Regression to the Mean

Regression to the Mean

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f interest to investors, advisors, and market analysts is determining "central tendency" of equity returns. Calculating mean return over some period of time is a typical approach.

Over the last 33 years, the mean return (geometric mean rather than arithmetic mean) of the U.S. equity market - as measured by the Dow Jones Wilshire 5000 Total Market Index - has been 11.6% (see Figure 1). As an aside, the arithmetic mean over the same time period was 13.1%. The geometric mean is always lower than (sometimes equal to) the arithmetic mean. Annualized returns which are reported by fund companies and data providers are always computed as a geometric mean. Geometric means represent a multiplicative function, whereas arithmetic returns are computed as an additive function. Inasmuch as money grows exponentially the geometric mean is the correct calculation.

Determining the variance around the mean is also important. Between 1971 and 2003 annual returns of the Wilshire 5000 Index ranged from a high of 38.5% to a low of -28.4%. The standard deviation of return over the 33 year period was 18.2%. The visual magnitude of such volatility is displayed in Figure 2.

The Wilshire 5000 Index had 24 positive annual returns and 9 negative returns during the 33 year period. Thus, this particular U.S. equity index reported a positive return about 73% of the time. The average positive return was 22.3% and the average negative return was -11.4%.

An interesting observation visible in Figure 2 is how seldom the annual returns were anywhere close to the 33 year mean return of 11.6%. In fact, over the 33 year period there were only 3 years in which the return was within 300 basis points of the mean return (1978, 1992, 1993). Thus, even though the fluctuating annual returns demonstrate a phenomenon known as "regression to the mean" it is clearly the case that such regression seldom produces mean-level returns in any given year. It is more accurate to think of regression to the mean as "a-change-in-direction-toward-the-mean-sooner-or-later", rather than a guiding force that produces mean-level returns on an annual basis. Regression to the mean tends to produce a pendulum effect that causes year-to-year equity returns to swing above and below the mean return, sometimes significantly so.

One of the challenges in managing investor expectations is reconciling long-term equity performance data (i.e. 11-12% mean annual return) with short-term performance which is often nowhere near 11%. One obvious approach is to encourage clients to take a longer view. But, that can be a hard sell when year-to-year equity returns are seldom near the "long-run" mean. Most of us theoretically believe in long-term, regression-to-the-mean performance, but in actual practice it's hard not to be adversely affected by volatility which can make it appear that mean returns are infrequently occurring (Figure 2).

One approach which renders a reasonable view of equity market returns and volatility of returns - without asking clients to find solace in 30+ year performance data - is to calculate performance via rolling returns. Rolling returns dampen annual volatility while still revealing central tendency. In spite of significant volatility in year-to-year returns (Figure 2) there is clearer "regression to the mean" performance demonstrated by rolling five-year return data (Figure 3).

Rolling five-year returns tend to highlight the central tendency of returns, whereas annual returns tend to highlight the volatility of returns. After all, focusing on annual returns (and their associated volatility) simulates one-year holding periods - which is hardly ideal investor behavior.

The first five-year rolling return of 2.3% (reported in Figure 1) is the average annualized five-year return from 1971-1975 (i.e. geometric mean). The second five-year rolling return is from 1972-1976, and so on. Interestingly, the arithmetic mean for the 29 five year rolling returns (where the rolling returns are calculated as geometric mean returns) was 13.1%, which is identical to the arithmetic average return for the 33 annual returns. However, the standard deviation of return of 6.9% for the 29 five-year rolling returns was 62% lower than the 18.2% standard deviation of the 33 annual returns.

Five year rolling returns demonstrate performance for holding periods of five years. Encouraging clients to hold positions at least five years is an excellent starting point, particularly when we observe that only 2 of the 29 five-year rolling returns were negative (and both of them were only slightly negative, -0.1% in 1977 and -0.9% in 2002).

These results suggest that holding a total-market equity position for at least five years generates results which are consistent with long-term performance expectations, but with dramatically reduced volatility compared to year-to-year performance. To this point, the five-year rolling returns were within 300 basis points of the long-term equity return of 11.6% on eight occasions, compared to only three times among the 33 annual returns.

Encouraging clients to "stay the course" is not new advice. Perhaps we need to re-think how the historical "course" is presented. Demonstrating historical performance in five-year rolling returns presents a more accurate picture of the central tendency of equity returns compared to the roller coaster behavior manifested by annual returns. For example, during the 19 year period between 1979-1997 the 5 year rolling returns demonstrated remarkable central tendency around the return of 15% (Figure 3). In comparison, there was no clear picture of central tendency (i.e. well-behaved regression to the mean) manifested by annual returns (Figure 2).

Showing clients 5 year rolling returns demands more commitment on their part. Only those who are willing to stay the course for reasonable periods of time (e.g. five years) will be rewarded with mean-like performance. Those who frequently jump in and out will experience first-hand the annual volatility of equity investing.

Year	Total Annual Return (%)	5 Year Rolling Geometric Mean Return (%)
1971	17.7	--
1972	18.0	--
1973	-18.5	--
1974	-28.4	--
1975	38.5	2.3
1976	26.6	3.8
1977	-2.6	-0.1
1978	9.3	6.0
1979	25.6	18.5
1980	33.7	17.7
1981	-3.8	11.4
1982	18.7	15.9
1983	23.5	18.8
1984	3.1	14.2
1985	32.6	14.0
1986	16.1	18.4
1987	2.4	14.9
1988	17.9	13.9
1989	29.2	19.1
1990	-6.2	11.2
1991	34.2	11.2
1992	9.0	15.9
1993	11.3	14.5
1994	-0.1	8.8
1995	36.5	17.3
1996	21.2	14.9
1997	31.3	19.3
1998	23.4	21.8
1999	23.6	27.1
2000	-10.9	16.7
2001	-10.9	9.7
2002	-20.9	-0.9
2003	31.6	0.4
33 Year Geometric Mean	11.6	--
33 Year Arithmetic Mean	13.1	13.1
33 Year Standard Deviation	18.2	6.9

Data source: Morningstar Principia

Figure 2. The Annual Roller Coaster

Figure 3. The Smoother Ride of a Rolling Five

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Craig L. Israelsen is an associate professor in the department of home and family living at Brigham Young University where he teaches Personal and Family Finance. His email is craig_israelsen@byu.edu