Nonlinear Value-At-Risk

Abstract

Consider a portfolio consisting of quantities x = (x ₁, x ₂, ⋯, x _n )′ of assets 1,2,⋯, n with time t values v = (v ₁, v ₂, ⋯ v _n )′.² Then the change in the price of the portfolio, V, over the next interval Δt is given by

$$ \Delta v = \sum\limits_{t = 1}^n {x_t \Delta v_t ,} $$

((1.1))

where Δ, (Δ V) denotes the change in the value of asset (the portfolio) over the interval t to t + Δt. The value-at-risk of the portfolio {bdx} for some defined probability level α, is defined as the level of loss, ΔV(α), such that the probability that ΔV ≤ ΔV is equal to α. When the joint distribution of the change in asset values can be taken as multivariate normal with a known mean and variance, the calculation of Δ V is straightforward. In many cases, however, and particularly when some of the assets are options, the assumption of multivariate normality will be inappropriate, even when appropriate for the underlying rates and prices.

We are grateful for comments and suggestions from participants at the 1997 EFA meetings inVienna and at 3 conference, “Empirical Research in Finance,” at the London School of Economics and also to Bill Farebrotlicr, Steven Satchcll, Caapar de Vries, and particularly, to Simon Benninga (the editor), Zvi Weiner (the referee), and Davide Menini.