# Calculating Ex-Post Sharpe Ratio

I'm trying to calculate an Ex-Post Sharpe Ratio for my portfolio and I would appreciate verification I'm doing it correctly.

I have my portfolio's daily returns in one column and my benchmark's returns (the S&P) in another. I calculate the differential between the two (Daily_return - S&P_return). Next I find the arithmetic average of the differential, which in my case is 0.15% over 87 trading days. I annualize the average return by (avg_daily_return+1)^252-1, for a value of 46%.

I then calculate the sample standard deviation of the differential returns, for a value of 0.62%. I annualize the standard deviation by multiplying this number by the square root of 252, for a value of 9.95%.

Finally, I divide the annualized daily average return by the annualized daily standard deviation (46%/9.95%) for a Sharpe of 4.59. I believe I'm performing the calculation correctly, but this value seems unreasonably high.

Thanks!

• The 'usual' way would be to compute a 'daily' Sharpe, $0.15 / 0.62 \approx 0.24$, then annualize it by multiplying by $\sqrt{252}$ to arrive at $3.8$. I agree, this is suspiciously high. (Note that it corresponds to a $t$ statistic of 2.2 due to the relatively short sample period.) – steveo'america Jul 11 '18 at 17:21
• Thanks Steve. So the method you suggested is more conservative in that it doesn't fully account for compound returns, but does scale the returns and standard deviation by the same multiple? My understanding is that annualizing the returns is the easy part, but it's difficult to know what standard deviation goes with the annualized returns. I also appreciate the t-statistic-- it's true that the Sharpe is suspiciously high, but it is net of fees as well, and I have realized returns of 19% over those 87 days with a beta of about 0.75.. tough to say at this point if it's luck or I'm good at this. – Tyler Jul 12 '18 at 3:55
• How to deal with annualization is somewhat controversial. For example, Rivin argues for annualization of returns and volatility in the way you computed the Sharpe. I am somewhat ambivalent on this. Regarding the $t$ test, you are kind of performing a paired test against the benchmark. Tests of this form can demonstrate high power, which I agree gives a false sense of confidence. The problem is that the test is not robust against model misspecification. – steveo'america Jul 12 '18 at 19:30
• Thank you for the link, I'll check it out. So by my model misspecification, you mean selecting a benchmark that's inappropriate for the portfolio? If this is the case, can you provide some guidance on selecting a better-fitting benchmark? My rationale for choosing the S&P was that it is generally considered representative of the market as a whole and there is easily available data, but I invest in a mixture of low-, mid-, and high-cap stocks that wouldn't all fall within the S&P. – Tyler Jul 13 '18 at 21:03
• model misspecification here means that we model your returns as correlated to the benchmark returns, with no omitted variables. See section 4.3 of short sharpe course for an example of the power of the paired Sharpe test as correlation goes to 1. – steveo'america Jul 13 '18 at 21:30