Everybody heard already that one should not expect high returns without high risk, or one should not expect safety without low returns.The goal of portfolio theory is to find the balance between maximizing the return and minimizing the risk.To do so we have to first understand and measure the risk.Naturally a good risk measure has to satisfy several properties - in theory and in practise.Markowitz suggested to use the variance as a risk measure in portfolio theory.
This led to the so called mean-variance model - for which Markowitz received the Nobel Prize in 1990.The model has been criticized because it is well suited for elliptical distributions but it may lead to incorrect conclusions in the case of non-elliptical distributions.Since then many risk measures have been introduced, of sukin body lotion woolworths which the Value at Risk (VaR) is the most widely used in the recent years.Despite of the widespread use of the Value at Risk there are some fundamental problems with it.It does not satisfy the subadditivity property and it ignores the severity of losses in the footjoy weste herren far tail of the profit-and-loss (P&L) distribution.
Moreover, its non-convexity makes VaR impossible to use in optimization problems.To come over these issues the Expected Shortfall (ES) as a coherent risk measure was developed.Expected Shortfall is also called Conditional Value at Risk (CVaR).Compared to Value at Risk, ES is more sensitive to the tail behaviour of the P&L distribution function.In the first part of the paper I state the definition of these three risk measures.
In the second part I deal with my main question: What is happening if we replace the variance with the Expected Shortfall in the portfolio optimization process.Do we have different optimal portfolios as a solution? And thus, does the solution suggests to decide differently in the two cases? To answer to these questions I analyse seven Hungarian stock exchange companies.First I use the mean-variance portfolio optimization model, and then the mean-CVaR model.The results are shown in several charts and tables.