# Sharpe Ratio understanding

Can someone help me with understanding Sharpe Ratio calculations. I thought I understood it, but the ratio on some of my systems just doesn’t make sense to me.

Here is my reasoning:

The formula for Sharpe Ratio is:

S(x) = ( rx - Rf ) / StdDev(x)

where

x is some investment

rx is the average annual rate of return of x

Rf is the best available rate of return of a “risk-free” return

StdDev(x) is the standard deviation of rx

StdDev(x) is always going to be a postive number, since this based on the Square of the distance from the mean. Then the only way the Sharpe Ratio can be negative is when ( rx - Rf ) is negative. With other words, when the sytem’s return is less than the return of a risk free instrument.

All of my system’s Sharpe Ratio’s seems to be lower than I think it should be (of course! )and some even have negative Sharpe Ratios. I am going to use CTS CLC for this discussion. It has a Sharpe Ratio of -0.492. This mean the system had to underperform (quite badly) a risk free return to get a negative Sharpe Ratio. The system is older than a year and annualized return is 16% with a 5% drawdown which I thought is a decent performance, especially since the system is designed to not trade frequently and to have stable returns year over year. If annualized 16% is underperforming a risk free investment, I would love to get my hands on that kind of risk free investment!

I can see if you calculate the Sharpe Ratio on weekly data for example, then systems which doesn’t trade actively will get low sharpe ratios as their return some weeks will be 0, which is less than risk free return. But by calculating the Sharpe on such short periods only, will put systems which doesn’t trade every week at a disadvantage. I think this way, there are many good systems slipping through the cracks and encourage active trading if vendors want to be noticed.

I am sure I am missing something. Can someone smarter than me (Jules?) please help me correct the flaws in my reasoning?

Regards

- Fanus

Thanks for the compliment, Fanus. I understand what you’re saying, but right now I’ve no idea how this can happen. I’ll look closer to it when I’ve time - unless someone else has already explained it by then.

Jules

In your case, quarterly returns, not weekly are being used, because the system is more than 1 year old (53 weeks.)

I see flat periods for several months at a time. That might be the reason for the negative Sharpe Ratio, as your system doesn’t trade actively and their return some quarters may be 0, which is less than the risk free return.

Fanus,

The true answer must be found in the way C2 computes it. Withouth this we can only guess, but then your explanation of weekly comparison is the simplest one. These are also some other points to consider (based on the first part of http://www.stanford.edu/~wfsharpe/art/sr/sr.htm):

1. Rf in your formula is not necessarily a ‘riskless’ return, it is more generally the return of some benchmark. We don’t know which benchmark C2 uses.

2. The formula for the ‘historic’ (ex post) Sharpe Ratio is not precisely as you describe it. You write that numerator should be the difference in average annual return, but Sharpe in the above link writes that it should be the average differential return across a series of time periods, and he explicitely recommends

"To maximize information content, it is usually desirable to measure risks and returns using fairly short (e.g. monthly) periods. For purposes of standardization it is then desirable to annualize the results."

So according to the first sentence it is good if C2 uses weeks, and the second sentence (if you also read the context in the rest of the link) says that the formula should then be adjusted (in the simplest case, with months, this would adjustment would mean that it is multiplied with SQRT (12).)

Note furthermore that in the case of C2 it is really unavoidable to use fairly short time periods. In order to compute the standard deviation, one should have a difference in return for each of many (at least two) time periods. If one would take 1 year as the length of a single time period, then most systems would not have a Sharpe Ratio at all, since they don’t live for more than 2 years. Obviously, if one uses months or weeks in the computation of the denominator (standard deviation), one should also use months or weeks in the numerator (average differential return).

3. However, point 2 alone is not sufficient to explain your problem. Suppose that your return in period i is x(i) and the benchmarks return in period i is b(i), then the numerator in your formula is

MEAN(x(i)) - MEAN(b(i))

while Sharpe writes

MEAN(x(i) - b(i)).

However, these two quantities are logically exactly equal, and moreover they are both equal to

{SUM(x(i)) - SUM(b(i))} / T

where T is the number of time periods. The numerator of this is just the difference in the total returns. Since T is a positive number, your Sharpe Ratio cannot be negative if the total return of your system is larger than the total return of the benchmark, irrespective of the length of the time periods.

This changes if compounding is taken into account. Suppose that the weekly percentage of return is used - which would be quite reasonable. Then this would imply that in the above formulas effectively weighted means are used. The weighted mean return can be negative while the unweighted mean (related to the total returns by the above formula) is positive. See also my answer to Hans earlier this week in the thread about ranking systems.

In summary, the fact that a slightly different formula should be used, combined with a correction for compounding, might explain your negative Sharpe Ratio. However, whether this is really the explanation remains to be seen - it depends on the formula that C2 uses. Since the return of your system doesn’t grow very fast it would be suprising if the correction for compounding (working with percentages) had such a huge effect. On the other hand, this also means that the numerator is already close to 0 and then a little change could perhaps make it negative.

Hope this helps a little - and perhaps some smarter people have much easier explanation!

Jules

It seems to me that the Sharpe calculation on C2 is off. I have calculated Sharpe ratios often enough to have a “feel” for it and it seems to me that all the C2 ratios are 1 below what the usual ratio calculation would result in. Maybe there is a little bug in the C2 formula.

Generally, a ratio of 1 means that you get the same return as with risk free investment when paralelled (adjusted) for volatility. Here on C2, it looks like that “break-even risk” point is set to 0 for some reason. I have seen several systems that clearly should be above 1 that were listed between 0 and 1.

Also, C2 should state on what time frame the sharpe ratio calculation is based. This is an important factor as ratios for different time frames differ significantly.

Nico Richter

"Generally, a ratio of 1 means that you get the same return as with risk free investment when paralelled (adjusted) for volatility."

Nicolai,

I would not consider myself an expert on Sharpe Ratios, but if I read the article of Sharpe in the aforementioned link then I conclude that a return that equals the return of a risk free investment should yield a Sharpe Ratio of 0, not 1.

Jules

PS to put it simply: It should be 0 because there is 0 reason to take the risk.

Fanus,

To put this long post a little easier: I presume that the negative Sharpe Ratio of your system can mean that someone who trades your system with a constant capital of \$100K (i.e. downsizing the trades after you made profit and upsizing after a drawdown) would in the end have a loss in comparison with a risk free investment. Again, I’m not sure of this.

Jules

Here on C2, it looks like that “break-even risk” point is set to 0 for some reason.

I would think that if we get the same return as with risk free investment, then the numerator would be 0, not 1.

>Also, C2 should state on what time frame the sharpe ratio calculation is based.

As I understand, initially C2 uses daily data, and when there are more data, it uses weekly, then monthly, then finally quarterly returns; so that when more data becomes available, the ratio in general tends to get better.

@Jules,

if you look at the way that the ratio is calculated, it can only become zero (when there are returns above the risk-free-return) if the Standard deviation is infinite. That seems not very probable for the data we have here, I would assume.

Maybe the equation given by Fanus initially (and that I also believe to be correct) is wrong somehow. But if it isn’t I have to stick to my view that something is out of line with the caluclations here.

@Anand, if the return were the same as the risk-free return, the Sharpe Ratio would of course be zero. For me (and most of us; I assume) that’s not the interesting part - the interesting part, I think, is: What with returns that are above the RfR but achieved with higher volatility? There, I think my heuristical approach should work.

(That is the case that I meant with “risk-adjusted”)

Waiting for more convincing evidence of being wrong Nico Richter

(Sorry to all for the confusion with my user names. The “Alex Richards” account is an old account of mine (not using my real name Nico Richter) that I set up a looong time ago. Long before it was clear if C2 would become the trustworthy platform that it has developed into today)

What with returns that are above the RfR but achieved with higher volatility?

If the returns are achieved with higher volatility, then the annualized StdDev of (Daily or weekly or monthly or quarterly) returns would be higher resulting in lower Sharpe Ratio, which by the way is the main drawback of the Sharpe Ratio in the sense that it penalizes both upside as well as the downside volatility.

Hence, in my view, the Sharpe Ratio should only be used after a method is evaluated in terms of its productivity (taking the Realism Factor into account.) In other words, when you find methods with similar expectancy (or Profit Factor, as these two are highly correlated), use the Sharpe Ratio to break a tie between them to see which method is more consistent in achieving those results.

In that sense, the Sharpe Ratio measures the integrity of a method, i.e., how loyal it is to the principles on which the method is based on. But it presupposes the existence of those principles, ie., Realism Factor and Productivity measured in terms of expectancy (or profit factor.) Otherwise the ratio would be meaningless, as one can be loyal to irrational principles or arbitrary notions; a mindless dictator faithfully executing his hatred of a particular group of people (as happened in world war II) is not an example of virtue.

Integrity means, loyalty not to a whim or a delusion, however strongly one feels they are true, but to one’s knowledge, to the conclusions one can prove logically to be practical and true.

Jules and everyone else who responded, thank you for the replies. I think I understand somewhat how the negative Sharpe Ratios can happen, but of course this doesn’t leave me any happier. Especially since Sharpe Ratio seems to be an important part in the ranking of the Best Systems Lists and I thought Grandma C2 would be happy with CLC and everyone with SnapBack. It does make me wonder though how many good systems might be hiding on C2, but never get highlighted and is very difficult to find, unless you ignore Sharpe Ratio.

Regards

- Fanus

Sorry, I meant to say Productivity measured in terms of Return of Equity, not Profit Factor or Expectancy as they measure the worth of a system.

ps: A method that returns 0.001% greater than the risk-free interest rate with zero drawdowns, and perfect consistency would have an infinite Sharpe Ratio, but I doubt many would want to invest in it, because even though the risk is very low, so are the returns, and real returns when adjusted by inflation would be negative (even the conservative Grandma would not be happy http://unicorn.us.com/trading/expectancy.html

My pet peeve with Sharpe is that it penalizes any big wins. It penalizes success. Do the math: A system that consistently returns \$1000/mo will have an excellent Sharpe. Huge, in fact. However, a system that has monthly returns of: \$1000, 1000, 1000, 1000, 20,000, 1000, 1000 will have a bad Sharpe. This is really dumb. Probably the most stupid stat on this website. There exists a better metric: Sortino It doesn’t penalize winners, just losers. I’ve mentioned this to Matthew, he seems a bit sympathetic, but I’ve not heard anything definitive from him. Why, why penalize success? Makes no sense. And, now with his new rankings, any long term trend following system has no chance, as the big winners that these systems occasionally have, will by definition, have bad Sharpe ratios, even if they don’t have significant drawdowns. Dumb, dumb, dumb. No, stupid! But that’s just my opinion. Ok, I’m usually not this vocal, adamant, about anything here, but this topic really sets me off. I suppose the fact that at one time me having a somewhat acceptable Sharpe, but then taking a \$14K profit in copper and seeing my Sharpe go in the toilet with this one transaction really bothered me. Would I have been better off not taking the trade? I’d have a better Sharpe ratio! I can’t imagine anybody saddled with the responsibility of running a business like this will take such an amateurish viewpoint. (Ok, Mr./Dr. Sharpe may have a Noble prize, but it couldn’t have been for this fatally flawed concept. Or else the Nobel judges didn’t finish kindergarden.)

FWIW, here is a Sortino definition pulled from google:

A variation of the Sharpe ratio which differentiates harmful volatility from volatility in general by replacing standard deviation with downside deviation in the denominator. Thus the Sortino Ratio is calculated by subtracting the risk free rate from the return of the portfolio and then dividing by the downside deviation. The Sortino ratio measures the return to “bad” volatility. This ratio allows investors to assess risk in a better manner than simply looking at excess returns to total volatility, since such a measure does not consider how often the price of the security rises as opposed to how often it falls. A large Sortino Ratio indicates a low risk of large losses occurring.

Nuff said!

OK, OK, I’ve settled down a bit now. Veins no longer popping out of forehead. Just wanted to apologize for the “stupid” word. Sorry.

But I think you can see how I could get fired up by seeing my stats ruined over a \$14K profit! And, yes, that one transaction made a big difference the day that I closed it out.

Sortino rules!!!

OK, this is really beating a dead horse, but I figured that I’d leave a thought here for those that are not necessarily mathematically inclined: It’s entirely possible that a trading system could have “0”, as in zero, drawdown, but a terrible Sharpe ratio!! Would you kick such a system out of bed? I didn’t think so…

Hans

A quick look at the Best Systems list confirm what you said about long term systems not standing much chance on C2. For Long Term Systems there are 3 listed, for Medium 7 and Fast there is 17! I doubt this is because Daytrade systems are so much better, especially since there are one with a 48% drawdown in the list. Unless proven wrong, I am sure this is because of the limitations of the Sharpe Ratio and the way it gets calculated on C2, that Longer Term systems get filtered out.

Off the topic, but another way where daytrading systems have the advantage is with the way equity charts are displayed. I am a full supporter that Open Drawdown need to be factored in and displayed, but for daytrading systems, their open drawdowns never appear on the equity charts since all their trades are closed each day when the charts are updated, where for Longer Term systems, the open drawdowns are displayed clearly. I suggested in the past that weekly and monthly equity charts in addition to daily be made available for long term systems, which will provide a clearer picture of the performance week over week and month over month.

I am a big fan of C2 and what MK is doing here, but I am afraid the way things are going on C2, that it will end up as a site featuring daytrading systems as this is just too difficult to get a long term system noticed.

Regards

- Fanus

Alex / Nicolai,

"if you look at the way that the ratio is calculated, it can only become zero (when there are returns above the risk-free-return) if the Standard deviation is infinite."

No, you’re wrong here. The standard deviation is only the denominator of the ratio. The ratio can also become zero if the numerator is zero. You explained this yourself to Pal, a few lines later, where you write that it can become zero. So I think you contradict yourself in this post.

Jules

Fanus, Pal, Hans, Nicolai,

I hope you understand that my postings where not meant as a defense of the Sharpe Ratio. I just tried to figure out how the negative outcome could happen, as Fanus asked. In general, I think that ‘risk’ is a multidimensional concept and that there is therefore no single unidimensional measure of it that will satisfy all.

Jules