DEPARTMENT OF ACTUARIAL MATHEMATICS AND STATISTICS

HERIOT-WATT UNIVERSITY

MSc FINANCIAL MATHEMATICS, QFRM, QFE F71CM – Credit Risk Modelling

Date: 8th May 2015

Copyright By cscodehelp代写 加微信 cscodehelp

Time: 09.00 to 11.00 (2 hours)

Question Marks 1 15

There are 5 questions in this paper, each worth 15 marks. Credit will be given for the best 4 answers.

“Formulae and Tables for Actuarial Examinations” and approved electronic calculators may be used.

1. This question concerns the pricing of a defaultable zero-coupon bond using the martingale modelling approach. The bond pays 1 unit at maturity T and the default time ⌧ of the bond is modelled using a hazard rate model under the risk-neutral measure Q. The interest rate is assumed to be constant and equal to r.

In answering the question you may use the fact that, for an integrable random

variable X,

X | H ) = I EQ(I{⌧>t}X), {⌧>t} t {⌧>t} Q(⌧ > t)

where Ht = ({I{⌧s} : s t}) is the information available to an investor at time t.

(a) Explain clearly the assumptions that are made in the recovery-of-treasury (RT) and recovery-of-face-value (RF) recovery models.

(b) Explain what is meant by a survival claim and derive a formula for the value of a survival claim in terms of the deterministic hazard function Q(t) under the risk-neutral measure and the interest rate r.

(c) Derive a formula for the price of the defaultable zero-coupon bond under the RT recovery model.

Suppose the risk-neutral hazard function is assumed to take the constant value ̄Q at all times.

(d) Derive a formula for the spread c(t, T ) of the defaultable bond in this case and simplify it as far as possible.

(e) Show that if the hazard rate ̄Q is small then, prior to default, the spread may

be approximated by c(t, T ) ⇡ ̄Q where is the loss-given-default.

(f) Briefly outline the limitations of hazard rate models for modelling defaultable bond prices.

[PLEASE TURN OVER] 2

[Total 15 Marks]

2. Suppose that rating transitions for an obligor are modelled using a discrete-time Markov chain (Rt), where t = 0,1,2,… and Rt takes values in S = {0,1,…,n}. The rating 0 may be taken to represent default and 1,…,n represent states of increasing credit quality.

(a) State mathematically the condition for the Markov chain to be stationary.

(b) State two drawbacks of using a discrete-time chain as opposed to a continuous- time chain for modelling rating transitions.

Now suppose that rating transitions are modelled using a continuous-time Markov chain. The transition rate between state j and state k is estimated to be jk for j 2 S {0}, k 2 S and j 6= k, where the state space S is as defined above.

(c) (i)Definetheelementsofthegeneratormatrix⇤ofthecontinuous-timeMarkov chain and (ii) give a formula relating the probability of ratings transitions in the interval [0, t] to the generator matrix ⇤.

(d) Give the steps of an algorithm for simulating rating transitions according to a continuous-time Markov chain model with generator matrix ⇤.

(e) Describe the empirical features of real rating migration data that are inconsis- tent with the Markov assumption.

(f) Rating agencies often claim that they operate a through-the-cycle approach to rating companies. Explain what this means.

[PLEASE TURN OVER] 3

[Total 15 Marks]

3. Let (X,d) denote a threshold model of default for m obligors with X an m- dimensional random vector and d an m-dimensional real-valued vector. Denote by Yi the default indicator random variable of obligor i.

(a) Show that the probability that an arbitrary subgroup of k obligors all default is given by an expression of the form

P(Yi1 = 1,…,Yik = 1) = Ci1,…,ik(pi1,…,pik)

showing how Ci1,…,ik and the quantities pi1,…,pik relate to X and d. State

the name of any theorem that you use.

Now assume that p1 = ··· = pm = ⇡ and that the copula C of X is an exchangeable copula of the form

C(u1,…,um)= 1(u1)+···+ m(um), where (t) = (1+t1/)1/✓ for ✓ 0 and 1.

(b) Derive a formula relating the probability that a subgroup of k obligors all default, ⇡k, to the default probability of any obligor, ⇡.

(c) Show that when ⇡ = 0.01, = 2 and ✓ = 0.5 the default correlation of any two obligors is approximately equal to 0.526.

(d) Assume that m = 500 and that ⇡, and ✓ are as in part (c). Compute the mean and standard deviation of the number of obligors that default in this model.

(e) Suppose we decide to replace the copula used in this question by an exchange- able Gaussian copula while holding the mean and standard deviation of the number of defaulting obligors fixed. Show that ⇢, the correlation parameter of the Gaussian copula, must satisfy the equation

Z 1 ✓ ✓1(⇡) + p⇢x◆◆2

p1⇢ (x)dx,

where and are the standard normal density and cumulative distribution function respectively.

[PLEASE TURN OVER] 4

[Total 15 Marks]

4. This question is concerned with asymptotic results for large portfolio losses. Let (ei)i2N, (i)i2N and (Yi)i2N be sequences of exposuPres, losses-given-default (LGDs) and default indicators respectively. Let L(m) = mi=1 Li, where Li = eiiYi and assume that the losses Li are conditionally independent given a standard normal variable . The exposures may be taken to be deterministic throughout the ques- tion.

(a) Define the asymptotic relative loss function `.

(b) Give an asymptotic formula linking the Value-at-Risk (VaR) of the portfolio loss L(m) to a quantile of the variable (under a few extra technical conditions which you need not state).

Suppose that there are two groups of obligors and let r(i) 2 {1, 2} give the group membership. Assume that defaults are conditionally independent and that

P(Yi=1| = )=(μr(i)+0.25),

where μ1 = 1.5 and μ2 = P2.5. The LGDs satisfy i = 0.75 for all i. Assume,

moreover, that

(c) Find an expression for the function `( ) in terms of and the function only.

P [4 Marks]

(d) Compute the approximate value of VaR0.95(L(m)) as a fraction of mi=1 ei when m is large. (You may need to look up tables for the standard normal distribu- tion.)

Now relax the deterministic assumption on the LGDs and assume that i=( )+(✏i),

where (✏i)i2N are iid standard normal random variables. You may assume that ✏i

and Yi are conditionally independent given and that ✏i is independent of .

i:r(i)=r ei

lim Pm =0.5, r=1,2.

m!1 i=1 ei

(e) Find an expression for the function `( ) in this case, in terms of

function only.

and the [2 Marks]

(f) Determine whether our estimate of VaR0.95 L(m) for m large would be larger or smaller than in the case of constant LGDs.

[PLEASE TURN OVER] 5

[Total 15 Marks]

5. This question concerns statistical estimation of credit risk models.

(a) Give a brief description of the approach that is typically used in industry port- folio models to calibrate a factor model for modelling the dependence between defaults. A good example to take would be the GCorr (Global Correlation) model of Moody’s Analytics.

(b) Suppose that we observe historical default data over n periods of time for a homogeneous group. For t = 1, . . . , n, let mt denote the number of observed companies at the start of period t and let Mt denote the number that default during the period.

Derive simple method-of-moments estimators that could be applied to the data to estimate a default probability and a default correlation for the homogeneous group.

[END OF PAPER] 6

[Total 15 Marks]

1. Parts (a), (b), (c) and (f) are bookwork. Parts (d) and (e) are similar to a tutorial

(a) RT: If ⌧ T the bondholder receives a default-free ZCB paying 1 at T ; in other words bondholder receives (1 )p0(⌧,T) at ⌧. RF: If ⌧ T the bondholder receives (1 ) at ⌧ .

(b) A survival claim pays a fixed amount (one say) at T provided ⌧ > T . Thus the pay-o↵ is I{⌧>T}. Its value is computed (using a formula given in class) to be

EQ p0(t,T)I{⌧>T} | Ht =

= I{⌧>t}p0(t,T)EQ I{⌧>t}I{⌧>T}

(c) The pay-o↵ of the bond is

EQ p0(t,T)I{⌧>t}I{⌧>T} | Ht Q(⌧ > t)

= I{⌧>t}p0(t,T)Q(⌧ > T) Q(⌧ > t)

= I{⌧>t}er(Tt)eRtT Q(s)ds. I{⌧>T} +(1)(1)I{⌧>T}

where the first term is the survival claim payo↵ and the second and third terms comprise the pay-o↵ of the recovery. This simplifies to I{⌧ >T } + (1 ) and hence the price of the bond is

p1(t,T) = EQp0(t,T)I{⌧>T} +(1)|Ht

= (1)p0(t,T)+EQp0(t,T)I{⌧>T} |Ht

where the second term is the value of a survival claim at time t. We thus obtain p1(t,T)=(1)p0(t,T)+p0(t,T)I{⌧>t}eRtT Q(s)ds.

(d) The spread is

c(t,T) = 1 ln✓p1(t,T)◆ R

When the hazard function under the risk-neutral measure Q is a constant

Q(t) = ̄Q we get

c(t,T)= 1 ln⇣1+I{⌧>t}e ̄Q(Tt)⌘.

Tt ⇣p0(t,T)

= 1 ln 1+I{⌧>t}e tT Q(s)ds .

(e) If⌧>tand ̄Q issmallwehave

c(t,T)⇡ 1 ln1 ̄Q(Tt)⇡ ̄Q

where we use the approximations exp(x) ⇡ 1 x and ln(1 x) ⇡ x for x

(f) The model can only model deterministic spreads prior to default. It cannot be used to price options on bonds. For these reasons we introduce doubly- stochastic random times.

2. The entire question is essentially bookwork, but this is new material this year.

(a) For stationarity we require

P(Rt+1 =k|Rt =j)=P(R1 =k|R0 =j), 8t2{0,1,…},j2S{0},k2S.

(b) The drawbacks are: (i) we may miss some transitions that occur within time periods, for example a firm that starts at AA and ends up at BBB, while passing through A during the period; (ii) it is dicult to calculate transition probabilities for a fraction of the time period.

(c) We define the elements ⇤jk of the generator matrix as follows. ⇤jk = jk for j6=kanPdj2S{0},k2S. ⇤0k =0fork2S(thisisthedefaultrow). ⇤jj = k6=j jK. The transition probabilities are computed using the matrix exponential according to

P (t) = exp(⇤t).

(d) An obligor starts in some rating class j and stays there for an exponentially distributPed period of time where the parameter of the exponential distribution

is j = j 6=k j k . When the obligor changes state it moves to state k with probability jk/j.

(e) Momentum: this refers to the fact that the risk of downgrade appears to be higher for companies that have recently been downgraded when compared with companies that have had the same rating for longer. Stickiness: this refers to the reluctance of rating agencies to downgrade companies in general.

(f) When assigning a rating they try to capture average credit quality of a firm over a complete business cycle of growth and recession. They don’t automatically downgrade all companies when economy goes into recession, just those whose quality appears weaker with respect to their peers.

3. Part (a) is bookwork. Thereafter a slightly tricky question but similar to tutorial questions.

(a) Defining indicators Yi = I{Xidi} for i = 1,…,m we use Sklar’s Theorem to show that

P(Yi1 = 1,…,Yik = 1) = P(Xi1 di1,…,Xik dik)

= Ci1,…,ik(P(Xi1 di1),…,P(Xik dik))

= Ci1,…,ik(pi1,…,pik) 8

where Ci1,…,ik is the copula of (Xi1 , . . . , Xik ).

(b) In the exchangeable case this formula takes the form

⇡k = P(Y1 = 1,…,Yk = 1) = C1,…,k(⇡,…,⇡).

⇡k = C1,…,k(⇡,…,⇡) = C(⇡,…,⇡,1,…,1)

(c) Default correlation is given by ⇢Y =⇢(Y1,Y2) =

Xk ! = 1(⇡)

i=1 = k(⇡✓ 1)

= 1 + k1/(⇡✓ 1)1/✓ .

1 + 21/(⇡✓ 1)1/✓ ⇡2

= andwhen=2,⇡=0.01and✓=0.5wegetthevalues⇡2 ⇡0.0053and

⇡(1⇡) ⇢Y ⇡0.526. P

(d) We compute moments of the random variable M = mi=1 Yi and we obtain E(M) = m⇡ = 500 ⇥ 0.01 = 5

var(M) = mvar(Y1)+m(m1)cov(Y1,Y2) = m var(Y1) + m(m 1)⇢Y var(Y1)

= ⇡(1⇡)(m+m(m1)⇢Y)⇡1304

(e) In the Gaussian exchangeable copula model we have Yi = I{Xidi} where di = 1(⇡) and p p

giving sd(M) ⇡ 36.1.

Xi= ⇢F+ 1⇢✏i for iid standard normal ✏i and F . This gives

⇡2 =P(Y1 =1,Y2 =1) = E(P(X1 d1,X2 d2 |F))

✓ d1 p⇢F ! d2 p⇢F ◆

= E P(✏1 p1⇢ ,✏2 p1⇢ |F) ✓1(⇡) p⇢F ◆2

which gives the required formula when we note that F and F have the same distribution.

4. Parts (a) and (b) are bookwork. The remaining computational parts follow the

pattern of tutorial questions and previous examination questions. These kind of

results are the basis of the Basel capital formula.

(a) The asymptotic relative loss function is given by

̄ 1 (m)

where we write am = Pmi=1 ei for m = 1,2,….

(b) The limiting result states that under technical conditions

✓L(m)◆ ̄ 1 VaR↵ a = `( (↵))

giving the large sample approximation

`()=lim EL | = m!1 am

which we apply in the remaining parts of the question.

(c) We have

̄ 0 . 7 5 0@

) ⇡ am`( (↵))

(d) We get 0.75(0.5(1.5 + 0.251(0.95)) + 0.5(2.5 + 0.251(0.95))) which is approximately 0.0587.

(e) With stochastic LGDs we have

1Xm ✓()+(✏i) ◆ `( ) = lim eiE Yi | =

`( ) = lim ei(μ1 +0.25 )+

m!1 am i:r(i)=1 i:r(i)=2

ei(μ2 +0.25 ) = 0.75(0.5(1.5 + 0.25 ) + 0.5(2.5 + 0.25 )).

m!1 am i=1 2

1 Xm m!1 am i=1

( )+E( (✏i))

ei E(Yi| = )

= lim = lim

( ) + 21 Xm 2am i=1

eiE(Yi| = )

) + 12 ◆(0.5(1.5 + 0.25 ) + 0.5(2.5 + 0.25 )) 2

(f) We need to compare 0.5((1(0.95)) + 12 ) and 0.75. The former evaluates to 0.725 and thus our large sample losses are estimated to be smaller under the stochastic LGD model.

5. This question is bookwork but comes at the very end of the course so is seldom examined. I anticipate that students who don’t like the calculation questions will attempt this.

(a) This is a very complete description of the approach taken from my textbook. Of course I expect much less detail. The model of interest takes the form

Xi=piF ̃i+p1i”i, i=1,…,m,

where F ̃i and “1, . . . , “m are independent standard normal variables, and where 0 i 1 for all i. The systematic variables F ̃i are assumed to be of the form F ̃i = a0iF where F is a vector of common factors satisfying F ⇠ Np(0, ⌦) with p < m, and where ⌦ is a correlation matrix. The factors typically represent country and industry e↵ects and the assumption that var(F ̃i) = 1 imposes the constraint that a0i⌦ai = 1 for all i. [4 marks for setting out the model and the terms to be estimated.]
Given time series (Xt)1tn of observations of X = (X1, . . . , Xm), which may be standardized asset returns or equity returns, the steps are as follows.
(i) We first fix the structure of the factor vector F so that, for example, the first block of components might represent country factors and the second block of components might represent industry factors. We then assign vectors of factor weights ai to each obligor based on our knowledge of the companies. The elements of ai may simply consist of ones and zeros if the company can be clearly identified with a single country and industry, but may also consist of weights if the company has significant activity in more than one country or more than one industry sector. For example, a firm that does 60% of its business in one country and 40% in another would be coded with weights of 0.6 and 0.4 in the relevant positions of ai.
(ii) We then use cross-sectional estimation techniques to estimate the factor values Ft at each time point t. E↵ectively the factor estimates Fˆt are constructed as weighted sums of the Xt,i data for obligors i that are exposed to each factor.
(iii) The raw factor estimates form a multivariate time series of dimension p. We standardize each component series to have mean zero and variance one to obtain (Fˆt)1tn and calculate the sample covariance matrix of the standardized factor estimates, which serves as our estimate of ⌦.
(iv) We then scale the vectors of factor weights ai so that the conditions a0i⌦ˆai = 1 are met for each obligor.
(v) Time series of estimated systematic variables for each obligor are then ˆ ̃ 0ˆ
constructed by calculating Ft,i = aiFt for t = 1, . . . , n.
(vi) Finally we estimate the i parameters by performing a time series regres-
sion of Xt,i on Ft,i for each obligor.
[4 marks for sketching the main steps.] 11
(b) Again this is taken from my textbook. For 1 t n, let Yt,1,...,Yt,mt be default indicators for the mt companies in the cohort. Suppose we define the
rv ✓Mt◆:= X Yt,i1 ···Yt,ik; k {i1 ,...,ik }⇢{1,...,mt }
this represents the number of possible subgroups of k obligors among the de- faulting obligors in period t (and takes the value zero when k > Mt). By taking

expectations we get and hence

E✓✓Mt◆◆ = ✓mt◆⇡k kk

⇡k = E✓✓Mt◆◆✓mt◆. kk

We estimate the unknown theoretical moment ⇡k by taking a natural empirical average constructed from the n years of data:

1Xn Mt 1Xn Mt(Mt1)···(Mtk+1) ⇡ˆ k = k = .

n mt n mt(mt 1)···(mt k+1) t=1 k t=1

For k = 1 we get the standard estimator of default probability ⇡ˆ=1Xn Mt,

and ⇢Y can obviously be estimated by taking ⇢ˆY = (⇡ˆ2 ⇡ˆ2)/(⇡ˆ ⇡ˆ2). The estimator is unbiased for ⇡k and consistent as n ! 1. [3 marks for describing the result on which the estimator depends; 4 marks for accurately describing the estimator itself.]

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