This is an interesting question posed by Victor Haghani in his new book, the Missing Billionaires.

First, some context to this question. Finance professionals often discuss returns in ‘real’ (i.e. after inflation) returns, while most investors instinctively target absolute returns: did I make or lose money? Nobel prize laureate Robert Merton makes a strong argument that we should be focusing on maximizing the utility we get from spending our wealth, rather than maximizing wealth itself.  This can only be done by growing your wealth relative to the rise in the cost of things.

Imagine a first scenario where the price of everything fell by half, and your total net-worth fell by -20%.  In the second scenario the price of everything doubles, and your total net-worth increases by +50%. Many investors would feel better in the second scenario. But in reality the first outcome is preferable. You’ve actually become richer because you can afford to buy more things, while in the second scenario you’re only able to buy half as many.

Merely keeping up with inflation isn’t enough. If someone from the US in 1900 kept spending the same amount adjusted for annual consumer price inflation each year, they would just be spending $5,000 per year in today’s dollars. They would conclude that their standard of living had declined over this long period, as per-capita spending in rich countries like Singapore or the United States is more than 10 times higher.

The failure to use a portfolio allocation framework that determines how much to allocate to multiple risky investments after inflation is the primary cause of the ‘missing billionaires’. Such a framework gives investors an objective guideline on how much of your portfolio you should allocate to the stock market, a specific stock like Tesla or Nvidia, or an emerging technology like crypto. The right allocation is the biggest determinant of your investment outcome, more than what you buy, when you buy it, or when you sell it.

To illustrate this with an easy example, the book uses a coin toss that comes up 60% heads and 40% tails.  At first glance you should always take this bet. However the ‘how much’ question comes into play. If your net-worth is one million dollars, and you’re offered this bet for $1,000 every year, you should take it every time for the rest of your life. But what if the bet is for one million? Rational people will not take a 40% risk of losing all their savings. What about $500,000? Even then most would pass.

What is the optimal sizing for this bet as a percentage of your net-worth? If your target was to maximise expected wealth without any other constraints, the correct answer is to bet all of your net-worth every time! This is because the possibility of flipping heads every time doubles your wealth each time; if you had 25 flips of the coin and you landed heads each time, your million would grow to $ 34 trillion! But the probability of getting totally wiped out during your journey is 99.9997%.  The high expected payout of this unlikely outcome skews the whole equation towards too much risk, an obvious conclusion to anyone considering this.

The right question should be what is the strategy that maximises expected wealth while avoiding ruin.

The answer may surprise most readers: it is 20% of your net-worth on every bet. This risk sizing produces the highest most likely (median) result. Both 10% and 30% risk sizing produce a lower result.  Betting your whole net-worth produces a most likely outcome of $0, a total loss, in line with our intuition.

Let’s go a step further, and tailor the formula to your personal risk aversion preference, defined as Constant Relative Risk Aversion utility (CRRA).  A CRRA utility of 2 would mean reducing the optimal bet by this number, resulting in a 10% risk sizing on each coin flip. We now have a formula which becomes useful. From answers to questions like the one at the beginning of this article, most investors’ CRRA utility is between 2 and 3, so the optimal risk sizing would be adjusted down accordingly.

Some spectacular train wrecks can be avoided by following these allocation guidelines. Take the example of crypto enthusiasts who believe the price of [insert coin of choice] will ‘go to the moon’. Even the most ardent enthusiast must acknowledge that there’s a non-zero chance of the whole investment being lost, if not by the value of the coin permanently collapsing, then by getting hacked, or forgetting one’s passkey to access their digital wallet.  Paradoxically if the price of crypto were really to ‘go to the moon’, the correct allocation of your net-worth should actually be smaller.  Putting all of your net-worth in Bitcoin is an attempt to maximise total net wealth, with a very real possible risk of losing everything instead. Instead by maximizing Expected Utility, the right allocation for most investors would be between 5 and 20%, depending on their individual risk tolerance.

How relevant is the coin toss example to how much you should allocate to the stock market? One of the key takeaways from the book’s use of Expected Utility is that the amount of compensation you should demand for bearing risk goes up by the square of the risk. The cost of risk of betting $10,000 on a coin toss should be 100 times the cost of betting $1,000. The conclusion is that investors tend to over-allocate to risky investments, and on top of this are not sufficiently compensated for the additional risk they’re taking.

Financial advice is almost exclusively focused on choosing what to invest in, rather than how much. In addition to calculating the optimal allocation to risky investments like the stock market, Expected Utility can also help make better decisions in broader financial questions such as:

• Should you rent or buy your home? If you buy, how should you finance it? How should your investment portfolio differ with or without the house?
• How much should you keep invested in your own business? If you have a buy-out offer, should you sell? Should you hedge against the downside risk in your business?
• How much should you pay to insure against death and disability?
• What is the optimal time to start taking money from your pension?
• When should you retire?

The writer is head of investments for Singapore at AlTi Tiedemann Global. The views and opinions expressed in this article are solely the author’s and do not reflect the views or positions of AlTi Tiedemann Global or its subsidiaries. This content is intended for informational purposes only and should not be considered as financial advice.