vasicek Single Factor Model
Vasicek Single Factor Model
Large Homogeneous Portfolio Approximation
Theorem
The portfolio loss rate in LHP converges in probability for N → ∞.
−1
√
Φ (p) − ρ · Y
P
√
L(N) → p(Y ) = Φ
1−ρ
Proof
For the large portfolio the variation of loss rate given Y tends to 0
Var (L(N) | Y ) =
≤
N
X
(N)
(ωn )2 · p(Y ) · (1 − p(Y ))
n=1
N
X
1
4
(N)
(ωn )2 −−−−→ 0
n=1
N→∞
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Vasicek Single Factor Model
Vasicek Single Factor Model
Large Homogeneous Portfolio Approximation
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 )) −−−−→ 0
N→∞
Convergence in L2 implies convergence in probability i.e. for all
> 0:
lim P L(N) − p(Y ) > = 0
N→∞
The law of L(N) converges weakly to the law of p(Y ), i.e.
P(L(N) ≤ x) −−−−→ P(p(Y ) ≤ x)
N→∞
for all x, where the distribution function of p(Y ) is continuous.
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Vasicek Single Factor Model
Vasicek Single Factor Model
Large Homogeneous Portfolio Approximation
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
−1
√
Φ (p) − ρ · Y
√
P(p(Y ) ≤ x) = P Φ
≤x
1−ρ
√
1 − ρ · Φ−1 (x) − Φ−1 (p)
=P Y ≤
√
ρ
√
1 − ρ · Φ−1 (x) − Φ−1 (p)
=Φ
√
ρ
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Vasicek Single Factor Model
Vasicek Single Factor Model
Large Homogeneous Portfolio Approximation
Approximative density of loss rate with p = 2%, ρ = 10%
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Vasicek Single Factor Model
Vasicek Single Factor Model
Large Homogeneous Portfolio Approximation
Properties of Loss Rate Distribution
(N)
E(p(Y )) = lim E(L
N→∞
) = lim
N→∞
N
X
(N)
ωn p = p
n=1
Because of convergence we can easily compute α-Quantiles of loss
rate distribution for large N
√
1 − ρ · Φ−1 (α) − Φ−1 (p)
(N)
P(L
≤ α) ≈ Φ
√
ρ
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Vasicek Single Factor Model
Vasicek Single Factor Model
Large Homogeneous Portfolio Approximation
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
N
1 X
Di → p
N
i=1
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Vasicek Single Factor Model
Vasicek Single Factor Model
Large Homogeneous Portfolio Approximation
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
(N)
lim 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
Bibliography
Vasicek : The Distribution of Loan Portfolio Value, Risk
(2002).
Martin, Reitz, Wehn : Kredit und Kreditrisikomkodelle,
Vieweg, (2006).
Schönbucher : Faktor Models: Portfolio Credit Risks When
Defaults are Correlated, Journal of Risk Finance (2001).
Elizalde : Credit Risk Models IV: Understanding and pricing
CDOs, discussion paper (2005).
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