# Credit Risk Modelling M.A. Fahrenwaldt Example sheet

**Credit Risk Modelling**

**M.A. Fahrenwaldt Example sheet**

- For
*t*= 0*,…,T*− 1,*j*∈*S*\ {0} and*k*∈*S*, let*N*denote the number of companies that are rated_{tj }*j*at time*t*and followed until time*t*+ 1 and let*N*denote the number of those companies that are rated_{tjk }*k*at time*t*+ 1. A discrete-time, stationary Markov chain is fitted to the data (*N*) and (_{tj}*N*)._{tjk}

Show that the maximum likelihood estimator of the transition probability *p _{jk }*is given by

* .*

- The matrix exponential of the Markov chain generator Λ ∈ R
^{(n+1)×(n+1) }can be calculated using a number of software packages. One case where it can be calculated by simple matrix multiplication occurs when the generator is diagonalizable, meaning that there exists an invertible matrix*A*∈ R^{(n+1)×(n+1) }such that*A*^{−1}Λ*A*=*D*where*D*= diag(*d*_{0}*,…,d*) is a diagonal matrix containing the eigenvalues of Λ. Show that, in this case, the matrix of transition probabilities_{n}*P*(*t*) for the interval [0*,t*] may be written

*P*(*t*) = *A *diag *.*

- In Merton’s model the debt of a company consists of a single-zero coupon bondwith face value
*B*and maturity*T*. The asset value process (*V*)_{t}_{t}_{≥0 }of a firm is modelled by the stochastic differential equation

*dV _{t }*=

*µ*+

_{V }V_{t}dt*σ*

_{V }V_{t}dW_{t}where *µ _{V }*∈ R and

*σ*0 are the drift and asset volatility respectively. Default occurs if

_{V }>*V*is less than

_{T }*B*. Assuming

*V*

_{0 }

*> B*, show that the probability of default is an increasing function of the volatility

*σ*.

_{V }- A bank uses a simple internal rating system in which there are only two ratings- A and B – as well as a default state D. You are given the information in the following table. There are a few missing entries in the table.

A B D

A 0.80 0.15 ? B 0.10 ? 0.20

D ? ? ?

Complete the table of transition probabilities and compute the probabilities that A-rated and B-rated obligors default over a two-year period.

- Create graphs to show how the credit spread in Merton’s model varies withthe relative debt lebel given by
*d*=*Bp*_{0}(*t,T*)*/V*. Experiment with different values for the time to maturity of the debt_{t}*T*−*t*and the asset volatility*σ*._{V } - This question will require the use of a computer package that contains the matrix exponential function. Estimate a generator matrix Λ for rating migrations by taking the average annual rating migration rates given in the table provided by Moody’s. Use Λ to derive the matrix of one-year transition probabilities
*P*=*P*(1).

Now assume that the credit-migration model is embedded in a firm-value model. This is done by introducing an asset value process (*V _{t}*) for each firm and thresholds

0 = *d*˜0 *< d*˜1 *< *··· *< d*˜*n < d*˜*n*+1 = ∞

such that *P*(*d*^{˜}* _{k }< V_{T }*≤

*d*

^{˜}

_{k}_{+1}) =

*p*. As noted in the lectures, the migration probabilities are invariant under simultaneous strictly increasing transformations of

_{jk}*V*and the thresholds

_{T }*d*

^{˜}

*. Let*

_{k}*X*=

_{T }*T*(

*V*) and

_{T}*d*=

_{k }*T*(

*d*

^{˜}

*) denote the transformed quantities. Find the values for the thresholds*

_{k}*d*

_{0}

*,…,d*

_{n}_{+1 }when

*X*∼

_{T }*N*(0

*,*1).

- The Gompertz model is widely used by actuaries in mortality modelling. Thedistribution function is given by

*.*

Caculate the hazard function and the cumulative hazard function of this distribution.

The Gompertz-Makeham model is an extension of the Gompertz model. If the hazard function of the Gompertz distribution is *γ*_{G}(*t*) the Gompertz-Makeham distribution has hazard function *γ*_{GM}(*t*) = *γ*_{G}(*t*) + *c *for some constant *c > *0. Calculate the distribution function of this model.

- Derive formulas for the credit spread of a defaultable zero coupon bond underboth the RT (recovery of treasury) and RF (recovery of face value) recovery models. Give simple expressions for these spreads in the case where the hazard function under the risk-neutral measure
*Q*is a constant*γ*(^{Q}*t*) = ¯*γ*.^{Q} - Suppose that the spread quoted in the market for a five-year CDS on a particular reference entity is 42 bp. Assuming regular quarterly payments, a loss-given-default of
*δ*= 0*.*6 and a constant annualized interest rate*r*= 0*.*02, calibrate a constant hazard model for the time to default*τ*of the reference entity under the risk-neutral probability measure*Q*. - Suppose that ¯
*γ*satisfies^{Q }

(1) Show that it must also satisfy

for *N *= 2*,*3*,…*.

Hence conclude that, when the CDS spread curve is flat, interest rates are constant and deterministic, and premium payments follow a regular schedule, a constant risk-neutral hazard rate ¯*γ ^{Q }*can be determined from a quoted market spread

*x*

^{∗ }by solving (1).

- Consider a portfolio containing
*m*= 1000 equally rated credit risks. Assume that for every obligor the exposure is*e*=_{i }*£*1*M*and the default probability is*p*= 1%. Calculate the expected value and standard deviation of the portfolio loss in the following situations._{i }- LGDs are modelled as deterministic and in all cases
*δ*= 0_{i }*.*Defaults are assumed to occur independently. - LGDs are modelled as deterministic as in (1) but defaults are assumed tobe dependent. Assume that in all cases the default correlation between pairs of default indicators is given by
*ρ*(*Y*) = 0_{i},Y_{j}*.*005 for*i*6=*j*. - Defaults are dependent as in (2) but LGDs are modelled as random variables ∆
satisfying ∆_{i }∼ Beta(_{i }*a,b*) where*a*= 9*.*2 and*b*= 13*.*Assume that LGDs are mutually independent across the portfolio and independent of the default indicator variables.

- LGDs are modelled as deterministic and in all cases

Note that the mean and variance of a Beta(*a,b*)-distributed random variable *X *are given by *E*(*X*) = *a/*(*a *+ *b*) and var(*X*) = *ab/*((*a *+ *b *+ 1)(*a *+ *b*)^{2}).

- In a threshold model (also known as a critical or latent variable model) thecritical variable for obligor
*i*is given by

where *F*_{1 }and *F*_{2 }are standard normally distributed factors which are correlated with correlation *ρ*. The constants *b*_{1 }and *b*_{2 }are weights or loadings with values in (0*,*1) and *Z*_{1}*,…,Z _{m }*are independent standard normal variables, which are also independent of

*F*

_{1 }and

*F*

_{2}. The rationale for this model is that obligors belong to two groups, for example two industrial sectors.

Derive expressions for the within-group and the between-group “asset correlations”.

- Consider a Gaussian threshold model (
*X,**d*) where*d*= (*d*_{1}*,…,d*)_{m}^{0 }is a vector of deterministic thresholds and*X*= (*X*_{1}*,…,X*)” is a vector of critical variables following the one-factor model given by_{m}

q

*X _{i }*=

*b*+ 1 −

_{i}F*b*

^{2}

_{i}Z_{i },and where *F,Z*_{1}*,…,Z _{m }*are iid

*N*(0

*,*1) random variables and −1

*< b*1 is a factor weight. By conditioning on

_{i }<*F*, show that the joint default probability for obligors {

*i*

_{1}

*,…,i*} ⊂ {1

_{k}*,…,m*} can be written as

Conclude that in an exchangeable default model the higher order default probabilities are given by

Is it possible to have negative asset correlation and/or negative default correlation in the general one-factor model and the exchangeable default model?

- Give a formula that relates higher-order default probabilities
*π*and default probability_{k }*π*in an exchangeable default model of threshold type that is based on the Gumbel copula. - Let
*Q*∼ Beta(*a,b*). Consider a portfolio of 1000 similar obligors whose default indicator variables*Y*are conditionally independent given_{i }*Q*such that*Y*|_{i }*Q*=*q*∼ Be(*q*). Assume that for every obligor the exposure is*e*= 1 and the LGD is_{i }*δ*= 100%. Show that the probability distribution (the probability mass_{i }

function) of the portfolio loss is

How should *a *and *b *be chosen so that the default probability of every obligor is *p _{i }*= 0

*.*01 and the default correlation is

*ρ*(

*Y*) = 0

_{i},Y_{j}*.*005 for

*i*6=

*j*?

- Suppose the random variable
*Q*follows a probit-normal mixing distribution with parameters*µ*∈ R and*σ >*In other words we have Φ^{−1}(*Q*) =*µ*+*σZ*for a standard normal variable*Z*. Derive the distribution function and the probability density function of*Q*. Note that it is not obvious how to calculate the mean of the distribution. - Now suppose we create an exchangeable Bernoulli mixture model for dependentdefaults by using the probit-normal mixing distribution. Show that the higher order default probabilities are given by

and that the joint probability function of the defaults is Explain why this model is equivalent to an exchangeable one-factor Gaussian threshold model with√

default probability *π *= Φ(*µ/ *1 + *σ*^{2}) and asset correlation *ρ *= *σ*^{2}(1 + *σ*^{2})^{−1}. Hence calculate the mean of a probitnormal distribution with parameters *µ *and *σ*^{2}.

- In a one-factor CreditRisk
^{+ }model the default of obligor*i*over a given time horizon is modelled as being conditionally Poisson with mean*k*given the realisation_{i}ψ*ψ*of an economic factor Ψ. The factor Ψ is taken to have a Ga(*α,*1) distribution for some parameter*α >*0 and*k*is specific to obigor_{i }*i*.

Show that this can be expressed as a Bernoulli mixture model with conditional default probability

*p _{i}*(

*ψ*) =

*P*(

*Y*= 1 | Ψ =

_{i }*ψ*) = 1 − exp(−

*k*)

_{i}ψ*.*Show that the probability density of

*Q*:=

_{i }*p*(Ψ) is given by

_{i}* .*

Why could this be very accurately approximated by the probability density of the beta distribution?

- Let
*N*∼ NB(*α,p*) be a negative binomial distribution. Derive the moment generating function (mgf)*M*(_{N}*t*) of*N*. Now consider a compound negative binomial variable . Derive the mgf of*Z*in terms of the mgf

*M _{X}*(

*t*) of the

*X*.

_{i}Now suppose that *X *∼ Ga(*θ,*1) for some prameter *θ > *0. Derive the mean and variance of *Z*?

For *j *= 1*,…,p*, let *N _{j }*∼ NB(

*α*), be independent negative binomial variables and let

_{j},p_{j}*X*,

_{ji}*i*= 1

*,*2

*,…*, be independent multinomial random variables satisfying

*P*(*X _{ji }*=

*x*) =

_{b}*q*= 1

_{jb}, b*,…,n,*

where = 1. Define the independent compound negative binomial variables . What is the moment generating function of *Z *=

P*pj*=1 *Z**j*?

- Consider a portfolio of
*m*= 1000 obligors with an exposure in every case of*e*= 1_{i }*M*$.Suppose we model dependent defaults in the portfolio using a 2factor CreditRisk^{+ }style of model. Assume that the default count variables*Y*^{˜}satisfy_{i }

(

*Y*˜* _{i }*| (Ψ

_{1}

*,*Ψ

_{2}) = (

*ψ*

_{1}

*,ψ*

_{2}) ∼ Poi(

*k*

*iψ*1)

*, i*= 1

*,…,*500

*,*

Poi(*k _{i}*(0

*.*5

*ψ*

_{1 }+ 0

*.*5

*ψ*

_{2}))

*, i*= 501

*,…,*1000

*,*

where *k _{i }*= 0

*.*01 for all obligors. Also assume that losses given default are 100% in all cases and that the factors Ψ

_{1 }and Ψ

_{2 }are independent gamm variables with unit mean and variance 2.

Compute the expected loss and the variance of the portfolio loss.

- Consider an exchangeable Bernoulli mixture model with conditional defaultprobabilities

*p _{i}*(

*ψ*) =

*P*(

*Y*= 1 | Ψ =

_{i }*ψ*) = 1 − exp(−

*κψ*)

*,*

where Ψ ∼ Ga(*α,*1) for parameters *α > *0 and *κ > *0. (This is the Bernoulli mixture model implied by an exchangeable one-factor version of CreditRisk^{+}.

Suppose we define ˜*π _{k }*=

*P*(

*Y*

_{1 }= 0

*,…,Y*= 0) for

_{k }*k*= 1

*,…,m*and we write

*π*˜ =

*π*˜

_{1}. Show that

wheredenotes the Clayton copula. Conclude that *π*_{2 }= *C*^{ˆ}(*π,π*) where *π*_{2 }and *π *have their usual interpretation and

*.*

- Over the years a retail banking division specialising in small commercial loanshas had a consistent lending policy. 50% of its loans have been for the amount of £5M and 50% of its loans have been for the amount of £ Moreover, 50% of both the larger and smaller loans have been rated as “risky” and have been assigned a default probability of 1% per annum, whereas the other 50% have been rated as “safe” and have been assigned a default probability of 0.1% per annum.

The bank uses a one-factor Gasussian threshold model for its portfolio and carries out a fully internal calculation for economic capital purposes. In the one-factor model the risky loans are assumed to be 80% systematic (i.e. 80% of the variance of the driving “asset value” variable is assumed to be explained by systematic factors) whereas the safe loans are assumed to be only 20% systematic. A deterministic loss-given-default of 0.6 is assumed.

The portfolio consists of 10000 individual loans and the bank decides to use a large portfolio argument to compute the 99.9% Value-at-Risk. Derive the form of the asymptotic conditional loss function ^{¯}*l*(*ψ*) under the assumption the portfolio is grown ad infinitum with the same lending policy. Use this to approximate the 99.9% VaR for the portfolio.

- Consider a Gaussian threshold model (
*X*)_{i},d_{i}_{1≤i≤m }where the critical variables follow the one-factor model

q

*X _{i }*=

*b*+ 1 −

_{i}F*b*

^{2}

_{i}Z_{i },where *F,Z*_{1}*,…,Z _{m }*are independent, identically distributed standard normal random variables and −1

*< b*1 is a loading coefficient. For

_{i }<*i*= 1

*,…,m*, write

*Y*= 1

_{i }_{{X}

*≤*

_{i}

_{d}*} for the default indicator variables and*

_{i}*p*=

_{i }*P*(

*Y*= 1) for the default probabilities.

_{i }- Show that the model is equivalent to a one-factor Bernoulli mixture mod-el for the default indicators where the common factor is Ψ = −
*F*and the conditional default probabilities take the form

*p _{i}*(

*ψ*) =

*P*(

*Y*= 1 | Ψ =

_{i }*ψ*) = Φ(

*µ*+

_{i }*σ*)

_{i}ψ*.*

Give the expressions for *µ _{i }*and

*σ*.

_{i}- Suppose there are 10000 obligors in the portfolio. 5000 of them have ex-posure
*£*2*M*, default probability*p*= 0_{i }*.*01 and factor loading*b*= 0_{i }*.*5000 of them have exposure*£*4*M*, default probability*p*= 0_{i }*.*05 and factor loading*b*= 0_{i }*.*8. Assume the loss-given-default (LGD) is 0.6 for all obligors. Use a large portfolio argument to compute an approximation for the 99% Value-atRisk of the portfolio loss. - Now suppose that a stochastic LGD ∆
depending on the economic factor Ψ is introduced into the model for every obligor_{i }*i*. It is assumed that (i) LGDs are conditionally independent given Ψ, (ii) they are independent of the default indicators given Ψ, and (iii) the expected LGD given Ψ satisfies*E*(∆| Ψ =_{i }*ψ*) = Φ(0*.*5 +*ψ*)*.*

Recompute the approximate 99% Value-at-Risk to incorporate the stochastic LGD.