Figure 11: A correlation matrix between the selected assets
Finally, we will have a look at the correlation matrix between the selected assets to illustrate the correlations between them. This should not be confused with the variance- covariance matrix described earlier, as it is a separate entity. From the correlation matrix it can be noted that ILS seemingly has a very low correlation with all the assets selected, with correlation values being in the singles or low 10s, percentage wise. Compare this to, for example, equity, and you can see that the linear relationship between equity returns and other asset returns seem to be a lot higher in comparison. None of the other assets displayed similarly low correlation to all assets, although U.S. bonds had negative correlations to both equity and private equity. From this, we can deduce that ILS returns have a seemingly low correlation to the returns of other assets, perhaps suggesting that they indeed do offer diversification benefits as the returns of ILS products are not as strongly linked to market movements as the returns of other asset classes.
that could be created with the selected assets. As this is the global minimum variance portfolio, any portfolio that lies underneath this point in the frontier should not be considered.
As the ILS index selected for this study has low returns, it should come as no surprise that the efficient frontier only includes ILS up to a certain point. As we went up by 0.5 percentage points for each datapoint, the program stopped including ILS somewhere between 4.5% and 5.0% expected return. This comes as no surprise considering the low return profile of the asset, even if it has a low correlation with the other assets. However, a more important question arises: is ILS included in the maximum Sharpe ratio portfolio?
For calculating the Sharpe ratio (Sharpe index) that was introduced in formula 17, you need the portfolio return, portfolio standard deviation and a risk-free rate. For this specific thesis, the risk-free rate selected was the 10 Year U.S. Treasury Rate, which was 1.51%
on June 14, 2021. The ratio essentially depicts that for each unit of risk that the investor takes, they are getting an X amount of return in compensation. It has also been called the Reward-to-Volatility ratio, which may be a more accurate description of what the measure is trying to achieve. (Bodie, Kane and Marcus 2014, p.134)
Interestingly, the highest Sharpe ratio was found in a portfolio containing 13.1% ILS, 44.7% U.S. bonds, 2.5% private equity, and 39.7% hedge funds, with a Sharpe ratio of 1.075. This portfolio had the expected return of 4.2% and the portfolio standard deviation of 25.5%. In comparison, when not including ILS in the optimization, a portfolio of 51.6%
U.S. bonds, 3.1% private equity and 45.3% hedge funds had the highest Sharpe ratio, with it being around 1.059. The portfolio expected return and standard deviation for this portfolio were 4.5% and 28.31%, respectively.
Figure 12: Efficient frontier of portfolios of the seven selected assets
It seems that ILS does contribute to a possibility of a higher return-to-volatility, at least in theory. Certainly, ILS having a low correlation with other assets does at least seem to be an important factor in the mean-variance analysis, which emphasizes variance- covariance between assets. As mentioned before, the efficient frontier only includes ILS up to a certain point, which may be hard to see in the traditional efficient frontier shown in figure 12. Thus, in figure 13, points where ILS is included were graphed, with the last point of reference being the one where both ILS and no ILS frontiers combine as ILS is no longer included into the efficient portfolios.
Highest Sharpe Ratio
0 0,02 0,04 0,06 0,08 0,1 0,12
0 0,05 0,1 0,15 0,2
Expected return %
Standard deviation %
Efficient Frontier with ILS Eurekahedge ILS Advisers Index MSCI World Investable Market Index S&P International Corporate Bond Index S&P U.S. Aggregate Bond Index FTSE Nareit US Real Estate Indexes S&P Listed Private Equity Index Eurekahedge Hedge Fund Index
Figure 13: Efficient frontier of portfolios of the seven selected assets until 5% expected return
As can be seen from the figure above, the efficient frontier of portfolios including ILS seems to have lower standard deviations for the same expected return levels until around 4.5% expected return, where the weighting of ILS is quite low, only around 7.8%. Thus, one could assume that while ILS seem to offer benefits in terms of diversification and lowering the volatility of a portfolio, only investors that are willing to settle for moderate returns may be inclined to include them in their portfolios. However, for these investors, ILS may prove to be an extremely valuable asset in the long run, as the global minimum variance portfolio of the efficient frontier including ILS has the standard deviation of only 20.8%, while the one included in the efficient frontier without ILS has a portfolio standard deviation of 24.5%. The expected returns for both portfolios are 3.34% and 3.76%, respectively. This could be considered very significant for many investors, as ILS as an asset class seems to be very potent in reducing volatility without sacrificing much in terms of expected returns, at least in low return portfolios.
0 0,01 0,02 0,03 0,04 0,05 0,06
0 0,005 0,01 0,015 0,02 0,025 0,03 0,035 0,04
Expected return %
Standard deviation %
Efficient Frontier without ILS Efficient Frontier with ILS
As safety-first measures are recommended for portfolio optimization, measuring the VaR related to these two maximum Sharpe portfolios should be considered. For this portion of analysis, arithmetic returns are once again used. The following assumptions are made: the investment into each portfolio is USD 1 million, we are estimating the monthly VaR for each portfolio, and the confidence intervals used are 5% and 1%. Only assets that were included in the two maximum Sharpe portfolios were considered, and the exact weightings for each portfolio were used. The results are shown in figure 14.
Figure 14: One-month VaR for the maximum Sharpe portfolio in both the ILS and non-ILS efficient frontiers
These results are read as following: with an investment of $1 million USD, the ILS investor can with a 95% confidence expect that the next month the portfolio will not lose more than $8665.89 USD, and with a 99% confidence they can expect that the portfolio will not lose more than $13679.39 USD over the next month. In comparison, the no ILS investor can expect that there is a 95% confidence that their portfolio will not lose more than $9790.09 USD over the next month, and a 99% confidence that the portfolio will not lose more than $15351.98 USD over the next month.
The results suggest that the portfolio with ILS has lower risk than the one without ILS, at least from the perspective of VaR. Of course, as the two portfolios selected were the maximum Sharpe portfolios, the assumed level of risk is not the same either. The no ILS portfolio has a higher mean portfolio return and a higher portfolio volatility, as is to be expected. However, the change is not linear, as the Sharpe ratio suggests. These results suggest that the downside risk of a portfolio with ILS included seems to be smaller.
However, one must remember that unlike cVaR, VaR makes no distinction concerning the
losses beyond the confidence interval, meaning that it is possible that the absolute downside risk may still be higher with the ILS portfolio compared to the no ILS portfolio.
In addition, one must note that VaR assumes normally distributed returns, and like we established previously, the returns ILS and many of the other assets in the study do not seem to be normally distributed. Unfortunately, although methods for measuring the risk of assets with abnormal skewness and excess kurtosis have been developed, the scope of the study limits how deep of an analysis will be done on the assets and portfolios in question. One should be mindful of the fact that these models may, for example, underestimate the risks involved with each portfolio.
To elaborate on this, figure 15 demonstrates the MVaR (modified VaR) for just the ILS index. Though previous literature has suggested to only use the measure until the confidence level of 95.84%, I have calculated the MVaR until the confidence level of 0.10% to illustrate how drastically the risk measure changes at different confidence levels compared to the classic VaR when an asset has a high negative skew and a high excess kurtosis. MVaR is designed to reward for positive skewness and low excess kurtosis, so it comes as no surprise that the risk measure reacts negatively, but the results are quite shocking.
Figure 15: One-month VaR and MVaR for ILS
Clearly, the MVaR is extremely sensitive to the negative skewness and excess kurtosis displayed by the returns of the ILS index. The confidence interval of 4.16% has been suggested to be used as any confidence interval below that does not seem to be in line with investors’ preferences. Nevertheless, a takeaway from this should be that the variance-covariance method of portfolio optimization and calculating Value at Risk may
not give an accurate representation of the risks involved with assets that have non- normally distributed returns, especially ones that exhibit as abnormal returns as the ILS index in question.