The Sharpe Ratio Rewards Smoothness, Which Is Not the Same as Safety

The Calculation

the sharpness ratio is the return above the risk-free rate divided by the standard deviation of the returns

A fund that returns 12 percent when cash pays 3 percent with a volatility of 9 percent has a Sharpe ratio of one. It earned one unit of excess return per unit of volatility

The goal is comparability. A strategy that returns 20 percent with huge swings and another that returns 8 percent with small swings cannot be compared only in terms of profitability. The relationship puts them on a common basis

Both contributions deserve more attention than they receive. The numerator is an excess return so choosing the risk-free rate changes the answer and comparisons made between periods with very different cash rates compare different amounts. The denominator is a volatility figure measured over whatever stretch of history the presenter has selected so the ratio inherits each weakness in that figure and then divides by it

Why It Became the Standard

It answers the right question. Returns without reference to risk are meaningless since any returns can be manufactured with enough leverage. Doubling the leverage roughly doubles the return and roughly doubles the volatility leaving the Sharpe ratio unchanged which is exactly the property you want in a measure of skill

Where Scale Invariance Stops Holding

The word so-so in that description is doing real work and it's worth knowing the places where it breaks

Leverage is not free. Borrowing occurs above the risk-free rate so the funding spread goes out of the numerator while the denominator increases completely. Therefore a twice leveraged strategy has a slightly lower ratio than the unleveraged version and a five times leveraged strategy has a noticeably lower ratio. Invariance is an approximation that degrades as leverage increases

Compounding breaks it in the other direction. The clean ratio holds for arithmetic returns. Once the returns are combined volatility affects the geometric result and two strategies with the same Sharpe ratio and different volatilities do not produce the same wealth over time. The higher volatility version compounds to less

And leverage has a floor that the ratio can't see. A leveraged position can be closed by a margin call on a drawdown that an unleveraged version would have survived ending the strategy rather than reducing its performance. The move treats that as another observation in the series if the series continues

There is also direct arithmetic pathology once the numerator becomes negative and it appears more often than it should because periods are lost. When the excess return is less than zero dividing by a larger denominator brings the ratio closer to zero so a strategy that lost the same amount with more volatility reports a better Sharpe ratio than one that lost it quietly. The ranking is reversed precisely when the ranking matters. Therefore comparisons between managers who lost money during the period are meaningless on this measure whichit is rarely mentioned in the accompanying footnote

Annualising by the Square Root of Time

Almost every Sharpe ratio cited is an annual figure calculated from monthly or daily data and the conversion is where much of the problem comes in

The arithmetic reason it works is that the mean and standard deviation scale differently. In twelve months the average excess return is multiplied by twelve. The standard deviation if each month is independent of the others grows only by the square root of twelve. Divide one by the other and the ratio is multiplied by the square root of twelve which is approximately 3.464

Thus a fund that earns a 0.5 percent excess return per month with a monthly standard deviation of 2 percent has a monthly Sharpe ratio of 0.25 and an annualized figure of 0.866. That number is the one on the presentation and was produced by a conversion based on an assumption that no one verified

The same conversion is done from daily data with a multiplier of about 15.87 the square root of about 252 trading days. Working from daily returns is common because it produces a larger sample and an estimate that appears more stable and it magnifies rather than reduces the problem since the dependence between consecutive observations is generally stronger with daily than with monthly frequency

Therefore frequency is a choice of information that carries a result. The same history annualized from daily returns monthly returns and quarterly returns produces three different Sharpe ratios and there is nothing to force anyone to reveal which one was used. When two funds are compared by this measure without both figures being calculated in the same way the comparison is between two conventions and not between two strategies

Why Autocorrelation Breaks the Conversion

The assumption is that each month is independent of the previous month. When it is not the square root of twelve is the incorrect multiplier and is incorrect in a constant direction

Positive autocorrelation means that a good month tends to follow a good month so monthly movements accumulate rather than partially canceling out. The annual variance grows faster than the independent case implies the correct annual standard deviation is larger than the shortcut produces and the annualized ratio turns out to be too high

The effect size is not small. For a series with a monthly autocorrelation of 0.3 the correct annualization factor is about 2.615 rather than 3.464. A standardly calculated annual Sharpe ratio of 1.0 is worth about 0.75 once dependence is taken into account. With an autocorrelation of 0.5 the factor falls to about 2.121 and about two-fifths of the reported figure arean artifact of the conversion

This makes the autocorrelation check the most useful diagnostic available from a series of results alone. Truly marked liquid returns show little relationship from one month to the next. Substantial positive autocorrelation says that reported volatility understates actual volatility and therefore that the Sharpe ratio is overstated before any questions are raised about the assets themselves

The First Problem: Symmetry

Standard deviation treats bullish and bearish deviation identically. A fund that occasionally returns 15 percent in a month receives exactly the same penalty as one that occasionally loses 15 percent

No investor considers them equivalent. The measure treats good surprises as risks

the sortino ratio addresses this by using only the downward deviation in the denominator. It is a genuine improvement and is cited much less mainly through conventions

The Second Problem: The Distribution

The standard deviation fully describes risk only for a normal distribution. Financial returns are not normally distributed and deviations are important

A strategy with many small wins and few large losses has a low measured volatility and a high Sharpe ratio up to the big loss. The ratio accurately describes the lull period and says nothing about the shape of the eventual failure

Negative skew is the property that the ratio is least prepared to report because the two inputs move in the wrong direction. Frequent small gains raise the numerator the absence of large moves in the sample keeps the denominator low and both effects raise the ratio while the exposure that defines the strategy is completely left out of the data used to calculate it

The Third Problem: It Can Be Manufactured

This is the serious one. Certain strategies produce high Sharpe ratios by construction rather than skill

Selling options generates a constant premium and reports low volatility until a big move arrives. Illiquid assets are marked infrequently and infrequent marking mechanically reduces the measured volatility since a price that is not updated cannot deviate. Any strategy whose losses come rarely and severely will look excellent on this measure during the interval between failures

The pattern is consistent: strategies that sell insurance report high Sharpe ratios because premium income is stable and claims are occasional. The ratio measures the frequency of payment not the magnitude of exposure

A Reported Ratio Is an Estimate

The last problem is the one that is almost never mentioned and that is that a Sharpe ratio calculated from a history is a statistic with a sampling error and the error is large

The standard error of an estimated Sharpe ratio depends on the length of the record. For a ratio of one three years of monthly data gives a standard error of about 0.20. Five years gives about 0.16. Ten years gives about 0.11

If a confidence interval is established around the three-year figure the range consistent with the data is between approximately 0.6 and 1.4. The evidence has not distinguished a manager who reports 1.2 and another who reports 0.8 over three years and presenting both figures to two decimal places suggests a precision that does not exist

This is why comparing managers based on small differences in ratio is a waste of effort and why the length of the record deserves as much attention as its value. This is also why a short track record with a spectacular ratio is weak evidence rather than strong evidence since a short window is exactly where it is easiest to produce a large estimate by luck

The Same Shape With a Different Denominator

The Sharpe ratio is a member of a family and knowing which member is listed is as important as the value

the information ratio maintains the structure and changes both inputs. The numerator becomes the return above a chosen benchmark rather than above cash and the denominator becomes the standard deviation of that difference usually called tracking error. It answers a different question: not how much return was earned per unit of total risk but how much was added per unit deviation from the benchmark

That makes it the right measure for a manager hired to beat an index and the wrong measure for a strategy without a natural benchmark. It also introduces a choice that can be made favorably. A benchmark selected because the strategy has beaten it produces a flattering amount of information and unlike the risk-free rate in a Sharpe calculation there is no obviously correct answer to argue against

the sortino ratio mentioned above is the same family with the denominator restricted to the downward deviation and the calm relationship replaces the denominator with a maximum drawdown which exchanges the statistical order for a number that an investor actually experiences

None of these are a solution to the above problems. They relocate the assumption rather than eliminate it as each still divides an average by a single spread summary. What they do offer is cross-checking and a strategy that looks great on one and mediocre on the other tells you exactly where your risk is hiding

Using It Properly

Compare within a strategy category rather than across categories as different strategies have structurally different return distributions. Look at the ratio over a long period that includes at least one stress event. Check the skewness and kurtosis on your side. And treat any Sharpe ratio well above the strategy's typical range as a question rather than an achievement

Three specific questions do most of the work. What is the monthly autocorrelation since it determines whether the annualization is honest? How many months is the record?Since that determines how much of the figure is noise. And what is the worst month since a ratio is an average ratio and the tail is what hides the average?

The Bottom Line

The Sharpe ratio measures excess return per unit of volatility and is really useful because it is not affected by leverage. Its weaknesses are that it treats edges as risk assumes that distribution returns do not follow and rewards strategies whose losses are rare and large. A very high number in a calm period is usually a description of a risk that has not yet materialized. Before it gets to that you have to check the annualization and the duration of the record because a ratio can be inflated by aa third or more without anyone doing anything with the underlying returns