Chapter 10, Problem 1P

a)

Summary Introduction

To determine: The expected return.

Introduction:

Expected return refers to a return that the investors expect on a risky investment in the future.

Answer to Problem 1P

The expected return is 5.5 percent.

Explanation of Solution

Given information:

The probability of a stock is 10% and the return from that stock is −25%; the probability of another stock is 20% and its return is −10%; the probability of a third stock is 25% and the return from that stock is 10%; the probability of a fourth stock is 30% and the return from that stock is 25%.

The formula to calculate the expected return on the stock is as follows:

$\text{Expected returns}=\left[\begin{array}{l}\left(\text{Possible returns}\left(R_1\right)\times \text{Probability}\left(P_1\right)\right)\\ +\left(\text{Possible returns}\left(R_2\right)\times \text{Probability}\left(P_2\right)\right)\\ +\mathrm{...}+\left(\text{Possible returns}\left(R_n\right)\times \text{Probability}\left(P_n\right)\right)\end{array}\right]$

Compute the expected return:

$\begin{array}{c}\text{Expected returns}=\left[\begin{array}{l}\left(\text{Possible returns}\left(R_1\right)\times \text{Probability}\left(P_1\right)\right)\\ +\left(\text{Possible returns}\left(R_2\right)\times \text{Probability}\left(P_2\right)\right)\\ +\left(\text{Possible returns}\left(R_3\right)\times \text{Probability}\left(P_3\right)\right)\\ +\left(\text{Possible returns}\left(R_4\right)\times \text{Probability}\left(P_4\right)\right)\end{array}\right]\\ =\left(-\frac{25}{100}\times \frac{10}{100}\right)+\left(-\frac{10}{100}\times \frac{20}{100}\right)+\left(\frac{10}{100}\times \frac{25}{100}\right)+\left(\frac{25}{100}\times \frac{30}{100}\right)\\ =\left(-0.25\times \left(0.10\right)\right)+\left(-0.10\times 0.20\right)+\left(0.10\times 0.25\right)+\left(0.25\times 0.30\right)\\ =\left(-0.025-0.02\right)+\left(0.025+0.075\right)\end{array}$

$\begin{array}{c}=-\text{}0.145+0.1\\ =0.055\text{ or 5}\text{.5\%}\end{array}$

Hence, the expected return is 5.5 percent.

b)

Summary Introduction

To determine: The standard deviation of the return.

Introduction:

Standard deviation refers to the variation in the actual returns from the expected returns.

Variance refers to the average difference of squared deviations of the actual data from the mean or average.

Answer to Problem 1P

The standard deviation of the return is 16.13%.

Explanation of Solution

Given information:

The probability of a stock is 10% and the return from that stock is −25%; the probability of another stock is 20% and its return is −10%; the probability of a third stock is 25% and the return from that stock is 10%; the probability of a fourth stock is 30% and the return from that stock is 25%.

The formula to calculate the standard deviation is as follows:

$\begin{array}{l}\text{Standard}\\ \text{deviation}\end{array}\}=\sqrt{\left(\begin{array}{l}\left[{\left(\text{Possible returns}\left(R_1\right)-\text{Expected returns}\right)}^2\times \text{Probability}\left(P_1\right)\right]\\ +\left[{\left(\text{Possible returns}\left(R_2\right)-\text{Expected returns}\right)}^2\times \text{Probability}\left(P_2\right)\right]\\ +\mathrm{...}+\left[{\left(\text{Possible returns}\left(R_n\right)-\text{Expected returns}\right)}^2\times \text{Probability}\left(P_n\right)\right]\end{array}\right)}$

Calculate the standard deviation:

$\begin{array}{c}\begin{array}{l}\text{Standard}\\ \text{deviation}\end{array}\}=\sqrt{\left(\begin{array}{l}\left[{\left(\text{Possible returns}\left(R_1\right)-\text{Expected returns}\right)}^2\times \text{Probability}\left(P_1\right)\right]\\ +\left[{\left(\text{Possible returns}\left(R_2\right)-\text{Expected returns}\right)}^2\times \text{Probability}\left(P_2\right)\right]\\ +\left[{\left(\text{Possible returns}\left(R_3\right)-\text{Expected returns}\right)}^2\times \text{Probability}\left(P_3\right)\right]\\ +\left[{\left(\text{Possible returns}\left(R_4\right)-\text{Expected returns}\right)}^2\times \text{Probability}\left(P_4\right)\right]\end{array}\right)}\\ =\sqrt{\left[\begin{array}{c}{\left(-\frac{25}{100}-\frac{5.5}{100}\right)}^2\times \frac{10}{100}+{\left(-\frac{10}{100}-\frac{5.5}{100}\right)}^2\times \frac{20}{100}\\ +{\left(\frac{10}{100}-\frac{5.5}{100}\right)}^2\times \frac{25}{100}+{\left(\frac{25}{100}-\frac{5.5}{100}\right)}^2\times \frac{30}{100}\end{array}\right]}\\ =\sqrt{\left[\begin{array}{c}{\left(-0.25-0.055\right)}^2\times 0.10+{\left(-0.10-0.055\right)}^2\times 0.20\\ +{\left(0.10-0.055\right)}^2\times 0.25+{\left(0.25-0.055\right)}^2\times 0.30\end{array}\right]}\\ =\sqrt{\left[\begin{array}{c}\left(0.093025\times 0.10\right)+\left(0.024025\times 0.20\right)\\ +\left(0.002025\times 0.25\right)+0.038025\times 0.30\end{array}\right]}\end{array}$

$\begin{array}{c}=\sqrt{0.0093025+0.004805+0.00050625+0.0114075}\\ =\sqrt{0.026}\\ =\text{0}\text{.1613 or 16}\text{.13\%}\end{array}$

Hence, the standard deviation of the return is 16.13%.