Expected Value, Variance, and Standard Deviation of Random Variables

Expected Value

The expected value of a random variable is the average of the possible outcomes of that variable, taking the probability weights into account. Therefore:

$$ E\left( X \right) =\sum _{ i=1 }^{ n }{ { X }_{ i }P\left( { X }_{ i } \right) } $$

Example: Expected Value

An analyst anticipates the following returns from an asset:

$$ \begin{array}{c|c} {\textbf{Return}} & {\textbf{Probability}} \\ \hline {5\%} & { 65\%} \\ \hline {7\%} & { 25\%} \\ \hline {8\%} & { 10\%} \\ \end{array} $$

The expected value of the investment is closest to:

Solution

$$ \begin{align*} \text{Expected return} & = 0.05 × 0.65 + 0.07 × 0.25 + 0.10 × 0.08 \\ & = 0.0325 + 0.0175 + 0.008 \\ & = 0.058 \\ \end{align*} $$

Variance

The variance of a random variable is the sum of the squared deviations from the expected value weighted by respective probabilities. Therefore:

$$ { \sigma }^{ 2 }\left( X \right) =\sum _{ i=1 }^{ n }{ { \left[ { X }_{ i }-E\left( { X } \right) \right] }^{ 2 }P } \left( { X }_{ i } \right) =\left\{ { \left[ X-E\left( { X } \right) \right] }^{ 2 } \right\} $$

Example: Calculating Variance

Using the data from the previous example, we can compute the variance of return:

$$ \begin{align*} { \sigma }^{ 2 }\left( X \right) & =0.65{ (0.05-0.058) }^{ 2 }+0.25{ (0.07-0.058) }^{ 2 }+0.10{ (0.08-0.058) }^{ 2 } \\ & = 0.000126 \\ \end{align*} $$

Standard Deviation

Variance is not easy to interpret because it has squared units. Therefore, we usually use the standard deviation, which has the same units as the expected value. To get the standard deviation, we use the square root of variance:

$$ \begin{align*} \text{Standard deviation} & = \sqrt{\text{Variance}} \\ &= \sqrt{0.000126} \\ & =0.01122 \text{ or } 1.12\% \\ \end{align*} $$

Note: You can always raise the variance to 0.5 power to get the same result.

$$ \begin{align*} \text{Standard deviation} & = \text{Variance}^{0.5} \\ &= 0.000126^{0.5} \\ & =0.01122 \text{ or } 1.12\% \\ \end{align*} $$

Question

You have been given the following data indicating the returns likely to be earned on a stock alongside the corresponding probabilities:

$$ \begin{array}{c|c} {\textbf{Return}} & {\textbf{Probability}} \\ \hline {4\%} & { 40\%} \\ \hline {5\%} & { 25\%} \\ \hline {6\%} & { 35\%} \\ \end{array} $$

The standard deviation of expected returns is closest to:

A. 0.00007475.

B. 0.0495.

C. 0.008646.

Solution

The correct answer is C.

The first step involves determining the expected return:

$$ \begin{align*} E(X) & = (0.04 × 0.4) + (0.05 × 0.25) + (0.06 × 0.35) \\ & = 0.0495 \\ \end{align*} $$

Next, we must compute the variance of returns:

$$ \begin{align*} { \sigma }^{ 2 }\left( X \right) & =0.4(0.04–0.0495)^{ 2 }+0.25(0.05–0.0495)^{ 2 }+0.35(0.06 – 0.0495)^{ 2 } \\ & = 0.00007475 \\ \end{align*} $$

Lastly, we find the square root of variance to get the standard deviation of expected return:

$$ { \sigma }= \sqrt{0.00007475} = 0.008646 $$