Credit Risk Modelling M.A. Fahrenwaldt Example sheet
1.Fort=0,…,T 1,j2S\{0}andk2S,letNtj denotethenumberof companies that are rated j at time t and followed until time t + 1 and let Ntjk denote the number of those companies that are rated k at time t + 1. A discrete-time, stationary Markov chain is fitted to the data (Ntj) and (Ntjk).
Show that the maximum likelihood estimator of the transition probability pjk is given by PT 1 Ntjk
t=0 pˆjk=PT 1N .
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2. The matrix exponential of the Markov chain generator ⇤ 2 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 di- agonalizable, meaning that there exists an invertible matrix A 2 R(n+1)⇥(n+1) such that A 1⇤A = D where D = diag(d0, . . . , dn) is a diagonal matrix con- taining the eigenvalues of ⇤. Show that, in this case, the matrix of transition probabilities P (t) for the interval [0, t] may be written
P(t) = A diag ed0t,…,ednt A 1.
3. In Merton’s model the debt of a company consists of a single-zero coupon bond with face value B and maturity T. The asset value process (Vt)t 0 of a firm is modelled by the stochastic di↵erential equation
dVt =μVVtdt+ VVtdWt
where μV 2 R and V > 0 are the drift and asset volatility respectively. Default occurs if VT is less than B. Assuming V0 > B, show that the probability of default is an increasing function of the volatility V .
4. 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.
first yeardefault so B 0.10 ? 0.20 Second yeardefault
Complete the table of transition probabilities and compute the probabilities that A-rated and B-rated obligors default over a two-year period.
5. Create graphs to show how the credit spread in Merton’s model varies with 二the relative debt lebel given by d = Bp0(t,T)/Vt. Experiment with di↵erent
values for the time to maturity of the debt T t and the asset volatility V .
6. This question will require the use of a computer package that contains the ma- trix exponential function. Estimate a generator matrix ⇤ for rating migrations by taking the average annual rating migration rates given in the table provided by Moody’s. Use ⇤ 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 (Vt) for each firm
ABD A 0.80 0.15 ?
and thresholds
such that P(d ̃ < V d ̃ ) = p . As noted in the lectures, the migration
probabilities are invariant under simultaneous strictly increasing transforma-
tionsofV andthethresholdsd ̃. LetX =T(V )andd =T(d ̃)denotethe T kTTkk
transformed quantities. Find the values for the thresholds d0, . . . , dn+1 when XT ⇠N(0,1).
7. The Gompertz model is widely used by actuaries in mortality modelling. The distribution function is given by
F(t)=1 exp⇣ ab ebt 1 ⌘, a>0,b>0,t 0.
Caculate the hazard function and the cumulative hazard function of this dis-
tribution.
The Gompertz-Makeham model is an extension of the Gompertz model. If the hazard function of the Gompertz distribution is G(t) the Gompertz-Makeham distribution has hazard function GM(t) = G(t) + c for some constant c > 0. Calculate the distribution function of this model.
0=d ̃
18. NowsupposewecreateanexchangeableBernoullimixturemodelfordependent defaults by using the probit-normal mixing distribution. Show that the higher order default probabilities are given by
⇡k =Z 1 k(μ+ x) (x)dx. 1
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 ⇡ = (μ/p1 + 2) and asset correlation ⇢ = 2(1 + 2) 1. Hence calculate the mean of a probitnormal distribution with parameters μ and 2.
19. In a one-factor CreditRisk+ model the default of obligor i over a given time horizon is modelled as being conditionally Poisson with mean ki given the realisation of an economic factor . The factor is taken to have a Ga(↵, 1) distribution for some parameter ↵ > 0 and ki is specific to obigor i.
Show that this can be expressed as a Bernoulli mixture model with conditional default probability
pi( )=P(Yi =1| = )=1 exp( ki ). Show that the probability density of Qi := pi( ) is given by
k ↵ 1 fQi(q)= i ( ln(1 q))↵ 1(1 q)ki 1 .
Why could this be very accurately approximated by the probability density of the beta distribution?
20. Let N ⇠ NB(↵,p) be aPnegative binomial distribution. Derive the moment generating function (mgf) MN(t) of N. Now consider a compound negative binomial variable Z = Ni=1 Xi. Derive the mgf of Z in terms of the mgf MX(t) of the Xi.
Now suppose that X ⇠ Ga(✓, 1) for some prameter ✓ > 0. Derive the mean and variance of Z?
For j = 1, . . . , p, let Nj ⇠ NB(↵j , pj ), be independent negative binomial vari- ables and let Xji, i = 1, 2, . . ., be independent multinomial random variables satisfying
P(Xji = xb) = qjb, b = 1,…,n,
where Pnb=1 qjb P= 1. Define the independent compound negative binomial
v = . What is the moment generating function of Z = p i=1
21. Consider a portfolio of m = 1000 obligors with an exposure in every case of ei = 1M$.Suppose we model dependent defaults in the portfolio using a 2- factor CreditRisk+ style of model. Assume that the default count variables Y ̃i satisfy
Y ̃ i | ( 1 , 2 ) = ( 1 , 2 ) ⇠ ( P o i ( k i 1 ) , i = 1 , . . . , 5 0 0 , Poi(ki(0.5 1 +0.5 2)), i=501,…,1000,
where ki = 0.01 for all obligors. Also assume that losses given default are 100% in all cases and that the factors 1 and 2 are independent gamm variables with unit mean and variance 2.
Compute the expected loss and the variance of the portfolio loss.
22. Consider an exchangeable Bernoulli mixture model with conditional default probabilities
pi( )=P(Yi =1| = )=1 exp( ),
where ⇠ Ga(↵, 1) for parameters ↵ > 0 and > 0. (This is the Bernoulli
mixture model implied by an exchangeable one-factor version of CreditRisk+. Suppose we define ⇡ ̃k = P(Y1 = 0,…,Yk = 0) for k = 1,…,m and we write
⇡ ̃ = ⇡ ̃1. Show that
⇡ ̃k = C1/↵(⇡ ̃,…,⇡ ̃), k = 1,…,m,
where CCl denotes the Clayton copula. Conclude that ⇡2 = Cˆ(⇡, ⇡) where ⇡2 1/↵
and ⇡ have their usual interpretation and
Cˆ(u1,u2)=u1 +u2 1+ (1 u1) 1/↵ +(1 u2) 1/↵ 1 ↵.
23. Over the years a retail banking division specialising in small commercial loans has 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 £1M. 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( ) 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.
24. Consider a Gaussian threshold model (Xi, di)1im where the critical variables follow the one-factor modelXi = biF + q1 b2i Zi ,
whereF,Z1,…,Zm areindependent,identicallydistributedstandardnormal random variables and 1 < bi < 1 is a loading coe cient. For i = 1,...,m, write Yi = 1{Xidi} for the default indicator variables and pi = P(Yi = 1) for the default probabilities.
a) Show that the model is equivalent to a one-factor Bernoulli mixture model for the default indicators where the common factor is = F and the condi- tional default probabilities take the form
pi()=P(Yi=1| = )= (μi+ i ). Give the expressions for μi and i.
b) Suppose there are 10000 obligors in the portfolio. 5000 of them have ex- posure £2M, default probability pi = 0.01 and factor loading bi = 0.6. 5000 of them have exposure £4M, default probability pi = 0.05 and factor loading bi = 0.8. 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-at- Risk of the portfolio loss.
c) Now suppose that a stochastic LGD i depending on the economic factor is introduced into the model for every obligor i. It is assumed that (i) LGDs are conditionally independent given , (ii) they are independent of the default
indicators given , and (iii) the expected LGD given satisfies E( i | = )= (0.5+ ).
Recompute the approximate 99% Value-at-Risk to incorporate the stochastic LGD.
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