question regarding sharpe ratio calculation

Hi All: I looked in the search icon for questions related to Sharpe Ratio but I couldn't find what I was looking for. My question is about how to calculate it. I will explain the scenario and then ask my 2 part question.

Suppose I have, during some month, M, daily returns $$X_i$$. So, roughly 20 returns. I want to calculate the sharpe ratio over the month. Suppose, for simplicity that all positions are long positions.

So, I can calculate the mean return, $$\bar{X}$$ of the 20 returns and the standard deviation of the 20 daily returns say $$\sigma$$. Then this allows me to calculate $$SR = \frac{\bar{X}}{\sigma}$$. ( not worrying about risk free rate ).

So, based on above, I have 2 questions.

A) Given what I've calculated, what does one call $$SR$$. Is it considered a daily sharpe ratio for that month ? or a monthly sharpe ratio ?

B) The 20 returns may have had different dollars allocated to them so they are not necessarily equally weighted. In other words, for one return, the trade may have had 20K allocated to it and for another return, the trade may have had 40K allocated to it. Does that matter when doing the sharpe ration calculation ?

EDIT: Adding a third question:

C) Assuming that what I calculated above is considered a daily sharpe ratio, then, if I did want the monthly sharpe ratio would I just multiply what I calculated by the square root of number of days in the month.

                                               Thanks.


2 Answers

The answers to your questions vary -- not because the answers are not precise but because Sharpe ratios are often used for marketing.

The Correct Estimator of a Sharpe Ratio

If we are trying to be correct (and not trying to fool ourselves), the expected Sharpe ratio $$S_P$$ for portfolio $$P$$ would be simply $$E(S_P) = E\left(\frac{r_P-r_f}{\sigma_P}\right)$$ for a risk-free rate $$r_f$$ (scaled for the time period). We often just plug in the estimated or average values yielding the realized estimator $$\hat{S}_P = \frac{\bar{r}_P-\bar{r}_f}{\hat\sigma_P}$$.

This is incorrect since, due to Jensen's Inequality, we should account for uncertainty (and skewness) in the estimate $$\hat\sigma_P$$. However, I'll leave that issue aside.

Annualizing a Sharpe Ratio

Sharpe ratios are like volatilities in that they are annualized unless stated otherwise. If you use daily returns, the $$S_{P,\text{daily}}$$ you calculate will be a daily Sharpe ratio and you would need to scale the mean daily returns and daily volatility for the number of trading days/year $$N$$ to get an annualized figure: $$\bar{r}_P\times N, \hat\sigma_P\times\sqrt{N}$$ or $$S_P=S_{P,\text{daily}}\sqrt{N}$$.

The number of returns used in calculating the mean does not matter for scaling; what matters for scaling is the period an average represents (daily, in your case).

To get a monthly Sharpe ratio, would would scale by the number of trading days per month; in your example, that would be $$S_{P,\text{monthly}}=S_{P,\text{daily}}\sqrt{20}$$. If you only have 7 days of data but could have traded the strategy for the entire month, then you should still take those 7 days as an estimator for the daily Sharpe ratio and get a monthly estimate by multiplying by $$\sqrt{20}$$.

Varying Dollar Amounts

If the 20 days had different dollar amounts, there are a number of options for computing the Sharpe ratio. The most correct approach would be to calculate the profits you made and express that as a return on the capital you used. Thus if you held some cash, you would (effectively) include that in the calculation. Slightly less correct is to ignore the cash and compute an internal rate of return -- and a weighted volatility. However, some people will also just cumulate the percentage returns as though no capital was infused or withdrawn during the period and compute the volatility on those unweighted returns.

This is where we start hitting the tension with marketing. The method used varies depending on which is most attractive (i.e. larger). Worse, marketing people may use the lowest of weighted or unweighted volatility with the highest of weighted or unweighted average returns.

Standard or Log-Returns?

That brings up another tension: though it is easier to compute a Sharpe ratio using log-returns, marketing people hate it because log-returns are slightly lower than standard returns. Standard returns (e.g. $$(p_{t+1}-p_t)/p_t$$) are preferred since they are slightly larger.

If you are computing a volatility, the returns averaged should be consistent with those used to compute the volatility. Guess what? Marketers do like volatilities computed from log-returns since those tend to be lower volatilities.

Other Skullduggery

We are not done with the truly reprehensible acts which may be committed in computing a Sharpe ratio. Some marketers will take the product of standard returns and then "average" by dividing the product by the number of days -- instead of doing the correct method of taking a geometric average. They may then divide that by a volatility computed from log-returns. Some will cherry-pick the time periods they get these figures from. They may also choose the minimal risk-free rate during the calculation period -- or the lower of a rate from the start or end of the period.

Finally, some strategies are "special situations" in that they exist for a short period of time and are not always open for investment. Examples include investing in distressed firms after bad news, earnings strategies, or index rebalances. If you cannot always hold a position in these strategies, some marketers will just scale the Sharpe ratio to an annualized figure -- even though the return and volatility you earn on an annual basis might be the same as that for the quarter in which the strategy was active.

If you are starting to feel unclean, remember this feeling every time you read a fund's claimed Sharpe ratio.

• Printing out and will go over carefully. Of course, if I have any questions, I'll holler. Thanks much for thorough and probably great answer. Aug 27, 2020 at 0:01
• That was quite useful and I think it's best for me to do it on capital. ( my capitals can vary pretty wildly ). So, I will calculate the average daily return based on the capital and the standard deviation of the daily capital. But, in that case, after I get some daily SR, I think it is still correct to multiply by the square root of 20, if I want to convert to monthly sharpe ? Thanks for confirmation and terrific answer. Aug 27, 2020 at 0:09
• Yes, multiplying by $\sqrt{20}$ would be the correct scaling to monthly. Also, you can fix the IRR method if you include the cash in the capital invested. However, if the capital invested changes because of the strategy, I would argue that is part of the strategy -- so just compute returns on your total capital base. Glad this helped! Aug 27, 2020 at 0:15
• It helped a lot. I was mostly worried about the capital issue which you explained nicely. Aug 27, 2020 at 0:24

OK, easy enough to answer with complicated formulae...

A- you've just given me a monthly Sharpe Ratio, calculated on daily returns. Or in fixed income jargon, a 21d1d Sharpe!

The obvious point being that you could give me a Sharpe Ratio based on rolling 3 day returns over the last 5 days. It might be completely meaningless to do so; but the ratio can be constructed on any horizons (plural) you wish, however sane or not.

Generally speaking, any 1m/3m/200d measurement will usually be made on daily observations. 12m will usually still be daily; but could be 52w. Once you get >5y, it's usually assumed by default you're looking at monthly observations. In between, can be a grey area that should be obvious, and disclosed; but inevitably not always is...

B- No conflict here. The Sharpe Ratio of the instrument you're tracking and the Sharpe of the portfolio you have that takes time-varying weights to that asset are two completely different things!

The SR of the asset(s) you're tracking won't change an iota given your exposure to them. The SR of your portfolio of those assets will absolutely depend on the timing of your weightings, as on the assets' cross-period metrics. But the distinction is like you saying that "bets on black" have a different payoff to "bets on black when I bet on black" ;-) That conditional is kinda important.

(C) Broadly speaking, yes. If you had say 25 daily returns with a standard deviation of 1%, then a monthly vol of 5%; a quarterly vol of 8%; or an annual vol of 16% would not be a totally unreasonable inference. [There being 21 trading days a year ~25; 65 a quarter ~64; or 261 a year ~256] This is a very standard assumption in most financial models on this kind of topic.

hope this helps.

• thanks. I will read carefully. not sure on why someone gave a -1. It looks useful at a glance to me. Aug 28, 2020 at 3:43
• Yeah, probably same troll downvoted the other answer (which I’m not putting down; I think we just at this from different angles)... Aug 29, 2020 at 1:21