# Calculate excess returns for Sharpe Ratio with today's or past risk free rate of return?

I am struggling with the calculation of the Sharpe ratio. I am wondering whether to calculate the daily excess returns with today's risk free rate of return or the risk free rates corresponding to the date of the return observations?

E.g., I have the annualized fund returns r_fund and the risk free rate of return, which is the daily 3-months US treasury bill rates r_f. Today's risk free rate of return is 0.27% p.a.

date           r_fund      r_f      excess(r_fund-r_f)
2016-01-05     0.200       0.25     -0.050
2016-01-07     0.800       0.26      0.540
2016-01-08     0.900       0.24      0.660
...


Thus, should I calculate the excess return as shown in the table above or just subtract today's r_f (0.27%) from each r_fund?

• You should use the return on the risk-free rate and not just the rate itself. Then, use create the excess return by subtracting the risk-free return from the fund's return over the same period. (The rf return in the first period is thus (0.25-0.26)/0.25 or ln(0.26/0.25)
– Tim
Mar 14, 2016 at 22:51

## 1 Answer

You can refer to Sharpe's paper. If you are computing an ex post Sharpe Ratio, you should calculate the excess return for each period as the return of the fund over the risk free rate return over that same period. Note that if, for example, each period is a month, you need to calculate the monthly risk free rate (and not use the annualised yield).

You then calculate the average excess return divided by the standard deviation of the excess returns.

• Thanks! Do I use today's risk free rate return to calculate the excess returns for each period or do I use for each date the corresponding risk free return (as shown in my example in the rf column)? Mar 15, 2016 at 17:26
• Sharpe said you use the historical. But if T-bill rate was 0.25% on 2016-01-05, then on that one day you will earn only 0.25/360 or 0.00069% on your money. The fund on the other hand could easily be up or down 1% on that day. Make sure you have the units right. Mar 15, 2016 at 17:43
• So for example a million bucks invested in Tbills earned 6.94 dollars on that day, the fund made or lost thousands of dollars. Mar 15, 2016 at 17:58
• @jeffrey you use each day rf return Mar 15, 2016 at 18:16
• Is r_f marked at the day's close? If so, does that mean it should indicate the NEXT day return? For example, the 0.00069% is earned on Jan 06 rather than on the fifth, therefore we should be subtracting the fund's return on the 6th by the daily rate as quoted on the 5th right? Mar 16, 2016 at 13:48