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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.




How far can we take this simple framework?

What expected return on equities would you need for your optimal allocation to be 75%? Recall that the required minimum return you said would make you indifferent to being 90% invested in equities, R90%, is also the expected return for which 45% is your optimal allocation. Optimal allocation varies in proportion to expected return, and so for it to be 75%, the expected return needs to be 75%/45% higher than your R90%. For our prototypical investor who has R90% of 3.6%, the expected return of equities at which a 75% allocation would be optimal is 6%.


What is the effect of over-investing? As can be seen from the diagram, if you invest double your optimal allocation, you’ve thrown away all the benefit of the investment, and things get worse at an increasing rate from there. This means we should err on the side of taking less risk in the face of uncertainty about the probabilities of future returns.


At the other extreme, what about small investments, for example a $100 coin flip for an investor whose net worth is $100,000? The ƒ2 rule of thumb suggests being indifferent at a side payment of about (0.0012) x 100,000 = 10 cents! That seems crazy. But if that investor has 45% in the stock market and the rest in the riskless asset, over about one hour his or her wealth will fluctuate by about $100 (with stock volatility of 20%). Using the same 3.6% excess return of equities as earlier, the expected gain in that hour is about 19 cents. So if optimally invested wealth is producing 19 cents for $100 of risk, he or she is roughly indifferent to a side payment of half that. The framework therefore suggests (if needed!) accepting smaller rewards for small risks than might at first seem intuitive.


Should a passive investor in the stock market have a static allocation to equities? Investment advisors typically subject their clients to a risk evaluation to determine an appropriate allocation to equities, often taking the form of the kind of indifference questions we posed. If you believe that the expected return and/or risk of the equity market change over time, then your optimal allocation to equities should also change.


Does horizon matter? To the extent investments are like coin flips, following a random walk, with the risk we care about (variance) and the compensation we’re being offered to accept that risk both growing proportionately with time, then, whether we choose a month, a year or a decade as the horizon for our investment won’t affect our choice of the optimal amount we should invest in it. Whether you’re flipping a biased coin once or 300 times, the amount you bet as a fraction of wealth should be the same.


How much is it worth to be able to invest in equities? This may sound like a somewhat strange question. Isn’t the positive expected return of equities simply the fair compensation required for us to hold them? ‘Fair’ isn’t a useful word in this context. As each of us will choose an allocation to equities, not at our indifference point, but at the point where we maximize the surplus of what we’re expecting to be paid versus what we need to get paid to accept that risk. That surplus is equal to half of the risk premium multiplied by our optimal allocation. So, for example, if we see equities as priced to deliver a 5% expected return above the risk-free rate, and our optimal allocation is 60%, then our surplus is ½ x 60% x 5% = 1.5% per year. This tidy sum should make us feel pretty good, even grateful, about the existence of the equity market and our ability to freely choose how much of it we’d like. We’re being invited to play a game with favourable odds, like betting on the flip of a coin that has a 60% heads bias, or playing in a poker game where the other players have more money than skill.


If we knew every investor’s risk aversion, or R90%, would we be able to calculate a fair expected return for equities? No, because we’d still need to know what fraction of everyone’s wealth needs to be allocated to equities in order for the whole equity market to be optimally owned. This casts the risk premium puzzle in a different light, and suggests that we can go through long periods where a historically low expected return for equities can be a stable equilibrium, if the size of the equity market is a small fraction of investable wealth.

What is ’Kelly’ betting, and how does it relate? The Kelly criterion, named after the scientist at Bell Labs credited with formulating it in 1956, tells us how to bet to maximize the expected growth rate of our wealth. Implicit in the Kelly criterion is a level of risk aversion that is half of what we’ve been using in these examples. So a Kelly investor in the stock market would need only 1.8% excess return to be indifferent to holding 90% of wealth in the market versus nothing -- equivalently he or she would only need to expect 3.6% to optimally be 90% invested in equities. This is a much more aggressive posture than most investors we have met seem comfortable with.


Conclusion

We hope this discussion has given you some simple but versatile tools for thinking about a broad range of investment-related questions. Seeing the risk-taking decision more like tuning the dial on a radio, rather than flipping an on-off switch, enables us to get the most out of any game where the odds are in our favor, including investing in the stock market. The trick is to tune your portfolio to the point where the cost of risk, which is increasing quadratically, is just about to grow faster than the expected return you’re being paid to take that risk, which is increasing linearly. We hope you’ve found some ‘utility’ in this brief summary, and that we haven’t taken too many liberties in attempting to distil what are some of the most valuable insights in finance from the last 400 years, from Daniel Bernoulli to Bob Merton, and many brilliant minds in between.



References:

Arrow, Kenneth J., Alternative Approaches to the Theory of Choice in Risk-Taking Situations, Econometrica,

Oct 1951.

Bernoulli, Daniel, Exposition of a New Theory on the Measurement of Risk (1738), Econometrica (1954).

Haghani, V. and Dewey, R., Rational Decision-Making under Uncertainty: Observed Betting Patterns on a Biased Coin, ssrn.com, 2016.

Merton, Robert C., Continuous-Time Finance. 1990.

Norstad, John, An Introduction to Utility Theory, www.norstad.org, 1999.

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