Chapter 10, Problem 1P

The figure on page informalfigure shows the one-year return distribution for RCS stock. Calculate

1. a. The expected return.
2. b. The standard deviation of the return.

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 stock return is -25%, probability of another stock is 20% and its return is -10%, the probability of a stock is 25% and the stock return is 10%, and probability of a stock is 30% and the stock return is 25%.

The formula to calculate the expected return on the stock:

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

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.

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 stock return is -25%, probability of another stock is 20% and its return is -10%, the probability of a stock is 25% and the stock return is 10%, and probability of a stock is 30% and the stock return is 25%.

The formula to calculate the standard deviation:

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

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%.