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I have a strategy that buys a set of stocks and holds them for 6 months then rebalances. I would like to apply a rolling window backtest to the following strategy, but am wondering what is the appropriate way to combine the results from the different starting point. For example I would have a backtest that start on month 1 and rebalances every month 7 and 1. Then I will also have a backtest that starts on month 2 and rebalances every month 8 and 2. I would have 6 of these, start on month 1, 2, 3, 4, 5, 6. How should I combine their results?

E.g. should I calculate the CAGR for each backtest, then get the arithmetic mean?

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Let's take the example of your rolling window with a 6 months holding periods and monthly rebalancing. You can apply a similar methodology with different holding periods and rebalancing frequency.

You can generate the performance of your strategy at each month and annualize it. This is equivalent to running 6 portfolios with the same strategy with each contributing 1/6 to the performance (so there is a strong assumption that they all have equal weight at any given rebalancing periods, which is likely wrong).

You can then use each monthly performance and annualize or average them (I would go with annualize) to evaluate your strategy and compare it to other strategies. Wrong because of the assumptions but useful and feasible. Also keep in mind that a backtest has many other assumptions so they are just a tool.

One important point with this approach is that the volatility of that strategy will be vastly underestimated because of the auto-correlation of the returns and if you want to get a more realistic Sharpe ratio you would need to adjust the volatility for the auto-correlation of your monthly returns. Newey-West type of methods, or otherwise.

This effect will have the most impact if you start comparing shorter holding periods volatility with the longer ones. Don't get fooled that the longer holding periods have higher Sharpe.

Definitely a lot of assumptions but something that can be useful to evaluate a strategy on multiple horizons.

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