Risk Management
Lecture 8
Jan Palczewski
University of Leeds
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 1 / 33
In today’s lecture
1 Credit Risk
Types of credit risk models
2 Mixture models
Bernoulli mixture model
3 Structural models
Threshold models
Merton’s model
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 2 / 33
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.
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 3 / 33
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.
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 4 / 33
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
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 5 / 33
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.
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 6 / 33
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.
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 7 / 33
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.
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 8 / 33
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.
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 9 / 33
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!
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 10 / 33
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.
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 11 / 33
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 .
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 12 / 33
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.
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 13 / 33
Structural models
Structural models (or Firm-value models)
Threshold models
Merton model (1974)
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 14 / 33
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.
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 15 / 33
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.
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 16 / 33
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
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 17 / 33
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 .
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 18 / 33
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).
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 19 / 33
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).
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 19 / 33
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 .
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 20 / 33
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.
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 21 / 33
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).
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 22 / 33
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).
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 23 / 33
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
)
.
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 24 / 33
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.
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 25 / 33
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.
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 26 / 33
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.
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 27 / 33
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.
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 28 / 33
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 .
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 29 / 33
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 .
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 30 / 33
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 .
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 31 / 33
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.
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 32 / 33
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?
Jan Palczewski (Univ. of Leeds) MATH5340 2023/24 33 / 33