![]() ![]() This all makes perfect sense according to the economic theory, as bonds should provide the most defensive allocation. In the beginning, we primarily allocate to the US Large Cap equity, which then changes to the US Small Cap towards the more risky portfolios. As we increase the risk level, the equity allocation increases as well. We see that the most risk-averse portfolio consists primarily of bonds with a minor allocation to Small Cap stocks. Let’s now observe the portfolio allocation for each frontier point. ![]() In the beginning, we have a large increase in return when allowing for a bit more risk, while in the end we gain nearly no increase in expected returns for the same increase in risk. We can observe that our efficient frontier looks similar to what we would expect. I use the brilliant Python library PuLP to formulate a linear optimization model, and iteratively find the optimal portfolio for different risk aversions. To illustrate the application of CVaR in a portfolio setting, I download data from Yahoo on 5 ETFs, tracking four equity markets and one aggregated bond market respectively. This effectively leads to the maximization of the Sharpe ratio in the mean-variance setting, and the STAR ratio for CVaR. This is represented by the shape of the efficient frontier.Īs risk and return are not linearly dependant, it makes sense to consider the marginal increase in expected return when increasing the risk. Hence, increasing the riskiness of a portfolio will not nessecarily result in an equal increase in expected returns. There exist a quadratic relationship between risk and expected return. If we introduce a risk aversion coefficient, then the mean-CVaR portfolio optimisation model can be written: Similar to the mean-variance model, we can construct a portfolio, which maximizes the expected return for some level of risk (in this case, expressed using CVaR). This means that we can easily integrate it in a portfolio optimization framework. So, Conditional Value at Risk is a superior measure of risk and can be mathematically expressed as defined asĬonditional Value at Risk is not only convinient as it better identifies the tail risk than VaR, but it also holds desirable numerical properties such as linearity. Intuitively, this does not make any sense and breaks the reason for diversifying a portfolio. In some special cases for the VaR metric, this statement becomes violated and it becomes mathematically possible to obtain a risk reduction by dividing a portfolio into two sub-portfolios. Here, subadditivity means that a portfolio’s risk cannot exceed the combined risks of the individual positions. ![]() In addition, the VaR measurement fails to be risk-coherent as it lacks subadditivity and convexity. ![]() the nessesity to assume normally distributed returns. The introduction of CVaR is justified by many numerical problems of using VaR in practice, e.g. The measure is a natural extention of the Value at Risk (VaR) proposed in the Basel II Accord. The metric is computed as an average of the % worst case scenarios over some time horizon. This post is about how to use the Conditional Value at Risk measure in a portfolio optimization framework.Ĭonditional Value at Risk (CVaR) is a popular risk measure among professional investors used to quantify the extent of potential big losses. For example, if a portfolio value drops by 10% then we would need to regain 11.1% to neutralize this loss. Here, the mitigration of large losses is of paramount importance, as gains and losses are asymmetric by nature. Constructing a portfolio with high risk-adjusted returns is all about risk management. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |