*Optimally Risk-Controlled Target Volatility Indexes*

The bespoke target volatility strategy aims to keep the risk profile of the strategy index at a predefined volatility level, which can be achieved by choosing the equity investment according to the response function (6). However, this strategy does not optimize the long-run Sharpe ratio of the strategy index.

The optimal risk-control strategy chooses the equity investment in such a way that the Sharpe ratio of the resulting index is maximized under the condition that the expected volatility of the index remains at a predefined target *T*.

As we show in Appendix 5 online, the optimal target volatility index uses an equity investment that is very similar to the target volatility strategy (6), except that the equity investment is inversely proportional to the variance *? ^{2}* of ?, i.e.,

with a proportionality factor given in Appendix 5 online. It is important to understand the economic difference between the target volatility strategy (6) and the optimal target volatility strategy (8): The target volatility strategy (6) is designed to keep the volatility of the investment scheme at the target level *T* at all times. However, the optimized response function (8) only targets the volatility level* T *on average and shows a stronger response to changes in volatility in the sense that in equity markets with relatively low levels of volatility, the equity investment is geared up such that investment portfolio is more volatile than average. On the other hand, when equity volatility is relatively high, the equity investment is reduced such that the investment portfolio is less volatile than average. To conclude, instead of keeping the volatility of the index portfolio at a constant level, the optimal risk strategy invests countercyclically: It takes more risk when equity markets are in a regime of low volatility but takes less risk when markets are at an above-average level of volatility. This countercyclical investment strategy achieves a better long-run Sharpe ratio than the pure target volatility strategy (6).

*Numerical Results*

To test the performance of the optimized risk-control mechanism presented above versus the nonoptimized methodologies, we use the Euro Stoxx 50 Index and the EONIA money market rate as portfolio components.

As a volatility measure, we will use a historical standard deviation of the returns of the past 60 trading days.

For estimating the return of the underlying equity index *µ* needed as input into (7), we use *µ *= the lower of either the annualized life-to-date return or the annualized return of the past 120 trading days of the index. The reason is that in the very long run, we expect *µ* to be (close to) constant. However, in the medium term, it is important to detect possible bear markets in the response function (7) to avoid situations where overly optimistic estimates of the underlying returns enter the investment scheme in the middle of a bear market.