# Modelling Multivariate Portfolio Risk With No Information About the Dependence Structure

Specialeforsvar: Marta Foldbo

Titel: Modelling Multivariate Portfolio Risk With No Information About the Dependence Structure

Abstract: The aim of this thesis is to find the best possible lower bound for multivariate portfolio risks with little to no information about the dependence structure. The main results are the standard bound and the dual bound. The standard bound is developed using the theory of copulas. By using copulas, one can find the joint distribution function, as a function of the marginal distributions. All copulas lie within the Frechet bounds, so by letting the joint distribution be equal to the lower Frechet bound, the distribution function of the portfolio risk is defined. This is called the standard bound. The distribution of the portfolio risk can also be described as a duality result. Assuming that the marginal distributions are identical, a lower bound is found for this duality result. This is called the dual bound. For two risk factors the dual bound and standard bound are the same.
For more than two risk factors the dual bound is an improvement of the standard bound. The better the lower bound of portfolio risk, the more accurate is the upper bound of the Value-at-Risk.
To illustrate the improvement of the standard bound, three numerical studies
were conducted, one using identically distributed data and two using relative returns of stock data. For all three examples the dual bound is an improvement of the standard bound, also for the non-identically distributed relative stock returns. We conclude that the best possible lower bound for the portfolio risk with no information on dependence has been improved, which leads to an improvement of the upper bound of the Value-at-Risk for the portfolio.

Vejleder: Jeffrey Collamore
Censor:   Camilla Schaldemose, TopDanmark