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# Sharpe Ratio Calculations - Using Daily Returns

1. Hi All,

Seeing if anyone is able to help me double check my Sharpe ratio calculations. For an example I've taken the daily returns of the S&P 500 Index (Data downloaded from Yahoo finance) from 1st February to 29th April (The time period i'm looking at).

For the risk free rate i'm using 0.16% which is the risk free rate IB use for their calculations.

- Since i'm doing daily returns I then divide this by 365 to get the daily risk free rate?

For the period the figures I get are;

Average excess return:0.1034%

Standard deviation:0.87%.

Dividing the average excess return by the standard deviation I then have an answer of 0.119

Since I'm using daily returns I then multiple this figure by the square root of 252( annual trading days) to get the final Sharpe ratio of 1.87.

Does this seem right? In most examples it says to just divide average excess return by the standard deviation and that's your answer.

The problem I have with this is that the Sharpe looks excessively low (in particular as the S&P performed very well during this time period). Interactive Brokers are also giving the Sharpe ratio for this time period at around 2.10 (relatively close to my figure).

Any help appreciated.

CG

2. Your figures are correct. But you have an incorrect opinion on what constitutes "low".

Bearing in mind that the long run Sharpe of the S&P 500 is about 0.20, then 1.87 is not "low".

GAT

3. Many thanks,

I agree 1.87 is quite a high Sharpe for the market, I wasn't very clear but I was referring to the 0.119 figure as being too low for this time period considering the strong gains in the S&P during this time.

Thanks again,
CG

4. there are 222 trading days not 252...

5. Dividing the 0.16% annual risk-free rate by 365 is not quite the right way. The correct conversion can be derived from here:
daily = (1+annual)^(1/365)-1 = (1+0.16)^(1/365)-1 = 0.000406713

Since you use 252 trading days for excess returns, you may as well use the corresponding risk-free rate adjusted for 252 days:
daily = (1+annual)^(1/252)-1 = (1+0.16)^(1/252)-1 = 0.000589142

daily = annual / 365 = 0.16 / 365 = 0.000438356

Since annual risk-free rate is so low, the resulting 3 answers above are not much different each other. In fact, for my purposes of calculating the Sharpe's ratio, I drop the risk-free rate from the equation altogether.

Other than that, you calcs appear to be correct. Verify with this:

Annualized Sharpe = sqrt(252) * [R / stdev(R)] = sqrt(252) * [(r-f) / stdev(r-f)]

where
r = daily asset (or index) return
f = daily risk-free return

If you drop the risk free rate, as I suggested, this simplifies to:
Annualized Sharpe = sqrt(252) * (r / stdev(r))

6. 252 is closer to being correct. 52 weeks*5 days in a week is 260. then subtract holidays. The NYSE lists 8 or 9 holidays. It depends on what day Christmas and New years land. 252 is correct.

7. To add to this topic, what is the norm for risk free calculation?

and they mention t-bill rates as a reference for risk free rate.

My confusion is that wouldn't this number change all the time? What is the norm for using this for backtesting a portfolio strategy? I use Amibroker and it has the option to set the risk free rate which is by default 5.

8. In his much referenced paper, our friend William Sharpe does not actually use a "risk-free" rate in his formula, but rather the "return on the benchmark portfolio", such as the S&P 500 index:
http://web.stanford.edu/~wfsharpe/art/sr/sr.htm

It's not clear to me how and when the industry decided to use the T-bill returns instead of the stock benchmark returns. I'd speculate this was done to artificially inflate their Sharpe's ratios. Since a large number of funds under-perform the S&P 500 index, their Sharpe's ratios would all be negative, if calculated as originally specified by William Sharpe!

10. Thanks for the article. Good read

11. This thread is really bringing interesting value.
I wish we had more like this.

12. One-year data may NOT be enough.

At least three or five years are needed to make some significant conclusion.

13. I use MarketXLS for this. It worked great for me.

14. any big difference in Std Dev of (R -- Rf) vs Std Dev of R alone? we are supposed to use the former? also is there a good measurement for S&P 500s current (2017 year end YTD) sharpe ratio?

15. I never understood the meaning of Sharpe ratio. If my results get better, the sharpe ratio gets worse????? The returns are hypothetical and just to show what I don't understand. More profit, no losing periods and still worse??? I would expect the opposite.

16. The Sharpe ratio penalizes the "fat tails", i.e. abnormally high and abnormally low returns. Under certain conditions, it may lead to what's known as a violation of the "first order stochastic dominance" rule. In laymen terms, it means that Sharpe ratio may lead to nonsensical results when evaluating performance.

Consider this example. Let's say we have two fund managers, A and B, with the following record of 10 monthly returns for each one:

A: {+1%, -1%,+1%, -1%,+1%, -1%,+1%, -1%,+1%, +6%}
B: {+1%, -1%,+1%, -1%,+1%, -1%,+1%, -1%,+1%, +20%}

Which one has better performance? Well, by all common sense, B is better than A, because in every single month fund B does either the same or better than fund A.

But Sharpe ratio of fund B is actually lower than that of fund A. This clear violation of common sense is well published and rightfully criticized as the idiosyncrasy of the Sharpe ratio.

17. In this era of ultra-low interest rates, the "risk-free" rate Rf is so low (virtually zero) that you can drop it from the equation.

There is another aspect to this. Sharpe ratio is supposed to be leverage-insensitive. But when Rf is high enough, it makes Sharpe ratio leverage-sensitive. For this reason, some people choose to drop Rf even when it's significantly different from zero.

18. Thanks, I am happy that my sharpe ratio is horribly bad!