Chapter 13, 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.

Answer to Problem 9QP

The expected return on the portfolio is 0.1308 or 13.08 percent.

Explanation of Solution

Given information:

Stock A’s return is 7 percent when the economy is booming, and 13 percent when the economy is in a bust cycle. The probability of having a boom is 65 percent, and the probability of having a bust cycle is 35 percent.

Stock B’s return is 15 percent when the economy is booming, and 3 percent when the economy is in a bust cycle. The probability of having a boom is 65 percent, and the probability of having a bust cycle is 35 percent.

Stock C’s return is 33 percent when the economy is booming, and (−6 percent) when the economy is in a bust cycle. The probability of having a boom is 65 percent, and the probability of having a bust cycle is 35 percent. All the above stocks carry an equal weight in the portfolio.

The formula to calculate the expected return on the stock:

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

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 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.

Compute the expected return on Stock A:

R₁ refers to the returns during a boom. The probability of having a boom is P₁.R₂ is the returns in a bust cycle. The probability of having a bust cycle is “P₂”.

$\begin{array}{c}\text{Expected returns}=\left[\begin{array}{l}\left(\text{Possible returns}\left(R_1\right)\times \text{Probability}\left(P_1\right)\right)\\ +\left(\text{Possible returns}\left(R_2\right)\times \text{Probability}\left(P_2\right)\right)\end{array}\right]\\ =\left(0.07\times 0.65\right)+\left(0.13\times 0.35\right)\\ =0.0455+0.0455\\ =0.091\end{array}$

Hence, the expected return on Stock A is 0.091 or 9.1 percent.

Compute the expected return on Stock B:

R₁ refers to the returns during a boom. The probability of having a boom isP₁.R₂ is the returns in a bust cycle. The probability of having a bust cycle is P₂.

$\begin{array}{c}\text{Expected returns}=\left[\begin{array}{l}\left(\text{Possible returns}\left(R_1\right)\times \text{Probability}\left(P_1\right)\right)\\ +\left(\text{Possible returns}\left(R_2\right)\times \text{Probability}\left(P_2\right)\right)\end{array}\right]\\ =\left(0.15\times 0.65\right)+\left(0.03\times 0.35\right)\\ =0.0975+0.0105\\ =0.108\end{array}$

Hence, the expected return on Stock B is 0.108 or 10.8 percent.

Compute the expected return on Stock C:

R₁ refers to the returns during a boom. The probability of having a boom is P₁.R₂ is the returns in a bust cycle. The probability of having a bust cycle is P₂.

$\begin{array}{c}\text{Expected returns}=\left[\begin{array}{l}\left(\text{Possible returns}\left(R_1\right)\times \text{Probability}\left(P_1\right)\right)\\ +\left(\text{Possible returns}\left(R_2\right)\times \text{Probability}\left(P_2\right)\right)\end{array}\right]\\ =\left(0.33\times 0.65\right)+\left(\left(-0.06\right)\times 0.35\right)\\ =0.2145+\left(-0.021\right)\\ =0.1935\end{array}$

Hence, the expected return on Stock C is 0.1935 or 19.35 percent.

Compute the portfolio expected return:

The expected return on Stock A is 9.1 percent (“E(R_{Stock A})”), the expected return on Stock B is 10.8 percent (“E(R_{Stock B})”), and the expected return on Stock C is 19.35 percent (“E(R_{Stock C})”).

It is given that the weight of the stocks is equal. Hence, the weight of Stock A is $\left(\frac{1}{3}\right)$ (x_{Stock A}), the weight of Stock B is $\left(\frac{1}{3}\right)$(x_{Stock B}), and the weight of Stock C is $\left(\frac{1}{3}\right)$(x_{Stock C}).

$\begin{array}{c}E\left(R_P\right)=\left[x_{\text{Stock A}}\times E\left(R_{\text{Stock A}}\right)\right]+\left[x_{\text{Stock B}}\times E\left(R_{\text{Stock B}}\right)\right]+\left[x_{\text{Stock C}}\times E\left(R_{\text{Stock C}}\right)\right]\\ =\left(\frac{1}{3}\times 0.091\right)+\left(\frac{1}{3}\times 0.108\right)+\left(\frac{1}{3}\times 0.1935\right)\\ =0.03033+0.036+0.0645\\ =0.1308\end{array}$

Hence, the expected return on the portfolio is 0.1308 or 13.08 percent.

b)

Summary Introduction

To determine: The variance of the portfolio.

Introduction:

Portfolio expected return refers to the return that the investors expect on a portfolio of investments. 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 0.0137 or 1.37 percent.

Explanation of Solution

Given information:

Stock A’s return is 7 percent when the economy is booming and 13 percent when the economy is in a bust cycle. The probability of having a boom is 65 percent, and the probability of having a bust cycle is 35 percent.

Stock B’s return is 15 percent when the economy is booming and 3 percent when the economy is in a bust cycle. The probability of having a boom is 65 percent, and the probability of having a bust cycle is 35 percent.

Stock C’s return is 33 percent when the economy is booming and (−6 percent) when the economy is in a bust cycle. The probability of having a boom is 65 percent, and the probability of having a bust cycle is 35 percent.

The expected return on Stock A is 9.1 percent, the expected return on Stock B is 10.8 percent, and the expected return on Stock C is 19.35 percent (Refer to Part (a) of the solution). Stock A and Stock B have a weight of 20 percent each and Stock C has a weight of 60 percent in the portfolio.

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 portfolioreturn during a boom:

The return on Stock A is 7 percent “R_{Stock A}”, the return on Stock B is 15 percent “R_{Stock B}”, and the return on Stock C is 33 percent “R_{Stock C}” when the economy is booming. 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}).

$\begin{array}{c}{R}_P\text{ during boom}=\left[x_{\text{Stock A}}\times R_{\text{Stock A}}\right]+\left[x_{\text{Stock B}}\times R_{\text{Stock B}}\right]+\left[x_{\text{Stock C}}\times R_{\text{Stock C}}\right]\\ =\left(0.20\times 0.07\right)+\left(0.20\times 0.15\right)+\left(0.60\times 0.33\right)\\ =0.014+0.03+0.198\\ =0.242\end{array}$

Hence, the return on the portfolio during a boom is 0.242 or 24.2  percent.

Compute the portfolioreturn during a bust cycle:

The return on Stock A is 13 percent “R_{Stock A}”, the return on Stock B is 3 percent “R_{Stock B}”, and the return on Stock C is (−6 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}).

$\begin{array}{c}{R}_P\text{ during bust}=\left[x_{\text{Stock A}}\times R_{\text{Stock A}}\right]+\left[x_{\text{Stock B}}\times R_{\text{Stock B}}\right]+\left[x_{\text{Stock C}}\times R_{\text{Stock C}}\right]\\ =\left(0.20\times 0.13\right)+\left(0.20\times 0.03\right)+\left(0.60\times \left(-0.06\right)\right)\\ =0.026+\text{0}\text{.006}+\left(-\text{0}\text{.036}\right)\\ =\left(-\text{0}\text{.004}\right)\end{array}$

Hence, the return on the portfolio during a bust cycle is −0.004 or (−0.4 percent).

Compute the portfolio expected return:

The expected return on Stock A is 9.1 percent (“E(R_{Stock A})”), the expected return on Stock B is 10.8 percent (“E(R_{Stock B})”), and the expected return on Stock C is 19.35 percent (“E(R_{Stock C})”).

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}).

$\begin{array}{c}E\left(R_P\right)=\left[x_{\text{Stock A}}\times E\left(R_{\text{Stock A}}\right)\right]+\left[x_{\text{Stock B}}\times E\left(R_{\text{Stock B}}\right)\right]+\left[x_{\text{Stock C}}\times E\left(R_{\text{Stock C}}\right)\right]\\ =\left(0.20\times 0.091\right)+\left(0.20\times 0.108\right)+\left(0.60\times 0.1935\right)\\ =0.0182+0.0216+0.1161\\ =0.1559\end{array}$

Hence, the expected return on the portfolio is 0.1559 or 15.59 percent.

Compute the variance:

R₁ refers to the returns of the portfolio during a boom. The probability of having a boom is P_1.R₂is the returns of the portfolio in a bust cycle. The probability of having a bust cycle is P_2. The expected return on the portfolio is 15.59 percent.

The possible returns during a boom are 24.2 percent and during a bust cycle is (−0.4 percent). The probability of having a boom is 65 percent and the probability of having a bust cycle is 35 percent.

$\begin{array}{c}\text{Variance}=\left(\begin{array}{l}\left[{\left(\text{Possible returns}\left(R_1\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_1\right)\right]\\ +\left[{\left(\text{Possible returns}\left(R_2\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_2\right)\right]\end{array}\right)\\ =\left[{\left(0.242-0.1559\right)}^2\times 0.65\right]+\left[{\left(\left(-0.004\right)-0.1559\right)}^2\times 0.35\right]\\ =\left[{\left(0.0861\right)}^2\times 0.65\right]+\left[{\left(-0.1599\right)}^2\times 0.35\right]\\ =\left(0.00741321\times 0.65\right)+\left(0.02556801\times 0.35\right)\end{array}$

$\begin{array}{c}=0.0048185865+0.0089488035\\ =0.0137\end{array}$

Hence, the variance of the portfolio is 0.0137 or 1.37 percent.