What is the variance of the returns on a security given the following information?
$\begin{array}{lcc}
\text { State of the Economy }&\text { Probability}&\text { E(R) }\
\text { Boom } & .25 & 16\% \
\text { Normal } & .45 & 10\% \
\text { Recession } & .30 & -8 \%
\end{array}$

A)48.18%
B)56.23%
C)64.38%
D)72.87%
E)90.99%

Question

What is the variance of the returns on a security given the following information?
$\begin{array}{lcc}
\text { State of the Economy }&\text { Probability}&\text { E(R) }\
\text { Boom } & .25 & 16\% \
\text { Normal } & .45 & 10\% \
\text { Recession } & .30 & -8 \%
\end{array}$

A)48.18%
B)56.23%
C)64.38%
D)72.87%
E)90.99%

Quizplus · Accepted Answer

The variance of the returns on a security can be calculated by finding the expected value of the squared deviations from the mean return, weighted by the probabilities of each state of the economy. First, calculate the expected return (E(R)):$$E(R) = (0.25 \times 16\%) + (0.45 \times 10\%) + (0.30 \times -8\%) = 4\% + 4.5\% - 2.4\% = 6.1\%$$Next, calculate the variance:$$Variance = (0.25 \times (16\% - 6.1\%)^2) + (0.45 \times (10\% - 6.1\%)^2) + (0.30 \times (-8\% - 6.1\%)^2)$$$$= (0.25 \times (9.9\%)^2) + (0.45 \times (3.9\%)^2) + (0.30 \times (-14.1\%)^2)$$$$= (0.25 \times 0.09801) + (0.45 \times 0.01521) + (0.30 \times 0.19881)$$$$= 0.0245025 + 0.0068445 + 0.059643$$$$= 0.09099$$Therefore, the variance of the returns on the security is 90.99%.

What Is the Variance of the Returns on a Security