# Calculating Ex Post Sharp Ratio's for decile portfolios

Dear Stack community,

I hereby would like to ask what the correct calculation is for calculating Ex Post Sharp Ratio's. If I am correct, I already know that I am supposed to divide the average excess return by the standard deviation of the excess returns.

For my data I am using monthly return series, and I am using one month treasury bills as the risk free rate,

I am comparing decile portfolio's which have twelve month buy and hold returns (which have been calculated as);

Cumulative n-period return =(1+r1) * (1+r2) (1+r3) ... (1+rn) - 1

I am now wondering whether I should do the same for the risk free rate to get the cumulative twelve month risk free rate. And then take the average of this difference (excess return) for every portfolio. And divide this by the standard deviation of the total excess returns within each portfolio.

Or whether instead I should take the difference on a monthly basis and annualize this number. And then take the standard deviation of these monthly excess returns and annualize it. To then divide the first by the latter.

Is any of the two correct? Or do you have any other suggestions.

Kind regards, Julien Maas

• Hi Julien! The standard is to work with monthly returns. Sort stocks based on your predictor into decile returns and record the monthly returns (even if you only rebalance annually). You then get 10 time series of monthly returns (one for each portfolio). From each of the ten return series, subtract the risk-free rate. Then you have ten excess returns. Now, calculate the mean and standard deviation of those time series. Their ratio is the monthly Sharpe ratio. If you want to annualise it, multiply by $\sqrt{12}$. Dec 10, 2023 at 17:41

Based on this calculation by morningstar;

https://awgmain.morningstar.com/webhelp/glossary_definitions/mutual_fund/mfglossary_Sharpe_Ratio.html

It is possible to report the annualized Sharpe ratio, by first calculating the excess returns each month and the standard deviation of these excess returns to then calculate the Sharpe ratio for each month.

Then we can annualize this number to get the annualized Sharpe ratio which puts the number in a twelve month context.

In this case, because I want to report these for each decile portfolio I can take the monthly average excess return of the decile portfolio as the numerator and the monthly standard deviation of the excess returns of the decile portfolio as the denominator to calculate monthly Sharpe ratio's for each decile and then annualize these to get the annualized Sharpe ratio's for each decile.

• Can you pls include details from the article in the link you have provided so that your question can still be understood even if the link is broken or the article is changed? Nov 15, 2023 at 17:40
• Of course, the post should now include a screenshot of the calculation by Morningstar. In essence the calculation is based on calculating the Sharpe Ratio at the monthly level and then annualizing this figure. Something I think is popular in practice. My main question was whether I should use this approach if I am looking at twelve month cumulative returns calculated by the formula in the original post above. Or whether it makes more sense to use a cumulative risk free rate as well calculated in the same way as the twelve month returns and then calculate the Sharpe ratio on this basis. Nov 16, 2023 at 20:31
• I hope everything is clear, if it isn't please let me know what you would like me to clarify or add. Nov 16, 2023 at 20:35
• Also the question has another layer to it, which is basically asking if and how the two methods above should be applied to decile portfolio's. In my anwser above I speculate that I can simply take the average excess return of the portfolio and divide it by the average standard deviation of excess returns of the portfolio. Yet then I am still wondering whether it makes more sense to calculate the Sharpe ratio at the monthly level and then annualize it, or to calculate it based on twelve month cumulative returns and twelve month cumulative risk free rates. Nov 16, 2023 at 20:43
• In practice, I tried both methods, and the results I get are very different. Calculating the Sharpe ratio at the monthly level with subsequent annualization leads to a larger variation in Sharpe ratio's when compared to calculating the Sharpe ratio based on twelve month cumulative returns and the twelve month cumulative risk free rate. It also changes which decile portfolio's outperform or underperform the others when looking at solely the Sharpe Ratio. Nov 16, 2023 at 21:45