The question
Here is a betting/investment puzzle to help introduce some tools for reasoning about risk.
You are offered a bet or risky investment that returns or pays-off X dollars for every dollar laid in, where X is a random variable distributed log-normally with parameters μ = -1, σ = 2. This is a bet with possible payoffs that look like the following distribution graph.
The x-axis is the returns for this bet and the y-axis is the density or likelihood of such values. The dashed vertical line at 1 is the break-even point where the bet pays back exactly what you put in, and the dashed line at 2.718 is the expected return for this bet’s distribution. This bet makes money on average.
We have right-truncated the graph for legibility, but most of the expected value of this bet is coming from the long-tail on the right (which has significant right-tail-mass much further than graphed). In fact 30.8% of the probability mass is above the break-even point. So a gambler has a just over 30% chance of having a profitable return, or alternately they lose money almost 70% of the time. This bet loses money most of the time.
We can try to roughly illustrate the probability mass to the right of break-even with a (somewhat non-standard) log-transform graph.
This type of distribution is technically considered “heavy tailed” (even though it does have all finite “power moments“) as the probability of large values is much larger than for the more familiar normal distribution (the distribution used to illustrate most investment advice and intuition). This is the technical way of saying that a lot of the value or risk of the bet is associated with rare events, and in particular “expected returns” may not be “typical returns.”
We can think of this bet as similar to laying a stake in a contested horse race: most of the time you lose, but the large payoffs offered seem to compensate. Taking the other side of the bet approximates writing an insurance policy: most of the time you collect a moderate fee, but once in a while you have to make a larger payout.
Our question is as follows: should you take the original bet? Is there a level of commitment or number of times to independently try the bet that makes sense for this situation? What factors (wealth, risk preferences, …) help drive a good decision here?
The trick or poison pill
Our bet has a feature that many risk managers don’t explicitly look out for: a negative expected log-return. The average value of the logarithm of the return for this bet is -1. In my opinion this feature is common and central to things like lotteries and insurance. You are going to run into this when managing risk, even if you don’t look for it.
Why is an expected negative log-return is dangerous? Because if we repeatedly lay all of our wealth on independent bets in the presence of negative expected log-return, we almost certainly lose almost all of our wealth. This is just the central limit theorem applied to the multiplicative returns (which are additive with a log-transform). So we have a bet that on average returns a profit, but almost certainly loses money if over-used.
The reason risk managers don’t routinely check for negative expected log return is many of the most popular modeling distributions (such as the Gaussian or normal distribution) don’t even support such calculations. Note: there are work-arounds, such as using truncated distributions.
Fixes
There are a number of ways to improve the performance of a bet. I will outline two: aggregation and de-leveraging.
Aggregation
One good fix for excess risk is aggregation. The idea is: making many low commitment identical independent bets preserves expected return and lowers variance or risk. Measuring and managing risk in terms of variance is the basis of modern portfolio theory and important investing ideas such as efficient frontier and investment diversification.
For example: suppose we could acquire k independent copies of this betting opportunity and we then laid a 1/k fraction of our wealth on each bet. This bet has the same expected value as the original distribution: returning on average 2.718 dollars for every dollar invested, and a much lower probability of losing money.
De-leveraging
Another good option for a risky bet is to only put a little money into the bet. This is the opposite of financial leverage. The following is a graph of the expected log returns for a portfolio where we hold out an (1 - f)-fraction of our wealth and gamble a f-fraction of our wealth on the bet.
It is a simple matter to search for the highest expected log return at f = 0.154, which we will just under 1/6th. This new de-leveraged portfolio retains an expected value above 1 and now also has a positive expected log-value.
The distribution of bet returns for this partial betting scheme or portfolio are pictured below.
Notice the losses are now lower-bounded by the portion of the wealth we are not investing. The log-transformed graph is given below.
There are criticisms of optimizing expected log-returns (or the Kelly criterion); but I find they come in two flavors: argumentum ad baculum and argumentum ad baculum from otherwise respected authorities.
Compounding returns
We can compound re-invest into the de-leveraged portfolio. The compounding plan being: re-investing in sequence just under 1/6th of our wealth in identical independent portfolios one after another.
We simulated betting just under 1/6th of our total worth 10 times in row (compounding wins or losses). We then repeated this simulation many times to get the distribution of possible outcomes of 10 compounded bets. 60% of the time this strategy was profitable. On average the partial bet strategy multiplied initial wealth by 9.8 times in 10 bets.
Longer compounding sequences lead to both higher returns, and reduced relative risk when the underlying portfolio has positive expected log returns. What we have done is combined a conservative move (hedging or de-leveraging the bet) with an aggressive move (compounding our bet) into a very effective strategy. This is in contract to accepting the original bet just because it has an expected return higher than the stake cost (or “positive expectation”, in this case meaning expected return greater than 1).
The “positive expectation is good enough” fallacy
I think the fallacy of judging bets on only expectation (or even just expectation and variance) stems from 3 sources:
- Conflating currency with utility.
- Neglecting the importance of bet size.
- Treating a bet as a once-only occurrence (ignoring aggregation or compounding).
- Compulsive or narcissistic behaviors around risk. From Berger, The Psychology of Gambling, Hill and Wang, 1957: a gambler patient is quoted as saying:
For me, gambling is a question I ask of Fate. The question is simple: “Am I your favorite?”
I think of this as the “downside risk is what might happen to others” attitude.
Estimating expected utility under uncertainty is always going to be a bit tricky. I am suggesting paying more attention to probabilities than to expected values, as I argued in “Choose by Probability, not by Expectation“.
Some history of expectation misconstrued as utility
The notion that an expectation 0 difference is a fair bet (and a positive expectation is simply an advantage) goes back to the early development of probability theory. However, “expectation” and “fair bet” are technical terms and should not be cudgels to force the acceptance of a bet.
An interpretation of subjective probabilities or beliefs through expected values goes back to the “problem of points” where Pascal, Fermat, Huygens, and others attempted to work out the value of an interrupted gambling game. Christiaan Huygens writes on the concept of “expected value” as follows.
AS a Foundation to the following Proposition, I shall take Leave to lay down this Self-evident Truth: That any one Chance or Expectation to win any thing is worth just such a Sum, as wou’d procure in the same Chance and Expectation at a fair Lay. As for Example, if any one shou’d put 3 Shillings in one Hand, without telling me know which, and 7 in the other, and give me Choice of either of them; I say, it is the same thing as if he shou’d give me 5 Shillings; because with 5 Shillings I can, at a fair Lay, procure the same even Chance or Expectation to win 3 or 7 Shillings.
Huygens is saying his ideal better would pay 5 shillings to play this game and would be willing to implement the game for another player if paid 5 shillings. That: he is asserting there is a payoff/fee number where an ideal player is indifferent to which side of the bet they are on. This gets misremembered as advice or as a rule: claiming one should value an equal chance at 3 or 7 shillings as being the same as holding 5 shillings.
In modern thinking this is considered unnatural. My preferred formulation is von Neumann – Morgenstern utility which can be used to encode and calculate over different preferences, such as preferring the 5 shillings to the shillings game, preferring the shillings game to 5 shillings, or considering them equivalent. Using expected return of currency does not have this flexibility, essentially disallows risk preferences, and can force some very bad decisions when used as a rule.
Conclusion
High expected values do not always automatically translate back into high probabilities of profits or success. Some steps should be taken to track and improve probabilities.
In finance terms: finding “alpha” (advantage) is hard. However it is market lore that the real winners better understand risk control. In this note we showed how de-leveraging a bet (only betting a fraction of our holdings) can convert a bad bet into good. Conversely, the market has demonstrated many times how over-leverage can turn a good bet into bad.
In general bets should be evaluated in context (utility, risk preferences, proposed use, and so on). There are very good analytic and numeric decision tools for doing this, it is just a matter of knowing and applying them. Our point is: if you are taking on a large number of negative log-return commitments, then you may have a much larger adverse risk exposure than is obvious.
Categories: Expository Writing Opinion Tutorials
Tagged as: expected log return expected value Kelly Criterion log-return probability risk Sharpe Ratio utility von Neumann and Morgenstern
