632598256894

considering the risk of a single asset held by itself and then move on to the case of a single asset held
in a portfolio of assets.
Stand-alone risk is the risk associated with an investment in a single asset held alone. In this case,
the investor has all of his or her wealth invested in this single asset and is entirely exposed to the
variability of returns of this asset, so we measure stand-alone risk as the risk of this asset by itself. To
do this we will look to the field of statistics, specifically the concept of standard deviation.
The standard deviation of returns for any investment, which is the square root of the variance of
returns for the investment, measures the dispersion or variability of returns around the mean return for
a period. A lower standard deviation of returns for an asset indicates a less variable and lower-risk
investment, and a higher standard deviation of returns for an asset indicates a more variable and
higher-risk investment.
Because of its predictive ability, standard deviation is particularly helpful in understanding the
variability and risk of an asset‟s returns. For any data series that follows a standard normal
distribution, the mean and standard deviation tell us how close or far the individual observations in the
series will be from the mean. Exhibit 5.3 presents this relationship between the mean (µ) and standard
deviation (σ ) for any standard normal data series. This relationship, along with knowledge of the
mean annual return and standard deviation of annual returns for any asset, allows us to predict the
distribution of returns and therefore the variability and risk of the asset.
Exhibit 5.3 The normal distribution.
The region in black on both side of the mean, plus or minus one standard deviation, accounts for
68.2% of the observations, so if the mean annual return of an asset is 15% and the standard deviation
of annual returns of the asset is 5%, then 68.2% of the asset‟s annual returns will be between 10% and
20%. The regions in dark gray and black on both sides of the mean, plus or minus two standard
deviations, account for 95.4% of the observations, so for the same asset 95.4% of the annual returns
will be between 5% and 25%. All three of the shaded regions on both sides of the mean, plus or minus
three standard deviations, account for about 99.7% of the observations, so for the same asset 99.7% of
the annual returns will be between 0% and 30%. This is the power of standard deviation in explaining
the variability and risk of an asset. If the asset has a high standard deviation of annual returns, then the
range of possible annual returns around the mean of the asset is very wide, and the asset is of higher
risk. If the asset, in contrast, has a low standard deviation of annual returns, then the range of possible
annual returns around the mean of the asset is very narrow, and the asset is of lower risk.
The standard deviation of returns for any asset is calculated in four steps: