###### Standard Error of the Sample Mean

When trying to estimate downside risk (i.e., returns below the mean), we can use the following measures:

**Semi-variance**: The average squared deviation below the mean.**Semi-deviation**(also known as semi-standard deviation): The positive square root of semi-variance.**Target semi-variance**: The sum of the squared deviations from a specific target return.**Target semi-deviation**: The square root of target semi-variance.

The target semi deviation, \(s_{\text {Target }}\), is calculated as follows:

$$s_{\text {Target }}=\sqrt{ \sum_{\text {for all } X_{i} \leq B}^{n} \frac{\left(X_{i}-B\right)^{2}}{n-1}}$$

Where \(B\) is the target and \(n\) is the total number of sample observations.

Yearly returns of an equity mutual fund are provided as follows.

$$

\begin{array}{c|c}

\textbf { Month } & \textbf { Return % } \\

\hline 2010 & 36 \% \\

\hline 2011 & 29 \% \\

\hline 2012 & 10 \% \\

\hline 2013 & 52 \% \\

\hline 2014 & 41 \% \\

\hline 2015 & 16 \% \\

\hline 2016 & 10 \% \\

\hline 2017 & 23 \% \\

\hline 2018 & -10 \% \\

\hline 2019 & -19 \% \\

\hline 2020 & 2 \% \\

\end{array}

$$

What is the target downside deviation if the target return is 20%?

**Solution**

$$

\begin{array}{c|c|c|c|c}

\textbf { Month } & \begin{array}{c}

\textbf { Return } \\

\%

\end{array} & \begin{array}{c}

\textbf { Deviation } \\

\textbf { from the 20% } \\

\textbf { target }

\end{array} & \begin{array}{c}

\textbf { Deviation } \\

\textbf { below the } \\

\textbf { target }

\end{array} & \begin{array}{c}

\textbf { Squared } \\

\textbf { deviations } \\

\textbf { below the } \\

\textbf { target }

\end{array} \\

\hline 2010 & 36.00 & 16.00 & – & – \\

\hline 2011 & 29.00 & 9.00 & – & – \\

\hline 2012 & 10.00 & (10.00) & (10.00) & 100 \\

\hline 2013 & 52.00 & 32.00 & – & \\

\hline 2014 & 41.00 & 21.00 & – & \\

\hline 2015 & 16.00 & (4.00) & (4.00) & 16 \\

\hline 2016 & 10.00 & (10.00) & (10.00) & 100 \\

\hline 2017 & 23.00 & 3.00 & – & \\

\hline 2018 & (10.00) & (30.00) & (30.00) & 900 \\

\hline 2019 & (19.00) & (39.00) & (39.00) & 1,521 \\

\hline 2020 & 2.00 & (18.00) & (18.00) & 324 \\

\hline {\text { Sum }} & {}&{}&{}&{\textbf{2,961}}\\

\end{array}

$$

Here \(n = 11 – 1 = 10\) so that:

$$\text{Target semi-deviation} = \left(\frac{2961 }{10}\right)^{0.5} = 17.21\%$$

The coefficient of variation, \(CV\), is a measure of spread that describes the amount of variability of data relative to its mean. It has **no units,** so we can use it as an alternative to the standard deviation to compare the variability of data sets that have different means. The coefficient of variation is given by:

$$ \text{CV} = \cfrac {S}{\bar{X}} $$

Where:

\(S\) = The standard deviation of a sample.

\(\bar{X}\) = The mean of the sample.

**Note****: **The formula can be replaced with \(\frac{σ}{μ}\) when dealing with a population.

**Procedure to Follow While Calculating the Coefficient of Variation**:

- Compute the mean of the data.
- Calculate the sample standard deviation of the data set, \(S\).
- Find the ratio of \(S\) to the mean, \(x̄\).

What is the relative variability for the samples 40, 46, 34, 35, and 45 of a population?

**Solution**

**Step 1:** Calculate the mean.

$$ \text{Mean} =\cfrac {(40 + 46 + 34 + 35 + 45)}{5} =\cfrac {200}{5} = 40 $$

**Step 2:** Calculate the sample standard deviation. (Start with the variance, \(S^2\).)

$$ \begin{align*} S^2 & =\cfrac {{(40 – 40)^2 + … + (45 – 40)^2 }}{4} \\ &=\cfrac {122}{4} \\ & = 30.5 \\ \end{align*} $$

**Note****: **Since it is the sample standard deviation (not the population standard deviation), we use \(n – 1\) as the denominator.

Therefore,

$$ S = \sqrt{30.5} = 5.52268 $$

**Step 3**: Calculate the ratio.

$$ \text{Ratio} =\cfrac {5.52268}{40} = 0.13806 \text{ or } 13.81\% $$

In finance, the coefficient of variation is used to measure the **risk per unit of return**. For example, imagine that the mean monthly return on a T-Bill is 0.5% with a standard deviation of 0.58%. Suppose we have another investment, say, Y, with a 1.5% mean monthly return and standard deviation of 6%, then,

$$ \text{CV}_{\text T-\text {Bill}} =\cfrac {0.58}{0.5} = 1.16 $$

$$ \text{CV}_\text{Y} =\cfrac {6}{1.5} = 4 $$

* Interpretation*: The dispersion per unit monthly return of T-Bills is less than that of Y. Therefore, investment Y is riskier than an investment on T-Bills.

QuestionIf a security has a mean expected return of 10% and a standard deviation of 5%, its coefficient of variation is

closestto:

- 0.005.
- 0.500.
- 2.000.

SolutionThe correct answer is

B.$$ \text{CV} = \cfrac {S}{\text x̄} = \cfrac {0.05}{0.10} = 0.5$$

Where:

\(S\) = The standard deviation of the sample.

\(x̄\) = The mean of the sample.

A is incorrect. It assumes the following calculation.$$CV=\frac{0.05}{10}=0.005$$

C is incorrect.It assumes the following calculation.$$CV=\frac{10}{5}=2$$