Introduction to Corporate Finance

PART 2: VAlUATION, RISK AND RETURN

and then divide the sum by 17.^9 The calculations show that the variance of share returns equals 396.1.

Interpreting the number 396.1 is a little tricky, because it is expressed in units of per cent squared.

Remember, to calculate the variance we worked with numbers in per cent form and then squared them.

What exactly does 396.1%^2 mean?

Fortunately, we don’t have to struggle to interpret these odd units. Instead, if we take the square root

of the variance, we are back in percentage units, and we have the standard deviation. The standard deviation

is just another measure of dispersion around the mean, but in the case of investment returns, it is easier

to interpret because it is expressed in percentage terms.

Standard deviation = variance = 396.^1 =19.9 %

9 You may wonder why we are dividing by 17 if we have 18 years of data. The reason is technical, and has to do with a statistical concept

known as degrees of freedom. The technical issue is not terribly important here, and with a very large sample, dividing by either n or n–1 will

make little difference in the variance calculation.

standard deviation
A measure of volatility equal
to the square root of the
variance

TABlE 6.4 ESTIMATING THE VARIANCE OF US SHARE RETURNS FROM 1993 TO 2010

To estimate the variance, first find the average return, 10.3% in this case. Next, take the difference between the actual

return in each year and the average return, and then square that difference. Add up the squared differences and divide the

sum by one less than the number of years in the sample. The standard deviation is the square root of the variance.

Year Return (%) Return(%) − 10.3 (Return(%) − 10.3)^2

1993 11.3 1.0 1.0

1994 0 –10.3 106.1

1995 36.4 26.1 681.2

1996 21.2 10.9 118.8

1997 31.3 21.0 441.0

1998 23.4 13.1 171.6

1999 23.6 13.3 176.9

2000 –10.9 –21.2 449.4

2001 –11.0 –21.3 453.7

2002 –20.9 –31.2 973.4

2003 31.6 21.3 453.7

2004 12.5 2.2 4.8

2005 6.4 23.9 15.2

2006 15.8 5.5 30.3

2007 5.6 –4.7 22.1

2008 –37.2 –47.5 2,256.3

2009 28.5 18.2 331.2

2010 17.1 6.8 46.2

Sum 184.7 6,733.0

Average 184.7 ÷ 18 = 10.3%

Variance (6,733.0 ÷ 17) = 396.1

Standard deviation 396.1=19.9%

Source: Elroy Dimson, Paul Marsh, and Mike Staunton, ‘Triumph of the Optimists,’ in Global Investment Returns Yearbook 2010. Published by ABN AMRO, London.

Updates provided by Dimson, et al. to 2009. Author’s estimate for 2010. Reprinted with permission.