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MATH5340 Credit Risk and Credit Risk Management

创建日期：2024-06-17 16:30:05

所属分类：数学mathematics

标签： MATH5340

Credit Risk and Credit Risk Management

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MATH5340

Credit Risk
Credit Risk and Credit Risk Management
Definition (Credit Risk)
Credit risk is the risk that the value of a portfolio changes due to default
of a counterparty or unexpected changes (downgrades) in the credit
quality of a counterparty.
Default = the counterparty cannot honour a financial commitment, for
example, repay the debt.
This is a relevant risk component in all portfolios.
Credit Risk Types of credit risk models
Static and dynamic models of credit risk
Static models:
Credit standing of a counterparty is assessed at the end of the
period over which the loss distribution is calculated
Used in credit risk management.
Dynamic models:
An action is performed at the time when the counterparty defaults.
Used in modelling and valuation of credit-risk derivatives.
We will mainly focus on static models for credit risk and credit risk
management.

Credit Risk Types of credit risk models
Modelling credit risk
Overview of the main types of models:
Reduced form models:
I Mixture models (Bernoulli mixture and Poisson mixture)
Structural models (or Firm-value models):
I univariate threshold models and Merton’s model
I multivariate threshold models
Challenges in credit risk modelling:
lack of public data
skewed loss distribution
dependence of defaults

Credit Risk Types of credit risk models
Overview of types of credit risk models
Reduced form models: the mechanism leading to default is not
specified. In static reduced form models the occurence of a default
is usually modelled by a Bernoulli random variable.
I Mixture models: Defaults occur independently given the values of
common (random) factors. Hence, defaults are not independent.
Structural models: Default occurs when the value of some random
variable (e. g. value of the company’s assets), falls below a certain
threshold.

Credit Risk Types of credit risk models
Notation
We consider a portfolio of m obligors
(where m is a positive integer).
T is a fixed time horizon.
We will focus on the binary outcomes of default and non-default
(we will ignore downgrading of the credit ranking of obligors).
We will denote by Y1,Y2, . . . ,Ym the default indicator random
variables of obligors 1, 2, . . . ,m, defined by:
Yi =
{
1, if obligor i defaults,
0, if obligor i does not default.
We denote by M the random variable corresponding to the number
of obligors which default, that is,
M = Y1 + Y2 + . . .+ Ym.

Mixture models
Mixture models
These are static reduced-form models.
The mechanism leading to default is left unspecified.
There are K common economic factors.
The default risk of each obligor is assumed to depend on the
common (random) economic factors.
Given a realisation of the factors, defaults of individual obligors are
assumed to be independent.
Examples:
I Bernoulli mixture model.
I Poisson mixture model.

Mixture models Bernoulli mixture model
Bernoulli mixture model
Definition
The random vector Y = (Y1,Y2 . . . ,Ym)′ follows a Bernoulli mixture
model if
there is a K -dimensional random vector (of risk factors)
Ψ = (Ψ1, . . . ,ΨK )
′, and
for every i = 1, . . . ,m, there is a function pi : RK → [0,1] such that,
conditionally on the values of Ψ, the components of Y are
independent Bernoulli random variables with
P(Yi = 1|Ψ = ψ) = pi(ψ),
P(Yi = 0|Ψ = ψ) = 1−pi(ψ),
for every i = 1, . . . ,m.

Mixture models Bernoulli mixture model
Default of a single obligor
Conditional on the realisation ψ of common economic factors Ψ we
have
P(Yi = 1|Ψ = ψ) = pi(ψ).
Remark
The default probability p¯i of a single obligor i is
p¯i = P(Yi = 1) = E(pi(Ψ)).
Defaults of different obligors are not idependent - they all depend on the
common economic factors Ψ!

Mixture models Bernoulli mixture model
Default for multiple obligors
Remark
For y = (y1,y2, . . . ,ym) ∈ {0,1}m:
P(Y = y|Ψ = ψ) =
m
∏
i=1
pi(ψ)yi (1−pi(ψ))1−yi
P(Y = y) = E
[ m
∏
i=1
pi(Ψ)
yi (1−pi(Ψ))1−yi
]
.
−→ Example : 2 companies with default indicators Y1 and Y2 and a
1-dimensional risk factor Ψ.

Mixture models Bernoulli mixture model
One-factor model: the homogeneous case
We consider the following particular case:
Ψ is univariate random variable, that is, only one factor.
The same function p for all m obligors, that is,
p1 = p2 = . . . = pm = p (homogeneous)
We define the random variable Q by:
Q = p(Ψ).
Theorem
Conditionally on Q = q, the number of defaults M = ∑mi=1 Yi has a
binomial distribution:
P(M = j|Q = q) =
(
m
j
)
q j(1−q)m−j .

Mixture models Bernoulli mixture model
One-factor model: the homogeneous case
Moreover:
If the random variable Q follows a discrete distribution with values
{q1, . . . ,qL}, then
P(M = j) =
(
m
j
) L
∑
n=1
q jn(1−qn)m−jP(Q = qn).
If the random variable Q follows a continuous distribution with the
density g(q), then
P(M = j) =
(
m
j
)∫ 1
0
q j(1−q)m−jg(q)dq.

Structural models
Structural models (or Firm-value models)
Threshold models
Merton model (1974)

Structural models Threshold models
Univariate threshold model
univariate = default of one counterparty
The default occurs when the value of a (random) state variable X1 lies
below a threshold d1, i.e., the default indicator Y1 is given by
Y1 =
{
0, X1 > d1,
1, X1 ≤ d1.
Examples:
Merton model: X1 denotes the firm value at time T , d1 = D is the
debt.
Credit rating model: X1 denotes a rating at time T taking values in
{0,1,2, . . . ,N} with 0 denoting bankruptcy and d1 = 0.

Structural models Threshold models
Different forms of default indicators
Depending on the interpretation of X1, the default indicator Y1 may take
the following forms:
Y1 = 1IX1≥d1, where X1 may be the debt-to-equity ratio,
or versions with strict inequalities:
Y1 = 1IX1d1.

Structural models Threshold models
Multivariate threshold model
There are m firms. Default of firm i occurs if Xi ≤ di , i.e., the default
indicator is
Yi = 1IXi≤di
or one of the forms from the previous slide.
The marginal distributions of Xi ’s are linked using a copula C.
Why this new model?
This model offers an alternative to mixture models.
Such models have been popular in the industry: CreditMetrics and
KMV model

Structural models Threshold models
Example of a multivariate threshold model
There are m obligors
The state variable Xi takes two values: 0 or 1
Threshold is di = 0
The dependence between state variables is described by a given
copula C
What is the probability that all counterparties default, i.e.
P(M = m) =?,
where
M =
m
∑
i=1
Yi .

Structural models Threshold models
CreditMetrics and KMV
There are m obligors (called firms)
The state variable Vi represents firm i ’s value
Threshold for firm i is di
(log(V1), . . . , log(Vm))∼ N(µ,Σ)
Default:
Vi ≤ di ⇐⇒ log(Vi)≤ log(di)
so there is an equivalent model with state variables Xi = log(Vi) with
(X1, . . . ,Xm)∼ N(µ,Σ)
and thresholds dˆi = log(di).
Structural models Threshold models
CreditMetrics and KMV
There are m obligors (called firms)
The state variable Vi represents firm i ’s value
Threshold for firm i is di
(log(V1), . . . , log(Vm))∼ N(µ,Σ)
Default:
Vi ≤ di ⇐⇒ log(Vi)≤ log(di)
so there is an equivalent model with state variables Xi = log(Vi) with
(X1, . . . ,Xm)∼ N(µ,Σ)
and thresholds dˆi = log(di).

Structural models Merton’s model
Merton’s model
Merton’s model is an extension of a univariate threshold model.
There is one firm (company, obligor).
T is the fixed time horizon.
The value of the firm’s assets at time t is denoted by V firmt .

Structural models Merton’s model
Debt
The structure of the firm’s debt is simple: it consists of D units of
zero-coupon bonds with maturity T which the firm issues at 0. In
other words, the firm owes the amount D to its bond holders.
At T , there are two situations:
I If V firmT > D, the firm does not default:
The firm pays D to its bond holders and the residual V firmT −D is left
for the shareholders.
I If V firmT ≤ D, the firm defaults:
The firm owes D but can repay only V firmT .
The bond holders receive V firmT , the shareholders receive nothing.

Structural models Merton’s model
Payoffs
In summary:
The amount received by the shareholders at time T is
0I{V firmT ≤D}+ (V
firm
T −D)I{V firmT >D} = (V
firm
T −D)+.
The amount received at T by the bondholders is
V firmT I{V firmT ≤D}+ DI{V firmT >D} = min(D,V
firm
T )
= D−max(0,D−V firmT )
= D− (D−V firmT )+.
Remark: Bondholders are "compensated" for the credit risk by the
so-called credit spread (cf. later in the slides).

Structural models Merton’s model
Dynamics of firm’s value
In Merton model, the value of the firm V firmt is modelled by a geometric
Brownian motion.
More precisely,
dV firmt = µV V
firm
t dt +σV V
firm
t dWt ,
where µV is the drift parameter and σV 6= 0 is the volatility parameter,
and (Wt) is a Brownian motion under P.
The explicit solution of the above SDE is:
V firmt = V
firm
0 e
(µV−σ2V/2)t+σV Wt ,
i.e.
log(V firmt )∼ N(log(V firm0 ) + (µV −σ2V/2)t, σ2V t).

Structural models Merton’s model
Probability of default
Question: What is the default probability of the firm in this model?
In other words, compute
P(V firmT ≤ D) =?
Answer: We use the same type of computations as the Black-Scholes
model to get:
P(V firmT ≤ D) = Φ
(
− log(V
firm
0 /D) + (µV −σ2V/2)T
σV
√
T
)
.

Structural models Merton’s model
Probability of default v. parameters
Question: How is the default probability affected by changes in the
parameters?
Answer: We can see from the formula for the default probability that:
The default probability increases, when the debt D increases.
The default probability decreases, when the initial value of the firm
V firm0 (at time 0) increases.
The default probability decreases, when the drift parameter µV
increases (upward tendency in the dynamics the process (V firmt )).
If V firm0 > D and µV ≥ σ2V/2, when the volatility parameter σV
increases, the default probability descreases.

Structural models Merton’s model
Derivatives on firm’s value
By using the tools developed for Black-Scholes option pricing model, we
can price derivatives contracts in Merton’s model when the underlying is
the value of the firm’s assets V firmT .
More specifically, let r be the risk-free interest rate (where r > 0).
Let X = h(V firmT ) be a pay-off payable at T (with h deterministic).
We have:
price0(X) = EQ(e
−rT h(V firmT )),
where Q denotes the risk neutral probability measure in this model.

Structural models Merton’s model
Risk neutral measure Q
Under the risk neutral measure the discounted firm’s value (Vˆt) is a
Q-martingale, i.e.,
V̂t = e
−rtV firmt
and for any t ≥ s ≥ 0 we have
EQ
(
V̂t |Fs) = V̂s,
where the filtration (Ft) is generated by the Brownian motion Wt .
Furthermore,
dV firmt = rV
firm
t dt +σV V
firm
t dW˜t ,
where (W˜t) is a Q-Brownian motion.

Structural models Merton’s model
Defaultable zero-coupon bond
A defaultable zero-coupon bond with maturity T and face value D
issued by the firm is a bond which
pays off D at the maturity T if the firm has not defaulted;
pays off V firmT at the maturity T if the firm has defaulted.
The amount received at T by the bondholders is
V firmT I{V firmT ≤D}+ DI{V firmT >D} = min(D,V
firm
T ) = D− (D−V firmT )+.
Notice that the amount received at T for a zero-coupon non-defaultable
bond would be D.

Structural models Merton’s model
Price of the defaultable zero-coupon bond
The payoff to the bondholders at time T is
D− (D−V firmT )+.
Its price at time 0 is:
EQ
(
e−rT (D− (D−V firmT )+)
)
(1)
= e−rT D−EQ(e−rT (D−V firmT )+) = e−rT D−pBS0 , (2)
where pBS0 denotes the Black-Scholes price at 0 of a put option on V
firm
T
with strike D and maturity T .

Structural models Merton’s model
Price of put option on the firm’s value
The price of the put is computed identically as in the Black-Scholes
model of the stock market:
pBS0 = e
−rT DΦ(−d2)−V firm0 Φ(−d1),
where
d1 =
log(V firm0 /D) + (r +σ2V/2)T
σV
√
T
,
d2 = d1−σV
√
T .

Structural models Merton’s model
Price per unit of debt
Hence, the pay-off at T of one unit of defaultable zero-coupon bond is
1
D
(D− (D−V firmT )+) = 1−
1
D
(D−V firmT )+.
The price at 0 of one unit of defaultable zero-coupon bond, denoted by
P(0,T ), is
P(0,T ) =
1
D
(e−rT D−PBS0 ) = e−rT −
1
D
pBS0 .
For comparison:
The pay-off at T of one unit of risk-free zero-coupon bond is 1.
The price at 0 of one unit of risk-free zero-coupon bond, denoted by
P free(0,T ), is e−rT .

Structural models Merton’s model
Credit spread
Definition (Credit spread)
The credit spread at time 0 (for maturity T ), denoted by Spread(0,T ),
is defined by:
Spread(0,T ) =− 1
T
(
log(P(0,T ))− log(P free(0,T ))
)
=− 1
T
log
( P(0,T )
P free(0,T )
)
where
P free(0,T ) is the price at time 0 of the default-free zero coupon
bond, and
P(0,T ) is the price at time 0 of the defaultable zero coupon bond.

Structural models Merton’s model
Credit spread in Merton’s model
In Merton’s model, we can compute:
Spread(0,T ) =− 1
T
log
(
Φ(d2) +
V firm0
DP free(0,T )
Φ(−d1)
)
.
Questions:
Why is there the logarithm in the definition of the spread?
How to derive the formula for the credit spead?

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