Investment lesson from a mathemetical puzzle

I attended a recent seminar about the St. Petersburg paradox. For those of you (like me) untrained in the history of obscure mathematical puzzles, this relates to a question posed in 1713 by an eminent mathematician of the era, Bernoulli, who happened to be residing in St. Petersburg at the time. (Don’t leave me at this point – there is an investment lesson from this story.) In essence the paradox concerns probability theory as follows: if you had the chance to participate in the following game, how much would you pay to do so?

The game involves tossing a (fair) coin such that if it lands heads, £1 is paid into the pot to be paid out, if it lands heads again £2, and heads again, £4 and so on; so each time it lands heads, the amount put into the pot (to be paid out to the participator in the game) is doubled. However, if it ever lands tails, the game is over and whatever is in the pot at that time is paid out. Without boring you with the maths, probability theory says that the expected value of participating in this game is infinite (ie a lot!), essentially because each throw produces a 50% chance of the pay – out doubling. However, if, in reality, we were offered the chance to play in this game, we probably wouldn’t pay very much to do so; hence the paradox: if the expected return (under probability theory) is infinite, why would wouldn’t we pay a large sum of money to participate?

I can hear some of you saying; well, it is obvious guv, you have a 50% chance of losing whatever you pay to participate, which doesn’t look very attractive odds. (This is essentially right in my view but per my seminar it doesn’t explain the maths). There have been various attempts to explain this paradox mathematically over the past three hundred years, but never to everyone’s satisfaction, until perhaps now, or at least 2011, when Professor Ole Peters proposed a resolution involving parallel universes (stay with me!). The resolution is essentially that probability theory may have limited application in some instances in the real world, since the real world is ‘path dependent’ or putting it another way, once the coin comes up tails, that is it; the game is over and we don’t get another chance. In parallel universes, however, the game can be played over and over by our parallel selves, although our parallel self can’t, unfortunately, if we have lost in the real world, compensate our real self with his or her winnings.

I took away two investment lessons from this seminar (it was supposed to be about investment). Firstly, high return/high risk investments need to be priced cheaply to make them attractive for all but the very largest investors; ie if there is a significant chance of a loss (tails coming up) one really doesn’t want to pay very much for participating (in plain terms be wary of shares with a high p/e and pay attention to valuation.) Secondly, it is good to be able to play the game over and over again; so, when making an investment or buying a share, it is sensible to hold back some of the capital that has been ear marked for the investment, in order to make a subsequent purchase (also reinvested dividends and regular savings are good ideas.)

The seminar also referred to a paper by Towers Watson related to this paradox, entitled ‘Time is irreversible or why you should not listen to Financial Economists’. To quote from this paper: ‘just as the tortoise beat the hare in Aesop’s fable, we have a hunch that a low-volatility, robust portfolio . . . will compound through time at a faster rate than a ‘racy’ portfolio . . . ‘. At IpsoFacto investor we think that our equity model portfolios and asset allocation approach provide just such robust portfolios, which don’t require living in parallel universes!