At the risk of most of my blogs linking to games or gambling, I think there are some fascinating – albeit rather intricate – relationships between gambling and investment strategies (and can’t resist writing another one). In this blog I seek to join the dots between different ways of describing investment returns, a betting strategy and an apparent paradox.
The following three statements are closely related:
- The St. Petersburg paradox is a theoretical game where you repeatedly toss a coin and get paid £2 if the first tail to appear occurs on the first toss, £4 if it appears on the 2nd toss, £8 if it appears on the 3rd toss, £16 on the 4th toss and so on. The table below calculates the expected payoffs, just by multiplying the chance with the payoff, for each possible outcome:
Toss number when first tail appears |
Chance |
Payoff |
Expected payoff = chance x payoff |
1 |
½ |
2 |
1 |
2 |
¼ |
4 |
1 |
3 |
1/8 |
8 |
1 |
4 |
1/16 |
16 |
1 |
5 |
1/32 |
32 |
1 |
… |
… |
… |
… |
The sum of the expected payoffs is 1 + 1 + 1 + 1 + 1… (forever) making playing this game appear priceless. But people are generally only willing to pay a relatively small amount to play. (How much would you pay?)
- For setting long-term investment strategy you should arguably focus on the expected rate of growth rather than expected growth itself
- Ideal long-term betting strategies should be in line with what is known as the “Kelly criterion”. This captures the idea that you should never bet the house on a wager, because if you always bet the house it is unlikely to be long before you lose everything. However if you are too timid you won’t grow your funds very quickly. The criterion tells you what percentage of your wealth you should bet to strike the right balance.
How on earth are these three linked? Starting with St. Petersburg paradox, one solution devised by the mathematician Daniel Bernouilli is to treat the happiness (or ‘utility’ as economists like to call it) of each extra £1 you own as decreasing as you get richer. As Bernouilli put it, “There is no doubt that a gain of one thousand ducats is more significant to the pauper than to a rich man though both gain the same amount.”
A common assumption is that you get the same increase in happiness from your wealth doubling from say £1,000 to £2,000 as you would from it doubling from say £100,000 to £200,000, despite the increase in the latter being much larger. Under this assumption, it’s the proportional increase that dictates how much better you feel. A fancy way to say this is that your utility is the logarithm of your wealth.
It also stands to reason that you should value a sure thing more than a small chance of a big payoff, even if their expected values are the same. On allowing for this you should theoretically pay only a small amount to play the game. The amount depends on how wealthy you currently are but even a billionaire should not pay more than about £31.
"Investors are not buying lottery tickets and should build a bare-minimum level of risk aversion into their decision making"
Moving onto the two types of return that investors may have heard of, focusing on maximising the average rate of growth (called the expected geometric return) of total wealth, rather than the average growth itself (called the expected arithmetic return) actually amounts to a similar strategy. To understand why, consider investing £1 for 100 years. The table below illustrates how wealth itself by the end of 100 years varies in a hugely skewed way depending on the rate of return achieved. Moving from 0% return to a 4% pa return improves your outcome by £54 but increasing a further 4% p.a. to 8% p.a. the outcome improves by a whopping £2,926. Taking the logarithm of wealth removes this skew and is as evenly spaced-out as the rates of return. (Readers who remember from school that logarithms are powers won’t find this too surprising).
Rate of return per annum (continuously compounded) |
End wealth = £1 compounded over 100 years |
Utility = Logarithm of wealth |
0% |
£1 |
0.0 |
4% |
£55 |
4.0 |
8% |
£2,981 |
8.0 |
Maximising the expected utility of wealth is consistent with maximising the expected rate of return. As such, paying attention to the expected geometric return can make a lot of sense: investors are not buying lottery tickets and may wish to build a bare-minimum level of risk aversion into their decision making.
The Kelly criterion actually amounts to the same thing – maximising the expected rate of growth – and is used by many professional gamblers. In practice many gamblers use greater levels of risk-aversion, such as “half-Kelly” (betting half what the Kelly criterion would tell you to), reflecting that they may have finite playing horizons. In a similar way, investors with finite time horizons are likely to be more prudent than seek to maximise their expected geometric return. Most investors will pay attention to downside outcomes, building in an additional layer of risk aversion and diversification.