Sharpe Ratio

Musing over the Sharpe Ratio, it is certainly an excellent tool.

But I think its main problem though, is for newer traders, trying for them to understand exactly what it means. I wonder how man experienced traders really understand exactly what it is measuring. Anyway, I was reading a publication this week, with an article called something like "Sharpe Ratio simplified". But even their simplifications still looks something like a calculus equation.

That is one reason for APD Ratio. Total Net Profits / Total Max DDs (after eliminating all "No Calc" rows). So easy, even a caveman can do itâ€¦

It would be nice if other measures could be simple. If newer traders cannot grasp Sortino, Sharpe or the many others, than how can they be expected to be confident in it, or to rely on it?

I agree. I think the main problem that they have is with the standard deviation; not with the mean in the numerator and not with the fact that it is a ratio.

If it is only for conceptual purposes that you can as well use any other risk measure in the denominator. However, there is some theory for the Sharpe ratio that does not necessarily generalize to other risk measures.

But remember how people reacted to your APD. Iâ€™m afraid that the problem is not in the exact formula, but in understanding the concept of â€śriskâ€ť.

A simple way to think about the Sharpe ratio is that a system with a Sharpe ratio > 2 has an expected profit greater than zero with 95% certainty. For a Sharpe ratio > 3 the expected profit is greater than zero with 99% certainty.

In some cases instead of "zero", a risk-free rate (e.g. T-bills) is used as the offset.

With these bounds you assume a normal distribution, I suppose, so this isnâ€™t exactly true. Nevertheless, Iâ€™m fine with that explanation. An even simpler desciption is that it is profit / risk and that high is good. But apparently such explanations arenâ€™t enough. Thatâ€™s why I think that the problem is with the proper understanding of â€śriskâ€ť and the consequences that it has for investing. After all, you could simply look to the equity curve and then you will also know whether it is smooth. So if this means anything to them they should still confine themselves to systems with a high Sharpe ratio.

There are some problems with looking to the charts though, namely that young systems have artificially smooth curves and that old DDs of old systems disappear in the linear view.

yes, itâ€™s an oversimplification. The problem I think with â€śprofit/riskâ€ť is that people might have problems with understanding risk in terms of a standard deviation. Iâ€™d guess when talking about risk they usually think about drawdown, the probability to lose a certain amount of capital etc. but not about the standard deviation.

So, they understand what it means if annual profits are â€śtwice the max drawdownâ€ť, or â€śthree times the average of the 5 worst days in a yearâ€ť, but not â€śtwo times the standard deviationâ€ť.

â€¦perhaps they should not be trading in the first place, but instead put their hard-earned money in a savings account?

If you assume a normal distribution, then perhaps a simpler explanation is this:

The Sharpe ratio corresponds to the mean return divided by the mean loss.

It is not the same as that ratio, but it is a strictly increasing function of it. I donâ€™t have the time to prove it rigorously now, but I just computed it for 100 randomly chosen positive means and standard deviations, and this makes clear that, under a normal distribution of returns, the mean return / mean loss depends on the Sharpe ratio only and is a strictly increasing function of it. Presumably somebody else has already proved this, because the relation is not really surprising if you understand normal distributions and Sharpe ratio.

So if I should devise a simple version of the Sharpe ratio, I suggest the mean return divided by the mean loss.

If the returns are normally distributed then this statistic is equivalent to the Sharpe ratio. In other cases it is (hopefully) easier to understand. Another advantage is that the expected loss is a so-called coherent risk measure.

A disadvantage is that, like the Sharpe ratio, it is based on one-period returns, so it does not capture the cumulation of losses in a drawdown. On the other hand, the sum of the losses is equal to the sum of the drawdowns (in \$value at least, not if you consider them as percentages). If the returns are normally distributed and independent of each other then I suspect that the expected drawdown is again a function of the expected loss, so alternatively you can - in theory - also generalize the Sharpe ratio to:

The mean return divided by the mean drawdown

One disadvantage of this is that it ignores the number of drawdowns, and a possibility to change that would be

The sum of the returns divided by the sum of the drawdowns.

but what do you mean by the "mean loss"? Is that the mean of all negative returns? What if a system has only positive returns (but still normally distributed)? Or only negative returns?

Yes, the mean of all negative returns. More exactly, the absolute value of that mean.

If the distribution is really (perfect) normal then there must be losses.

If the empirical distribution has no negative returns, then the ratio is undefined. And should be, because only these two cases can happen:

- Either the probability of a loss is really 0 and then you have riskfree investment (with perhaps some variation in the profits, but that isnâ€™t really â€śriskâ€ť)

- Or the true probability of a loss is larger than 0 but, since there are no losses in the sample, you have no idea how large a loss will be when it happens. So there are insufficient data to estimate the risk.

Note that we can make a difference between the theoretical definition and the way how you estimate it. In the theoretical definition I would use the expected loss. A simple practical estimate for that is the mean observed loss. But I can think of other estimates, e.g. based on extreme value theory. That would not change the concept though.

BTW, I said that the expected loss is a coherent risk measure, but I confused it with expected shortfall. I donâ€™t know if expected loss is coherent.

You can compute it from the advanced statistics (except that the riskfree return will be subtracted from it first; this means that you perceive a loss as something where you get less than the riskfree investment). Use the Statistics related to Sortino ratio.

The mean return is: Upside part of mean + Downside part of mean.

The mean loss is:

- Downside part of mean * (N nonnegative terms + N negative terms) / N negative terms.

(Actually both are multiplied by sqrt(n periods in year), but that doesnâ€™t matter for the ratio).

I suggest to use the daily values, full history. You can use the excess return rates or the log return rates, whatever you want. In this example I use the excess returns:

For Trend Plays #1 the values of today are

Upside part of mean 1.940

Downside part of mean -1.208

N nonnegative terms 136

N negative terms 209

So the mean return is 1.940 -1.208 = 0.732 and the mean loss is 1,208 * (136 + 209) / 209 = 1.994. Then the mean return / mean loss is 0.732 / 1.994 = 0.367.

If you make the same computation for extreme-os:

Upside part of mean 1.530

Downside part of mean -0.702

N nonnegative terms 326

N negative terms 472

So the mean return is 0.828 and the mean loss is 1.187, so their ratio is 0.698.

Now compare this with Sharpe ratios of the systems:

System, Sharpe ratio, mean return / mean loss

Trend Plays #1, 2.303, 0.367

Extreme-os, 3.878, 0.698

So what you see is that the system with the highest Sharpe ratio is also the system with the highest mean return / mean loss. This will not always be the case, but under a normal distribution that is presumably a mathematical necessity.

and this discussion somewhat illustrates the problem. When 2 smart guys are dissecting the simplification of it, then something is lost in newer traders trying to use it.

I prefer measures that I could explain to my grandmother.

I see what you mean and it all makes sense to me.

Frankly, I donâ€™t see why you think that ratio is more difficult than your APD. The mean return divided by the mean loss, is that so much harder to understand than your mean profit divided by mean drawdown? Actually it is the same as your APD applied to days instead of trades, so if your grandma canâ€™t understand this then she canâ€™t understand your APD either.

His grandma may not even understand him
Period She purposely tunes him out because of his obnoxious attitude. I surely donâ€™t blame her. Who would want to face that kind of attitude every day?

The problem is that RossÂ’ question wasnÂ’t very clear in the first place: did he want a simpler explanation of the Sharpe ratio, or a new measure, or both? ST and I tried both. If weÂ’re trying to find a simplified explanation of the Sharpe ratio then it must be a correct explanation and so we will have to discuss that. People who donÂ’t understand the Sharpe ratio wonÂ’t understand that discussion because it is about the Sharpe ratio and that is what they donÂ’t understand. Requiring that such a discussion doesnÂ’t take place is like saying that a math teacher shouldnÂ’t prepare his lesson with colleagues, or that his pupils should be able to understand all discussions in that preparation. I canÂ’t work that way. ST asked a question and I answered on his level. If grandma asks a question I will try to answer at her level. Right now that canÂ’t be an explanation like Â“go over the account values and do this and thatÂ” because these account values are not displayed at C2. I described how it can be computed from the advanced statistics, but these werenÂ’t designed for grandma, and that answer wasnÂ’t directed at her either.

Jules,

Definitely correct me if Iâ€™m wrong here.

For those that arenâ€™t willing or able to follow statistics such as ADP or Sharpe, all one needs to do is look at an equity curve.

Most statistics simply try to mathematically describe the equity curve.

That said, ADP was introduced because C2 doesnâ€™t show INTRADAY equity curves. In other words, a day-trading system could try to â€śmartingaleâ€ť itsâ€™ way out of a losing trade, without that ever showing. Therefore, ADP is valuable on C2 for day-trading systems, because it shows data that canâ€™t be seen on the displayed equity chart.

Iâ€™d say more, but thereâ€™s not much point. In short, if you donâ€™t understand the various measures (Sharpe, ADP, etc.) just look at the equity curve â€“ and avoid day-trading systems.

Dustin,

Basically I agree. But let me be more specific.

I agree with what you write about the Sharpe ratio. If you donâ€™t understand it, look at the equity curve. I think that is what I answered in the first place. The same is true for the â€śadvanced statisticsâ€ť. If you look into the guide that I wrote then you will see that I named it â€śstatistics program for equity curvesâ€ť. That says it all: Everything you see there is derived from the equity curve.

I use the Sharpe ratio mainly as sort criterion. You canâ€™t do that with charts. I also use it when I trade different systems to decide how much I put in each system.

Iâ€™m not sure what Ross had in mind when he introduced the APD, but what you write is how I view it.

The APD belongs to a different class of statistics, together with win%, P/L per unit, etc. You canâ€™t compute them from the equity curve. These statistics provide valuable information on top of the equity curve, because you will have to deal with slippage, volume, etc. In theory a proper â€śrealism correctedâ€ť equity curve would tell you enough about that too. In addition these statistics tell you something about trading style, averaging down, leverage, etc., which allows you to theorize about risks that are not yet shown in the curve. In theory you will see these risks realized in the equity curve if you wait long enough before you subscribe.

So in addition to your â€śjust look at the equity curve â€“ and avoid day-trading systems.â€ť I would say: consider only old systems and inform yourself about the slippage that subscribers get.

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I think the broader issue raised in this thread is if an easy to calculate and understand measure is sufficient for choosing between the different systems. E.g. a measure that Rossâ€™ grandma would understand (without making any further speculations about her IQ).

The problem I have with that - from a theoretical perspective - is that if that is indeed possible, weâ€™d all pick the best trading system without much effort and get rich quickly. Taking it one step further, we would quit our jobs, convince our bank that we found a way to make money so simple that even our grandma can do it, and borrow a ton of money to invest in the system highlighted by this easy-to-understand indicator.

It sounds too good to be true though. To me a much more plausible theory is that thereâ€™s no free lunch and no easy way to pick the â€śrightâ€ť system. Perhaps picking a system is purely a matter of luck, or perhaps there is indeed some skill involved on the side of the subscriber. If you believe the latter, I would expect that on average subscribers with better knowledge of trading, statistics and performance measures would be at an advantage compared to those who are lacking such knowledge. A logical conclusion would then be that we should educate those lacking the knowledge and skills and convince them to educate themselves rather than giving them an illusion that some easy to understand measure will pave the way to financial freedom.

"The problem I have with that - from a theoretical perspective - is that if that is indeed possible, weâ€™d all pick the best trading system without much effort and get rich quicklyâ€¦"

Itâ€™s very true that there is no free lunch in the trading world. Thus, even if you could pick the best system spot on and be right on direction all the time, you would end up with smaller and smaller profits, simply because slippage and spreads steal a portion of your profits as more people want in on the action. Itâ€™s almost a double whamy when you experience a losing trade as everyone else is exiting with you through that narrow door at the same time. Therefore, liquidity is a big key for a highly succesful system with many participants.

But all these things arenâ€™t captured by the Sharpe ratio either. The question (as I understand it) was not for a general rating that includes all facets, but only another version of the Sharpe.

I think that one reason why people donâ€™t understand the Sharpe is that they try to understand it the wrong way. With most statistics there is what I call a â€śstructuralâ€ť and a â€śfunctionalâ€ť way of understanding. Structural understanding means that you understand the formula. Functional understanding means that you know how to use the outcome.

If you use the Sharpe to select a trading system then it is enough that you understand it functionally. That is very simple: Select the system with the highest Sharpe.

If it is true what Ross says, that people donâ€™t use it because they donâ€™t understand it, then apparently these people donâ€™t want to believe the experts who say that you should use the Sharpe; they want to verify this themselves. But everybody who isnâ€™t mentally disabled can understand it if he puts enough time and effort in it. So the problem is that they donâ€™t understand it easily enough.

Itâ€™s okay if someone doesnâ€™t want to spend much time in understanding it, but then it is pretty stupid to say that you donâ€™t believe the experts who were boring enough to spend a lot of time in it. If you donâ€™t use the Sharpe ratio because you donâ€™t understand it, that is like saying that you wonâ€™t use the car because you donâ€™t understand the engine, and that you donâ€™t want to understand the engine because youâ€™re too lazy for that. Will it help if we replace the engine by a steam engine? Probably not.

This is why I said this in the first place: it is not the formula that is the problem, but the perception of risk. You can get a pretty good idea of the risk if you look at the equity curve.