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One place not to use the Sharpe ratio

Having worked in finance I am a public fan of the Sharpe ratio. I have written about this here and here.

One thing I have often forgotten (driving some bad analyses) is: the Sharpe ratio isn’t appropriate for models of repeated events that already have linked mean and variance (such as Poisson or Binomial models) or situations where the variance is very small (with respect to the mean or expectation). These are common situations in a number of large scale online advertising problems (such as modeling the response rate to online advertisements or email campaigns).

Photo “eggs in a basket” copyright MicoAssist appropriate CC license

In this note we will quickly explain the problem.

The Sharpe ratio is an attempt to take risk into consideration when valuing actions or investments.

The idea is: even if we use money as our notion of linear utility (so two million dollars is considered twice as desirable as one million dollars and not subject to any sort of diminishing returns or, as an alternative, threshold to buy the house you want) a rational actor should look at more than just expected values and avoid uncompensated risk. They should prefer a 5% chance at two million dollars to a 2.5% chance at four million dollars. These two alternatives have the same expected value (one hundred thousand dollars) so without the risk adjustment they have the same utility (by assumption!). However the second alternative is riskier: it is worth nothing 97.5% of the time. The Sharpe ratio is an attempt to adjust a given utility to account for risk in the following way: price value at expected utility divided by the square root of the variance. So our two alternatives are:

Scenario Win Probability Win Value Expected Value Sharpe Ratio
1/20 chance at $2,000,000 0.05 $2,000,000 $100,000 0.229
1/40 chance at $4,000,000 0.025 $4,000,000 $100,000 0.160

This is because an event that is value V with probability p (and 0 otherwise) has expected value pV and variance p(1-p)V^2 . So the Sharpe ratio is sqrt(p/(1-p)) (independent of V, which cancels out). So far this is mostly good: the Sharpe ratio is discounting rare payoffs (as we want).

This is also not quite a correct application of the Sharpe ratio. The Sharpe ratio is a dimensionless quantity (in our case it is a ratio of dollars to dollars), so it should be not used to price overall investments but instead to price the marginal value of buying a dollars worth of a given investment. In fact the argument for the Sharpe ratio works is based on a portfolio pricing argument: you can change the payoff ratio of any investment by leverage or borrowing money to invest. This makes an investment look like it has higher risks and rewards, but it doesn’t change the Sharpe ratio (as mean and sqrt(variance) scale together with investment size). So there is never any reason (in mean-value portfolio theory) to move to lower Sharpe ratio: even if you have a high risk tolerance it is better to use leverage to simulate more risk on high Sharpe ratio portfolios than to move to truly inferior investments. This is also one of the reasons diversification is important: it lowers risk without direct cost- increasing Sharpe ratio.

A problem arises when moving to repeated events. Suppose instead of two events as above we instead of many events as below. We have two marketing campaigns. Each campaign represents 10,000 advertising exposures and campaign 1 has one chance in 20 of being worth $2 on each exposure and campaign 2 has one chance in 40 of being worth $4 on each exposure. Take our campaign size as k (right now 10,000) as a variable and let’s attempt to value the campaigns using the Sharpe ratio:

Scenario Expected Value Variance Sharpe Ratio
Campaign 1 k * $2 / 20 = $0.1k k * (1/20) * (1-1/20) * $2^2 = 0.19 k ($^2) 0.229 sqrt(k)
Campaign 2 k * $4 / 40 = $0.1k k * (1/40) * (1-1/40) * $4^2 = 0.39 k ($^2) 0.160 sqrt(k)

The issue is: the ratio of Sharpe ratios is as before and independent of k. The first campaign looks like it is to be greatly preferred, even if the second campaign paid a bit more than it does, and no matter how long we run the campaigns. This is a wrong determination.

In fact the two campaigns are almost identical. They both have an expected return of $0.1k, and as k gets large they both have tiny variances ( 0.43*sqrt(k) and 0.62*sqrt(k) respectively, both tiny compared the expected values) and unbounded Sharpe ratios. There is no real reason to prefer the first campaign over the second once k is large (and in this setting 10,000 is certainly large). These are both “safe investments,” not the sort of risky investments the Sharpe ratio is used to price. What is fooling the mean/variance analysis is for distributions like Poisson, Binomial, or sums of same the mean and variance are linked (you know one and you know the other) so there isn’t any possibility of finding a variation that has the same expected value and lower variance (the essence of the mean/variance portfolio analysis- pricing changes in variance independent of changes in mean or expectation). And the Sharpe ratio is designed to value risky investments, exceedingly large Sharpe ratios are not the routine subject of mean/value portfolio theory.

Our pragmatic (non-theoretical advice) is: once you have k large enough that risk isn’t a real factor (that is sqrt(variance) is small compared to expected value) then it is no longer appropriate to use multiplicative risk adjustments. You can go back to picking based on expected value alone. Or you can try to keep a bit of risk in your calculations by using an additive (not multiplicative) ad-hoc risk adjustment such as valuing each campaign as something like “expected value minus sqrt(variance)” which (assuming normality) values each campaign at roughly its lower 15% quantile. Of course discounting campaigns of different sizes and ages is a bit trickier (as you don’t want to introduce a bias that excludes all new or small campaigns) which is why online testing or “bandit problems” take a bit more work than just having a convenient “discount formula.”

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