A couple of weeks ago, I received an email from an old friend with the subject: Sound Familiar? It contained a link to an article in Financial Advisor Magazine where the author describes “the free lunch in investing” by quoting a 1992 article in the Financial Analysts Journal written by David Booth and Eugene Fama…
“The portfolio compound return is greater than the weighted average of the compound returns on the assets in the portfolio.”
This sounds familiar all right!
This idea is central to Intech’s investment philosophy; our founder actually published this observation as a mathematical theorem in 1982 – ten years before the Booth and Fama FAJ article appeared! But, as Booth and Fama point out, it’s not exactly a free lunch. It requires some statistics, and some discipline. While our investment team uses some intense math, most investors can grasp the basic mathematical concepts by digging just a little deeper into the concept of return.
Fernholz, R., & Shay, B. (1982). Stochastic Portfolio Theory and Stock Market Equilibrium. The Journal of Finance, 37(2), 615–624. https://doi.org/10.1111/j.1540-6261.1982.tb03584.x.
How Can the Sum be Greater Than the Parts?
At first glance, the whole idea may seem a bit strange. How can a portfolio’s return be greater than the weighted-average return of the stocks it holds? As you may know, Markowitz (1952) formalized the concept that a portfolio’s return over a single period is simply the weighted-average return of the stocks in a portfolio; that was, however, only true for the arithmetic return. The same does not apply for a portfolio’s compound (or geometric) return.
To understand the difference between arithmetic and compound returns for a portfolio, it’s important to review and define return calculations more generally. While this may seem like a trivial exercise, its answer has repercussions on risk and portfolio management.
The classic definition of the (arithmetic) return on an investment is simply the difference between the final value and the initial value, divided by the initial value (also written as: return = final value / initial value – 1). This formula is fine for a single-period, but most investors have long-term time horizons and are interested in a multi-period, compound return that combines the n annual returns of r1, r2, . . . , rn . The common methods for calculating compound returns – arithmetic, geometric, and logarithmic – all have different characteristics.
Arithmetic Average Returns
These returns are the sum of all the annual returns, divided by the number of years (where n = number of years, and r = each year’s return):
Arithmetic returns are widely used in Modern Portfolio Theory (MPT). The calculation, however, has an upward bias as an estimator of expected long-term growth to an extent that can easily reach absurd proportions. For example, consider a +100% return in year one followed by a -50% return the following year. Here, the arithmetic average return over the two-year period is 25%, but in reality, such an investment would have zero growth over the two years.
Geometric Average Returns
These returns are the n-th root of the product of the annual returns, where n is the number of years.
This is the most commonly used method by investment professionals. It helps to avoid some of the absurd results demonstrated in the example of arithmetic average returns given above, and this gives the calculation some scientific gloss. Unfortunately, the geometric return is awkward to work with, and it too has an upward bias as an estimator of expected growth over the long term.
Logarithmic Average Returns
These returns are the sum of the logarithms of the annual returns (plus one), divided by the number of years.
Unlike both arithmetic and geometric returns, logarithmic returns are the only calculation that is unbiased as an estimator of expected long-term growth. What’s more, logarithmic average returns are the most conservative estimator of expected long-term growth:
Consequently, for single-period returns, arithmetic returns are sufficient, but to examine accurately a portfolio’s compound return, we recommend using stocks’ logarithmic returns. They’re an unbiased estimation of expected compound returns and they produce the most reliable results.
Relating Single-Period and Compound Stock Returns
The stock attribute that drives the difference between single-period and compound returns is volatility. As a stock’s volatility increases, the negative impact on long-term compounding also increases, resulting in a lower compound return. The log-return of a stock is approximately equal to its average arithmetic return minus half its variance. Many refer to the latter term as “volatility drag.”
Relating Stock Returns to Portfolio Returns
When applying the above relationship to a combination of stocks in a portfolio, the stocks’ portfolio contributions become interesting because of their potential diversification effects. The contribution of a stock to a portfolio’s compound return can be greater than its own compound return, because the contribution of a stock to the portfolio’s variance is less than its own variance.
The incremental portfolio return contribution of a stock increases to the extent that diversification reduces overall portfolio risk. Therefore, the compound return of a portfolio is not simply the weighted-average compound return of its constituents – it’s potentially greater!
Fernholz and Shay (1982) described the amount by which a portfolio’s compound return exceeds the contribution of compound returns from the underlying stocks as “an excess growth rate.” It is a critical component to a portfolio’s compound return and measures the extent to which the volatility of the portfolio is less than that of its constituent stocks. That is to say, it represents diversification.
In other words, you can uncover alpha by optimizing portfolio diversification: combining volatile, uncorrelated stocks, thereby improving a portfolio’s diversification, you can increase this source of alpha.
Intech has sought to capitalize on these diversification effects for over 30 years. While most active managers focus on the stock effects by trying to forecast stock returns, we focus on diversification effects, due to the interactions between stocks.
Harnessing stock price volatility to optimize diversification is important for both growth and risk control. By systematically replenishing diversification, we attempt to preserve portfolio efficiency and capture a trading profit – a rebalancing premium.
This is a vastly different paradigm of investing and it offers a framework that can help explain many so-called “anomalies.” For instance, in our paper, Diversification, Volatility, and Surprising Alpha, which was recognized among the Best Factor Investing Papers by Savvy Investor, we offered a factor-free explanation as to why many simple, systematic investment strategies beat cap-weighted indexes. You can dig deeper here:
Fernholz, E. R. (2002). Stochastic Portfolio Theory, Volume 48 of Applications of Mathematics (New York). Springer-Verlag, New York. Stochastic Modelling and Applied Probability.
Fernholz, R. and B. Shay (1982). Stochastic portfolio theory and stock market equilibrium. Journal of Finance 37 (2), 615–624.
Markowitz, H. (1952). Portfolio selection. Journal of Finance 7 (1), 77–91.
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