Everyone agrees on the expected values at the extremes – it’s the middle that confuses people. For a single period, the expected value is the arithmetic average. Over a very long time (infinite), the expected value is the geometric average. But in-between the expected value becomes less obvious. Let’s shed some light on this problem.
The Math of Random Momentum
In the prior post on momentum I showed how a portfolio of random return can behave in the exact same way as “momentum” in stocks. The expected value of the portfolio should rise to start, and then fall backwards over time.
I want to show the basic math behind this phenomenon. In the prior post, I used the following game:
Bet $100. Flip a coin, heads you win 50% of the bet (receive $150 back, for a gain of $50), tails you lose 36.7% of your money (get $63.30 back, for a loss of $36.70).
You play two games simultaneously. Each game is then played with the full payout from the previous game rolled into the next round. The game repeats without rebalancing.
Average Compound Return
The expected value of a single trial is straight forward.
We find the average compound return for one coin flip equal to the arithmetic return. Nothing surprising here.
With two flips the math gets more complicated:
The compound return is the actual return received, rooted to the # of flips. Example: after 2 flips, use the square root. After 4 flips, use the 4th root, etc. The average compound return is just the average of all possible compound returns.
You can see that after two flips, the average compound return has already fallen from the arithmetic return, and is now in-between the arithmetic and the geometric return.
For 3 flips, the table would grow to 64 lines, and produce an average compound return of 1.033 and an expected return of 1.103. The average compound return has fallen further still.
I’ve summarized further results in the table to the left.1 The expected return curves because it moves from the higher arithmetic average to the lower geometric one. Each flip takes the average compound return closer and closer to the geometric return.2
The geometric return is the average compound return for an infinite number of flips.
Therefore, this process connects the two values most agree on. One period, use the arithmetic average for the expected return. Infinite periods, use the geometric average. In-between, use the average compound growth rate.
Portfolio’s Effect on Convergence.
Adding more coins to our game slows the convergence from the arithmetic return to the geometric return. The convergence still happens, but slower. The same is directionally true with stocks, although the rate of convergence is different due to the continuous distribution of returns. A portfolio of 100 stocks will converge far slower than a couple stocks.
The portfolio’s size can cause confusion when evaluating the expected return. The return on a portfolio of 100 stocks over the next week is reasonably estimated by the arithmetic return. But the return of those same stocks over the next 10 years is best estimated as a value somewhere between the arithmetic and the geometric returns. Over the next 100 years, go with the geometric return.
Often studies run Monte Carlo simulations with hundreds of thousands of repetitions in order to show that “expected return” is still the arithmetic average. But averaging thousands and thousands of samples simulates holding a portfolio of thousands and thousands of assets. That portfolio should track the arithmetic mean for a very long time. In the real world, you can’t construct a portfolio of thousands and thousands of similar assets needed to “expect” this return.
A True Understanding of Expected Return
The arithmetic return is applicable over the short term. Everyone seems to understand this. Most grasp that over a long enough time, you should expect to receive the geometric return. In-between though, most studies still claim the arithmetic return is “expected”.
But you can see this has to be a false conclusion. Between the very short term, and the extreme long term, the expected return must slowly move between the arithmetic and geometric return. I’ve shown the math to calculate this value with coin flips. In investment markets, the math is more complicated, but the overriding concept is the same. The longer the investment goes through time, the closer the expected returns will trend toward the geometric return.
1- I only calculated these out through the 12th flip, and then projected the results from there. The math was getting very long.
2- There are actually two forces at work here. The first, is when the portfolio converges from the portfolio arithmetic return to the original portfolio’s geometric return, which in this example is near 1. This convergence mostly happens first. The second effect comes from the portfolio getting further and further unbalanced, similar to the effect shown here with market indexes. As the portfolio becomes unbalanced, the geometric mean of the portfolio itself falls towards the geometric mean of the individual assets.
In the “One Flip” table, in the first two rows, for H+H and H+T, the “Math” column shows “1.5/2+.0633/2” for both, but I believe it should be “1.5/2+1.5/2” and “1.5/2+0.633/2” respectively.
Your right. Thanks for noticing.
I’ve been reading from the beginning and I’m finding all of this fascinating
Glad to hear it.
So if you had a portfolio of n assets where n approached infinity, you would stay as close as possible to the arithmetic return as long as possible?
Love your blog. Many thanks.
Yes, this is correct. More assets means it takes more repetitions (time) for the returns to degrade to the geometric return. This is of course assuming they are uncorrelated.
Why not include a strategy like managed futures? I know you aren’t a trend fan – but CTAs add 40-70 markets and additional diversification due to taking both longs and shorts.
The “rebalancing yield” would improve from ~1% to 2% or so as well.
I’ve looked into it. Previous period correlation and volatility doesn’t seem as predictive as they are active strategies, so it has to be mixed on the side, where ultimately it doesn’t help the portfolio as the returns aren’t typically high enough, nor negatively correlated enough to improve the geometric return. If I found the correct strategy I’d be open to adding it but I haven’t yet.
These are very insightful and informative post. Needless to say, volatility can make / break your future and another critical element is the timing of it . Whether you hit by a black swan event in your 20s or closed to retirements.
Fundamental question is what’s the best way to handle volatility without sacrificing the returns ( remember geometric mean works both ways .. leaving upside can also push you back ) .
here is something interesting : https://caia.org/sites/default/files/2013-aiar-q1-comparison.pdf
Agree there is a tradeoff, and I feel the ideas on this blog is the best way, as I’m not trying to just minimize volatility.
I think the ideas of this post provide a good framework, as over a single period, the arithmetic return is the average compound growth rate is the arithmetic return, but as time increases, the average compound growth rate approaches the geometric return. With enough repetitions, the average compound growth rate will be nearly equal to the geometric return, so you can target that goal without giving up any upside.