Portfolio Optimization with CVaR

In times of great insecurity and turbulence on every major stock exchange, it is evident that controlling the risks in ones investment strategies is an important issue for the entire global economy. Perhaps there is no such thing as a golden rule on how to manage a portfolio, but history shows that focusing too much on the return is risky business. In the end of last decade, a risk-measure called Conditional Value-at-Risk (CVaR) was introduced to the market. It was the successor of a measure called Value-at-Risk (VaR), which caught the interest of the market, but has faced problems not being sub-additive, which is an important feature in the financial world. CVaR, does not lack this property, and has therefore been gaining ground in the recent years. The method has since then been used by insurance companies, mutual funds and other players in the financial market who have the need of evaluating their risks. This thesis is the implementation of a theory proposed in an article by (Krokhmal et al. [2001]) in the beginning of this decade. Rather than focusing on optimizing a portfolio with a fix rate of return, the approach of this report is instead an optimization with a given amount of risk tolerance. The formula for CVaR is in a linearized form used together with linear programming to create the best portfolio from available assets. The method itself is not dependent of any pricing models, but open to a very large variety of scenarios and assets. This has made it possible for us to create the scenarios from Monte Carlo-simulations and find the optimal portfolios using large-scale opti- mization. The linear constraints of the article are displayed with graphs of efficient frontiers and tables of the portfolio compositions, with explanations of their effects. A small case study is also performed with four different market scenarios to further shed light on the method’s functionality.