Thursday, October 19, 2023

#1 The Missing Billionaires - The Merton Share


I’m going to start a new series on The Missing Billionaires by Haghani and White. This is a brilliant piece of work that fills a very important gap in financial decision making that bridges the theoretical maths and actual investment decisions. I though perhaps reading the book a second time at a much slower pace of a chapter a day would be beneficial as I reform some of my training materials to be more consistent with this piece of work. 

I’m going to introduce the concept of a Merton Share, which is an elaboration of the Kelly Criterion I teach in my ERM classes. 

Imagine you have a coin that flips heads 60% of the time and you are given a chance to bet on heads coming up 25 times. What percentage of your capital should you pick to take these bets?

If you are gunning for the highest expected payout, you should bet all of your wealth into each flip 25 times. But that’s intuitively dumb because a single tail will make you lose all your wealth, so it does not make sense to maximise the expected value of the bets.

So, instead, there is a more rational approach. If you bet 20% of your wealth, you will maximise the median profit, which divides the more favourable and less favourable outcomes into two equal parts. But this is still quite stressful for humans as there is still about a 15% chance of losing more than half of your starting wealth.

So, most human beings are advised to bet 10% of their existing wealth before each coin flip.

In this simple illustration, we captured what it means to be a human being. We do not optimise for the highest expected or median outcomes. We take bets based on what allows us to sleep at night.

Translated to the field of investments, we can calculate our allocation into equities and risky assets using the Merton share, which looks like this :

It means that we increase our proportion linearly into a risky asset when the excess returns go up but reduce our proportion of the same assets based on the square of the standard deviation.

Interestingly, the Greek alphabet lambda measures our personal risk tolerance, which looks like λ. When lambda is 1, the term collapses to the Kelly Criterion, which is how much we will bet optimally. The problem arises when banks and wealth houses measure the lambda of their clients. Even the most savvy investors have a lambda of 2, which neatly falls into the half-Kelly bets that professional poker players bet. Lay people are often three times more risk averse at 3. 

This has quite interesting applications to making personal finance decisions. In my program, we do estimate the returns and standard deviations by our backtesting exercises, but I do not have a scheme to show students how to allocate between this portfolio and the risk-less asset because I imagine a lot of their wealth is already stuck in CPF and physical property. 

More interestingly, the role of a financial advisor is to shift a client's lambda from a starting position of 3 into something closer to 2. 

From a practical perspective, if you meet a frugal housewife who is always hunting for the best fixed-deposit rates by switching different banks, it may be justified to get her to allocate her assets to some local stocks or REITs, too.

The sad thing is many of our financial advisors may not even understand what a standard deviation is, much less a Merton share, and this equation should become a foundation knowledge if we launch a personal finance course nationwide. 


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