Chapter 13, Problem 9QP

Returns and Variances [LO1] Consider the following information:

a. What is the expected return on an equally weighted portfolio of these three stocks?

b. What is the variance of a portfolio invested 20 percent each in A and B and 60 percent in C?

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.

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 expected return on the portfolio is 11.42 percent.

Explanation of Solution

Given information:

Stock A’s return is 6 percent when the economy is booming and 11 percent when the economy is in a bust cycle. The probability of having a boom is 75 percent, and the probability of having a bust cycle is 25 percent.

Stock B’s return is 15 percent when the economy is booming and (4 percent) when the economy is in a bust cycle. The probability of having a boom is 75 percent, and the probability of having a bust cycle is 25 percent.

Stock C’s return is 25 percent when the economy is booming and (8 percent) when the economy is in a bust cycle. 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 on the stock:

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

The formula to calculate the portfolio expected return:

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

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”.

$\begin{array}{c}\text{Expected returns}\left[E\left(R\right)\right]=\left[\begin{array}{l}\left(\text{Possible returns}\left(R_1\right)\times \text{Probability}\left(P_1\right)\right)+\\ \left(\text{Possible returns}\left(R_2\right)\times \text{Probability}\left(P_2\right)\right)\end{array}\right]\\ =\left(0.06\times 0.75\right)+\left(0.11\times 0.25\right)\\ =0.045+0.0275\\ =0.0725\text{ or }7.25\%\end{array}$

Hence, the expected return on Stock A is 7.25 percent.

Compute the expected return on Stock B:

“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”.

$\begin{array}{c}\text{Expected returns}\left[E\left(R\right)\right]=\left[\begin{array}{l}\left(\text{Possible returns}\left(R_1\right)\times \text{Probability}\left(P_1\right)\right)+\\ \left(\text{Possible returns}\left(R_2\right)\times \text{Probability}\left(P_2\right)\right)\end{array}\right]\\ =\left(0.15\times 0.75\right)+\left(\left(0.04\right)\times 0.25\right)\\ =0.1125+\left(0.01\right)\\ =0.1025\text{ or }10.25\%\end{array}$

Hence, the expected return on Stock B is 10.25 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”.

$\begin{array}{c}\text{Expected returns}\left[E\left(R\right)\right]=\left[\begin{array}{l}\left(\text{Possible returns}\left(R_1\right)\times \text{Probability}\left(P_1\right)\right)+\\ \left(\text{Possible returns}\left(R_2\right)\times \text{Probability}\left(P_2\right)\right)\end{array}\right]\\ =\left(0.25\times 0.75\right)+\left(\left(0.08\right)\times 0.25\right)\\ =0.1875+\left(0.02\right)\\ =0.1675\text{ or }16.75\%\end{array}$

Hence, the expected return on Stock C is 16.75 percent.

Compute the portfolio expected return:

The expected return on Stock A is 7.25 percent (“E(R_{Stock A})”), the expected return on Stock B is 10.25 percent (“E(R_{Stock B})”), and the expected return on Stock C is 16.75 percent (“E(R_{Stock C})”).

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

$\begin{array}{c}E\left(R_P\right)=\left[x_{\text{Stock A}}\times E\left(R_{\text{Stock A}}\right)\right]+\left[x_{\text{Stock B}}\times E\left(R_{\text{Stock B}}\right)\right]+\left[x_{\text{Stock C}}\times E\left(R_{\text{Stock C}}\right)\right]\\ =\left(\frac{1}{3}\times 0.0725\right)+\left(\frac{1}{3}\times 0.1025\right)+\left(\frac{1}{3}\times 0.1675\right)\\ =0.0242+0.0342+0.0558\\ =0.1142\text{ or }11.42\%\end{array}$

Hence, the expected return on the portfolio is 11.42 percent.

Summary Introduction

To determine: The variance of the portfolio.

Answer to Problem 9QP

The variance of the portfolio is 0.00957 percent.

Explanation of Solution

Given information:

Stock A’s return is 6 percent when the economy is booming and 11 percent when the economy is in a bust cycle. The probability of having a boom is 75 percent, and the probability of having a bust cycle is 25 percent.

Stock B’s return is 15 percent when the economy is booming and (4 percent) when the economy is in a bust cycle. The probability of having a boom is 75 percent, and the probability of having a bust cycle is 25 percent.

Stock C’s return is 25 percent when the economy is booming and (8 percent) when the economy is in a bust cycle. The probability of having a boom is 75 percent, and the probability of having a bust cycle is 25 percent.

The expected return on Stock A is 7.25 percent, the expected return on Stock B is 10.25 percent, and the expected return on Stock C is 16.75 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:

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

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:

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

Compute the portfolio return during a boom:

The return on Stock A is 6 percent “R_{Stock A}”, the return on Stock B is 15 percent “R_{Stock B}”, and the return on Stock C is 25 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}).

$\begin{array}{c}{R}_P\text{ during boom}=\left[x_{\text{Stock A}}\times R_{\text{Stock A}}\right]+\left[x_{\text{Stock B}}\times R_{\text{Stock B}}\right]+\left[x_{\text{Stock C}}\times R_{\text{Stock C}}\right]\\ =\left(0.20\times 0.06\right)+\left(0.20\times 0.15\right)+\left(0.60\times 0.25\right)\\ =0.012+0.03+0.15\\ =0.1920\text{ or }19.20\%\end{array}$

Hence, the return on the portfolio during a boom is 19.20 percent.

Compute the portfolio return during a bust cycle:

The return on Stock A is 11 percent “R_{Stock A}”, the return on Stock B is (4 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}).

$\begin{array}{c}{R}_P\text{ during bust}=\left[x_{\text{Stock A}}\times R_{\text{Stock A}}\right]+\left[x_{\text{Stock B}}\times R_{\text{Stock B}}\right]+\left[x_{\text{Stock C}}\times R_{\text{Stock C}}\right]\\ =\left(0.20\times 0.11\right)+\left(0.20\times \left(0.04\right)\right)+\left(0.60\times \left(0.08\right)\right)\\ =0.022+\left(0.008\right)+\left(0.048\right)\\ =\left(0.034\right)\text{ or }\left(3.4\%\right)\end{array}$

Hence, the return on the portfolio during a bust cycle is (3.4 percent).

Compute the portfolio expected return:

The expected return on Stock A is 7.25 percent (“E(R_{Stock A})”), the expected return on Stock B is 10.25 percent (“E(R_{Stock B})”), and the expected return on Stock C is 16.75 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}).

$\begin{array}{c}E\left(R_P\right)=\left[x_{\text{Stock A}}\times E\left(R_{\text{Stock A}}\right)\right]+\left[x_{\text{Stock B}}\times E\left(R_{\text{Stock B}}\right)\right]+\left[x_{\text{Stock C}}\times E\left(R_{\text{Stock C}}\right)\right]\\ =\left(0.20\times 0.0725\right)+\left(0.20\times 0.1025\right)+\left(0.60\times 0.1675\right)\\ =0.0145+0.0205+0.1005\\ =0.1355\text{ or }13.55\%\end{array}$

Hence, the expected return on the portfolio is 13.55 percent.

Compute the variance:

“R₁” refers to the returns of the portfolio during a boom. The probability of having a boom is “P₁”. “R₂” is the returns of the portfolio in a bust cycle. The probability of having a bust cycle is “P₂”. The expected return on the portfolio is 13.55 percent. The possible returns during a boom are 19.2 percent and during a bust cycle is (3.4 percent).

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

$\begin{array}{c}\text{Variance}=\left(\begin{array}{l}\left[{\left(\text{Possible returns}\left(R_1\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_1\right)\right]+\\ \left[{\left(\text{Possible returns}\left(R_2\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_2\right)\right]\end{array}\right)\\ =\left[{\left(0.192-0.1355\right)}^2\times 0.75\right]+\left[{\left(\left(0.034\right)-0.1355\right)}^2\times 0.25\right]\\ =0.00239+0.00718\\ =0.00957\end{array}$

Hence, the variance of the portfolio is 0.00957.