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 10.885%.

Explanation of Solution

Given information:

The rate of return of Stock A is 7 percent, Stock B is 1 percent, and Stock C is 27 percent when the economy is in booming condition. The rate of return of Stock A is 12 percent, Stock B is 19 percent, and Stock C is −5 percent when the economy is in busting condition.

The probability of having a boom is 75 percent, and the probability of having a bust cycle is 25 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.

$\begin{array}{c}\text{Expected returns}=\left[\begin{array}{l}\left(\text{Possible returns}\left(R_1\right)\times \text{Weights}\left({\text{W}}_{\text{1}}\right)\right)\\ +\left(\text{Possible returns}\left(R_2\right)\times \text{Weights}\left({\text{W}}_{\text{2}}\right)\right)\\ +\left(\text{Possible returns}\left(R_3\right)\times \text{Weights}\left({\text{W}}_{\text{3}}\right)\right)\end{array}\right]\\ =\left(0.07\times \frac{1}{3}\right)+\left(0.01\times \frac{1}{3}\right)+\left(0.27\times \frac{1}{3}\right)\\ =0.023+0.0033+0.09\\ =0.1163\text{or}11.63\%\end{array}$

Hence, the expected return of the boom economy is 11.63%.

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

$\begin{array}{c}\text{Expected returns}=\left[\begin{array}{l}\left(\text{Possible returns}\left(R_1\right)\times \text{Weights}\left({\text{W}}_{\text{1}}\right)\right)\\ +\left(\text{Possible returns}\left(R_2\right)\times \text{Weights}\left({\text{W}}_{\text{2}}\right)\right)\\ +\left(\text{Possible returns}\left(R_3\right)\times \text{Weights}\left({\text{W}}_{\text{3}}\right)\right)\end{array}\right]\\ =\left(0.12\times \frac{1}{3}\right)+\left(0.19\times \frac{1}{3}\right)+\left(\left(-0.05\right)\times \frac{1}{3}\right)\\ =0.04+0.0633-0.0167\\ =0.0866\text{or}\text{8}\text{.66\%}\end{array}$

Hence, the expected return of the bust economy is 8.66%.

Compute the portfolio expected return:

$\begin{array}{c}E\left(R_P\right)=\left[x_1\times E\left(R_{boom}\right)\right]+\left[x_2\times E\left(R_{bust}\right)\right]\\ =\left(0.75\times 0.1163\right)+\left(0.25\times 0.0866\right)\\ =0.0872+0.02165\\ =0.10885or10.885\%\end{array}$

Hence, the expected return on the portfolio is 10.885%.

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 0.00746.

Explanation of Solution

Given information:

The rate of return of Stock A is 7 percent, Stock B is 1 percent, and Stock C is 27 percent when the economy is in booming condition. The rate of return of Stock A is 12 percent, Stock B is 19 percent, and Stock C is −5 percent when the economy is in busting condition.

The probability of having a boom is 75 percent, and the probability of having a bust cycle is 25 percent and invested 20% in A and B and 60% in C.

All the above stocks carry equal weight in the portfolio.

Compute the portfolio return during a boom:

$\begin{array}{c}{R}_P\text{ during boom}=\left[x_{\text{Stock A}}\times R_{\text{Stock A}}\right]+\left[x_{\text{Stock B}}\times R_{\text{Stock B}}\right]+\left[x_{\text{Stock C}}\times R_{\text{Stock C}}\right]\\ =\left(0.20\times 0.07\right)+\left(0.20\times 0.01\right)+\left(0.60\times 0.27\right)\\ =0.014+0.002+0.162\\ =0.178\end{array}$

Hence, the return on the portfolio during a boom is 0.178 or 17.8%.

Compute the portfolio return during a bust cycle:

$\begin{array}{c}{R}_P\text{ during bust}=\left[x_{\text{Stock A}}\times R_{\text{Stock A}}\right]+\left[x_{\text{Stock B}}\times R_{\text{Stock B}}\right]+\left[x_{\text{Stock C}}\times R_{\text{Stock C}}\right]\\ =\left(0.20\times 0.12\right)+\left(0.20\times 0.19\right)+\left(0.60\times \left(-0.05\right)\right)\\ =0.024+0.038-0.03\\ =-0.032\end{array}$

Hence, the return on the portfolio during a bust cycle is −3.2% or− 0.032.

Compute the portfolio expected return:

$\begin{array}{c}E\left(R_P\right)=\left[x_1\times E\left(R_1\right)\right]+\left[x_{\text{2}}\times E\left(R_2\right)\right]\\ =\left(0.75\times 0.178\right)+\left(0.25\times -0.032\right)\\ =0.1335-0.008\\ =0.1255\end{array}$

Hence, the expected return on the portfolio is 0.1225 or 12.55%.

Compute the variance:

The probability of having a boom is 75 percent, and the probability of having a bust cycle is 25 percent.

$\begin{array}{c}\text{Variance}=\left(\begin{array}{l}\left[{\left(\text{Possible returns}\left(R_1\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_1\right)\right]+\\ \left[{\left(\text{Possible returns}\left(R_2\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_2\right)\right]\end{array}\right)\\ =\left[{\left(0.178-\left(0.75\times 0.178\right)\right)}^2\times 0.75\right]+\left[{\left(\left(-0.032\right)-0.1225\right)}^2\times 0.25\right]\\ =\left[{\left(0.178-0.1335\right)}^2\times 0.75\right]+\left[{\left(-0.1545\right)}^2\times 0.25\right]\\ =\left({\left(0.0445\right)}^2\times 0.75\right)+\left(0.0239\times 0.25\right)\end{array}$

$\begin{array}{c}=\left(0.00198\times 0.75\right)+0.005975\\ =0.001485+0.005975\\ =0.00746\end{array}$

Hence, the variance of the portfolio is 0.00746.