With reference to this post.

Unknown commenter pointed out the issue with portfolio E in particular, that although it had an expected gain of 5% per year, investors who persist with this portfolio over the long term would probably lose more in the bad years than they would gain in good ones. Sounds contradictory? Not quite. If you do the sums, you will see that the expected gain over a long sequence of years is generated from a very small probability of a extremely large gain, together with a very large probability of losing almost all your initial investment. The distribution of wins and losses is binomial (which tends towards Gaussian for a lot of years) but in order to come out ahead the investor needs to get lucky roughly 3 out of 5 years, and the probability of this happening will shrink exponentially (in the long term) as the number of years increases because it's moving further and further into the tail of a Gaussian.

As an extreme version of this, consider being invited to place a sequence of bets on a coin toss where the result of a T means you lose whatever your stake was, but H means you get back 3 times your stake (ie you win 2x stake, plus get your stake back - odds of 2:1 in betting parlance). This bet clearly has positive expectation, each pound bet has an expected return of £1.50, so if you want to maximise your expected wealth then rationally this bet is a great offer. If you start with a pound in the pot and do this 20 times in a row, betting your entire pot each time, you either end up with 3^20 pounds (with a 1 in a million probability, when you get 20 heads) or else you lose everything (with 999,999 in a million probability, when a tail turns up at any time). (2^20 is actually 1,048,576 which is close enough to a million for many purposes and can be a useful rule of thumb to remember). The expected gain at the end of the 20 bets is about £3400 but the vast majority of players will end up with nothing. Would any of my readers pay £1000 for the right to take part in this game?

In fact, for most people, most of the time, increasing wealth by a factor of 10 doesn't really make life 10 times better, but most people would be very averse to a bet where they could lose everything they own, including their house and the clothes off their back, even if the expected return was positive (eg betting the farm on the coin toss as above). A standard approach to account for this is to evaluate uncertain outcomes in terms of expected utility rather than expected value, and a utility function which is the logarithm of value is a plausible function to use. One typical implication would be that the subject would be ambivalent about taking a bet where they might either double or halve their wealth with equal probability. The expected value of the bet is positive of course, but expected utility (compared to the prior situation) is zero. It should be noted that no-one really behaves as a fully rational utility-maximiser in realistic testing, but it's a plausible starting point widely used for rational decision theory.

This logarithmic utility maximisation idea leads naturally to the Kelly Criterion for choosing the size of the stake in betting games like the coin toss above. The point is that by betting a proportion of your wealth (rather than all of it) you can improve your return in terms of expected utility. Note that the log of 0 is infinitely negative, so losing all you own is best avoided! In 1956, Kelly proposed a formula for the stake which gives the maximum expected gain in logarithmic terms. The Kelly formula of (p(b+1)-1)/b, where p is probability of winning and b is odds in the traditional sense, implies a stake of (0.5*3-1)/2 = 0.25, ie you should bet a quarter of your wealth on each of the "triple or nothing" coin tosses. After the first bet, you will have either 0.75 or 1.5 pounds etc, so you either gain 50% or lose 25% and if you were to have an equal number of wins and losses you will more than triple your money in 20 bets. A smaller win in absolute terms, but a much better outcome in terms of expected utility and the majority of players who follow this strategy will make a profit.

So what does this have to do with the investment portfolios? Returning to the investments, each portfolio can be considered a bet where you stake a proportion of your wealth with a particular odds and 50% chance of winning. Eg with portfolio E the investor is betting 0.48 of their wealth with odds of (1.06/0.48 - 1):1 = 1.21:1. Kelly says that with such odds and a 50% win chance, you should really bet only about 9% of your wealth, which would return either 0.91 or 1.11 which gives a small gain in log terms. Of course the investor doesn't get to choose their stake here, but it still provides an interesting framework for comparison. The 5 investments have the following implied odds, stakes, geometric mean returns and Kelly-optimal stakes respectively:

A 1.6 0.07 1.02 0.18

B 1.7 0.10 1.03 0.12

C 1.6 0.16 1.02 0.18

D 1.4 0.27 1.00 0.14

E 1.2 0.48 0.91 0.09

C has a better return than A (having the same odds and a closer to optimal bet) but the rounding conceals it. B is better than either due to having better odds and a near-optimal stake. D is useless and E is worse than useless in these terms, implying a massive bet on rather poor odds which means most of the time you'll actually lose money in the long run.

It is fair to say that not everyone necessarily wants to maximise the expected log of their wealth, but I was surprised to see investment strategies proposed that were actually loss-making in log space. It's also true that investment E has the largest gain in purely expected value terms, but it would require an extraordinary appetite for risk to take it (rather than tolerance or indifference). And this wasn't a single accident, the other similar question had no fewer than 3 out of 6 options having the same property. I actually wonder if it's partly due to a cognitive error due to presentation. One of the questionees said that they wouldn't be bothered by a 40% loss one year if they could expect a 60% gain the next. If that was written as dividing their investment by a factor of 1.7 one year and then multiplying it by 1.6 the next, it might seem less attractive!

## 1 comment:

Without stating probabilities the chance of a recession has dramatically increased.

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