 Asset Allocation | Investment Manager Research

# Advisors Who Understand Rate of Return Math May Produce Better Client Outcomes

June 25, 2020

As an investment advisor, forecasting investment returns is both critical and challenging for a myriad of reasons. Whether it’s setting client portfolio-level return expectations, building probability distributions of likely outcomes, developing individual asset class return expectations over varying time horizons, or running Monte Carlo simulations based on client expected cash flow projections, understanding and properly applying the arithmetic vs. geometric return math to the correct situation is important. In this paper, we share a new formula we derived (arithmetic vs. geometric return over varying time periods) that could be helpful as you apply forecasted returns to various real work client situations.

About a decade ago, we wrote a white paper titled “The Low Volatility Tailwind” illustrating how you could increase the expected return of a portfolio by lowering its volatility. The exhibit used to illustrate the point was rather straightforward. Assume your client loses 50 percent this year then gains 50 percent next year. The arithmetic 2-year return would be 0 percent over the period, but the geometric return would be -25 percent (or -13.4 percent annualized). So, reducing a portfolio’s volatility by adding a low-correlating investment can improve a portfolio’s expected long-term return, even if the new investment does not have a higher expected return than the investment it replaced. How is it possible for a portfolio to have a higher expected return than the sum of its parts? The first step to understanding the Low Volatility Tailwind is to differentiate arithmetic from geometric investment returns.

The arithmetic and geometric returns are only equal when a portfolio has the same return every year (or no volatility of returns). Otherwise, the geometric return for any time period is always less than the arithmetic return. The higher the volatility, the greater the difference between a portfolio’s arithmetic and geometric annual returns. Simply put, investment losses drain a portfolio’s long-term return more than investment gains of the same percentage boost it (in geometric terms).

The geometric return of a portfolio can be estimated by subtracting [(standard deviation)2/2] from the arithmetic return. So, if a hypothetical investment were expected to generate a 10 percent annual arithmetic return with a 17 percent annual standard deviation, the annual geometric return is expected to be 8.6 percent [10%-(17%)2/2] over an infinite time horizon.

Therefore, any action that would reduce a portfolio’s volatility, but maintain the same expected arithmetic return, would increase a portfolio’s geometric returns. Since geometric returns reflect financial reality (or the investment return your client can spend), maximizing time-horizon geometric returns should be the objective.

Exhibit II compares the relationship between arithmetic returns, geometric returns and volatility. The analysis shown graphically in Exhibit II makes the assumption that an investment has a 10 percent arithmetic return. It also illustrates the relationship between arithmetic and geometric returns based on volatilities or standard deviations) ranging from of 0 percent or 100 percent. The higher the volatility, the lower the corresponding expected geometric return and, consequently, the larger the difference between the two. Well, not so fast. The problem with this geometric return forecast (if derived from the arithmetic return forecast and volatility) is that it is based on an infinite time horizon. Of course, our clients are often mortal and have investment horizons that are significantly less than infinite. So, using some common sense and Monte Carlo simulations, we were able to derive a novel formula for how to convert geometric expected returns to arithmetic returns (or vice versa) for varying time horizons. It’s a fairly simple solve, so we’re surprised nobody in academia has derived it before. After running millions of simulations and crunching some numbers, the following is what I came up with:

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