an introduction to credit risk modeling phần 4 docx

In the limit we obtain a portfolio loss variablepY describing the fraction of defaulted obligors in an infinitely fine-grained credit portfolio.. In this case, the absolute size-mportfo

account that we are in a Bernoulli framework,

So for portfolios with a sufficiently large portfolio size m satisfyingAssumption 2.5.2, the percentage quote of defaulted loans for a givenstate of economy Y = y is approximately equal to the conditionaldefault probability p(y) In the limit we obtain a portfolio loss variablep(Y ) describing the fraction of defaulted obligors in an infinitely fine-grained credit portfolio

We now want to derive the cumulative distribution function and theprobability density of the limit loss variable p(Y ), Y ∼ N (0, 1), withp(·) as in (2 54) Denote the portfolio’s percentage number of defaults

in an infinitely fine-grained portfolio (again assuming constant LGDs

of 100%) by L We then have for every 0 ≤ x ≤ 1

In the sequel we will denote this distribution function by

Fp,%(x) = P[L ≤ x] (x ∈ [0, 1]) (2 56)The corresponding probability density can be derived by calculatingthe derivative of Fp,%(x) w.r.t x, which is

FIGURE 2.5

The probability density fp,% for different combinations of p and

% (note that the x-axes of the plots are differently scaled)

50 100 150 200 250

bps bps

p

0 0.2 0.4 0.6 0.8 1 100

200 300 400 500

% 99 99

5 10 15 20

% 5

%

Figure 2.5shows the loss densities fp,% for different values of p and %.

It could be guessed fromFigure 2.5that regarding the extreme casesw.r.t p and % some reasonable limit of fp,% should exist Indeed, onecan easily prove the following statement:

2.5.7 Proposition The density fp,% admits four extreme cases duced by the extreme values of the parameters p and %, namely

in-1 % = 0 :

This is the correlation-free case with loss variables

Li = 1{ri=Zi< N−1 [p]} ∼ B(1; p) ,taking (2 48) into account In this case, the absolute (size-m)portfolio loss P Li follows a binomial distribution, Pm

i=1Li ∼B(m; mp), and the percentage portfolio loss Lm converges by ar-guments analogous to Proposition 2.5.4 (or just by an application

of the Law of Large Numbers) to p almost surely Therefore, fp,0

is the density22 of a degenerate distribution23 concentrated in p.This is illustrated by the first plot inFigure 2.5, where an almostvanishing correlation (%=1 bps) yields an fp,%, which is almostjust a peak in p=30 bps

2 % = 1 :

In this case one has perfect correlation between all loss variables

in the portfolio (see also Section 1.2, where the term “perfectcorrelation” was mentioned the first time) In this case we canreplace the percentage portfolio loss Lm by L1 ∼ B(1; p), which

is no longer dependent on m Therefore, the limit (m → ∞)percentage portfolio loss L is also Bernoulli B(1; p), such thatP[L = 1] = p and P[L = 0] = 1 − p The case of (almost) perfectcorrelation is illustrated in the fourth plot (p=30 bps, %=99.99%)

ofFigure 2.5, clearly showing the shape of a distribution trated in only two points, yielding an “all or nothing” loss

concen-3 p = 0 :

All obligors survive almost surely, such that P[L = 0] = 1

22 More precisely, it is a delta distribution.

23 More explicitly, we are talking about a Dirac measure.

portfo-2.5.8 Proposition For any given level of confidence α, the α-quantile

qα(L) of a random variable L ∼ Fp,% is given by

By definition (seeSection 1.2) the Unexpected Loss (UL) is the dard deviation of the portfolio loss distribution In the following propo-sition the UL of an infinitely fine-grained uniform portfolio is calcu-lated

stan-2.5.9 Proposition The first and second moments of a random able L ∼ Fp,% are given by

vari-E[L] = p and V[L] = N2N−1[p], N−1[p]; % − p2 ,where N2 is defined as in Proposition 2.5.1

Proof That the first moment equals p follows just by construction

of Fp,% Regarding the second moment, we write V[L] = E[L2] − E[L]2

We already know E[L]2= p2 So it only remains to show that E[L2] =

N2[N−1[p], N−1[p]; %] For proving this, we use a typical “conditioning

trick.” For this purpose, let X1, X2 ∼ N (0, 1) denote two dent standard normal random variables, independent from the randomvariable

We write gµ,σ2 for the density of X Then, we can write E[L2] as

E[L2] = E[p(Y )2] = E[N (X)2]

X1− X and X2− X, we conclude E[L2] = N2[N−1[p], N−1[p]; %] 2The next proposition reports on higher moments of Fp,%

2.5.10 Proposition The higher moments of L ∼ Fp,% are given by

E [Lm] = Nm(N−1[p], , N−1[p]), C%where Nm[· · · ] denotes the m-dimensional normal distribution functionand C%∈ Rm×m is a matrix with 1 on the diagonal and % off-diagonal.Proof The proof relies on the same argument as the proof of Propo-sition 2.5.9 A generalization to m ≥ 2 is straightforward 2

24 Shifting and scaling a random variable in order to achieve mean zero and standard ation one.

devi-TABLE 2.2: Economic Capital EC α for an infinitely fine-grained portfolio (portfolio loss

L ∼ Fp,% ) w.r.t.pand%, for α = 99.5%.

TABLE 2.3: Economic capital EC α for an infinitely fine-grained portfolio (portfolio loss

L ∼ Fp,%) w.r.t.pand %, forα = 99.98%.

TABLE 2.4: Unexpected loss UL for an infinitely fine-grained portfolio (portfolio loss

L ∼ Fp,% ) w.r.t.pand%.

Given a uniform one-year average default probability p and a form asset correlation %, Tables 2.2 and 2.3 report on the EconomicCapital (EC) w.r.t confidence levels of α = 99, 5% and α = 99, 98% for

uni-an infinitely fine-grained portfolio (described by the distribution Fp,%),hereby assuming an LGD of 100% (seeSection 1.2.1 for the definition

of EC) Analogously, Table 2.4 shows the Unexpected Loss for a givenpair (p, %)

Figure 2.6 illustrates the sensitivity of the EC w.r.t the chosen fidence level It can be seen that at high levels of confidence (e.g., from99,9% on) the impact of every basispoint increase of α on the portfolio

con-EC is enormous

Another common portfolio-dependent quantity is the so-called ital multiplier (CMα); see also Chapter 5 on capital allocation It isdefined as the EC w.r.t confidence α in units of UL (i.e., in units ofthe portfolio standard deviation) In pricing tools the CM is sometimesassumed to be constant for a portfolio, even when adding new deals to

cap-it The contribution of the new deal to the total EC of the enlargedportfolio is then given by a multiple of the CM In general, the CMheavily depends on the chosen level of confidence underlying the ECdefinition Because for given p and % the CM is just the EC scaled bythe inverse of the UL, Figure 2.6 additionally illustrates the shape ofthe curve describing the dependency of the CM from the assumed level

of confidence

For example, for p=30 bps (about a BBB-rating) and %=20% (theBasel II suggestion for the asset correlation of the benchmark riskweights for corporate loans) the (rounded!) CM of a portfolio with lossvariable L ∼ Fp,% is given by CM99%≈ 4, CM99,5% ≈ 6, CM99,9%≈ 10,and CM99,98% ≈ 16 (in this particular situation we have an UL of 59bps, as can be read from theFigure 2.4)

Now, as a last remark in this section we want to refer back to tion 1.2.2.2, where the analytical approximation of portfolio loss distri-butions is outlined The distribution Lp,%, eventually combined withsome modifications (e.g., random or deterministic LGDs), is extremelywell suited for analytical approximation techniques in the context ofasset value (or more generally latent variable) models

Sec-2.5.2 The CreditRisk+ One-Sector Model

We already discussed CreditRisk+inSection 2.4.2and will come back

to it in Chapter 4 Therefore this paragraph is just a brief

“warming-FIGURE 2.7

Negative binomial distribution with parameters (α, β) = (1, 30)

up” for the next paragraph where we compare the uniform portfolioloss distributions of CreditMetricsTM respectively KMV with the cor-responding distribution in the CreditRisk+ world

Assuming infinitely many obligors and only one sector, we obtain asituation comparable to the uniform portfolio model of CreditMetricsTM

and KMV

Under these assumptions, the portfolio loss is distributed ing to a negative binomial distribution N B(α, β) due to a gamma-distributed random intensity The derivation of the negative binomialdistribution in the CreditRisk+ framework is extensively discussed inChapter 4 Denoting the portfolio loss by L0 ∼ N B(α, β), the lossdistribution is determined by

1 + β

n

, (2 58)where α and β are called the sector parameters of the sector; see For-mula (4 26) The expectation and the variance of L0 are given by

E[L0] = αβ and V[L0] = αβ(1 + β) , (2 59)

as derived in Formula (4 27) Figure 2.7 illustrates the shape ofthe probability mass function of a negative binomial distribution, here

with parameters α = 1 and β = 30 The expected loss in a portfolioadmitting such a loss distribution is

EL = E[L0] = 1 × 30 = 30,and the unexpected loss (volatility of the portfolio loss) is

UL = pV[L0] = p1 × 30 × (1 + 30) = 30.5

We are now prepared for the next section

2.5.3 Comparison of One-Factor and One-Sector ModelsRecalling the discussion about general mixture models at the be-ginning of this chapter one could say that in this section we compareBernoulli and Poisson mixture models by means of a typical example

As a representative for the Bernoulli mixture models we choose therandom variable L ∼ Fp,% describing the percentage loss of an infinitelyfine-grained portfolio with uniform default probability p and uniformasset correlation %; see (2 55) Such portfolios typically arise in analyt-ical approximations in the CreditMetricsTM respectively KMV frame-work

The one-sector model of CreditRisk+ as described in the previousparagraph will serve as a representative for Poisson mixture models

A very natural way to calibrate the two models on a common basis is

by moment matching One problem we face here is that L takes place

in the unit interval and L0 generates random integers We overcomethis problem by fixing some large m, say 20,000, such that the tailprobability P[L0 > m] is negligibly small, and transforming L0 into avariable

TABLE 2.5: Comparison of Bernoulli and Poisson mixture models by means of one- factor respectively one-sector models.

applying Proposition 2.5.9 As a last step we solve (2 59) for α and

β One always has

α = m × E[ ˜L0]2

m × V[ ˜L0] − E[ ˜L0] , β =

m × V[ ˜L0] − E[ ˜L0]E[L˜0] , (2 60)e.g., for p=30 bps, %=20%, and m=20,000 we apply 2.5.9 for

V[L] = N2N−1[0.003], N−1[0.003]; 0.2 − 0.0032 = 0.000035095 The unexpected loss of L therefore turns out to be UL=59 bps Ap-plying Formulas (2 60), we get

α = 0.26 and β = 232.99 ,

so that the distribution of ˜L0 is finally determined

InTable 2.5high-confidence quantiles of factor respectively sector models with different parameter settings are compared It turnsout that the Bernoulli mixture model always yields fatter tails than

the Poisson mixture model, hereby confirming our theoretical resultsfrom Section 2.3 A more detailed comparison of the KMV-Model andCreditRisk+ can be found in [12].

Copula functions have been used as a statistical tool for ing multivariate distributions long before they were re-discovered as avaluable technique in risk management Currently, the literature onthe application of copulas to credit risk is growing every month, sothat tracking every single paper on this issue starts being difficult ifnot impossible A small and by no means exhaustive selection of pa-pers providing the reader with a good introduction as well as with avaluable source of ideas how to apply the copula concept to standardproblems in credit risk is Li [78,79], Frey and McNeil [45], Frey, McNeil,and Nyfeler [47], Frees and Valdez [44], and Wang [125] However, the basicidea of copulas is so simple that it can be easily introduced:

construct-2.6.1 Definition A copula (function) is a multivariate distribution(function) such that its marginal distributions are standard uniform

A common notation for copulas we will adopt is

of an infinitely fine-grained portfolio So we implicitly already metcopulas in previous paragraphs

However, in the next section we are going to show how to use copulas

in order to construct portfolio loss variables admitting a stronger taildependency than induced by the normal copulas

But before continuing, we want to quote a Theorem by Sklar [113,114], saying that copulas are a universal tool for studying multivariatedistributions

2.6.2 Theorem (Sklar [113]) Let F be a multivariate m-dimensionaldistribution function with marginals F1, , Fm Then there exists acopula C such that

Summarizing Theorem 2.6.2 and Proposition 2.6.3, one can say thatevery multivariate distribution with continous marginals admits a uni-que copula representation Moreover, copulas and distribution func-tions are the building blocks to derive new multivariate distributionswith prescribed correlation structure and marginal distributions Thisimmediately brings us to the next section

2.6.1 Copulas: Variations of a Scheme

Here, we are mainly interested in giving some examples of how thecopula approach can be used for constructing loss distributions withfatter tails than it would be for normally distributed asset value log-returns In this book we restrict ourselves to normal and t-copulas,because they are most common in the credit risk context For othercopulas we refer to Nelsen [99]

For our example we look at a Bernoulli mixture model but replace themultivariate normal asset value log-return vector as used in the models

of CreditMetricsTMand KMV by a multivariate t-distributed log-returnvector For the convenience of the reader we first recall some basic testdistributions from statistics (see, e.g., [106]):

The Chi-Square Distribution:

The χ2-distribution can be constructed as follows: Start with an i.i.d.sample X1, , Xn ∼ N (0, 1) Then, X2

1 + · · · + X2

n is said to be χ2distributed with n degrees of freedom The first and second moments

-of a random variable X ∼ χ2(n) are

E[X] = n and V[X] = 2n

In some sense the χ2distribution is a “derivate” of the gamma-distribution(see 2.4.2), because the χ2(n)-distribution equals the gamma-distributionwith parameters α = n/2 and β = 2 Therefore we already know theshape of χ2-densities from Figure 2.2

The (Student’s) t-distribution:

The building blocks of the t-distribution are a standard normal able Y ∼ N (0, 1) and a χ2-distributed variable X ∼ χ2(n), such that Yand X are independent Then the variable Z defined by Z = Y /pX/n

vari-is said to be t-dvari-istributed with n degrees of freedom The density of Z

FIGURE 2.8

t(3)-density versus N (0, 1)-density

For large n, the t-distribution is close to the normal distribution Moreprecisely, if Fn denotes the distribution function of a random variable

Zn ∼ t(n), then one can show that Fn converges in distribution tothe distribution function of a standard normal random variable Z ∼

N (0, 1); see [106]

This convergence property is a nice result, because it enables us tostart in the following modification of the CreditMetricsTM/KMV modelclose to the normal case by looking at a large n By systematicallydecreasing the degrees of freedom we can transform the model step-by-step towards a model with fatter and fatter tails

In general the t-distribution has more mass in the tails than a normaldistribution Figure 2.8illustrates this by comparing a standard normaldensity with the density of a t-distribution with 3 degrees of freedom.The multivariate t-distribution:

Given a multivariate Gaussian vector Y = (Y1, , Ym) ∼ N (0, Γ)with correlation matrix Γ, the scaled vector ΘY is said to be multi-variate t-distributed with n degrees of freedom if Θ = pn/X with

X ∼ χ2(n) and Θ is independent of Y We denote the distribution ofsuch a variable ΘY by t(n, Γ) The matrix Γ is explicitly addressed asthe second parameter, because ΘY inherits the correlation structure