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Optimal Bet Sizing in a Game with Favourable Odds: The Stock Market |Victor Haghani and Andy Morton

Victor Haghani and Andy Morton 2016

A thought experiment

You can invest your wealth in only two assets: a risk-free one and a market portfolio of all public equities. Your investment choices however are limited to: A) put 100% in the risk-free asset, or B) 10% in the risk-free asset and 90% in equities. You cannot mix A) and B) -- you must choose one or the other. What is the lowest return (above the risk free rate) you would need to expect from equities for you to choose B?

Do you have your number in mind? Great. Now, imagine you’re still in this two asset world, and you wake up one day and you determine that the expected return of equities is in fact exactly equal to your answer above. But now you’re no longer limited to just options A and B; you’re completely free to invest however much you like in equities, from 0% to 100% or more. How would you invest now?

We’ve asked about a dozen friends this question, all financial professionals. If you’re like them, you’ve likely read the question twice, trying to understand exactly what we’re getting at. You may feel like you just answered that question, and isn’t 90% the answer?

Well, no, we don’t think it is. Please read on as we try to explain why we think 45% is about the right answer, and how we can use the perspective of this problem to answer some other interesting questions.

You can get an immediate intuition for the problem and its solution by replacing our first question with: how much ketchup on your fries would be so much that you’d be indifferent between no ketchup and that much ketchup? And then replace our follow-up question with: how much ketchup is your optimal amount? Our first question was calibrating your risk aversion via indifference points, and the follow-up question involved choosing an optimal point anywhere in between. The halfway point is a pretty good estimate, and exact in some investing models.

Putting a price on risk

The properties of risk aversion are central to this problem, so let’s analyse the simplest type of risk, a 50/50 coin flip. A side payment is needed to make a typical, risk-averse person indifferent to risking a fraction ƒ of his wealth on such a flip. How should this required side payment vary with ƒ? There is a very good reason to suggest that it should be proportional to ƒ2, meaning we should demand four times the side payment for twice the bet size. To see this, compare a single flip risking √2% of your wealth with two consecutive flips each risking 1%. In both cases the mean and variance of total wealth change are the same (0 and 2). It seems reasonable that the total side payment should be the same too, which is only the case if the side payment is proportional to ƒ 2.

Thus the problem of side payments for coin flips boils down to choosing the specific multiple of ƒ 2 as your indifference point. Although it’s a matter of individual preference, we suggest a reasonable range for this multiple is 1 to 2. Let’s use 1, meaning you are indifferent to an offer of 1% compensation to take on a ±10% of wealth flip, or 4% to take on a ±20% flip and so on. You would accept coin flip offers with higher side payments than this, and decline those with lower.

To connect the coin example to the stock market, let’s simplistically model investing 90% of savings in the stock market for one year as risking 18% of wealth on the flip of a coin, meaning over a year we’ll either lose 18% or gain 18%. The rule suggests that we’d require a side payment of 3.2% (i.e. 0.182) of our wealth, or 3.6% on the 90% we have invested in equities, to be indifferent. You can see that in answering the question about what expected return you would need to be indifferent between putting 90% of your savings into equities or 0% in equities, you were calibrating your function of risk aversion.

Let’s now turn to the question of how much you should invest given the freedom to choose any bet size you like. Consider what happens as you increase your investment from 0. Your expected gain goes up proportionally, but your risk and hence the required reward for bearing it, goes up with the square of the investment. Adding these two effects gives the diagram below, illustrating why the optimal point is halfway between the indifference points of 0% and 90%. So, someone who is indifferent to investing 90% in equities if they had a 3.6% expected return would optimally invest 45% of his or her wealth in equities at that expected return. In general, optimal size is half the indifference point size.