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Empirical Risk Parity

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The PsyQuation’s research team doesn’t stop its risk management research and search for the best diversification methods. In order to better understand what is being discussed in this article, I recommend you read our first article if you have not done that yet. In that paper, we proved the effectiveness of the Risk Parity and showed its advantages over the equal-weight portfolio.

Let’s recall: Risk Parity diversification focuses on risk allocation, seeking to generate both higher and more consistent returns. Risk parity builds on an assumption that we have selected the best funds and we do not want to prefer one asset to the other just to equally split the risk between assets.

This time our goal was to implement risk parity algorithm. However, we didn’t calculate CVaR as a measure of portfolio risk by using the covariance matrix, but as an empirical value (we used only returns of each portfolio asset).

CVaR.

Conditional value at risk (CVaR), also called expected shortfall (ES), is a risk measure — a concept used in the field of financial risk measurement to evaluate the market risk or credit risk of a portfolio. The “expected shortfall at q% level” is the expected return on the portfolio in the worst q% of cases.

Empirical CVaR.

Calculate empirical CVaR is quite a simple task. To do this, we no longer have to calculate the covariance matrix as before. To do this, we don’t need to calculate the covariance matrix as before. We have returns of each portfolio asset for different moments of time. We need to multiply asset’s return by the weight of this asset, and then calculate the sum of these products for all portfolio items for each time t.

 
Portfolio return calculating.

Empirical CVaR equal to the mean value of 5%-th quantile of the resulting values set.

A similar problem is an empirical standard deviation of portfolio returns. The difference from empirical CVaR is that we don’t calculate the mean value of 5%-th quantile, but the standard deviation of the whole obtained set.

The optimization task is the same. Only y* looks different. Now R(y) is an empirical CVaR (or empirical STD).

The optimization program.

The solution of our problem, as in the previous article, can be obtained with the following scaling

Scaling of y*.

where y* is weights, obtained from minimizer.

So what?

In order to understand the fundamental difference between these approaches, we offer you a comparison of empirical CVaR, empirical STD and CVaR risk parity.

We investigated returns of the top 10 PsyQuation’s traders, calculated the weights for portfolio components using all three algorithms (empirical CVaR, empirical STD and CVaR risk parity), compared obtained weights, profits and Sharpe ratios.

  • The graph below shows the weights obtained using empirical CVaR (blue), empirical STD (yellow) and CVaR risk parity (green).

The weights obtained using empirical CVaR (blue), empirical STD (yellow) and CVaR risk parity (green).

We see that all algorithms give co-proportional weights to the same accounts. It’s noticeable, however, that non-empirical CVaR has assigned significantly larger weights for the last two accounts than two other algorithms. We suggest to see how it affected the profit of the portfolio.

  • We compared profits of the portfolio, obtained with different weights for a predetermined start amount (we have $1000000). The graph below shows the profit obtained with weights calculated using empirical CVaR (blue), empirical STD (yellow) and CVaR risk parity (green).

Profit obtained with weights calculated using empirical CVaR (blue), empirical STD (yellow) and CVaR risk parity (green)

We can see that empirical CVaR causes higher profits. At the same time, in periods when the portfolio gives losses, the reduction of performance is not as great as in two other cases. This is a positive consequence of the fact that the weight produced by empirical CVaR is more optimal than other weights.

  • Sharpe ratio is a way to examine the performance of an investment by adjusting for its risk. Generally, the greater the value of the Sharpe ratio, the more attractive the risk-adjusted return.

Sharpe ratio for portfolios created using empirical CVaR, empirical STD and CVaR risk parity.

The screenshot shows a calculated Sharpe ratio for returns produced by three different algorithms: empirical CVaR, empirical STD and CVaR risk parity. We see that in the first version of CVaR risk parity, it’s negative (this may be due to the fact that the selected accounts are the best on our criteria, but may be bad in combination, for example, some of them can be strong-correlated, which affects the covariance matrix and the resulting weights), but empirical algorithms yielded better result. The best result has empirical CVaR.

Conclusion

Thus, empirical algorithms yield significantly better results than the previous version of CVaR risk parity: empirical CVaR has the highest Sharp ratio, at the same time empirical algorithms yield more optimal weights. This is confirmed by the fact that the reduction in performance is less, therefore the resultant profit is greater, the losses are less. Also, a significant advantage of empirical algorithms is that for their work we use only returns, that is, we don’t perform additional calculations, such as the covariance matrix. Sometimes covariance had a big condition number, so it was a big problem in the first version of risk parity.

But this is another story, for another time, when we publish it take a look here.

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