### How high of a Sharpe ratio is implausibly high for a low-frequency equity strategy?

• I am looking to convince someone that an annualized Sharpe Ratio of 7 is 'extremely high' for a low frequency (daily rebalancing, say) long-short technical strategy on U.S. equities. I was hoping for a published source (preferably a journal article or conference paper) that either

1. Provides a scale for interpreting Sharpe (e.g. "> 1 is good, > 2 is excellent, ... "), based on observed Sharpe ratios of, say, active managers, or some such. (I can imagine this being rejected as "biased" or "underinformed")
2. Preferrably, presents statistics on achieved Sharpe ratios of Hedge Funds and other active managers, perhaps by strategy class, with long-ish histories, even with some back-fill bias, that would allow one to estimate what quantile a given Sharpe ratio would fall at. (e.g. "the cutoff for top 1% of Convertible Arb. funds achieved Sharpe is 1.5" (I am making that up.))

edit: I reiterate that I have no doubts the number is bogus, but am trying to convince someone else, someone without much market experience, that this is way outside of normal.

7 is too high to be true for a low frequency strategy. I do not believe it and nor should your investors. Is this off the back of back tests or is this your risk adjusted performance of real trades, trades you put on in a fully funded (not simulated) trading account?

@Freddy I don't believe it either, and find it absurd. The number is from a backtest performed by a third party. My job is to convince someone that this figure is suspiciously high. If I just tell them I think it is too high, it is my word against someone else's. This is why I am looking for a published account that, presumably, has been vetted and is representative of achieved performance.

Well, that it comes from a "backtest" tells it all. Ask them to send you the Sharpe ratio of audited real trading track records. You will see that your 7 Sharpe ratios will compress to possibly 1+ or 2.x (if that at all, someone stating 7 sharpes for low frequency strats lost every and all credibility)

The tag "harpe-ratio" misses an "s" ... I don't have the privilege to edit the tag.

I remember one of my mentors years ago was trying to explain to a junior colleague why a high Sharpe ratio in a particular low-frequency backtest he had run was unbelievable. He said, "if this were true, we'd put all of our money into this strategy." Then he pointed to the converts desk and said, "And we'd put all of *their* money into this strategy." Also worth noting: Peter Muller, who used to run PDT at Morgan Stanley, has said that a realized Sharpe ratio of 2 should be considered amazing.

is it 7 without transaction costs?

Another possible mistake besides transaction costs: computing return volatilities on portfolio notionals, but returns on portfolio capital.

I have seen Sharpe ratios like these and yes, it is extremely likely to be utter BS. You have some good answers so I will just add my 2 cents. It will be far more valuable for you and everyone else involved if you prove WHY the Sharpe ratio is so wrong instead of just claiming it is wrong. It shouldn't be hard to find. Start looking at these likely culprits; Data mining issues, no transaction costs, slippage, survivorship bias, extreme curve fitting, etc... With a small amount of effort, you should be able to find some serious data sins that help explain the results.

Just curious. Can you quantify what "low frequency" is? For example, if I made one trade a month that increased my portfolio by 1% each month, is that low frequency? What about portfolio turnover and portfolio holding time? If that 1% came from a different stock each month, is that less plausible to you? Just asking the infinitesimal case so I understand your question.

• Here are couple references. Especially the first link to Andy Lo's paper contains a list of Sharpe ratios of popular mutual and hedge funds:

The Statistics of Sharpe Ratios

Dow Jones Credit Suisse Hedge Fund Index

Generalized Sharpe Ratios and Portfolio Performance Evaluation

I would go with the first paper.

None of these three links are currently working. Can you update or provide some summary statistics?

@Jared, updated, though a google search would have immediately brought up the results, too.

• The answer your are looking for might be the story in "Benchmarking Measures of Investment Performance with Perfect-Foresight and Bankrupt Asset Allocation Strategies", by Grauer (Journal of Portfolio Management).

While this work main concerns are the differential ranking of various performance measures and with negative betas for market timing strategies, its analysis of perfect foresight allocation is relevant to the point you want to make.

The punch line is that even perfect foresight strategies that grow an investment more than trillion-fold over ~60 years have a sharpe ratio that is barely in excess of 1.

The table below describes summarily the low frequency strategies considered (I believe monthly, but it might be quarterly) and reports the wealth accumulated from 1934 to 1999 assuming an initial investment of 1 dollar.

Some selected performance measures for this strategies are in the next table:

The "Industry No Margin" perfect foresight strategy multiplies the initial investment by a factor of $\mathbf{1.4x10^{14}}$ over 65 years, yet it achieves a Sharp ratio of 1.14.

These observations don't settle the question, but they should instill enough doubts about any claim of a 7+ sharpe ratio for a low frequency strategies.

This is one thing that burns my butter: Sharpe ratios published without units! I cannot tell from Exhibit 2 shown above whether the SRs are monthly, quarterly, or annualized. It matters! (Although in this case, not terribly).

• I would even stick to the original paper by Sharpe (1966):

Mutual Fund Performance. The Journal of Business Vol. 39, No. 1, Part 2 pp.119--138

If you look at the numbers on Page 6 you can see that the funds sharpe ratios roughly are between $0$ and $1$.

Since the Sharpe ratio already adjusts for the risk-free rate, you cannot really argue about its change. And if you do, you have to take into account that markets have become more efficient since 1966 (computers) so one would suspect the Sharpe ratio to have a tendency to be lower.

If you know facts about the calculation methodology of the backtest (which timeseries are involved) you could also look for signs of bias (look-ahead?) or to re-calculate the strategy for yourself.

Totally agree to your answer. Everything above 1 is questionable - especially in a back test.

@Richard, this is not what Vanguard said, and you are quite incorrect, generally Sharpe ratios in back tests are better than Sharpe ratios measured on real returns.

@Richard, I quite disagree with your post, in fact empirical evidence points to the fact that you are incorrect in your assertion. Sharpe ratios have gone up over time because hedge funds and mutual funds alike have moved to new asset classes which added diversification effects and thus improved risk adjusted returns over time. Also short selling and hedging skills have improved over time, adding value as well. If anything then Sharpe ratios have somewhat increased over time (please take a look at my referenced papers in my own answer).

@Freddy, I will have a look at your references. Maybe my post was unclear. What I want to say: I often see high Sharpe ratios in back tests but the Sharpe ratio when a strategy goes live is most of the time much lower. Some back tests are misleading. E.g. it depends on the assumption of which prices to take. I saw back tests on option stategies where prices between entry and maturity were interpolated. This can reduce volatility estimates. As a summary: when somebody shows me a Sharpe ratio $> 1$ on a back test then I usually have a lof of questions. That's why I agree to vanguard2k.

@Freddy, furthermore the first paper that you cite shows that volatiltiy estimates can be too low if autocorrelations are not taken into account. This can also (wrongly) drive up Sharpe ratios. I don't have a reference with me right now but especially in the case of hedge funds I find volatility estimation difficult as there may be auto-correlations, sparse pricing and other problems with their NAV publication.

@Freddy Hi! Thank you freddy for the interesting references (i like the first one)! The main point i was making is: One should be careful with sharpe ratios $>1$. In the first paper there are four hedge fund sharpe ratios "slightly" $>1$. In fact in your $2^{nd}$ link only one sharpe ratio is $>1$. In the third paper as far as i can see hedge fund indices are being used which are known to have at least _some_ bias. In my opinion it is still unclear if the sharpe ratios generally increased over time or not. But all data casts doubt on ratios greater than (lets be optimistic here) $1.25$).

I think one has to be careful what to attach Sharpe Ratios to. Andy Lo and some others try to make their points by assigning Sharpe Ratios to whole fund companies rather than to individual strategies. Fact is that generally performance degrades the more notional is put on. Thus, if Sharpe ratios are calculated over a whole set of sub-funds then the ratios naturally turn out lower. Many hedge funds (if not most) can pride themselves in employing a few PMs who run individual strategies at sometimes 3-4 Sharpe ratios. Its totally not unheard of...

...however it will be very seldom that such strategies employ a significant enough notional portion to significantly kick up the whole fund's ratio. I think one should keep in mind the distinction between Sharpe ratios of individual strategies vs the one of whole fund companies. Also a distinction must be made between Sharpe ratios of single strategies that employ different amounts of notional.

• This is a very common and serious problem among academic papers and with some hedge fund marketing materials, I can almost guarantee that the high ratio of 7 was without transaction costs and that when these are included this 7 will drop down somewhere between 0 and 1.

do this as a simple exercise, calculate the sharpe ratio for a simple strategy, with and with out the risk-free , and with and without transaction costs see what happens

I was referring to this claim: " ... very common and serious problem among academic papers".

would you like me to dig up a few papers all with good sharpe ratios that when transaction costs are added become useless?

I'd like factual statements to be substantiated with evidence. In my experience, top finance journals routinely require transaction cost analysis as a robustness check to any finding.

well maybe your experience has been different from mine

I would like to stress that the statement in the previous comments “anything above 1 is questionable” is not necessarily true. I have personally known fund managers with a great and stable track record of Sharpe ratios above 1.5 (below 2) annually, maybe with a low risk free, ok, but now in the Eurozone with such rates you can’t take a risk-free of 3% for your Sharpe! I have also built quant portfolios doing very well from the point of view of their Sharpe in their live trading, so I would say a Sharpe below 2 is still possible (albeit rare!). 7 is questionable, absolutely yes!!

@Fr1 You're either not using an appropriate benchmark or looking at an overfit backtest and NOT realized returns. A 7 sharpe is easy when your benchmark is libor.

@pyCthon yes indeed I said "7 is questionable, absolutely yes!! " because 7 is something absurd.. I was saying that 1.5

• Pardon the lack of an actual link, and the formatting, but in footnote 6 of "Alpha is Volatility times IC times Score", Grinold, Richard C.,
Journal of Portfolio Management, Summer 1994 v20 n4 p9(8)
, Grinold suggests that "a truly outstanding manager" might have an information ratio of 1.33:

(6) A rough guideline for determining the required IC comes from Grinold !1989^. If you have N stocks, then a truly outstanding manager who has an information ratio of IR = 1.33 (corresponding to a t-stat of 3 over five years) will need an IC (for each stock!) given approximately by IC = {IR}/!(# of Stocks).sup.1/2^ = 1.33/!(500).sup.1/2^ = 0.06. Top quartile might have (let's be generous) an information ratio of IR = 0.90 (t-stat of 2 over five years); thus the IC of 0.04 = 0.9/!(500).sup.1/2^. These numbers are rough guidelines. The guideline can tell us that for 500 stocks and a quality manager ICs of 0.3 or 0.001 are out of range. The rough guideline will not help us tell if 0.03 or 0.04 is a better choice.

• Perhaps check out Poti and Levich (2009), or in a different setting but from one of the same authors, Poti and Wang (2010) "The coskewness puzzle" in JBF. They directly address the issue of what level of SR is plausible.