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I've been a member for many years, yet still have Foolish questions! Long story made short: In order to calculate the Sharpe Ratio, we need the arithmetic mean of the (usually, yearly) returns R (call it <R>). Also the risk-free return (either every year, or an average; subtract that from <R>, then divide the whole thing by the SD.

However, the standard calculation for any screen gives the CAGR and the GSD, which is fine. It also gives the SR, BUT it is not at all clear how that is found! (Unless a separate calculation of the ROI and the SD are made, and not shown). Could someone be kind enough to write down the mathematical expression for getting the SR when only the CAGR and GSD are calculated?

It would also be useful to know whether (and how) it is possible to obtain the arithmetic return (ROI) from the CAGR. I could expand the mathematical expressions for the logarithms, and obtain an approximation, but that seems like a lot of work!

First, it's incorrect to call the average of annual returns ROI (Return on Investment). It is no such thing. It is simply an average of annual returns, which means nothing in terms of your long term return on investment. Numerically, the arithmetic average and the CAGR are usually pretty close, but only one has real meaning - the CAGR.

As for SD vs. GSD. I think Jamie calculates the Sharpe Ratio using GSD and not SD. It's hard to argue with a renowned professor like William Sharpe, but IMHO using the ordinary SD of annual returns in his calculation is just as invalid as using the arithmetic average of returns. IMO the method used in the backtester to calculate the Sharpe ratio is just fine. It has a more solid statistical foundation, and there's no need to engage in any effort to precisely reconstruct Sharpe's formula.

If you do want to calculate the Sharpe ratio according to Sharpe, take the raw annual returns from the backtester and perform the calculation yourself. It's pretty straight forward.

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