Modeling Expected Returns: The Future Is Not What It Used to Be

 
Statistical methods that most practitioners get wrong
Do you use financial planning software which projects future returns to assist you with asset allocation?

Next month I will moderate several sessions at the Society of Actuaries (SOA) 2011 Annual meeting for the SOA’s Investment Section. A consistent thread running through several of these sessions will echo what people have been saying about “the new normal”: we probably face a future with lower long-term equity expectations than we’ve been told to expect in our professional lives.

However, not all the views are due to dim forecasts of slow economic growth. Some of them are due to statistical methods that most practitioners get wrong.

The Investment Section’s first session will feature UCSD’s Alex Kane, who is probably best known as co-author of the textbook Investments by Bodie, Kane, and Marcus. Professor Kane will present a method of estimating long-term returns, a method that was unknown or at least unpublished when I attended business school and when I obtained the CFA charter.

If you studied Investments prior to 2005, chances are you weren’t exposed to this in class because it wasn’t widely known. I’ll start by retracing what has been well known to most practitioners.

Common Knowledge: Annualized Return is Lower than Arithmetic Mean Return
Most practitioners know that the average annualized return, also known as the Geometric Mean return, is lower than the Arithmetic Mean of discrete returns that generated it. The well-known approximation for the difference between the two is half the variance.

Also, recognize that since you cannot lose more than 100% of your capital, returns are approximately log-normal. That means in order to analyze historical or model expected annual returns, you should first transform them into their natural log equivalents. For example, a return of 10% has a lognormal equivalent of .09531 ( = 0.09531.)
 
The natural log equivalent of the arithmetic mean return over T observations, is

=

 
The natural log equivalent of the geometric mean return over a sample period T, is

= ,

where is the initial Value at time 0 and is the terminal Value at time T.
 
The difference between the Arithmetic Mean and Geometric Mean return is roughly half the variance:

–

 
Restated, the natural log equivalent of the Arithmetic Mean from time 0 to the end of horizon, time H, is approximately

+

OK, so how do we use this to estimate future expected returns?
What this common knowledge relationship doesn’t tell you, however, is which estimate to use when modeling expected returns. Many practitioners conservatively apply the Geometric Mean, since it is lower.

The authors pointed out that errors in your estimation will be biased upward, because the impact of a positive error will be greater than the impact of a negative error of the same magnitude.

A more accurate formula: The longer the horizon, the lower the projection
Documenting this bias, Kane and two co-authors, Eric Jacquier and Alan Marcus, identified a formula for an unbiased estimate of expected return over horizons of different lengths, given a known probability distribution of returns.

It might surprise some to learn that the forecasting formula is a weighted average of the Arithmetic mean and the Geometric mean of the sample return data, and that the weights depend on the ratio of the length of the future horizon, H, to the length of the sample period, T:

Expected log return over H periods =

(1- )

In other words, if your horizon is only one year, the Arithmetic Mean is very close to the unbiased estimate. The longer the horizon, the greater the weight on Geometric mean.

Here’s where many practitioners over the age of 35 will be surprised: when the forecast horizon is longer than the sample period, the unbiased estimate is LOWER than the sample Geometric Mean return: the weight on the Geometric Mean is greater than 1.0, and the weight on the Arithmetic Mean is negative.

This table, from the Jacquier 2003 article, tells you how much the Arithmetic Mean return will overestimate the expected portfolio value over different time horizons, depending on how many years of sample data you use and on the volatility.

	Horizon Years
	10	20	30	40
	A. Sample period: 75 years
15%	1.015	1.062	1.145	1.271
20%	1.027	1.113	1.271	1.532
25%	1.043	1.181	1.455	1.948
30%	1.062	1.271	1.716	2.612
	B. Sample period: 30 years
15%	1.038	1.162	1.401	1.822
20%	1.069	1.306	1.822	2.906
25%	1.110	1.517	2.554	5.294
30%	1.162	1.822	3.857	11.023

Ratio of forecasted to true expected portfolio value, from Table 1 of Geometric or Arithmetic Mean: A Reconsideration, Financial Analysts Journal, Eric Jacquier, Alex Kane, and Alan J. Marcus, November/December 2003, Vol. 59, No. 6: 46-53, (doi: 10.2469/faj.v59.n6.2574)

 
To use an interactive version of this table, visit the Terminal Wealth Forecast Bias Worksheet.

What are the Punch Lines?

• The longer the projection horizon, the lower the estimate of expected return
• If the Horizon is longer than the sample period of observed returns, the unbiased estimate is even lower than the sample annualized rate of return

Note that this formula applies only when you know the probability distribution of expected returns. When you don’t (as is typically the case) the expected return is possibly even lower. More to come in a future post.

Jacquier, E., A. Kane, and A. Marcus. (2003). Geometric or Arithmetic Mean: A Reconsideration. Financial Analysts Journal 59(6), 46–53.