6: The Trade-Off Between Risk and Return6-3b THE VARIABIlITY OF EQUITY RETURNS
Every normal distribution has two key characteristics: its mean and its variance. As you may recall from
your study of statistics, the variance measures the dispersion of observations around the mean of the
distribution. To be more precise, the variance is the expected value (or the average value) of squared
deviations from the mean. In equations, variance is usually noted by the Greek symbol σ 2. Suppose we
are estimating the variance of share returns using n years of historical data. The return in any given year
t is rt, and the average return is r–. We estimate the variance using the equation below:
Eq. 6.3 Variance==
1
σ^2
(rt r)^2
t=1n
∑n
Table 6.4 illustrates a variance calculation using stock returns in the US from 1993 to 2010. Over
this period, the average annual return equals 10.3%, about one percentage point less than the 11.4%
historical average from 1900–2010. In the table’s third column, we subtract the average return from
the actual return in each year. The fourth column squares that difference. We square deviations from
the mean so that both positive and negative deviations contribute to the variance calculation. If we
simply added up positive and negative deviations from the mean, then the resulting sum would be zero
by virtue of the definition of a mean. To find the variance, add up the numbers in the fourth column
variance
A measure of dispersion of
observations around the mean
of a distribution; it is equal to
the expected value of the sum
of squared deviations from the
mean divided by one less than
the number of observations in
the sampleFIGURE 6.5 HISTOGRAM OF NOMINAL RETURNS ON US EQUITIES, 1900–2010The figure illustrates the performance of stocks in the US in every year from 1900−2010. For example, stock returns were between −20% and −30% in
1907, 1930, 1974 and 2002. The figure suggests that the historical distribution of stock returns is at least roughly approximated by a bell curve, also
known as a normal distribution.
Per cent return in a given year1999
2009
1998
1996
1989
1983
1992
2005
2007
1979
1987 1976
1984 1967
1978 1963
1970 1993
2004
2006
2010
1961
1960 1988 1955
1956 1986 1951 2003
1994 1953 1982 1950 1997
1990 1948 1972 1949 1995
2001 1981 1947 1971 1944 1991
2000 1977 1939 1968 1943 1985
1973 1966 1934 1965 1938 1980
1969 1946 1926 1964 1925 1975
1962 1941 1923 1959 1924 1945
1957 1940 1916 1952 1919 1936
2002 1929 1932 1912 1942 1909 1928
1974 1920 1914 1911 1921 1905 1927 1958
1937 1930 1917 1913 1906 1918 1904 1922 1935 1954
1931
2008
1907 1903 1910 1902 1901 1900 1915 1908 1933
< –30 –30 to –20 –20 to –10 –10 to 0 0 to 10 10 to 20 20 to 30 30 to 40 40 to 50 > 50Source: Elroy Dimson, Paul Marsh and Mike Staunton, in ‘Triumph of the Optimists,’ Global Investment Returns Yearbook 2010. Published by ABN AMRO, London.
Updates provided by Dimson, et al. to 2009. Reprinted with permission.