Chapter 7, Problem 1PS

Expected return and standard deviation A game of chance offers the following odds and payoffs. Each play of the game costs $100, so the net profit per play is the payoff less $100.

What are the expected cash payoff and expected rate of return? Calculate the variance and standard deviation of this rate of return.

Summary Introduction

To compute: The expected payoff and expected rate of return.

Explanation of Solution

The formula to calculate expected payoff is as follows:

$\begin{array}{c}\text{Expected payoff}=\left({\text{Probability}}_{\text{1}}{\text{×Payoff}}_{\text{1}}\right)+\left({\text{Probability}}_2{\text{×Payoff}}_2\right)\\ +\left({\text{Probability}}_3{\text{×Payoff}}_3\right)\end{array}$

The computation of expected payoff is as follows:

$\begin{array}{c}\text{Expected payoff}=\left(.10\times \$500\right)+\left(.50\times \$100\right)+\left(.40\times \$0\right)\\ =\$100\end{array}$

Hence, the expected payoff is $100

The formula to calculate rate of return is as follows:

$\text{Rates of return}=\left(\frac{\text{Payoff}-\text{Cost}}{100}\right)$

The calculation of rate of return is as follows:

$\begin{array}{l}\begin{array}{l}\frac{\left(\$500–100\right)}{\$100}=400\%\hfill \\ \frac{\left(\$100–100\right)}{\$100}=0\%\hfill \\ \frac{\left(\$0–100\right)}{\$100}=–100\%\hfill \end{array}\\ \end{array}$

The formula to calculate expected rate of return is as follows:

$\begin{array}{c}\text{Expected rate of return}=\left({\text{Probability}}_{\text{1}}\times {\text{Rate of return}}_{{}_{{}_{\text{1}}}}\right)+\left({\text{Probability}}_1\times {\text{Rate of return}}_2\right)\\ +\left({\text{Probability}}_3\times {\text{Rate of return}}_3\right)\end{array}$

The calculation of expected rate of return is as follows:

$\begin{array}{c}\text{Expected rate of return }=\left(.10\times 400\%\right)+\left(.50\times 0\%\right)+\left(.40\times –100\%\right)\\ =0\%.\end{array}$

Hence, the expected rate of return is 0%.

Summary Introduction

To compute: The variance and standard deviation of rate of return.

Explanation of Solution

The formula to calculate variance is as follows:

$\begin{array}{c}\text{Variance}={\text{Probability}}_1\times {\left({\text{Rate of return}}_1-{\text{Expectedreturn}}_1\right)}^2+\\ {\text{Probability}}_2\times {\left({\text{Rate of return}}_2-{\text{Expectedreturn}}_2\right)}^2+\\ {\text{Probability}}_3\times {\left({\text{Rate of return}}_3-{\text{Expectedreturn}}_3\right)}^2\end{array}$

The computation of variance is as follows:

$\begin{array}{c}\text{Variance}=.10{\left(400\%–0\right)}^2+.50{\left(0\%–0\right)}^2+.40{\left(–100\%–0\right)}^2\\ =20,000\end{array}$

Hence, the variance is 20,000.

The formula to compute standard deviation is as follows:

$\text{Standard deviation}=\sqrt{\text{Variance}}$

The computation of standard deviation is as follows:

$\begin{array}{c}\text{Standard deviation}=\sqrt{20,000}\\ =141.42\%.\end{array}$

Hence, the standard deviation is 141.42%.