Multivariate Hypernormal Densities

The recently proposed family of hypernormal density functions possess the analytically convenient and computationally efficient property of closed form moments and anti-derivatives in the univariate case. While this result allows many univariate applications to be solved faster and/or more accurately, multivariate results are not currently available. Such results would be useful in applications such as portfolio analytics or macro simulation involving several control variables. In this paper we define and analyze the n-variate hypernormal density, which converges to the n-variate normal, and we derive closed form expressions for its moments and integrals,. From these results it follows that computations involving n-variate normals can be supplanted with analogous and much more rapid computations using the hypernormal. Further, due to the closed form expressions for moments and integrals of the hypernormal density, the dimensionality of the problem does not affect the speed of computation in recursive applications. We first illustrate the usefulness of these results in the context of portfolio Value at Risk. Our method generates significantly more accurate confidence bands regarding the expected losses at any significance level than those produced by models relying on normal or t-distribution assumptions. Next we examine the computational implications for a comparative evaluation of a recursive macro model using standard numerical integration versus using the hypernormal. Here the gain in calculation speed is proportional to the number of recursions incurred. Multivariate hypernormal densities thus emerge as an appealing tool for a variety of multivariate financial and economic applications.