Portfolio Risk Measurement

Abstract

The main goal of this thesis is computing portfolio risks and finding the optimal portfolios. Four types of popular risk measures: variance (Var), semi variance (SVar), Value at Risk (VaR), and Average Value at Risk (AVaR), are reviewed and their computation process are preformed for portfolios consisting of single and multiple primary assets, as well as general portfolios consisting of primary and secondary assets. Finding the distribution for financial loss plays an important role here. In the general portfolio case, moment generating function (MGF) is computed first, then we apply Fourier inversion to obtain the cumulative distribution function (CDF) of the portfolio loss. Following Kevin Dowd's research, Cornish-Fisher approximation is applied to obtain VaR; Cornish-Fisher value at risk 2 (CFVaR2) is an approximation of VaR with one term and Cornish-Fisher value at risk 3 (CFVaR3) with two terms. Two numerical experiments are performed, with the aim to quantify risk in a portfolio of options, and to explore the effect of correlation on risk measurements.

We also find optimal portfolios in a fairly general setting. The Var optimal portfolio, and the CFVaR2 optimal portfolios are obtained by means of quadratic programming. A numerical experiment shows that the optimal CFVaR2 portfolio and the minimal Var portfolio are very similar due to the mean-variance type formula of CFVaR2. However, different optimal portfolios are obtained by minimizing CFVaR3.