## Wednesday, November 18, 2009

### Risk tolerance and the Sharpe ratio

Today I’m going to offer some insights from Jacques Pézier’s chapter in the monumental three-volume, 1301-page work The Professional Risk Managers’ Handbook: A Comprehensive Guide to Current Theory and Best Practices, edited by Carol Alexander and Elizabeth Sheedy (PRMIA Publications, 2004).

Let’s start with the curvature of the classic utility function to describe risk-seeking, risk-neutral, and risk-averse attitudes. A convex, upward-sloping curve describes a risk-seeking attitude, a straight-line reflects a risk-neutral attitude, and a concave, downward curve expresses risk aversion. Our task is to determine the risk tolerance of an individual trader. We ask her the following questions:

1. If you were offered a once-in-a-lifetime opportunity to win x dollars or to lose x/2 dollars with equal chances, for what value of x would you hesitate between taking the gamble and letting the opportunity go by? We could follow up this question with a few others involving simple gambles before pushing toward the extremes, as in the next two questions.

2. If you were offered (a) a lottery ticket to win \$10 million with some probability p, and nothing otherwise, or (b) a sure prize of \$500,000, for which probability p would you be indifferent between taking the lottery or settling for the sure prize?

3. If you were asked to pay a \$250,000 insurance premium to insure against a potential \$3 million loss, what would be the minimum probability of loss that would justify this premium?

Suppose, the author continues, the trader answers 33% to the second question and 5% to the third, we can calculate the utilities and plot them on the trader’s utility curve. (The math’s not important here.) The trader, not surprisingly, has a convex, upward-sloping utility curve.

* * *

One way to obtain a quick estimate of the risk tolerance of the trader is to invoke the mean-variance criterion. In its simplest form, for a gamble X (for instance, a portfolio return) with the expected value E[X] and variance Var(X), the mean-variance criterion for choosing among mutually exclusive gambles is maximize {E[X] – Var(X)/2λ}, where λ denotes the risk tolerance of the trader. Consider the opportunity to gain x or to lose x/2 with equal probabilities. (Click on figures to enlarge.)

If the value x is negligible compared to the expected value, the risk is worth taking. However, if we increase x, the variance increases faster than the expected value; at some point the risk is not worth taking. In this case the point at which the trader is indifferent to the risk is 9x/8.

Before you sign on to this way of estimating risk tolerance, consider its obvious downside.

In the above figure all four gambles have equal expected values and variances, but they are clearly not all equally attractive. Everyone would be happy with A because there’s nothing to lose; the other three have different degrees of risk. And yet all four have the same expected value of 5 and the same standard deviation of 15!

I hope that by now some bells are going off, which is why I’ve taken you down this long path. Sharpe ratio anyone? In fact, the Sharpe ratio, the author asserts, can be deduced from a special application of the mean-variance criterion. As a result I suspect that a trading system may have a favorable Sharpe ratio and nonetheless give the trader ulcers.

1. Hi Brenda,

Yes, giving equal weight to the variance of both profits and losses is one of the problems of the Sharpe ratio. Although, I'm sure you will soon discuss additional methods, Kaufman discusses better ways to measure risk/reward in New Trading Systems and Methods (including the - literally - Ulcer Index, Var, Schwager's average maximum retracement etc.).

Also, Curtis Faith, in his - in my opinion - not-so-great Way of the Turtle, has a couple of worthwhile chapters on risk/reward measures, including his R-cubed.

Thank you for keeping up the blog. Best trading,

Jorge

2. Jorge,

I know some of the standard criticisms of the Sharpe ratio; in fact, in this comment I’ll even excerpt two and give a url for another. But for me at least seeing a visual of wildly varying risk situations that have identical expected values and variances was a stunner.

And now for the pedantry:
“Although it is essential to consider fund returns in the context of fund risks, the Sharpe ratio is a bit of a blunt instrument to measure risk-adjusted returns. Past returns don't predict future returns. And although relative risks among funds have a good deal of consistency over time, standard deviation is only a rough proxy for a concept as elusive as risk. Further, weighting risk as equal to return in importance in the formula is completely arbitrary. Here is the reality of investing, as I see it: An extra percentage point of standard deviation is meaningless, but an extra percentage point of return is priceless. Large differences in risk are extremely important - there is a difference between a stock portfolio and a bond portfolio - but the expedient of weighing risk and return equally, in a simple formula, leaves much to be desired. In the final analysis, risk-adjusted returns, like beauty, may be in the eye of the beholder.” (John C. Bogle, Common Sense on Mutual Funds, p. 150)

“The issue here is that the Sharpe ratio is only appropriate for ranking investment/trading alternatives whose performance is completely defined by the first two moments of the distribution. If some alternatives have skew or kurtosis (but identical mean and S.Dev.), then Sharpe ratio won't distinguish among them. Yet most economic agents would prefer positive skew and low kurtosis. I'm sure you could extend the Sharp ratio concept to additional moments using analogous arguments of a tangent figure between the risk-free alternative and the multidimensional efficient frontier on the space of moments”. (post on Wilmott Forums)

And finally, “A Critique of the Sharpe Ratio” by David Harding, Winton Capital Management (http://www.edge-fund.com/Hard02.pdf).

Thanks for your comment. You always stretch my mind.

Best,
Brenda