Use Period-Matched Risk-Free Returns in an Ex Post Sharpe Ratio

Summary

The question asks whether excess returns for a Sharpe ratio should use the current risk-free rate or the rate that applied when each fund return was observed. The answer recommends matching each observation with the risk-free return for that same period. This makes the excess-return series reflect the historical return available from a low-risk alternative during each measurement interval, rather than applying a rate from a different date.

After calculating period-matched excess returns, the answer says to take their average and divide by their standard deviation. The risk-free return must also be expressed over the observation interval: for monthly returns, use a monthly rate rather than subtracting an annualized yield directly. The exchange is a concise calculation rule, not a full treatment of Sharpe-ratio conventions. It does not discuss choices such as annualization, compounding, or which risk-free instrument best fits a particular portfolio.

Key ideas

• For an ex post Sharpe ratio, match each fund return with the risk-free return from the same period.
• Convert quoted yields to the return interval used for the observations.
• Compute the ratio from the mean and standard deviation of the period-by-period excess returns.
• A current risk-free rate does not represent the rate applicable to historical observations.

Tags

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


# 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`?

## Answer by assylias (score 3)

https://quant.stackexchange.com/a/24881

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.