Expected Values, Variances, and Standard Deviations

Expected Value

Expected value is an essential quantitative concept investors use to estimate investment returns and analyze any factor that may impact their financial position.

Mathematically, the expected value is the probability-weighted average of the possible outcomes of the random variable. For a random variable \(X\), the expected value of \(X\) is denoted \(E(X)\). More specifically,

$$ \begin{align*}
E(X) &=P(X_1) X_1+P(X_2) X_2+\cdots+P(X_n) X_n \\
&=\sum_{i=1}^n P(X_i) X_i \end{align*} $$

Where,

\(X_i\)= One of \(n\) possible outcomes of the discrete random variable \(X\).

\(P(X_i)=P(X_i = x_i)\) = Probability of \(X\) taking the value \(x\).

Note that the expected can be a forecast (looking into the future) or the true value of the population mean.

The sample mean differs from the expected value. The sample mean is a central value for a specific set of observations, calculated as an equally weighted average of those observations.

Example: Calculating 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

Recall that,

$$ \begin{align*}
E(X)&=\sum_{i=1}^n P(X_i) X_i \\
& =0.05 \times 0.65+0.07 \times 0.25+0.10 \times 0.08 \\
& =0.0325+0.0175+0.008 \\
& =0.058=5.8\%
\end{align*} $$

Variance and Standard Deviation

Consider expected value as a forecast of an investment’s outcome. Then, variance and standard deviation measure the risk of an investment, which is the dispersion of outcomes around the mean.

The variance of a random variable is the expected value (the probability-weighted average) of squared deviations from the random variable’s expected value. Denoted by \(\sigma^2 (X)\) or \(Var(X)\), its formula is given by:

$$ \begin{align*}
Var(X) & =E[X-E(X)]^2 \\
&=P(X_1) [X_1-E(X)]^2+P(X_2) [X_2-E(X)]^2+\cdots\\ & +P(X_n) [X_n-E(X)]^2 \\
\Rightarrow Var(X) & =\sum_{i=1}^n P(X_i) [X_i-E(X)]^2 \end{align*} $$

Since variance is in squared terms, it can take any number greater than or equal to \(0 (Var (X) \ge 0)\). Intuitively, if \(Var (X) = 0\), there is no risk (dispersion). On the other hand, if \(Var (X) \gt 0\), it signifies the dispersion of outcomes.

Moreover, \(Var(X)\) is a quantity given in square units of \(X\). That is, if the \(X\) is given in percentage, then \(Var(X)\) is given in squared percentage.

The standard deviation is the square root of variance:

$$ \sigma(X)=\sqrt{\sigma^2 (X)}=\sqrt{Var(X)} $$

The standard deviation is given in the same units as the random variance, making it easy to interpret.

Question

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 variance and standard deviation of the investment are closest to:

Solution

We know that,

$$ \begin{align*}
Var(X) &=\sum_{i=1}^n P(X_i) [X_i-E(X)]^2\\
& =0.65(0.05-0.058)^2+0.25(0.07-0.058)^2 \\ & +0.10(0.08-0.058)^2 \\
& =0.000126
\end{align*} $$

For standard deviation,

$$ \sigma(X)=\sqrt{Var(X)}=\sqrt{0.000126}=0.0112 $$