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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?

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    $\begingroup$ The risk free rate (1 year Tbills) is quite low nowadays, closer to 0.5% than to 5% and it probably will not go back to 5% for a long time, if ever. $\endgroup$ – Alex C Jul 30 '16 at 5:32
  • $\begingroup$ The Sharpe Ratio is usually expressed in annual terms, I have never seen it expressed in two-yearly or three-yearly terms. $\endgroup$ – Alex C Jul 30 '16 at 5:33
  • $\begingroup$ Thanks for the help. If you have say 16 months is it still the general practice to multiply it by the sqrt(12) to make it annualized? $\endgroup$ – shell Jul 30 '16 at 7:35
  • $\begingroup$ Yes: if you have 16 months you work with monthly values and at the end you multiply by sqrt(12) to annualize. $\endgroup$ – Alex C Jul 30 '16 at 13:13
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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.

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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.

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  • $\begingroup$ Then for daily returns should the overnight fed funds rate be used? And on a related note, what if said rate changed over the course of the period I'm looking at? $\endgroup$ – Andre Terra Sep 17 '17 at 22:41
  • $\begingroup$ @AndreTerra - The "risk free" rate should be as close to "risk free" as possible. Treasuries are not risk free (the US government certainly may default, at least in theory). The Fed funds rate is even less risk free - it's the rate at which I, super-strong bank, will lend to you, a super-strong bank - NOT to the US Treasury. But, I don't think there is any "more risk-free" daily rate; the Fed funds rate will have to do. Caveat (regarding 30-day T-Bills too): the rate must be expressed as a DAILY (resp. MONTHLY) rate; it is usually expressed in annualized terms. $\endgroup$ – mathguy Sep 18 '17 at 4:19
  • $\begingroup$ @AndreTerra - If the "said" rate changed over the course of the period, that doesn't change anything in the computations. The rate to use is the rate as stated at the beginning of the period. If I invest \$100 at a monthly rate of 0.25%, and if the instrument is indeed risk-free, I will receive \$100.25 at the end of one month, no matter what happens to monthly interest rates during the month. So, the answer to the question "what if said rate changed..." is, nothing. Nothing happens/changes; the rate changing during the period has no relevance to the calculation. $\endgroup$ – mathguy Sep 18 '17 at 4:22
  • $\begingroup$ I fully agree with both your statements. I'm well aware that the US can indeed default, but we have nothing closer to a true risk-free rate so we use the best available proxy for it. As for the changing interest rate, my only issue is that I often see folks applying a single nominal value for the interest across the entire period, regardless of how it has fluctuated. This may not be the case in the world of quants where everyone is being precise about calculations, but at any given MBA course everyone will just throw in a single number for the rate and call it a day. $\endgroup$ – Andre Terra Sep 18 '17 at 14:28
  • $\begingroup$ @AndreTerra - I understand the second question now. If we compute the SR from monthly returns over a five year period, the risk-free rate to subtract for the first month must be the T-Bill rate as it was at the beginning of that month. If at the beginning of the second month the T-Bill rate is different, subtract the NEW rate from the return for the second month. So the "risk-free rate" applied to each month may vary over time. That would be correct. You didn't mean "how does the calculation of the first month difference change" - that would not have been correct, but it wasn't your question. $\endgroup$ – mathguy Sep 18 '17 at 14:39
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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.

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