Quantile of a Mixture with Application to Model Risk Assessment

Abstract We provide an explicit expression for the quantile of a mixture of two random variables. The result is useful for finding bounds on the Value-at-Risk of risky portfolios when only partial dependence information is available. This paper complements the work of [4].


Introduction
Consider a mixture S = IX+ ( − I)Y where I is a Bernoulli distributed random variable with parameter q and where the components X and Y are independent of I. In this paper, we aim at nding an explicit expression for the quantiles of S as a function of the quantiles of the variables X and Y. Here, the quantile at level p ( < p < ) of a given distribution F is de ned¹ as By convention, inf{∅} = ∞ and inf{R} = −∞, so that the quantile is properly de ned by (1) for all p ∈ [ , ]. In a risk management context, one often considers X (distributed with F X ) as a loss variable in which case F − X (p) can be broadly interpreted as the maximum loss ("Value-at-Risk") one can observe with p−con dence.
Expressing F − S (·) in terms of F − X (·) and F − Y (·) can sometimes be relatively straightforward. For example, when F X and F Y are strictly increasing with unbounded support², in which < α * < and < β * < are uniquely de ned and satisfy qα * + ( − q)β * = p. However, formula (2) does not cover the general case. We need a more general version of it, which requires a careful examination of all possible cases for the distributions F X and F Y (Theorem 1). This result is of some probabilistic interest, carole.bernard@grenoble-em.com 1 Note that the de nition of quantile is not unique. Here, we consider the lower quantile, which is the prevalent way to de ne quantiles in a risk management context; See e.g., Section 4.4 and De nition A.20 in [7] as well as the works of [1] and [11]. 2 Note that under similar conditions on the distribution functions the two-dimensional case can be extended to n dimensions in a straightforward way; it is actually su cient that the distributions of the components involved in the mixture are invertible and have a common support. Another notable case dealing with distributions that have a disjoint support was treated in [5]. In this paper we focus further on providing a general formula for the quantile of a mixture in the two-dimensional case.
as it makes it possible to reduce the dimensionality of the problem of assessing a quantile of a mixture. It also has a clear application in risk management that we further explain as follows: In the risk assessment of high dimensional portfolios X := (X , X , . . . , X d ) the variable of interest is typically the portfolio sum d i= X i and the risk measure used in the industry is a quantile. From Sklar's theorem, the evaluation of quantiles of d i= X i is at most a numerical issue once the marginal distributions of the variables X i as well as their dependence (copula) is completely speci ed. Unfortunately, estimating copulas is a di cult task and the assessment of X is thus prone to model misspeci cation. [4] assume that candidate models are consistent with the following distributional properties of X: In other words, the joint distribution of X is only fully speci ed on the subarea F of R d and the quantiles of d i= X i cannot be computed (unless p F = ). It is then of interest to nd among all possible distributional models for X the one that yields the highest (resp. lowest) possible outcome for the desired quantile of d i= X i . The maximum and minimum possible values can be obtained using a mixture representation. Speci cally, consider the indicator variable I corresponding to the event "X ∈ F" It is then clear that for any choice of joint distribution for X that is consistent with properties (i), (ii) and (iii), there exists a multivariate vector (Z , Z , ...Z d ) that we can take independent of I, such that the portfolio sum can be represented as a mixture where "= d " denotes equality in distribution. Here, for i = , , ..., d, where U =R d \F. From the properties (i), (ii) and (iii) it follows that the marginal distributions F Z i of Z i are known, but the copula of (Z , Z , ...Z d ) remains unspeci ed. In the given context, we thus e ectively aim at nding copulas that yield maximum and minimum value for quantiles of a mixture like in (4). In this regard, an explicit formula for the quantile of a mixture is useful, as it avoids that one has to resort to lengthy (nested) simulations; see Section 3 and the discussion that follows Proposition 2 in particular.
The explicit computation of the quantile of a mixture is presented in Section 2. Its application to model risk assessment is developed in Section 3.

Quantile of a mixture
We formulate the following result for the quantile of a mixture. qα and let β * = p−qα * −q ∈ [ , ]. Then, for p ∈ ( , ) , This maximum can be computed explicitly by distinguishing along the four following cases for F X (·) and for F Y (·) : Case (1) F X is continuous at sp and for all z < sp, F X (z) < F X (sp) Case (2) F X is continuous at sp and there exists z < sp such that F X (z) = F X (sp) Case (3) F X is discontinuous at sp and for all z < sp, F X (z) < F X (s − p ) Case (4) F X is discontinuous at sp and there exists z < sp such that F X (z) = F X (s − p ) Case (a) F Y is continuous at sp and for all z < sp, We have summarized the computations of sp in Table 1 for the sixteen possible combinations. Table 1: Summary of all cases for the quantiles of a mixture where sp = F − S (p). In all cases, α * is de ned by (7) and β * =

Impossible
Proof. Since X and Y are independent of I we nd for the distribution of S = IX+ ( − I)Y , Let p ∈ ( , ) and denote F − S (p) by sp, In what follows, when considering α, β ∈ ( , ) we always assume that they satisfy qα + ( − q)β = p. Note that we de ne α * as and β * = p−qα * −q . The proof consists in verifying that sp can always be expressed as From Table 1, it is clear that (8) is proved. Let us now make the calculations case by case to prove Table 1.
In this case we always have that sp = F − X (F X (sp)). Hence, we only need to show that α ). It is also clear that for α < F X (sp) and thus β > F Y (sp), one has that F − X (α) < F − Y (β). Hence, as per de nition of α * , one has α (1c): F Y has a discontinuity at sp and for all z < sp, Case 2: F X is continuous at sp and there is a z < sp , However, for all α > F X (sp) and thus β < is excluded as it implies that F − S (p) < sp should hold (similar to the case (2b)) which is a contradiction with the de nition of sp.
Case 3: F X has a discontinuity at sp and for all z < sp, F X (z) < F X (s − p ) In this case, sp = F − X (F X (sp). This situation is merely identical to previous cases.
(3a): it is the same as (1c) by changing the role of X and Y.
(3b): it is the same as (2d) by changing the role of X and Y.
We also know that F S (s − p ) p F S (sp) and there are two possibilities: In the case when F S (s − p ) < p, then there exists α ∈ (F X (s − p ), F X (sp)) and β ∈ (F Y (s − p ), F Y (sp)) so that qα + ( − q)β = p and F − In the case when F S (s − p ) = p, then qF X (s − p )+( −q)F Y (s − p ) = p and one has that F − Case 4: F X has a discontinuity at sp and there exists z < sp, F X (z) = F X (s − p ) By changing the role of X and Y we have that the case (4a) corresponds to (1d), the case (4b) corresponds to (2d) and the case (4c) corresponds to (3d). Finally the case of (4d) is treated as follows. In the case (4d), both F X and F Y are discontinuous at sp, and there exists z and z such that F X (z ) = F X (sp) and F Y (z ) = F Y (sp) so that F X is constant on (z , sp) and F Y is constant on (z , sp). Then F − S (p) min(z , z ) < sp which contradicts the de nition of sp = F − S (p). This case is thus impossible.
It is clear that in many cases F − X (α * ) = F − Y (β * ). For example, by inspection of Table 1 we nd it is su cient for F X and F Y to be strictly increasing with unbounded support (Case (1a)).

Application: Bounds on Quantiles of Portfolios
Let X := (X , X , ..., X d ) be some random vector of interest having nite mean and de ned on an atomless probability space. In what follows we interpret X as a portfolio of risks that a nancial institution is exposed to. Its distribution is not fully known but complies with properties (i), (ii) and (iii) for a given F ⊂ R. Hence, S := d i= X i can be represented as a mixture of the type (4) and its risk assessment is intimately connected with the analysis of extreme dependence among Z i in (4).
A special role in this analysis is played by the comonotonic dependence, i.e, when all Z i are increasing in each other. For this particular dependence we denote Z i by Z c i . Formally, we write for some uniformly distributed random variable U that we take independent of I.
To assess the risk of S, it is standard to compute F − S (p) for some < p < that is typically close to (e.g., p = . as in Solvency III and Basel II regulation). In this context, a quantile is typically called a VaR. Precisely, we denote, by VaRp(S) the VaR of S at level p, In At rst, the role of the variables H i and L i may seem odd. However, note that the variables Z i that appear in the general mixture (4) can also be expressed as Z i = VaR U i (Z i ) for some uniformly distributed random variable U i . Clearly, the VaR of d i= VaR U i (Z i ) is bounded by its TVaR. Furthermore, TVaR is maximized in the case of a comonotonic dependence and VaR and TVaR are additive, we thus obtain that the VaR of d i= VaR U i (Z i ) is bounded by the VaR of the comonotonic sum d i= H i . When there is full uncertainty, i.e., when U = R d , then I = , and we recover the VaR bounds of the portfolio, as provided in Theorem 2.1 of the [2].
In general the bounds mp and Mp are not known in analytic form and their numerical evaluation is not straightforward to do. Speci cally, while it is easy to simulate possible realizations for (Z c , Z c , ..., Z c d ) the realizations for (L , L , ..., L d ) and (H , H , ..., H d ) do not follow immediately, which leads to nested simulations when computing mp and Mp . Indeed, the simulation of a single realization for L i (or for H i ) requires, for each simulated value u of the uniformly distributed variable U, a large number of draws from the variable Z i in order to estimate LTVaR and TVaR at the level U = u. In this respect, the formula for the quantile of a mixture is convenient, as it allows to develop an alternative formulation of the VaR bounds. This alternative formulation makes use of an auxiliary variable T, for the same uniform random variable U used in the de nition of Z c i in (10). Hence, T is a random variable independent of I with distribution F i X i |(X ,X ,...,X d )∈F (x). We formulate the following proposition. Proposition 3 (Alternative formulation of the VaR Bounds). Let (X , X , ..., X d ) be a random vector that satis es properties (i), (ii) and (iii), and let I, (Z c , Z c , ..., Z c d ) and T be de ned as in (3), (10) and (12). Recall that p F = P(I = ). De ne and α = min , p p F .Then, for p ∈ ( , ) , where β The expression for the lower bound mp is obtained by replacing, in the above statements,"TVaR" with "LTVaR".
Proof. The proof follows as a direct application of Theorem 1. Consider X = T with distribution F X , and It is clear that the cdf F Y of Y is continuous and strictly increasing on its support. First, let < F Y (Mp) < . By inspection of the table displayed in Theorem 1, we are in the situation of the cases (1a), (2a), (3a) and (4a). We observe that β * = F Y (Mp) and Thus, for all < β * < , or, equivalently, Proposition 3 may look more complicated than Proposition 2; however, it is now easier to use Monte Carlo simulations for estimating VaR bounds because nested simulations can be avoided. Remark 4 (Best-possible bounds). In nite dimensions, the inequalities in (11) are typically strict so that the bounds mp and Mp are not best-possible in general. Indeed, it not straightforward to nd a vector (Z , Z , ..., Z d ) with given marginal distributions as in (5) such that for a given < p < , I d i= X i + ( − I) d i= Z i and I d i= X i + ( − I) d i= H i have the same same p-quantile. This situation would be obtained when the vector (Z , Z , ..., Z d ) is such that its sum d i= Z i has a at quantile function on the appropriate interval (β * , ). The literature refers to this situation as "joint mixability" (for a homogeneous portfolio this concept is known as "complete mixability"), a concept that can essentially be traced back to a paper of [8] and has been extensively studied in a series of papers including [15], and [13,14]. In these papers it is shown, among other results, that in several theoretical cases of interest one can construct a dependence among the risks that lead to mixability. Remark 5 (High dimensions). A large class of distributions exhibits asymptotic mixability implying that in high-dimensional problems the bounds mp and Mp that are stated in Proposition 2 and Proposition 3 are expected to be approximately best-possible; see e.g., [14] and [12]. Remark 6 (Rearrangement Algorithm). [4] show that the Rearrangement Algorithm (RA) of [6] can be conditionally applied to obtain approximations of the best-possible VaR bounds. There is numerical evidence that these approximations typically yield results that do not di er a lot from the bounds we investigate here; see [2] and [3] for illustrations.
The following example illustrates Proposition 3 with a multivariate Student's t distribution as a benchmark model.

. Example (Multivariate Student's t distribution)
We consider a random vector X with standard Student's t distributed marginals that follows a multivariate standard Student's t distribution on a trusted area F; see also [4]. Speci cally, the density of X on F is given by Here, ν is the number of degrees of freedom and |R| is the determinant of the correlation matrix R satisfying R i,j = ρ (− /(d − ) < ρ < ) for all pairs (X i , X j ) with i ≠ j (homogeneous portfolio); see also [9]. When ν > , the covariance matrix Σ exists and is given by Σ = ν ν− R. As for the subset F we take the ellipsoid where c(p F ) is the appropriate cuto value³ corresponding to P(X ∈ F) = p F . We further consider a portfolio of d = risks and consider a multivariate Student's t distribution with ν = degrees of freedom. The VaR bounds reported in Table 2 were obtained within a few minutes, using 3,000,000 Monte Carlo simulations.⁴ We make the following observations. First, model risk is clearly present even when the dependence is "mostly" known (i.e., p F is large). Furthermore, the precise degree of model error depends highly on the level of the probability p that is used to assess VaRp. Let us consider the benchmark model with ρ = (the risks are uncorrelated and standard Student's t distributed) and p F = (no uncertainty). We nd that VaR % i= X i = . and, similarly, VaR . % i= X i = . , VaR . % i= X i = . . However, if p F = %, then p U = %, and the benchmark model might overestimate the %−VaR by (8.1-7.9)/7.9=2.5% 3 To determine c(p F ), one can use the fact that the scaled squared Mahalanobis distance d XR − X t follows a F−distribution with parameters d and ν (i.e., d XR − X t ∼ F(d, ν)). 4 When no information on the dependence is available (p F = %), the upper and lower bounds stated in Proposition 2 reduce to A = d i= TVaRp (Xi) and B = d i= LTVaRp (Xi) (see [2]) and can be computed exactly ( [10]). . ) ( . , ) ( . , ) ( -. , ) p= . % ρ = .
. ( . , . ) ( . , ) ( . , ) ( -. , ) or underestimate it by (9-8.1)/9 =10%. However, when using the . %−VaR, the degree of underestimation may rise to (56.6-14.2)/56.6=75%, whereas the degree of overestimation is equal only to (14.2-13.4)/13.4=6.0%. Hence, the risk of underestimation is sharply increasing in the probability level that is used to assess VaR. Finally, note that when very high probability levels are used in VaR calculations (p = . %; see the last three rows in Table 2), the constrained upper bounds are very close to the unconstrained upper bound, even when there is almost no uncertainty on the dependence (p F = %). The bounds computed by [6] are thus nearly the best possible bounds, even though it seems that the multivariate model is known at a very high con dence level as F nearly contains all R d . This implies that any e ort to t a multivariate model accurately will not reduce the model risk on the assessment of Value-at-Risk at very high con dence levels.

Final remarks
In this paper, we provide an explicit expression for the quantile of a mixture of two random variables and provide an application to nding VaR bounds of risky portfolios when only partial dependence information is available. We leave it to future research to extend the results to the general n−dimensional case.