Solution to 3. Table 3-21 summarizes the results from Part 2 for standard deviation and incorporates the results for MAD from Example 3-10.

Fund |
Standard Deviation |
Mean Absolute Deviation |

SLASX |
16.7% |
14.1% |

PRFDX |
10.0% |
6.9% |

Note that the mean absolute deviation is less than the standard deviation. The mean absolute deviation will always be less than or equal to the standard deviation because the standard deviation gives more weight to large deviations than to small ones (remember, the deviations are squared).

Because the standard deviation is a measure of dispersion about the arithmetic mean, we usually present the arithmetic mean and standard deviation together when summarizing data. When we are dealing with data that represent a time series of percent changes, presenting the geometric mean—representing the compound rate of growth—is also very helpful. Table 3-22 presents the historical geometric and arithmetic mean returns, along with the historical standard deviation of returns, for various equity return series. We present these statistics for nominal (rather than inflation-adjusted) returns so we can observe the original magnitudes of the returns.

Geometric |
Arithmetic |
Standard | |

Return Series |
Mean |
Mean |
Deviation |

I. Ibbotson Associates Series: 1926-2002 | |||

S&P 500 (Annual) |
10.20% |
12.20% |
20.49 |

S&P 500 (Monthly) |
0.81% |
0.97% |
5.65 |

II. Dimson et al. (2002) Series (Annual): 1900-2000 | |||

Australia |
11.9% |
13.3% |
18.2% |

Belgium |
8.2% |
10.5% |
24.1% |

Canada |
9.7% |
11.0% |
16.6% |

Denmark |
8.9% |
10.7% |
21.7% |

France |
12.1% |
14.5% |
24.6% |

Germany |
9.7% |
15.2% |
36.4% |

Ireland |
9.5% |
11.5% |
22.8% |

Italy |
12.0% |
16.1% |
34.2% |

Japan |
12.5% |
15.9% |
29.5% |

Netherlands |
9.0% |
11.0% |
22.7% |

South Africa |
12.0% |
14.2% |
23.7% |

Spain |
10.0% |
12.1% |
22.8% |

Sweden |
11.6% |
13.9% |
23.5% |

Switzerland |
7.6% |
9.3% |
19.7% |

United Kingdom |
10.1% |
11.9% |
21.8% |

United States |
10.1% |
12.0% |
19.9% |

Source: Ibbotson EncorrAnalyzer; Dimson et al.

Source: Ibbotson EncorrAnalyzer; Dimson et al.

7.5 Semivariance, Semideviation, and Related Concepts

An asset's variance or standard deviation of returns is often interpreted as a measure of the asset's risk. Variance and standard deviation of returns take account of returns above and below the mean, but investors are concerned only with downside risk, for example returns below the mean. As a result, analysts have developed semivariance, semideviation, and related dispersion measures that focus on downside risk. Semivariance is defined as the average squared deviation below the mean. Semideviation (sometimes called semistandard deviation) is the positive square root of semivariance. To compute the sample semivariance, for example, we take the following steps:

i. Calculate the sample mean.

ii. Identify the observations that are smaller than the mean (discarding observations equal to and greater than the mean); suppose there are n* observations smaller than the mean.

iii. Compute the sum of the squared negative deviations from the mean (using the n* observations that are smaller than the mean).

iv. Divide the sum of the squared negative deviations from Step iii by n* - 1. A formula for semivariance is

for all xt<x

To take the case of Selected American Shares with returns (in percent) of 16.2,20.3,9.3, -11.1, and -17.0, we earlier calculated a mean return of 3.54 percent. Two returns, -11.1 and -17.0, are smaller than 3.54 (n* = 2). We compute the sum of the squared negative deviations from the mean as (-11.1 - 3.54)2 + (-17.0 - 3.54)2 = -14.642 + -20.542 = 214.3296 + 421.8916 = 636.2212. With n* - 1 = 1, we conclude that semi-variance is 636.2212/1 = 636.2212 and that semideviation is V636.2212 = 25.2 percent, approximately. The semideviation of 25.2 percent is greater than the standard deviation of 16.7 percent. From this downside risk perspective, therefore, standard deviation understates risk.

In practice, we may be concerned with values of return (or another variable) below some level other than the mean. For example, if our return objective is 10 percent annually, we may be concerned particularly with returns below 10 percent a year. We can call 10 percent the target. The name target semivariance has been given to average squared deviation below a stated target, and target semideviation is its positive square root. To calculate a sample target semivariance, we specify the target as a first step. After identifying observations below the target, we find the sum of the squared negative deviations from the target and divide that sum by the number of observations below the target minus 1. A formula for target semivariance is

for all Xi<B

where B is the target and n* is the number of observations below the target. With a target return of 10 percent, we find in the case of Selected American Shares that three returns (9.3, -11.1, and -17.0) were below the target. The target semivariance is [(9.3 -10.0)2 + (-11.1 - 10.0)2 + (-17.0 - 10.0)2 ]/(3 - 1) = 587.35, and the target semi-deviation is V587.35 = 24.24 percent, approximately.

When return distributions are symmetric, semivariance is a constant proportion (one-half) of variance and the two measures are effectively equivalent. For asymmetric distributions, variance and semivariance rank prospects' risk differently.32 Semivariance (or

32 For negatively skewed returns, semivariance is greater than one-half variance; for positively skewed returns, semivariance is less than one-half variance. See Estrada (2003). We discuss skewness later in this chapter.

semideviation) and target semivariance (or target semideviation) have intuitive appeal, but they are harder to work with mathematically than variance.33 Variance or standard deviation enters into the definition of many of the most commonly used finance risk concepts, such as the Sharpe ratio and beta. Perhaps because of these reasons, variance (or standard deviation) is much more frequently used in investment practice.

7.6 chebyshev's The Russian mathematician Pafnuty Chebyshev developed an inequality using standard inequality deviation as a measure of dispersion. The inequality gives the proportion of values within k standard deviations of the mean.

• Definition of Chebyshev's Inequality. According to Chebyshev's inequality, the proportion of the observations within k standard deviations of the arithmetic mean is at least 1 - 1II? for all £ > 1.

Table 3-23 illustrates the proportion of the observations that must lie within a certain number of standard deviations around the sample mean.

k |
Interval Around the Sample Mean |
Proportion |

1.25 |
X ± 1.25s |
36% |

1.50 |
X ± 1.50» |
56% |

2.00 |
X±2 s |
75% |

2.50 |
X ± 2.50s |
84% |

3.00 |
X±3s |
89% |

4.00 |
X ± As |
94% |

Note: Standard deviation is denoted as s.

Note: Standard deviation is denoted as s.

When k = 1.25, for example, the inequality states that the minimum proportion of the observations that lie within ± 1.25.? is 1 - 1/(1.25)2 = 1 - 0.64 = 0.36 or 36 percent.

The most frequently cited facts that result from Chebyshev's inequality are that a two-standard-deviation interval around the mean must contain at least 75 percent of the observations, and a three standard deviation interval around the mean must contain at least 89 percent of the observations, no matter how the data are distributed.

The importance of Chebyshev's inequality stems from its generality. The inequality holds for samples and populations and for discrete and continuous data regardless of the shape of the distribution. As we shall see in the chapter on sampling, we can make much more precise interval statements if we can assume that the sample is drawn from a population that follows a specific distribution called the normal distribution. Frequently, however, we cannot confidently assume that distribution.

The next example illustrates the use of Chebyshev's inequality.

33 As discussed in the chapter on probability concepts and the chapter on portfolio concepts, we can find a portfolio's variance as a straightforward function of the variances and correlations of the component securities. There is no similar procedure for semivariance and target semivariance. We also cannot take the derivative of semivariance or target semivariance.

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