#4 The Missing Billionaires - The St Petersberg Paradox.

[This series is a slow chapter-by-chapter review of the book The Mission Billionaires by James White and Victor Haghani. It's getting special treatment on this blog because it is novel and can potentially result in a new Dr Wealth course. To understand it all, the best option is to read the book. Otherwise, you may need to read from the first installment of this series here.]

The St Petersberg paradox illustrates the differences between mathematical theory and street-level practice.

Suppose you are offered a bet where you toss a coin. The first time you get heads, you get $2. The second time you get heads, you get $4. This keeps doubling until you get a tail. Then the game ends, and you keep your winnings. So the payoff is 2 to the power of (consecutive heads+1).

The question is, how much will you pay for this game?

If you use mathematics, the expected winning amount is infinity, as there is a 50% chance of getting $2, a 25% chance of getting $4, and so on.... sums to infinity. 

Theoretically, a rational mathematician should bet his entire net worth to play this game. But in practice, few would pay more than $15 to play this game.

So what gives?

The answer is that we get diminishing marginal utility from our wealth. Halving our wealth can be as painful as the joy we get from doubling it. In such a  case, we should not consider the expected returns but the expected logarithm of the wealth from a bet. Some mathematical gymnastics later, this bet's street price is closer to $18. 

A class of equations model this kind of risk aversion called constant relative risk aversion or CRRA that covers a spread of different risk tolerances or lambda. 1 is for an alien SBF kind of trader. 2 is for a savvy investor. and 3 is for a layman.  

The book has two practical applications of a theory like this.

• If you work for a startup and anticipate an IPO because you are optimising the utility of your wealth and not its expected value, in any case, you may want to liquidate your shares and cash out for more certainty than entertain the possibility of loss after the lockup period. 
• The second application is the idea that Goals-based investing can be dangerous. Economists run computer simulations and found that insisting on meeting a financial goal, like earning 50% of your capital ( get rich or die trying ), can result in wealth loss compared to strategically sizing your investment bets to maximise utility. This is because bets get more desperate and aggressive the further you are away from your goal. 

I wonder if the second application will be helpful in Singapore as our financial advisors are probably too busy getting MDRTs to understand Goal-based investing. 

They know their goal is to earn a commission.