Choosing Risk-Free Returns and Annualizing the Sharpe Ratio
Summary
The discussion covers two linked choices when calculating a Sharpe ratio from monthly returns: selecting a risk-free return that matches the measurement horizon and converting the ratio to an annual scale. One answer suggests using a short-term Treasury rate for monthly excess returns, while another emphasizes matching the risk-free instrument’s horizon to the return interval. The thread also cautions that a one-year bill is not risk-free over a one-month holding period.
For annualization, the commonly used square-root-of-time rule multiplies a monthly Sharpe ratio by the square root of twelve; longer samples use the square root of the number of months if expressed as a multi-year scale. The replies disagree about the usefulness and interpretation of annualized Sharpe ratios, and note that the scaling relies on assumptions about how returns and volatility grow over time. Results can differ when computed directly from annual returns, so frequency and horizon choices should be stated clearly.
Key ideas
- The risk-free return used for monthly excess returns should reflect a comparable short-term horizon.
- The conventional annualization factor for monthly observations is the square root of twelve.
- The square-root scaling assumes returns accumulate linearly with time and volatility scales with its square root.
- Annualized Sharpe ratios can differ from ratios computed directly using annual returns.
- Comparisons require consistent return frequencies and evaluation periods.
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Full text
# What value should the risk free monthly return rate be (Sharpe ratio calculation)?
# What value should the risk free monthly return rate be (Sharpe ratio calculation)?
In calculating an annualized Sharpe ratio using monthly returns, what is commonly used as the value for the risk free rate? I am using this formula:
```
excess return = monthly returns - risk free rate
Sharpe ratio = (average(excess returns) / std(excess returns)) * sqrt(12)
```
Multiplying by the sqrt(12) in order to make the result annual.
My understanding is that a common yearly risk free rate is roughly equal to 5%, is this true? Would the monthly risk free rate then be equal to 5% / 12 or .4167%?
Secondary question, if you are dealing with more than one year of monthly returns, such as 2 or 3 years, would you still multiply by the sqrt(12), or would it be for:
```
2 years = multiply by sqrt(24)
3 years = multiply by sqrt(36)
```
and so on?
## Answer by David C (score 2, accepted)
https://quant.stackexchange.com/a/28414
The Daily Treasury Yield Curve Rates are a commonly used metric for the "risk-free" rate of return. Currently, the 1-month risk-free rate is 0.19%, and the 1-year risk-free rate is 0.50%.
Annualizing your Sharpe ratios depends on the time unit you are using to calculate your returns. You simply multiply your calculated Sharpe ratio by the following (unit-less) factor:
$$\sqrt{\frac{1\ year}{1\ time\ unit}}$$
So in this case, if you are using monthly returns, you multiply by $\sqrt{12}$.
For 2 and 3 year normalization, you'd want to substitute the numerator "1 year" with "2 years" and "3 years". As you guessed, this means you will want to multiply by $\sqrt{24}$ and $\sqrt{36}$, respectively.
## Answer by user20429 (score 1)
https://quant.stackexchange.com/a/28410
First, it never made sense to me to "annualize" the Sharpe ratio if the input data is monthly. But yes, if you want to annualize you multiply by sqrt(12), and for three years you should multiply by sqrt(36).
To explain my view: The Sharpe ratio really has no meaning by itself. It only has meaning in comparisons (this portfolio vs. the other), and only if it over the same period. (The annual Sharpe ratio of a portfolio over 1971-1980 compared to the annual Sharpe ratio of the same portfolio over 2001-2010 makes no sense whatsoever.) In these comparisons, what's the benefit of multiplying everything by sqrt(12), or sqrt(36)? (Hint: NONE!) What is always important to state is that the inputs were monthly returns (as opposed to annual returns, etc.)
And, to be clear: If you compute a Sharpe ratio from ANNUAL returns of the same portfolio(s), you may get results VERY DIFFERENT from taking the SR computed from monthly returns and multiplying by sqrt(12). I hope the idea of "annualizing" is not based on a misunderstanding of this trivial fact!
Now to your actual question: If the input data is monthly (monthly returns), then the risk-free rate should also be on instruments that are risk-free at the one-month horizon - 30 day T-Bills most likely. A one-year T-Bill is not risk-free over one-month horizons.
## Answer by Ami44 (score 1)
https://quant.stackexchange.com/a/29528
The squareroot rule stems from the assumption, that the returns scale linear in time and the standard deviation scale with the squareroot of time. That means for the return of 1 year, that it should equal 12 times the monthly return (at least in some average sense). That assumption is certainly not fullfilled, if you use different risk free rates for yearly and monthly returns. In order for the annualized sharpe ratio to be independent from the fact that it is calculated from monthly, yearly or any other frequency of returns you have to use the yearly risk free rate. Another possibility seems to be to use the risk free rate that matches the maturity of your assets, whatever that exactly means.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.