Elm AI summary:
- Investors often estimate an asset’s future return from its past. We simulate that assumption: ten ordinary stocks whose expected return tracks trailing performance almost all end up at zero or the moon, averaging an impossible ~100% a year.
- Expected returns should come from forward-looking cash flow expectations, not a rearview mirror. When you can’t estimate one confidently, we think abstaining beats assuming the past will repeat.
One of the central themes of our writing and research at Elm is the importance of investment sizing, and the analysis of investment problems which involve trade-offs between expected returns and risk. Longtime readers will have seen ‘expected return’ play a starring role in many of our notes and as a central input to many of our tools and calculators. For understandable reasons, many investors find it simple, convenient, and appealing to estimate the expected return of an investment from its own historical returns, especially when they’ve been good. We’ve always preferred a more forward-looking approach – and to see why that’s our preference, it’s helpful to be precise about what we mean by ‘expected return,’ and how and why we use it.
The highest-horsepower tools in our toolbox don’t actually need an expected return estimate at all – they take as an input an investment’s complete ‘return distribution’ – the full spectrum of return outcomes an investment might plausibly produce over a given horizon, along with the probabilities associated with each outcome. Sometimes we need all that power, but often it’s both reasonable and practical to use some ‘summary’ metrics for the return distribution instead. ‘Expected return’ is the most important summary metric – it’s the probability-weighted average of the full spectrum of possible returns – the mean of many different potential outcomes, not a point estimate of a specific outcome. So it’s not our “best guess” of the future return, or even the return we “expect” in a natural-language sense; instead ‘expected return’ is a richer metric which incorporates the potential for outstanding outcomes, average outcomes, bad outcomes, etc. along with their associated probabilities. When we equate historical return and expected return we throw out all that richness, and potentially introduce a considerable bias as well, by ignoring the probabilities of a range of future outcomes which are entirely possible but materially different from the historical experience.
Reductio ad absurdum
So we can see that expected return and historical return aren’t the same thing, but we don’t know for sure that historical return doesn’t happen to be a reasonable proxy for the true expected return. What if it is?
An amusing – and (we think) persuasive – argument against estimating expected return from historical return takes the form of reductio ad absurdum.1 Our first step is to entertain the idea that historical return is a good estimate for expected return. Then we’ll look and see whether, given that assumption, we can get plausible dynamics for investment assets, or are they pretty plainly ‘absurd’?
Let’s find out! The chart below shows ten stocks with simulated price paths under exactly this assumption. Each stock’s expected return, defined precisely as above, continuously resets to match its own trailing one-year realized return. A stock’s actual return for a given day is its expected return plus a random shock (i.e., volatility).2 As you can see, nearly every stock emphatically heads in the direction of zero or the moon. The chart’s y-axis is in log-scale, and the numbers to the right of each line are the ending price and the annual compound return of that stock.
Note that the stocks don’t show any unusual short-term volatility – day to day, they behave like ordinary stocks – but each stock picks up a large, self-reinforcing drift, up or down, that compounds relentlessly. A stock that has a good year becomes expected to keep having good years, which pushes the price up, which produces a higher likelihood of another good year, and so on – and the mirror image on the way down. Occasionally there’s a shock that’s strong enough to push the stock in the opposite direction, but as you can see in the chart, that’s rare, occurring in just one of the ten cases.
This isn’t what real stocks (or investment assets more generally) look like. The behavior of the portfolio of these ten stocks also looks absurd: the average of the ending prices is about $101,000, which corresponds to an annual compound return of roughly 100% over ten years – a ridiculous figure for equity returns by any standard, historical or theoretical (though it would be wonderful if true!).3 For visualization purposes we’ve only modeled ten stocks here, but the results would be equivalent if we model hundreds or thousands of stocks instead. Given these dynamics, the assumption that historical return is a reasonable proxy for expected return can’t be right.
Compare these results to the chart below, which shows the same ten stocks, with the same volatility, but with a constant expected return that doesn’t follow trailing performance. The average ending price here is $225, for an average annual compound return of 8.4%.4
Reality may sit somewhere between these two extremes, but it plainly sits much closer to the second chart than the first. Even a heavily dampened version of the “expected return equals trailing return” rule produces distorted, unstable behavior. Below, each stock’s expected return is a 75/25 blend of its trailing one-year return and its original starting expectation. The self-reinforcing spiral is softer and slower to develop, but the same basic pathology is still there: extreme, bimodal outcomes rather than a sensible spread of results. The compound return of the average price here is 29% per annum, still way too high.
Connecting the dots
Expected returns should be estimated by thinking through the full distribution of plausible future outcomes and, wherever possible, by working from long-term cash flow expectations for the asset in question – not by reading off whatever happened to occur in the past.
We recognize it’s not always possible to estimate an asset’s expected return with enough confidence to justify a meaningful investment in it, which is for example how we feel about investing in commodities or digital currencies like Bitcoin. When we’re at a loss to estimate the expected return of an asset, we think the better response is to abstain from investing in that asset altogether, rather than to fall back on the assumption that future returns will simply mirror past returns. As they say in the US West where water can be scarce: better to go thirsty than drink muddy water.
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This content is intended only to provide observations and views of the author(s) at the time of writing, both of which are subject to change at any time without prior notice. The information contained in the commentaries is derived from sources deemed by Elm Wealth to be reliable, but its accuracy and completeness cannot be guaranteed. This material does not have regard to specific investment objectives, financial situation and the particular needs of any specific reader. Any views regarding future prospects may or may not be realized. Past performance is no guarantee of future results.
- A reasoning technique in which an argument is disproven by following its premise to its logical conclusion, showing that the premise leads to an absurd, impossible, or otherwise untenable result.
- More precisely, it’s assumed that next-period stock returns are normally distributed with the mean return matching the trailing one-year realized return.
- It is interesting that the average of the compound returns of the ten stocks is -6%.
- We’d expect the average compound return to be roughly the 7% arithmetic starting return less the “volatility drag” of just 0.3% (one-half the variance of the portfolio, or 0.5 x 0.082), i.e. around 6.7%. With a sample of only ten paths, though, there’s plenty of room for sampling error to push the realized average well above or below that figure.