Chapter 8, Problem 1PS

Portfolio risk and return* Here are returns and standard deviations for four investments.

Calculate the standard deviations of (the following portfolios.

1. a. 50% in Treasury bills, 50% in stock P.
2. b. 50% each in Q and R, assuming the shares have
3. • Perfect positive correlation.
4. • Perfect negative correlation.
5. • No correlation.
6. c. Plot a figure like Figure 8.3 for Q and R, assuming a correlation coefficient of .5.
7. d. Stock Q has a lower return than R but a higher standard deviation. Does that mean that Q’s price is too high or that R’s price is too low?

Summary Introduction

To determine: Standard deviation of 50% Treasury bills and 50% in stock P.

Explanation of Solution

Given information:

Calculation of standard deviation:

$\begin{array}{c}\text{Standard}\text{deviation}\left(\text{σ}\right)=\left(0.50\times .0\right)+\left(0.50\times .14\right)\\ =0.7\\ =7\%\end{array}$

Therefore, the standard deviation is 7%

Summary Introduction

To determine: Standard deviation of 50% each in Q and R in the following situations.

Explanation of Solution

With a perfect positive correlation:

$\begin{array}{c}\text{Standard}\text{deviation}\left(\text{σ}\right)={\left[\left({0.5}^2\times {0.28}^2\right)+\left({0.5}^2\times {0.26}^2\right)+2\left(0.5\times 0.50\times 1\times 0.28\times 0.26\right)\right]}^{0.50}\\ =0.27\\ =27\%\end{array}$

Therefore, the standard deviation in a perfect positive correlation is 27%

With a perfect negative correlation:

$\begin{array}{c}\text{Standard}\text{deviation}\left(\text{σ}\right)={\left[\left({0.5}^2\times {0.28}^2\right)+\left({0.5}^2\times {0.26}^2\right)+2\left(0.5\times 0.50\times \left(-1\right)\times 0.28\times 0.26\right)\right]}^{0.50}\\ =0.01\\ =1\%\end{array}$

Therefore, the standard deviation in a perfect positive correlation is 1%

With no correlation:

$\begin{array}{c}\text{Standard}\text{deviation}\left(\text{σ}\right)={\left[\left({0.5}^2\times {0.28}^2\right)+\left({0.5}^2\times {0.26}^2\right)+2\left(0.5\times 0.50\times 0\times 0.28\times 0.26\right)\right]}^{0.50}\\ =0.191\\ =19.1\%\end{array}$

Therefore, the standard deviation in a perfect positive correlation is 19.1%

Summary Introduction

To graph: Figure showing the stocks of Q and R by assuming a correlation coefficient of 0.5

Explanation of Solution

Summary Introduction

To discuss: If Q has a low return than R but with a higher standard deviation whether this mean that price of Q’s stock is too high or price of R’s stock is too low.

Explanation of Solution

When stock Q has lower return that stock R but, higher standard deviation, thus this doesn’t mean that price of Q’s stock is too high or price of R’s stock is too low. Because the risk factor is measured by beta not by the standard deviation.

Standard deviation measures the total risk whereas, beta measures non-diversifiable risk and inventors are solely compensated with a risk premium in holding the non-diversifiable risk.