Chapter 11, Problem 10QP

a)

Summary Introduction

To determine: The portfolios expected return

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 0.1061 or 10.61%.

Explanation of Solution

Given information:

The probability of having a boom, good, poor, and bust economy are 0.15, 0.50, 0.25, and 0.10 respectively. Stock A’s return is 35 percent when the economy is booming, 12 percent when the economy is good, 1 percent when the economy is poor, and −11 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, 2 percent when the economy is poor, and −25 percent when the economy is in a bust cycle.

Stock C’s return is 33 percent when the economy is booming, 17 percent when the economy is good, −5 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.

Formulae:

The formula to calculate the return on portfolio during a particular state of economy:

$R_P=\left[x_1\times R_A\right]+\left[x_2\times R_B\right]+\mathrm{...}+\left[x_n\times R_C\right]$

Where,

R_p refers to the return on portfolio

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

R₁ to R_n” refers to the rate of return on each asset from 1 to “n” 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 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 return on portfolio during a boom:

$\begin{array}{c}{R}_P=\left[x_1\times R_A\right]+\left[x_2\times R_B\right]+\left[x_3\times R_C\right]\\ =\left(0.30\times 0.35\right)+\left(0.40\times 0.45\right)+\left(0.30\times 0.33\right)\\ =0.105+0.18+0.099\\ =0.384\end{array}$

Hence, the return on portfolio during a boom is 0.384 or 38.4%.

Compute the return on portfolio during a good economy:

$\begin{array}{c}{R}_P=\left[x_1\times R_A\right]+\left[x_2\times R_B\right]+\left[x_3\times R_C\right]\\ =\left(0.30\times 0.12\right)+\left(0.40\times 0.10\right)+\left(0.17\times 0.30\right)\\ =0.036+0.04+0.051\\ =0.127\end{array}$

Hence, the return on portfolio during a good economy is 0.127 or 12.7%.

Compute the return on portfolio during a poor economy:

$\begin{array}{c}{R}_P=\left[x_1\times R_A\right]+\left[x_2\times R_B\right]+\left[x_3\times R_C\right]\\ =\left(0.30\times 0.01\right)+\left(0.40\times 0.02\right)+\left(0.30\times \left(-0.05\right)\right)\\ =0.003+0.008+\left(-0.015\right)\\ =-0.004\end{array}$

Hence, the return on portfolio during a poor economy is −0.004 or −0.4%.

Compute the return on portfolio during a bust cycle:

$\begin{array}{c}{R}_P=\left[x_1\times R_A\right]+\left[x_2\times R_B\right]+\left[x_3\times R_C\right]\\ =\left(0.30\times \left(-0.11\right)\right)+\left(0.40\times \left(-0.25\right)\right)+\left(0.30\times \left(-0.09\right)\right)\\ =-0.033+\left(-0.1\right)+\left(-0.027\right)\\ =-0.16\end{array}$

Hence, the return on portfolio during a bust cycle is −0.16 or −16%.

Compute the expected return on portfolio:

$\begin{array}{c}E\left(R_P\right)=\left[x_1\times E\left(R_1\right)\right]+\left[x_2\times E\left(R_2\right)\right]+\mathrm{...}+\left[x_n\times E\left(R_n\right)\right]\\ =\left(0.15\times 0.384\right)+\left(0.50\times 0.127\right)+\left(0.25\times 0.004\right)+\left(0.10\times -0.16\right)\\ =0.0576+0.0635+0.001-0.016\\ =0.1061\end{array}$

Hence, the expected return on the portfolio is 0.1061 or 10.61%.

b)

Summary Introduction

To determine: The standard deviation and 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.

Standard deviation refers to the variation in the actual returns from the expected returns of the assets. The square root of variance gives the standard deviation.

Answer to Problem 10QP

The variance of the portfolio is 0.02346 and the standard deviation of the portfolio is 0.1532 or 15.32%.

Explanation of Solution

Given information:

The probability of having a boom, good, poor, and bust economy are 0.15, 0.50, 0.25, and 0.10 respectively. Stock A’s return is 35 percent when the economy is booming, 12 percent when the economy is good, 1 percent when the economy is poor, and −11 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, 2 percent when the economy is poor, and −25 percent when the economy is in a bust cycle.

Stock C’s return is 33 percent when the economy is booming, 17 percent when the economy is good, −5 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.

Formulae:

The formula to calculate the variance of the portfolio:

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

The formula to calculate the standard deviation:

$\text{Standard deviation}=\sqrt{\text{Variance}}$

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.

$\begin{array}{c}\text{Variance}=\left(\begin{array}{l}\left[{\left(\text{Possible returns}\left(R_1\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_1\right)\right]\\ +\left[{\left(\text{Possible returns}\left(R_2\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_2\right)\right]\\ +\left[{\left(\text{Possible returns}\left(R_3\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_3\right)\right]\\ +\left[{\left(\text{Possible returns}\left(R_4\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_4\right)\right]\end{array}\right)\\ =\left[\begin{array}{l}\left[{\left(0.384-0.1061\right)}^2\times 0.15\right]+\left[{\left(0.127-0.1061\right)}^2\times 0.50\right]\\ +\left[{\left(0.004-0.1061\right)}^2\times 0.25\right]+\left[{\left(\left(-0.16\right)-0.1061\right)}^2\times 0.10\right]\end{array}\right]\\ =\left[{\left(0.2779\right)}^2\times 0.15\right]+\left[{\left(0.0209\right)}^2\times 0.50\right]+\left[{\left(-0.1021\right)}^2\times 0.25\right]+\left[{\left(-0.2661\right)}^2\times 0.10\right]\\ =\left[0.0772\times 0.15\right]+\left[0.00044\times 0.50\right]+\left[0.0104\times 0.25\right]+\left[0.0708\times 0.10\right]\end{array}$

$\begin{array}{c}=0.01158+0.0022+0.0026+0.00708\\ =0.02346\end{array}$

Hence, the variance of the portfolio is 0.02346.

Compute the standard deviation:

$\begin{array}{c}\text{Standard deviation}=\sqrt{\text{Variance}}\\ =\sqrt{0.02346}\\ =0.1532\end{array}$

Hence, the standard deviation of the portfolio is 0.1532 or 15.32%.