Chapter 13, Problem 23QP

Portfolio Returns and Deviations [LO2] Consider the following information about three stocks:

a. If your portfolio is invested 40 percent each in A and B and 20 percent in C, what is the portfolio expected return? The variance? The standard deviation?

b. If the expected T-bill rate is 3.80 percent, what is the expected risk premium on the portfolio?

c. If the expected inflation rate is 3.50 percent, what are the approximate and exact expected real returns on the portfolio? What are the approximate and exact expected real risk premiums on the portfolio?

Summary Introduction

To determine: The expected return on the portfolio, the variance, and standard deviation.

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 23QP

The expected return on the portfolio is 13.70 percent. The variance of the portfolio is 0.02734. The standard deviation of the portfolio is 16.53 percent.

Explanation of Solution

Given information:

The probability of having a boom, normal, and bust economy is 0.25, 0.60, and 0.15 respectively. Stock A’s return is 21 percent when the economy is booming, 17 percent when the economy is normal, and 0 percent when the economy is in a bust cycle. Stock B’s return is 36 percent when the economy is booming, 13 percent when the economy is normal, and (28 percent) when the economy is in a bust cycle.

Stock C’s return is 55 percent when the economy is booming, 9 percent when the economy is normal, and (45 percent) when the economy is in a bust cycle. The weight of Stock A and Stock B is 40 percent each, and the weight of Stock C is 20 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 return on the portfolio during a boom:

$\begin{array}{c}{R}_P=\left[x_1\times R_A\right]+\left[x_2\times R_B\right]+\left[x_3\times R_C\right]\\ =0.40\times 0.21+0.40\times 0.36+0.20\times 0.55\\ =0.3380\text{ or }33.80\%\end{array}$

Hence, the return on the portfolio during a boom is 33.80%.

Compute the return on the portfolio during a normal economy:

$\begin{array}{c}{R}_P=\left[x_1\times R_A\right]+\left[x_2\times R_B\right]+\left[x_3\times R_C\right]\\ =0.40\times 0.17+0.40\times 0.13+0.20\times 0.09\\ =0.1380\text{ or }13.80\%\end{array}$

Hence, the return on the portfolio during a normal economy is 13.80%.

Compute the return on the portfolio during a bust cycle:

$\begin{array}{c}{R}_P=\left[x_1\times R_A\right]+\left[x_2\times R_B\right]+\left[x_3\times R_C\right]\\ =0.40\times 0+0.40\times \left(0.28\right)+0.20\times \left(0.45\right)\\ =\left(0.2020\right)\text{ or }\left(20.20\%\right)\end{array}$

Hence, the return on the portfolio during a bust cycle is (20.20%).

Compute the expected return on the portfolio:

$\begin{array}{c}E\left(R_P\right)=\left[x_1\times E\left(R_1\right)\right]+\left[x_2\times E\left(R_2\right)\right]+\mathrm{\dots }+\left[x_n\times E\left(R_n\right)\right]\\ =0.25\times 0.3380+0.60\times 0.1380+0.15\times \left(0.2020\right)\\ =0.1370\text{ or }13.70\%\end{array}$

Hence, the expected return on the portfolio is 13.70%.

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 normal economy. The probability of having a normal economy is “P₂”. “R₃” is the returns of the portfolio in a bust cycle. The probability of having a bust cycle is “P₃”.

$\begin{array}{c}\text{Variance}=\left(\begin{array}{l}\left[{\left(\text{Possible returns}\left(R_1\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_1\right)\right]+\\ \left[{\left(\text{Possible returns}\left(R_2\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_2\right)\right]+\\ \left[{\left(\text{Possible returns}\left(R_3\right)-\text{Expected returns }E\left(R\right)\right)}^2\times \text{Probability}\left(P_3\right)\right]+\end{array}\right)\\ =\left[\begin{array}{l}\left[{\left(0.3380-0.1370\right)}^2\times 0.25\right]+\left[{\left(0.1380-0.1370\right)}^2\times 0.60\right]+\\ \left[{\left(\left(0.2020\right)-0.1370\right)}^2\times 0.15\right]\end{array}\right]\\ =0.02734\end{array}$

Hence, the variance of the portfolio is 0.02734.

Compute the standard deviation:

$\begin{array}{c}\text{Standard deviation}=\sqrt{\text{Variance}}\\ =\sqrt{0.02734}\\ =16.53\%\end{array}$

Hence, the standard deviation of the portfolio is 16.53%.

Summary Introduction

To determine: The expected risk premium of the portfolio.

Answer to Problem 23QP

The expected risk premium of the portfolio is 9.90 percent.

Explanation of Solution

Given information:

The Treasury bill rate is 3.80%. The expected portfolio return is 13.70%.

Compute the expected risk premium:

The expected risk premium is the difference between the portfolio return and the risk-free rate. Hence, the expected risk premium is 9.90% $(13.70\%-3.80\%)$.

Summary Introduction

To determine: The approximate and exact real returns of the portfolio and the real risk premiums.

Answer to Problem 23QP

The approximate and exact real returns from the portfolio are 10.20% and 9.86% respectively. The approximate and exact real risk premiums are 9.90% and 9.57% respectively.

Explanation of Solution

Given information:

The Treasury bill rate is 3.80%. The expected portfolio return is 13.70%. The inflation rate is 3.5%.

The formula to calculate the approximate real rate:

$\text{Nominal rate}=\text{Real rate}+\text{Inflation rate}$

The formula to calculate the real rate using Fisher’s relationship:

$1+R=\left(1+r\right)\times \left(1+h\right)$

Where,

“R” is the nominal rate of return

“r” is the real rate of return

“h” is the inflation rate

Compute the approximate real return of the portfolio:

$\begin{array}{c}\text{Nominal rate}=\text{Real rate}+\text{Inflation rate}\\ 0.1370=\text{Real rate}+0.035\\ \text{Real rate}=0.1020\text{ or }10.20\%\end{array}$

Hence, the approximate real rate of return is 10.20%.

Compute the exact real return of the portfolio:

$\begin{array}{c}1+R=\left(1+r\right)\times \left(1+h\right)\\ r=\frac{1+R}{1+h}-1\\ =\frac{1+0.1370}{1+0.035}-1\\ =0.0986\text{ or }9.86\%\end{array}$

Hence, the real return of the portfolio is 9.86 percent.

Compute the approximate real risk-free rate:

$\begin{array}{c}\text{Nominal rate}=\text{Real rate}+\text{Inflation rate}\\ 0.038=\text{Real rate}+0.035\\ \text{Real rate}=0.003\text{ or }0.30\%\end{array}$

Hence, the approximate real risk-free rate is 0.30%.

Compute the exact real risk-free rate:

$\begin{array}{c}1+R=\left(1+r\right)\times \left(1+h\right)\\ r=\frac{1+R}{1+h}-1\\ =\frac{1+0.038}{1+0.035}-1\\ =0.0029\text{ or }0.29\%\end{array}$

Hence, the real risk-free rate is 0.29 percent.

Compute the approximate real risk premium:

The expected real risk premium is the difference between the real portfolio return and the real risk-free rate. Hence, the expected risk premium is 9.90% $(10.20\%-0.30\%)$.

Compute the exact real risk premium:

The expected real risk premium is the difference between the real portfolio return and the real risk-free rate. Hence, the expected risk premium is 9.57% $(9.86\%-0.29\%)$.