Arithmetic Versus Geometric Mean Conundrum
Although I liked the article “You Are Not a Monte-Carlo Simulation”, by Corey Hoffstein, there was a conundrum in it that needs to be addressed because I think that is important to do since it is the source of confusion. This article is intended to serve as starting point of a discussion about this conundrum. I am sure Corey Hoffstein is aware of this but omitted details in the article to keep things simpler and make a point. However, as it turns out, this is basic problem and needs to be addressed.
The article starts with a hypothetical scenario as follows:
Each week, you earn 0.65% (such that over a year you earn 40%), but there is a 1-in-200 chance that you lose -95%.
Then, the article talks about the difference between the “ensemble average”, also referring to it as a Monte Carlo simulation, and the geometric mean. Specifically, the ensemble average, also known as the expectation, or mean of distribution, in case of sufficient samples (A. Papoulis, p. 138), is calculated as follows in the article:
The math here is simple: 99.5% of the time we make 1.0065x our money and 0.5% of the time, we end up with 0.05x our money. On average, then, we end up with 1.0017x, or 1.092x annualized.
The above calculation does not reflect the expectation in general and can lead to a strawman argument if not placed in the right context, as discussed below. Then, the article showed how the geometric mean of this strategy points to decay of wealth:
It is not the arithmetic mean we are after, but rather the geometric mean which will account for the effects of compounding. Calculating the geometric mean — 1.0065^99.5% x 0.05^0.5% — leaves us with a value of 0.9915, i.e. our wealth is expected to decay over time.
The following question is then raised in the article, which is a strawman as we will show below:
How is it possible that on average the strategy is a winner if each and every path is expected to decay over time?
The strawman can be seen if we note the difference in the two calculations:
In the ensemble average (mean or expectation, since sufficient samples are assumed) the value the random variable X takes is percentage, 0.65% or -95%.
This is the formula for the mean µ of a random variable X:
µ = ∑XP( X)
If we assume that X takes percentage values, 0.65% or -95% in this case, then the mean is an expected percentage equal to 0.17175% in this case. But if we use percentages for the values of the random variable X, then this is not the correct expectation that allows calculating increase in wealth and we cannot determine the final wealth by multiplying the result by the number of periods and adding it to initial wealth. Therefore, a positive ensemble average as a percentage does not necessarily imply a positive final wealth. This would be necessarily true only if the ensemble value was expressed in dollar terms and it was positive.
On the other hand, in the geometric mean formula, the path is irrelevant since the order of multiplication of returns does not affect the result. In the specific example, there is no difference between 1.0065^99.5% x 0.05^0.5% and 0.05^0.5% x 1.0065^99.5%.
Therefore, in this particular example mentioned in the article by Corey Hoffstein, the average based on percentage values for the random variable that was compared to the geometric mean led to the strawman expressed in the form of the above question. I have seen this strawman before several times. At any rate, the geometric mean result prepares the investor in this particular example for surprises unless luck plays its role and final wealth target is reached before a tail event.
In other words, if the ensemble average is expressed in dollar terms, then a positive value always guarantees equity increase in a given period but if it is expressed as percentage, the final equity may increase or decrease depending on variance, as may be seen from the first order approximation
G = A — V/2
where G, A, V are the geometric mean, arithmetic mean and variance, respectively.
Thus, use of examples with percentages to contrast mean and geometric returns can lead to strawman arguments that evaporate when the expectation is expressed in the right form to allow fair comparisons. Note that in general it is not correct to say that the expectation can be positive but the geometric mean results in decay due to volatility. This is true only when we are comparing arithmetic returns to geometric return, as the above approximation shows. In addition, if the values of the random variable X are in dollars and some are negative, the geometric mean cannot be calculated without some ad hoc modifications. In this case it can be shown that the geometric mean asymptotically converges to the arithmetic mean.
Overall, the paper by Corey Hoffstein is good and useful but there is the conflation of the arithmetic mean based on percentages and geometric mean. Actually the article could stand without any reference to ensemble averages and arithmetic mean because the points about risks and log utility are valid.
Next, there is something not directly related to the article but it is interesting: there are conflicting statements in the literature about the relative importance of arithmetic and geometric means.
Some claim the geometric mean is the true longer-term mean, for example H. Markowitz is his paper “Mean-Variance Approximations To The Geometric Mean”.
However, others claim that the arithmetic mean is an unbiased and consistent estimate of the compound return while the geometric mean is a negatively biased estimate. I have noticed these conflicting claims several times but I am not surprised since finance and probability and statistics is a very complex domain. At any rate, practitioners do not care much about these differences. In fact, all that practitioners have to know is that expected alpha is almost never realized, as explained by N. N. Taleb in this Bloomberg video and actually this is all they need to know for investing purposes.
Finally, if you are now more confused about ensemble averages, geometric means and how they relate, you should not worry. Actually, it is the academic community that is confused, not the practitioners, as it is the case in other subjects to. If you treat problems on a case by case basis, the generalizations are not required. Academia has the tendency to generalize, but this is also its weakness.
Reference: You Are Not a Monte-Carlo Simulation
If you have any questions or comments, happy to connect on Twitter:@mikeharrisNY
This article was originally published in Price Action Lab Blog
About the author: Michael Harris is a trader and best selling author. He is also the developer of the first commercial software for identifying parameter-less patterns in price action 17 years ago. In the last seven years he has worked on the development of DLPAL, a software program that can be used to identify short-term anomalies in market data for use with fixed and machine learning models. Click here for more.