Expected Log-Return as a Risk Control – Win Vector LLC
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Expected Log-Return as a Risk Control
By John Mount on August 10, 2026 • ( Leave a comment )
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...