Financial Mathematics Text

Sunday, July 28, 2013

Follow-up On Investment Poll

In an earlier blog I presented a poll with three alternatives (if you haven't taken the poll, see here). There's some observations at the above blog as well as at Fool and OSV.

Here's a description of the options with some basic stats:

Figure 1 - Summary Statistics

According to a basic Mean-Variance view, there should be no preference for these since all three scenarios have the same mean and variance (sort of... see below).

So what was the point of this?

Part of my interest was to see what preference there was for skewness. According to standard utility theory (and I'm a little fuzzy on the details; they seem to be suggesting that the 4th and 5th derivatives of the utility function ought to be related with skewness and kurtosis, respectively), investors should prefer high skewness and low kurtosis.

There's an interesting article on the question: Do Investors Prefer Negative Skewness?

One observation with this is that Scenario B (which is the high skewness scenario) is that people prefer options where there is a small chance of a huge windfall or "longshot bias". In this respect it's like the lottery (you pay $ \$1 $ or $ \$ 2$ for the small chance to be a millionaire.)

On the other hand, Taleb argues that there's a preference for negative skewness. After all, in my Scenario C you would have outperformed the expected return 90% of the time if you picked C. That's a pretty good psychological boost.

Most of the empirical literature which I have read has found evidence that investors require a higher rate of return for negative skew returns compare to the positive skew returns. So I'm not sure what to make of it. Should this preference imply that negative skewness is a risk factor? Or is another kind of preference altogether?

One problem with my poll is that in order to perform the test, you have to control for other variables. That's not easy because (1) it's not even clear what all the other possible variables are and (2) I had hard enough time trying to get three scenarios with the same mean/variance but different skewness.

For example, one factor that was brought up by several people is that B is favorable because there is no chance of loss. I made a few attempts to make it so that all three scenarios would have a loss factor but I didn't have much success in constructing that. I'm pretty sure it could be done but it would need to be tinkered with.

Initial Results of the Poll

The initial results of the poll, however, indicate that most of the folks that took the survey preferred B the best, A next and C the least. So perhaps Taleb is mistaken here.

Granted, C has a much higher kurtosis as shown in Figure 1 so perhaps that's a factor as well. (There's more to this story, however, see below.)

What do returns look like over longer time periods?

So I spent some time playing with the scenarios and ran some trials (10,000 to be exact) for what the distribution of returns would look like for 10 time periods. So this is just a bunch graphs and summary statistics.

There are two interesting things to note.
  1. Returns aren't identical. This is due to a difference between arithmetic and geometric mean. But they are so close as the difference is probably not noticeable. 
  2. Variance of the 10 period returns are actually different. B has the lowest variance and C has the highest variance. 
So I ran the three scenarios for 10 periods and I did so 10,000 times which sounds like a lot but it was required to get some of the stats to appear to be converging to some extent.

About the portfolios. There were 10 different portfolios.  Three of the portfolios just contained either A, B or C. Three of the portfolios contained two independent identical type A's, B's and C's respectively. Three of the portfolios were a mix of two independent different types. And the last portfolio was an equal weighted of all three (again assuming independence).

Obviously the independence assumption is questionable (at least in real markets) since there tends to be a good amount of correlation between different asset classes.

Figure 2 - Summary Statistics on 10 Period Trials

One interesting thing here is that (and I ran the scenarios several times) the volatility of C is the highest while B has the lowest volatility. I suspect part of that is due to the difference between geometric and arithmetic statistics but I haven't explored it further.

One advantage to Scenario A is that it's more likely to have returns greater than 3% than the other two. Scenario B is more likely to underperform the expected return than the other two.

Different combinations of the also had interesting features.

Here's a look at what the overall distribution of returns looked like for the different portfolios:

Portfolio Distributions

So I don't know if these pictures would change your opinion about the different investments. But I definitely think there's more to the story as far as risk and preference goes (compared with the standard mean/variance variety.) Looking at risk more broadly is beneficial in my opinion.

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