Trabajos previos

In this section we analyse the implications for the distribution of the portfolio loss and for the associated risk measures, ie, the value-at-risk and the average value-value-at-risk, if the SRF model is replaced by the GSRF model with non-Gaussian latent risk factors Z ∼ FZ

and Y ∼ FY. For this purpose, the distribution functions F_Zand F_Y are only restricted by setsFZ⊆F0,1andFY ⊆F0,1, where generally FZ≠FY is possible. In this chapter, the special caseFZ =FY=F0,1

is analysed, whereas further restrictions of the sets are also possible (eg, continuous distribution functions, symmetric distributions, uni-modal distributions or distributions with finite moments of a certain order).

VaR, AVaR and model risk sets

The model risk, or, more precisely, the misspecification risk associ-ated with the assumption of Gaussian distributed latent risk factors, can be quantified by means of model risk sets as defined by Equa-tion 3.1. For this reason, model risk sets for the value-at-risk and the average value-at-risk of a credit portfolio with respect to the risk-factor distribution functions F_Zand F_Y, each restricted only by the setF0,1, are given in the next three problems. These problems seem not to have been considered in the literature so far, either for homogeneous portfolios with a small number of obligors or for the asymptotic portfolio loss V∞in the GHSRF model.

Problem 3.1 (GSRF model). Let 0 < α < 1 be a given probability level. Suppose the conditions of the GSRF model given in Defini-tion 3.10 hold. Then the distribuDefini-tion of the loss variable Vndepends on the vectors v := (v1, . . . , vn), p := (p1, . . . , pn)and
:= (1, . . . , n) of potential losses, default probabilities and correlations.

Which properties do the model risk sets

Vα,v,p,
:= {VaRα[V_n]| FZ∈F0,1, F_Y∈F0,1}

and

Aα,v,p,
:= {AVaRα[Vn]| FZ∈F0,1, FY∈F0,1}

have for the value-at-risk and the average value-at-risk with respect to FZ ∈F0,1and FY ∈ F0,1? More particularly, what are the worst-case value-at-risk supVα,v,p,
, the best-case value-at-risk infVα,v,p,
, the worst-case average value-at-risk supAα,v,p,
and the best-case average value-at-risk infAα,v,p,
?

Problem 3.2 (GHSRF model: finite case). Let 0 < α < 1 be a given probability level. Suppose the conditions of the GHSRF model given in Definition 3.12 hold. Then the distribution of the loss variable Vn

depends on the number of obligors n, on the default probability π and on the correlation .

Which properties do the model risk sets

Vα,n,π,:= {VaRα[Vn]| FZ∈F0,1, FY∈F0,1} and

Aα,n,π,:= {AVaRα[V_n]| FZ∈F0,1, F_Y∈F0,1}

have? More particularly, what are the worst-case value-at-risk supVα,n,π,, the best-case value-at-risk infVα,n,π,, the worst-case average value-at-risk supAα,n,π,and the best-case average value-at-risk infAα,n,π,?

Problem 3.3 (GHSRF model: asymptotic loss distribution). Let 0 < α < 1 be a given probability level. Suppose the conditions of the GHSRF model given in Definition 3.12 hold and let the random variable V∞ be defined by Equation 3.17. Then the distribution of V∞depends on the default probability π and on the correlation .

Which properties do the model risk sets

Vα,∞,π,:= {VaRα[V∞]| FZ∈F0,1, F_Y∈F0,1} and

Aα,∞,π,:= {AVaRα[V∞]| FZ∈F0,1, FY∈F0,1}

have? More particularly, what are the worst-case value-at-risk supVα,∞,π,, the best-case value-at-risk infVα,∞,π,, the worst-case average value-at-risk supAα,∞,π, and the best-case average value-at-risk infAα,∞,π,?

The special case FZ = FY = Φ ∈ F0,1 shows that the six sets defined in Problems 3.1–3.3 are not empty. We conjecture, but are not able to prove in full generality, that the six sets are convex, ie, are intervals. The derivation of the worst-case and best-case risks men-tioned above can be based on the extremal risk-factor distributions presented in the following subsection.

Extremal risk-factor distributions

A prerequisite for finding extremal risk-factor distributions is given in the next lemma, where bounds for the lower and upper quantiles for all standardised random variables are stated.

Lemma 3.15. Let hold for each distribution function F ∈ F0,1 with lower quantile function F⁻(cf Equation 3.14) and upper quantile function

F⁺(u) := inf{x | F(x) > u}, 0 < u < 1

These lower and upper bounds for the lower and upper quantiles can be linked to the class of dichotomous distributions with mean 0 and variance 1. Dichotomous distributions concentrate their proba-bility mass on exactly two different points so that the corresponding distribution functions are step functions with two steps.

Lemma 3.16. Let aθand bθbe defined as in Lemma 3.15.

(i) All dichotomous distributions with distribution functions in F0,1are given by

Pr(X= aθ)= θ and Pr(X = bθ)= 1 − θ (3.21) with 0 < θ < 1.

(ii) For a fixed 0 < θ < 1, the distribution function FX(x) :=

Pr(X  x) of the random variable X defined in Equation 3.21 is

The lower quantile function of X is

F⁻_X(α)=

⎧⎨

⎩

aθ if 0 < α θ

bθ if θ < α < 1 (3.23) and the upper quantile function of X is

F⁺_X(α)=

⎧⎨

⎩

aθ if 0 < α < θ

bθ if θ α < 1 (3.24) In particular, the lower and upper θ-quantiles of X are given by

F⁻_X(θ)= aθ and F⁺_X(θ)= bθ (3.25) The class of dichotomous distributions with mean 0 and variance 1 can be characterised by G := {Gθ | 0 < θ < 1} ⊂F0,1 with distri-bution function Gθdefined in Equation 3.22. The subsetG specifies discrete distributions which are extremal distributions in the setF0,1

in the following sense.

Theorem 3.17. Let 0 < θ < 1 be a fixed probability level and let X be a random variable with distribution function Gθ defined in Equa-tion 3.22 and with lower θ-quantile F⁻_X(θ) and upper θ-quantile F⁺_X(θ)given in Equation 3.25. Then the distribution of X minimises the lower θ-quantile in the classF0,1, ie

min{F⁻(θ)| F ∈F0,1} = F⁻_X(θ) and maximises the upper θ-quantile in the classF0,1, ie

max{F⁺(θ)| F ∈F0,1} = F⁺_X(θ)

These properties qualify the class of dichotomous distributions with mean 0 and variance 1 (characterised by the setG) in the search for worst-case and best-case risks, eg, the worst-case value-at-risk and the worst-case average value-at-risk. It should be noted that these distributions are used only to derive bounds for extreme sce-narios and are not used as substitutes for the true but unknown distributions of the risk factors or as benchmark scenarios.

Worst-case VaR and AVaR

In this subsection, Problems 3.2 and 3.3 are addressed. In the GHSRF model, the value-at-risk VaRα[Vn] and the average value-at-risk

AVaRα[Vn]defined in Equations 3.6 and 3.7 can be approximated by VaRα[V∞]and AVaRα[V∞]given in Equations 3.18 and 3.19, respec-tively. Since the aim is to find the corresponding worst-case scenar-ios of these risk measures, the quantile maximising and minimis-ing properties given in Theorem 3.17 for the class of dichotomous distributions with mean 0 and variance 1 are used.

Lemma 3.18. Suppose the conditions of the GHSRF model given in Definition 3.12 hold. Let θ= 1 −√

1− π and FZ= FY = Gθ, where Gθis defined by Equation 3.22.

(i) If√

1− π(1 − (1 −√

1− π)ⁿ) < α <1, then

VaRα[V_n]= AVaRα[V_n]= 1 (3.26)

(ii) If√

1− π < α < 1, then

VaRα[V∞]= AVaRα[V∞]= 1 (3.27) where the random variable V∞is defined in Equation 3.17.

Lemma 3.18 holds for relevant parameter values of π and α, eg, for probability level α = 99. 5%, default probability π = 1%, any positive correlation less than 1 and risk-factor distribution func-tions F_Z = FY = Gθ with θ= 1 −√

1− π. Since the random vari-ables Vnand V∞are bounded by the unit interval, the risk measures VaRα[V_n], AVaRα[V_n], VaRα[V_∞]and AVaRα[V_∞]have the upper bound 1. Lemma 3.18 shows that this upper bound is taken for some parameter constellations. An immediate consequence of Lemma 3.18 and ofG ⊂ F0,1is the next theorem, which gives a partial answer to the questions of Problems 3.2 and 3.3.

Theorem 3.19. Suppose the conditions of the GHSRF model given in Definition 3.12 hold.

(i) If√

1− π(1 − (1 −√

1− π)ⁿ) < α < 1, then the worst-case value-at-risk and the worst-case average value-at-risk of the portfolio loss Vnreach the maximum value 1, ie

supVα,n,π,= max Vα,n,π,= 1 supAα,n,π,= max Aα,n,π,= 1

⎫⎬

⎭ (3.28)

The maximums in Equation 3.28 are reached by the worst-case

1− π < α < 1, then the worst-case value-at-risk and the worst-case average value-at-risk of the asymptotic portfolio loss V∞reach the maximum value 1, ie

supVα,∞,π,= max Vα,∞,π,= 1 supAα,∞,π,= max Aα,∞,π,= 1

⎫⎬

⎭ (3.30)

The maximums in Equation 3.30 are reached by the worst-case asymptotic portfolio loss distribution

This theorem has the following implications for the GHSRF model with latent systematic and idiosyncratic risk factors. If the only restriction for the distributions of the latent risk factors is the stan-dardisation, then the risk measures at-risk and average value-at-risk are only bounded by the trivial upper bound 1 for relevant parameter constellations. To get sharper bounds, further restrictions of the allowed risk-factor distributions are necessary, eg, continuous, symmetric or unimodal distributions with mean 0 and variance 1.

These restrictions appear to be arbitrary if the risk factors are latent and not observable. The upper bounds in Equations 3.28 and 3.30 show impressively that the values for the value-at-risk and for the average value-at-risk derived in the context of the common HSRF model depend substantially on the assumption of normality for the latent risk factors.

SUMMARY

This chapter focuses on the quantification of the model risk arising from a potential misspecification of the distributions of latent risk

factors in a special class of credit-portfolio models. For this purpose, new measures of model risk such as the model risk sets are intro-duced. Since model risks can only be quantified if the underlying model is defined, the SRF model with Gaussian distributed latent risk factors and a special case, the HSRF model, with homogeneous potential losses, default probabilities and correlation parameters, are discussed. In addition, generalisations of both models are devel-oped, where the distributional assumptions for the latent risk fac-tors are reduced. More precisely, in the generalised single-risk-factor model and in the generalised homogeneous single-risk-factor model the latent risk factors are assumed only to be standardised (mean 0 and variance 1) instead of having a standard normal distribution.

Lower and upper bounds for the lower and upper quantiles for all standardised random variables are given, and the correspond-ing class of distributions which reach these bounds is presented.

Using these extremal distributions, it is shown that in the GHSRF model with standardised systematic and idiosyncratic risk-factor distributions for relevant parameter constellations no upper bound less than 1 exists for the value-at-risk and the average value-at-risk.

The results highlight the fact that the values for the value-at-risk and for the average value-at-risk usually calculated with the common HSRF model depend substantially on the assumption of normality for the latent risk factors.