Tuesday, January 22, 2013

A Happy Skewness Delusion

There are several papers asserting that investors like positive skew to their returns. This is because empirically investors tend to be highly undiversified, and have a bias towards highly volatile stocks, and so they seem to want big lottery-type payoffs (incidentally, this is the exact opposite to what Nassim Taleb states is common, that people prefer payoffs that have high modes and low means--negative skew). Anyway, In the early days of portfolio theory, reconciling the desire to take such big risks was considered side by side with minimizing variance, as Markowitz looked at both approaches back in 1952 but eventually just sided with the total variance approach, and preference for positive skew was abandoned as an exception to a mean-variance rule.

Alas, it keeps cropping up, as in papers by Kraus and Litzenberger (1976), Harvey and Siddique (2000), or Statman and Shefrin (2000). I think Statman and Shefrin do the best job here, notingg that this isstrictly irrational, in that if there is a skewness preference and a risk aversion preference, then the skewness preference will be pretty tame, and as Post, van Vliet and Levy (2006) note, second order relative to the equity risk premium (eg, less than 1%). Rational risk averse investors would not take a lot of risk with some of their assets, at least, not enough to affect pricing much. That's why I like the Statman and Shefrin model, because they explicitly say this doesn't make sense if investors are rational risk averse investors (at least, with risk conventionally defined), unlike Harvey and Siddique who try to cram this into a fully rational model and ignore how it all fits together.

In any case, I was looking at this and came across an interesting take by Brunnermeier, Gollier and Parker (2008).  The model gets pretty convoluted but the intuition is very interesting.  The benefits of being ex ante biased towards skewed assets outweighs the ex post costs of having inefficient portfolios.  These highly skewed assets make it possible to conceive of an aspirational state of much greater wealth, as such an event is possible, though improbable.  Your excessive optimism makes you better off because you anticipate a small probability payoff incorrectly, but since your happiness is the discounted value of these beliefs, you are a happier deluded self. Think about all the time you have spent considering what you would do if you won the lottery, as surely everyone has done that at least once. The cost can be pretty low (for a lottery ticket, $1), but the benefit of visualizing that fantasy pretty large, and it doesn't keep me in a debilitating stupor because I don't do this much (moderation in all things).  

Now, as an explanation of the low vol effect, I think this Brunnermeier et al model is inferior to the Shefren and Statman approach because it pretends to be a rational, general equilibrium model, when its not (funny how general equilibrium models built on fundamental preferences are usually quite parochial because of their limited assumptions).  But I like the intuition, or find it intriguing.  It's nice to think we are on a journey with sealed orders with some higher purpose that we aren't privy to, but it exists and is important. It may not be true, but it's comforting to think so, and doesn't really cost much. 


J said...

Arthur Koestler compares being "on a journey with sealed orders with some higher purpose that we aren't privy to, but it exists and is important" with the alternative of sailing with no orders and no idea of what is the purpose of the trip. Koestler says that the difference between the two situation is important but doesnt say why. You say that the difference is in the confort (the illusion of being part of higher plan). It is interesting that we keep thinking the same ideas that were thought by all generations before embarking or surgery.

Michael James said...

I've seen preference for both positive and negative skewness in the same person. He bought lottery tickets regularly (positive skewness) and invested in corporate bonds (negative skewness). He focused on the small-probability positive payoff with the lottery tickets and ignored the "trivial" few dollars they cost. Reaching for higher yield with corporate bonds instead of government bonds was based on a feeling that it was essentially free extra money and that the likelihood of default was very low. So, he focused on the low probability result in one case and ignored it in the other.

Anonymous said...

two comments. First, I think you would find Milton Friedman published a paper in the AER in the late 60s which established that this behavior, buying a lotto ticket, was rational. It all depends on how one defines the utility function. Second, why this behavior surprises anyone is beyond me. It is exhibited by just about everyone who ever started a business, i.e., it is what entrepreneurship is about.

Eric Falkenstein said...

Actually, Friedman and Savage back in 1948 wrote a paper on this, and that is what Markowitz was really responding to in 1952. The problem is that these two things are impossible to put into a standard representative agent.

Tel said...

There was some guy tried to convince me that buying a lottery ticket gives a positive return from a mathematical basis, even after considering that from a probability point of view the expectation is always negative.

The theory was that for most middle-classed people, spending $5 per week has no difference on your lifestyle, but winning the lottery allows you to do things you would never be able to do by just working a job. Thus the effective value of the dollar is different depending on context -- living a normal lifestyle, vs having a small but real chance of living an exceptional lifestyle.

I was never convinced by this, and I don't buy lottery tickets either, but the argument has haunted me a bit because it seems like rubbish but there's no obvious way to prove it is rubbish. I think that all those $5 per week can add up, over a lifetime to make a difference, like going out somewhere nice with the family once twice a year or something like that.