Chapter 11, Problem 9QP

a)

Summary Introduction

To determine: The expected return on the portfolio of equally weighted Stock A, Stock B, and Stock C.

Introduction:

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

Portfolio expected return refers to the return that the investors expect on a portfolio of investments.

Answer to Problem 9QP

The expected return on the portfolio is 11.67 percent.

Explanation of Solution

Given information:

The rate of return of Stock A is 15 percent, Stock B is 2 percent, and Stock C is 34 percent when the economy is in booming condition. The rate of return of Stock A is 3 percent, Stock B is 16 percent, and Stock C is −8 percent when the economy is in busting condition. The probability of having a boom is 60 percent, and the probability of having a bust cycle is 40 percent.

All the above stocks carry equal weight in the portfolio.

The formula to calculate the expected return of the portfolio in each state of the economy:

$\text{Expected returns}=\left[\begin{array}{l}\left(\text{Possible returns}\left(R_1\right)\times \text{Weights}\left({\text{W}}_{\text{1}}\right)\right)\\ +\left(\text{Possible returns}\left(R_2\right)\times \text{Weights}\left({\text{W}}_{\text{2}}\right)\right)\\ +\left(\text{Possible returns}\left(R_3\right)\times \text{Weights}\left({\text{W}}_{\text{3}}\right)\right)\end{array}\right]$

The formula to calculate the portfolio expected return:

$E\left(R_P\right)=\left[x_1\times E\left(R_1\right)\right]+\left[x_2\times E\left(R_2\right)\right]+\mathrm{...}+\left[x_n\times E\left(R_n\right)\right]$

Where,

E(R_P) refers to the expected return on a portfolio

“x₁ to x_n” refers to the probability of each asset from 1 to “n” in the portfolio

“E(R₁) to E(R_n) ” refers to the expected return on each asset from 1 to “n” in the portfolio

Compute the expected return of the portfolio of the boom economy:

R₁ refers to the rate of returns of Stock A. R₂ refers to the rate of returns of Stock B.

R₃ refers to the rate of returns of Stock C.

$\begin{array}{c}\text{Expected returns}=\left[\begin{array}{l}\left(\text{Possible returns}\left(R_1\right)\times \text{Weights}\left({\text{W}}_{\text{1}}\right)\right)\\ +\left(\text{Possible returns}\left(R_2\right)\times \text{Weights}\left({\text{W}}_{\text{2}}\right)\right)\\ +\left(\text{Possible returns}\left(R_3\right)\times \text{Weights}\left({\text{W}}_{\text{3}}\right)\right)\end{array}\right]\\ =\left(0.15\times \frac{1}{3}\right)+\left(0.02\times \frac{1}{3}\right)+\left(0.34\times \frac{1}{3}\right)\\ =0.05+0.00667+0.11333\\ =0.17\end{array}$

Hence, the expected return of the boom economy is 0.17 percent.

Compute the expected return of the portfolio of the bust economy:

$\begin{array}{c}\text{Expected returns}=\left[\begin{array}{l}\left(\text{Possible returns}\left(R_1\right)\times \text{Weights}\left({\text{W}}_{\text{1}}\right)\right)\\ +\left(\text{Possible returns}\left(R_2\right)\times \text{Weights}\left({\text{W}}_{\text{2}}\right)\right)\\ +\left(\text{Possible returns}\left(R_3\right)\times \text{Weights}\left({\text{W}}_{\text{3}}\right)\right)\end{array}\right]\\ =\left(0.03\times \frac{1}{3}\right)+\left(0.16\times \frac{1}{3}\right)+\left(\left(-0.08\right)\times \frac{1}{3}\right)\\ =0.01+0.053333-0.02667\\ =0.0367\end{array}$

Hence, the expected return of the bust economy is 0.0367 percent.

Compute the portfolio expected return:

$\begin{array}{c}E\left(R_P\right)=\left[x_1\times E\left(R_{boom}\right)\right]+\left[x_2\times E\left(R_{bust}\right)\right]\\ =\left(0.60\times 0.17\right)+\left(0.40\times 0.0367\right)\\ =0.102+0.01468\\ =0.1167\end{array}$

Hence, the expected return on the portfolio is 0.1167 or 11.67 percent.

b)

Summary Introduction

To determine: The variance of the portfolio

Introduction:

Portfolio variance refers to the average difference of squared deviations of the actual data from the mean or expected returns.

Answer to Problem 9QP

The variance of the portfolio is 0.014292 percent.

Explanation of Solution

Given information:

The rate of return of Stock A is 15 percent, Stock B is 2 percent, and Stock C is 34 percent when the economy is in booming condition. The rate of return of Stock A is 3 percent, Stock B is 16 percent, and Stock C is −8 percent when the economy is in busting condition. The probability of having a boom is 60 percent, and the probability of having a bust cycle is 40 percent.

The formula to calculate the portfolio expected return:

$E\left(R_P\right)=\left[x_1\times E\left(R_1\right)\right]+\left[x_2\times E\left(R_2\right)\right]+\mathrm{...}+\left[x_n\times E\left(R_n\right)\right]$

Where,

E(R_P) refers to the expected return on a portfolio

“x₁ to x_n” refers to the weight of each asset from 1 to “n” in the portfolio

“E(R₁) to E(R_n) ” refers to the expected return on each asset from 1 to “n” in the portfolio

The formula to calculate the variance of the portfolio:

$\text{Variance}=\left(\begin{array}{l}\left[{\left(\text{Possible returns}\left(R_1\right)-\text{Expected returns}E\left(R\right)\right)}^2\times \text{Probability}\left(P_1\right)\right]\\ +\mathrm{...}+\left[{\left(\text{Possible returns}\left(R_n\right)-\text{Expected returns}E\left(R\right)\right)}^2\times \text{Probability}\left(P_n\right)\right]\end{array}\right)$

Compute the portfolio return during a boom:

The return on Stock A is 15 percent “R_{Stock A}”, the return on Stock B is 2 percent “R_{Stock B}”, and the return on Stock C is 34 percent “R_{Stock C}” when there is a boom cycle. It is given that the weight of Stock A is 20 percent (x_{Stock A}), the weight of Stock B is 20 percent (x_{Stock B}), and the weight of Stock C is 60 percent (x_{Stock C}).

$\begin{array}{c}{R}_P\text{ during boom}=\left[x_{\text{Stock A}}\times R_{\text{Stock A}}\right]+\left[x_{\text{Stock B}}\times R_{\text{Stock B}}\right]+\left[x_{\text{Stock C}}\times R_{\text{Stock C}}\right]\\ =\left(0.20\times 0.15\right)+\left(0.20\times 0.02\right)+\left(0.60\times 0.34\right)\\ =0.03+0.004+0.204\\ =0.2380\end{array}$

Hence, the return on the portfolio during a boom is 0.2380 or 23.80 percent.

Compute the portfolio return during a bust cycle:

The return on Stock A is 3 percent “R_{Stock A}”, the return on Stock B is 16 percent “R_{Stock B}”, and the return on Stock C is (−8 percent) “R_{Stock C}” when there is a bust cycle. It is given that the weight of Stock A is 20 percent (x_{Stock A}), the weight of Stock B is 20 percent (x_{Stock B}), and the weight of Stock C is 60 percent (x_{Stock C}).

$\begin{array}{c}{R}_P\text{ during bust}=\left[x_{\text{Stock A}}\times R_{\text{Stock A}}\right]+\left[x_{\text{Stock B}}\times R_{\text{Stock B}}\right]+\left[x_{\text{Stock C}}\times R_{\text{Stock C}}\right]\\ =\left(0.20\times 0.03\right)+\left(0.20\times 0.16\right)+\left(0.60\times \left(-0.08\right)\right)\\ =0.006+0.032-0.048\\ =-0.01\end{array}$

Hence, the return on the portfolio during a bust cycle is −0.01percent or −1 percent.

Compute the portfolio expected return:

$\begin{array}{c}E\left(R_P\right)=\left[x_1\times E\left(R_1\right)\right]+\left[x_{\text{2}}\times E\left(R_2\right)\right]\\ =\left(0.60\times 0.2380\right)+\left(0.40\times -0.01\right)\\ =0.1428-0.004\\ =0.1388\end{array}$

Hence, the expected return on the portfolio is 0.1388 or 13.88 percent.

Compute the variance:

The probability of having a boom is 60 percent, and the probability of having a bust cycle is 40 percent.

$\begin{array}{c}\text{Variance}=\left(\begin{array}{l}\left[{\left(\text{Possible returns}\left(R_1\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_1\right)\right]+\\ \left[{\left(\text{Possible returns}\left(R_2\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_2\right)\right]\end{array}\right)\\ =\left[{\left(0.2380-\left(0.60\times 0.2380\right)\right)}^2\times 0.60\right]+\left[{\left(\left(-0.01\right)-0.1388\right)}^2\times 0.40\right]\\ =\left[{\left(0.2380-0.1428\right)}^2\times 0.60\right]+\left[{\left(-0.1488\right)}^2\times 0.40\right]\\ =\left({\left(0.0952\right)}^2\times 0.60\right)+\left(0.02214\times 0.40\right)\end{array}$

$\begin{array}{c}=\left(0.00906\times 0.60\right)+0.008856\\ =0.005436+0.008856\\ =0.014292\end{array}$

Hence, the variance of the portfolio is 0.014292.