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## Monday, 18 August 2014

### A Practioner's Approach to CVaR Portfolio Optimization

Conditional Value-At-Risk (CVaR) is defined as expected loss beyond a certain threshold. On a return distribution, CVaR is the expected return calculated over a certain area of the left tail. It is well known that CVaR has several properties which makes it a much more attractive risk measure than Value-At-Risk (VaR). Especially in the context of portfolio construction, it has been shown that CVaR optimization can be transformed to a linear optimization problem under restrictions. Many algorithms exist to solve such problems, even with thousands of assets.
This allows the construction of efficient frontiers in return/CVaR space. This addresses the old concern of practitioners and researchers that volatility might not be a suitable risk measure due to its property that both positive as well as negative surprises increase measured risk. A major advantage of volatility is that portfolio volatility can be calculated from asset exposures and asset risk with a rather simple formula...

\$\sigma_{p}^{2} = w \cdot \Sigma \cdot w' \$

With \$\sigma_{p} \$ as portfolio volatility, \$w \$ as asset weights and \$\Sigma \$ as the asset covariance matrix. Asset covariances can be further split into a correlation matrix \$\Omega \$ and a vector of asset weights \$\sigma\$.
A simple formula for calculating portfolio CVaR from asset tail risk information does not exist. But if we replace the vector of asset volatilities \$\sigma\$ with a vector of asset CVaR figures and process it with asset correlations \$\Omega \$, a matrix \$\Sigma^{*} \$ results which can be interpreted as an asset "coCVaR" matrix. This coCVaR matrix can be used to calculate an approximation for portfolio CVaR which performs surprisingly well in practical applications...

\$CVaR_{p}^{2} \approx w \cdot \Sigma^{*} \cdot w' \$

This approximation allows us to calculate return/CVaR efficient frontiers with standard mean variance optimizers or the critical line algorithm with a simple adjustment of our asset covariance matrix.
A simple two asset example illustrates the approach: below, we summarize the relevant risk and return characteristics for CGBI WGBI WORLD ALL MATS TR and JPM EMBI+ BRADY BROAD TR, both measured in USD and calculated from monthly data January 1991 to December 2012 (non-annualized)...

The higher moments of WGBI are close to zero, indicating that it can be approximated with a normal distribution. EMBI, on the other hand, exhibits significant tail risk beyond a normal distribution: negative skewness creates a long tail and positive excess kurtosis a fat tail. Together with the higher volatility, 99% CVaR of EMBI by far exceeds the corresponding WGBI value.
The correlation between this WGBI and EMBI data is 12.65%, which indicates that there exist potential diversification benefits between WGBI and EMBI in a portfolio context.
Below, we show the composition of the efficient portfolios on the mean/volatility frontier...

If we now calculate the mean/CVaR frontier based on the above approximate portfolio CVaR formula, we can compare the true and approximative CVaRs of the frontier portfolios...

As we expect, the approximation works very well. For investment purposes, it is interesting to compare the investment advice implied by volatility risk and tail risk as measured by CVaR. The impact of non-normal tail risk captured by CVaR can be seen by plotting the composition of the efficient portfolios on the mean/CVaR frontier against their volatility...

Taking into account explicitly the non-normal tail risk characteristics of emerging market debt, investment advice based on CVaR generates lower EM debt exposures despite the rather large diversification potential implied by low correlation values.