Reducing model risk via positive and negative dependence assumptions

We give analytical bounds on the Value-at-Risk and on convex risk measures for a portfolio of random variables with fixed marginal distributions under an additional positive dependence structure. We show that assuming positive dependence information in our model leads to reduced dependence uncertainty spreads compared to the casewhere onlymarginals information is known. Inmore detail, we show that in our model the assumption of a positive dependence structure improves the best-possible lower estimate of a risk measure, while leaving unchanged its worst-possible upper risk bounds. In a similar way, we derive for convex risk measures that the assumption of a negative dependence structure leads to improved upper bounds for the risk while it does not help to increase the lower risk bounds in an essential way. As a result we find that additional assumptions on the dependence structure may result in essentially improved risk bounds. © 2014 Elsevier B.V. All rights reserved. 1. Preliminaries and motivation The problemof assessing themodel risk associatedwith the risk measurement of a high dimensional portfolio has recently gathered a lot of interest in the actuarial and financial literature. To set a mathematical framework, we assume that a financial institution holds a d-dimensional risk portfolio over a fixed time period. This risk portfolio is represented by a random vector X = (X1, . . . , Xd) on a standard atomless probability space (Ω, F , P). The total loss exposure associated with X is given by the sum X d = X1 + · · · + Xd. Using a risk measure ρ, the aggregate random position X d is mapped into the real value ρ(X d ), to be interpreted as the regulatory capital to be reserved in order to be able to safely hold X . In this paper, we mainly deal with the case where ρ is a convex risk measure or the case where ρ is the Value-at-Risk (VaR). The evaluation of ρ(X d ) is mainly a numerical issue once the joint ∗ Corresponding author. E-mail addresses: [email protected] (V. Bignozzi), [email protected] (G. Puccetti), [email protected] (L. Rüschendorf). distribution of X has been chosen or statistically evaluated. Estimating a multivariate distribution is a challenging task which is usually performed in two steps: first, d individual models Fj for the marginal loss exposures Xj are independently developed. Then, the marginal distributions are merged into a joint distribution using a dependence structure. In fact, banks/insurance companies typically have better methods/more data for estimating a one-dimensional distribution for each risk type Xj than they have to estimate the overall dependence structure of X . It is therefore reasonable to assume that the marginal distributions F1, . . . , Fd are known,while FX , the joint distribution of X , varies in Fd(F1, . . . , Fd), the so-called Fréchet class of all possible joint distributions having the fixed marginal models F1, . . . , Fd. The choice of a single distribution in Fd(F1, . . . , Fd) can lead to themiscalculation of the reserve ρ(X d ). The impliedmodel risk is referred to as dependence uncertainty. A natural way tomeasure dependence uncertainty and, inmore generality, model risk consists in finding the minimum and maximumpossible values of the riskmeasure ρ evaluated over the class of candidate models; this is the approach taken in Cont (2006). In our framework, we define the smallest and biggest capitals to be held coherently with the given marginal distributions as ρ(X d ) = inf  ρ(X d ); FX ∈ Fd(F1, . . . , Fd)  , (1.1) http://dx.doi.org/10.1016/j.insmatheco.2014.11.004 0167-6687/© 2014 Elsevier B.V. All rights reserved. 18 V. Bignozzi et al. / Insurance: Mathematics and Economics 61 (2015) 17–26