Investing and Gambling- A Taylor Series Problem
Team Latte
April 19, 2007

Most of us think that the world is divided into two classes of people - investors and gamblers. Living in Hong Kong, we know the second category only too well. Horse racing, lottery tickets, you name it and everyone, from the cab driver to the noodle shop owner is willing to bet their life savings on them in the hope that they make a fortune. Then you have the investors - or shall we say the savvy gamblers - who pour millions into stocks, bonds, derivatives, real estate, and what not, and most do so through their fund managers or bankers. Win or lose, markets up or down, the fund manager is always happy, he gets to skin the cat both ways (unless of course, you are a hedge fund manager and you are dumb enough to lose your investors money pretty early on in the game).

Investors invest money in any asset with the belief that the expected return will be high and variance of the returns (volatility) will be low. That is they implicitly believe in a high first moment and a low second moment of the normal distribution (assuming of course that returns follow a normal distribution). Another way to look at it is through Taylor series expansion which will express any function as a polynomial combination of the moments of an arbitrary distribution. Any investor's utility can be approximated as a polynomial function around a mean portfolio return as:

If we take the expectation of the above equation and ignore the higher order terms as well as approximate the derivatives of the utility with constants we get:

Therefore, an investor's expected utility - or you may say in simple terms as "possible benefit" from a certain investment - depends on mean, variance, skewness and kurtosis (the first, second, third and the fourth moment respectively).

Investors want a high (positive) mean and low variance from any investment, but what about skewness and kurtosis? Taylor series expansion shows that the third and the fourth terms are also pertinent in impacting the utility of an investor. But classical investment theory ignores these two terms and analyzes the entire investment problem in the light of only the first two terms (hence the "mean-variance" analytic framework).

In reality, skewness and kurtosis play an important role in determining an investment's outcome from the market. A high mean should also mean a high (positive) skewness. But positive skewness means frequent small gains (profits) and infrequent large losses. This means over the long run if an investor has a positive skew, because of the fact that he desires high expected return, somewhere along the line he might be taken to the cleaners by the market. But it is now an expected norm that under weak assumptions investors desire high odd moments (the first and the third term in the above expansion) and low even moments (the second and the fourth term in the above expansion).

This view of investors preferring high odd moments and low even moments is completely at odds with the behaviour of the gamblers. What is the underlying rationale of gambling? Of course, this phrase is an oxymoron, still why does the can driver buy the lottery ticket religiously every week? Why does the noodle shop owner put more faith in the horses than her dumplings and fried rice?

In the case of a lottery ticket, or any gambling exercise for that matter, the expected return is very low; in fact it is usually negative. The chances very high are you will lose all that you put into the gamble. And obviously by deduction the variance of the return is very high. If we assume that the Taylor series captures an investor's - in this case the gambler's - utility correctly then we have to say that the second and the third term above, i.e. the skewness and the kurtosis should play some role in this as well. A low or negative mean certainly means low or negative skewness (both first and third moments will behave in a similar fashion) and a high variance will mean a high kurtosis (both the second and the fourth moment will behave in a similar fashion).

Thus gamblers are seeking low mean and negative skew with high positive variance and kurtosis. In fact, the negative skew trade means that gamblers are indeed betting on the fact that they will keep losing small money periodically and continuously - weekly lottery for the cab driver or bi-weekly horse race for the noodle shop owner - but once in a while, over a certain longer period he will make a fortune. This is at odds with the behaviour of our conventional investor.

In that sense gamblers behave more like option buyers.

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