Chapter 13, Problem 10QP

a)

Summary Introduction

To determine: The expected return on the portfolio.

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 10QP

The expected return on the portfolio is 9.96%.

Explanation of Solution

Given information:

The probability of having a boom, good, poor, and bust economy are 0.15, 0.55, 0.25, and 0.05 respectively. Stock A’s return is 35 percent when the economy is booming, 16 percent when the economy is good, (−1 percent) when the economy is poor, and (−12 percent) when the economy is in a bust cycle.

Stock B’s return is 45 percent when the economy is booming, 10 percent when the economy is good, (−6 percent) when the economy is poor, and (−20 percent) when the economy is in a bust cycle.

Stock C’s return is 27 percent when the economy is booming, 8 percent when the economy is good, (−4 percent) when the economy is poor, and (−9 percent) when the economy is in a bust cycle. The weight of Stock A and Stock C is 30 percent each, and the weight of Stock B is 40 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.

Compute the return on portfolio during a boom:

$\begin{array}{c}{R}_P=\left[x_1\times R_A\right]+\left[x_2\times R_B\right]+\left[x_3\times R_C\right]\\ =\left(0.30\times 0.35\right)+\left(0.40\times 0.45\right)+\left(0.30\times 0.27\right)\\ =0.105+0.18+0.081\\ =0.3660\end{array}$

Hence, the return on portfolio during a boom is 36.60%.

Compute the return on portfolio during a good economy:

$\begin{array}{c}{R}_P=\left[x_1\times R_A\right]+\left[x_2\times R_B\right]+\left[x_3\times R_C\right]\\ =\left(0.30\times 0.16\right)+\left(0.40\times 0.10\right)+\left(0.30\times 0.08\right)\\ =0.048+0.04+0.024\\ =0.1120\end{array}$

Hence, the return on portfolio during a good economy is 11.20%.

Compute the return on portfolio during a poor economy:

$\begin{array}{c}{R}_P=\left[x_1\times R_A\right]+\left[x_2\times R_B\right]+\left[x_3\times R_C\right]\\ =\left(0.30\times \left(-0.01\right)\right)+\left(0.40\times \left(-0.06\right)\right)+\left(0.30\times \left(-0.04\right)\right)\\ =\left(-0.003\right)+\left(-0.024\right)+\left(-0.012\right)\\ =\left(-0.0390\right)\end{array}$

Hence, the return on portfolio during a poor economy is (−3.90%).

Compute the return on portfolio during a bust cycle:

$\begin{array}{c}{R}_P=\left[x_1\times R_A\right]+\left[x_2\times R_B\right]+\left[x_3\times R_C\right]\\ =\left(0.30\times \left(-0.12\right)\right)+\left(0.40\times \left(-0.20\right)\right)+\left(0.30\times \left(-0.09\right)\right)\\ =\left(-0.036\right)+\left(-0.08\right)+\left(-0.027\right)\\ =\left(-0.1430\right)\end{array}$

Hence, the return on portfolio during a bust cycle is (−14.30%).

Compute the expected return on portfolio:

$\begin{array}{c}E\left(R_P\right)=\left[x_1\times E\left(R_1\right)\right]+\left[x_2\times E\left(R_2\right)\right]+\left[x_3\times E\left(R_3\right)\right]+\left[x_4\times E\left(R_4\right)\right]\\ =\left(0.15\times 0.3660\right)+\left(0.55\times 0.1120\right)+\left(0.25\times \left(-0.039\right)\right)+\left(0.05\times \left(-0.1430\right)\right)\\ =0.0549+0.0616+\left(-0.00975\right)+\left(-0.00715\right)\\ =0.0996\end{array}$

Hence, the expected return on the portfolio is 9.96%.

b)

Summary Introduction

To determine: The variance and standard deviation 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 10QP

The variance of the portfolio is 0.018475. The standard deviation of the portfolio is 0.1359 or 13.59%.

Explanation of Solution

Given information:

The probability of having a boom, good, poor, and bust economy are 0.15, 0.55, 0.25, and 0.05 respectively. Stock A’s return is 35 percent when the economy is booming, 16 percent when the economy is good, (−1 percent) when the economy is poor, and (−12 percent) when the economy is in a bust cycle.

Stock B’s return is 45 percent when the economy is booming, 10 percent when the economy is good, (−6 percent) when the economy is poor, and (−20 percent) when the economy is in a bust cycle.

Stock C’s return is 27 percent when the economy is booming, 8 percent when the economy is good, (−4 percent) when the economy is poor, and (−9 percent) when the economy is in a bust cycle. The weight of Stock A and Stock C is 30 percent each, and the weight of Stock B is 40 percent 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 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 good economy. The probability of having a good economy is P₂. R₃ is the returns of the portfolio in a poor economy. The probability of having a poor economy is P₃. R₄ is the returns of the portfolio in a bust cycle. The probability of having a bust cycle is P₄.

$\begin{array}{c}\text{Variance}=\left(\begin{array}{l}\left[{\left(\text{Possible returns}\left(R_1\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_1\right)\right]\\ +\left[{\left(\text{Possible returns}\left(R_2\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_2\right)\right]\\ +\left[{\left(\text{Possible returns}\left(R_3\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_3\right)\right]\\ +\left[{\left(\text{Possible returns}\left(R_4\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_4\right)\right]\end{array}\right)\\ =\left[\begin{array}{l}\left[{\left(0.3660-0.0996\right)}^2\times 0.15\right]+\left[{\left(0.1120-0.0996\right)}^2\times 0.55\right]\\ +\left[{\left(\left(-0.039\right)-0.0996\right)}^2\times 0.25\right]+\left[{\left(\left(-0.1430\right)-0.0996\right)}^2\times 0.05\right]\end{array}\right]\\ =\left[\begin{array}{c}\left({\left(0.2664\right)}^2\times 0.15\right)+\left({\left(0.0124\right)}^2\times 0.55\right)+\left({\left(-0.1386\right)}^2\times 0.25\right)\\ +\left({\left(-0.2426\right)}^2\times 0.05\right)\end{array}\right]\\ =\left[\begin{array}{c}\left(0.0007096896\times 0.15\right)+\left(0.00015376\times 0.55\right)+\left(0.01920996\times 0.25\right)\\ +\left(0.05885476\times 0.05\right)\end{array}\right]\end{array}$

$\begin{array}{c}=0.010645344+0.000084568+0.00480249+0.002942738\\ =0.018475\end{array}$

Hence, the variance of the portfolio is 0.018475.

Compute the standard deviation:

$\begin{array}{c}\text{Standard deviation}=\sqrt{\text{Variance}}\\ =\sqrt{0.018475}\\ =0.1359\end{array}$

Hence, the standard deviation of the portfolio is 0.1359 or 13.59%.