Chapter 7, Problem 12CP

a.

Summary Introduction

To compute: (i) Expected return on portfolio, (ii) Covariance of ABC stock return with initial return and (iii) standard deviation of new portfolio.

Introduction: An investor may invest in various stocks to reduce the risk of losses. Such a theory is called correlation theory. It is believed that an investor takes a lot of risk to achieve higher returns on their investment portfolio.

Explanation of Solution

Let,

OP = Original (initial) portfolio

ABC = New stock

NP = New portfolio

Given:

  $\begin{array}{l}\text{OP = \$900,000}\hfill \\ \text{ABC = \$100,000}\hfill \\ \text{E}\left({\text{r}}_{\text{OP}}\right)\text{ = 0}\text{.67}\hfill \\ \text{E}\left({\text{r}}_{\text{ABC}}\right)\text{ = 1}\text{.25}\hfill \end{array}$

Weights can be computed as:

Total = $900,000 + $100,000

Total = $1,000,000

Weight of Original portfolio:

  $\begin{array}{l}{\text{w}}_{\text{OP}}\text{ = }\frac{\text{\$900,000}}{\text{\$1,000,000}}\hfill \\ {\text{w}}_{\text{OP}}\text{ = 0}\text{.9}\hfill \end{array}$

Weight of ABC:

  $\begin{array}{l}{\text{w}}_{\text{ABC}}\text{ = }\frac{\text{\$900,000}}{\text{\$1,000,000}}\hfill \\ {\text{w}}_{\text{ABC}}\text{ = 0}\text{.1}\hfill \end{array}$

(i) Expected return on NP (new portfolio) that includes ABC (new) stock has been computed below:

  $\begin{array}{l}{\text{E(r}}_{\text{NP}}{\text{) = w}}_{\text{op}}\text{E}\left({\text{r}}_{\text{op}}\right){\text{ + w}}_{\text{ABC}}\text{E}\left({\text{r}}_{\text{ABC}}\right)\hfill \\ {\text{E(r}}_{\text{NP}}\text{) = }\left(\text{0}\text{.9}\times \text{0}\text{.67}\right)\text{ + }\left(\text{0}\text{.1}\times \text{1}\text{.25}\right)\hfill \\ {\text{E(r}}_{\text{NP}}\text{) = 0}\text{.603 + 0}\text{.125}\hfill \\ {\text{E(r}}_{\text{NP}}\text{) = 0}\text{.728\%}\hfill \end{array}$

(ii) Covariance of ABC stock return with return on original portfolio has been computed below:

  $\begin{array}{l}{\text{Cov = ρ × σ}}_{\text{OP}}{\text{ × σ}}_{\text{ABC}}\hfill \\ \text{Cov = 0}\text{.40 × 2}\text{.37 × 2}\text{.95}\hfill \\ \text{Cov = 2}\text{.7966 or 2}\text{.8}\hfill \end{array}$

(iii) Standard deviation of new portfolio that includes ABC stock has been computed below:

  ${\text{σ}}_{\text{NP}}\text{ = }{\left[\left({\text{w}}_{\text{op}}{}^{\text{2}}{\text{ × σ}}_{\text{OP}}{}^{\text{2}}\right)\text{ + }\left({\text{w}}_{\text{ABC}}{}^{\text{2}}{\text{ × σ}}_{\text{ABC}}{}^{\text{2}}\right)\text{ + }\left({\text{2 × w}}_{\text{op}}{\text{ × w}}_{\text{ABC}}\text{ × Cov}\right)\right]}^{\text{1/2}}$

  $\begin{array}{l}{\text{σ}}_{\text{NP}}\text{ = [(0}{\text{.9}}^{\text{2}}\text{ × 2}{\text{.37}}^{\text{2}}\text{) + (0}{\text{.1}}^{\text{2}}\text{ × 2}{\text{.95}}^{\text{2}}\text{)+(2 × 0}\text{.9 × 0}\text{.1 × 2}{\text{.8)]}}^{\text{1/2}}\\ {\text{σ}}_{\text{NP}}\text{ =[4}\text{.5497 + 0}\text{.0870 + 0}{\text{.504]}}^{\text{1/2}}\\ {\text{σ}}_{\text{NP}}\text{ = 5}{\text{.1407}}^{{}^{\text{1}}\text{/2}}\\ {\text{σ}}_{\text{NP}}\text{ = 2}\text{.2673 or 2}\text{.27}\end{array}$

b.

Summary Introduction

To compute: (i) Expected return on portfolio that includes government securities, (ii) Covariance of government securities return with initial return and (iii) standard deviation of new portfolio that includes government securities.

Introduction: An investor may invest in various stocks to reduce the risk of losses. Such a theory is called correlation theory. It is believed that an investor takes a lot of risk to achieve higher returns on their investment portfolio.

Explanation of Solution

Let,

OP = Original (initial) portfolio

GS = Government securities

NP = New portfolio

Given:

  $\begin{array}{l}\text{OP = \$900,000}\hfill \\ \text{ABC = \$100,000}\hfill \\ \text{E}\left({\text{r}}_{\text{OP}}\right)\text{ = 0}\text{.67}\hfill \\ \text{E}\left({\text{r}}_{\text{GS}}\right)\text{ = 0}\text{.42}\hfill \end{array}$

Weights can be computed as:

Total = $900,000 + $100,000

Total = $1,000,000

Weight of Original portfolio:

  $\begin{array}{l}{\text{w}}_{\text{OP}}\text{ = }\frac{\text{\$900,000}}{\text{\$1,000,000}}\hfill \\ {\text{w}}_{\text{OP}}\text{ = 0}\text{.9}\hfill \end{array}$

Weight of ABC:

  $\begin{array}{l}{\text{w}}_{\text{GS}}\text{ = }\frac{\text{\$900,000}}{\text{\$1,000,000}}\hfill \\ {\text{w}}_{\text{GS}}\text{ = 0}\text{.1}\hfill \end{array}$

(i) Expected return on NP (new portfolio) that includes ABC (new) stock has been computed below:

  $\begin{array}{l}{\text{E(r}}_{\text{NP}}{\text{) = w}}_{\text{op}}\text{E}\left({\text{r}}_{\text{op}}\right){\text{ + w}}_{\text{ABC}}\text{E}\left({\text{r}}_{\text{GS}}\right)\hfill \\ {\text{E(r}}_{\text{NP}}\text{) = }\left(\text{0}\text{.9×0}\text{.67}\right)\text{ + }\left(\text{0}\text{.1×0}\text{.42}\right)\hfill \\ {\text{E(r}}_{\text{NP}}\text{) = 0}\text{.603 + 0}\text{.042}\hfill \\ {\text{E(r}}_{\text{NP}}\text{) = 0}\text{.645\%}\hfill \end{array}$

(ii) Covariance of ABC stock return with return on original portfolio has been computed below:

  $\begin{array}{l}{\text{Cov = ρ × σ}}_{\text{OP}}{\text{ × σ}}_{\text{GS}}\hfill \\ \text{Cov = 0 × 2}\text{.37 × 0}\hfill \\ \text{Cov = 0}\hfill \end{array}$

(iii) Standard deviation of new portfolio that includes ABC stock has been computed below:

  ${\text{σ}}_{\text{NP}}\text{ = }{\left[\left({\text{w}}_{\text{op}}{}^{\text{2}}{\text{ × σ}}_{\text{OP}}{}^{\text{2}}\right)\text{ + }\left({\text{w}}_{\text{GS}}{}^{\text{2}}{\text{ × σ}}_{\text{GS}}{}^{\text{2}}\right)\text{ + }\left({\text{2 × w}}_{\text{op}}{\text{ × w}}_{\text{GS}}\text{ × Cov}\right)\right]}^{\text{1/2}}$

  $\begin{array}{l}{\text{σ}}_{\text{NP}}\text{ = [(0}{\text{.9}}^{\text{2}}\text{ × 2}{\text{.37}}^{\text{2}}\text{) + (0}{\text{.1}}^{\text{2}}{\text{ × 0}}^{\text{2}}\text{)+(2 × 0}\text{.9 × 0}{\text{.1 × 0)]}}^{\text{1/2}}\\ {\text{σ}}_{\text{NP}}\text{ =[4}{\text{.5497 + 0 + 0]}}^{\text{1/2}}\\ {\text{σ}}_{\text{NP}}\text{ = 4}{\text{.5497}}^{{}^{\text{1}}\text{/2}}\\ {\text{σ}}_{\text{NP}}\text{ = 2}\text{.1330 or 2}\text{.13}\end{array}$

c.

Summary Introduction

To state: If systematic risk of new portfolio with government securities would be more than the original portfolio.

Introduction: An investor may invest in various stocks to reduce the risk of losses. Such a theory is called correlation theory. It is believed that an investor takes a lot of risk to achieve higher returns on their investment portfolio.

Explanation of Solution

The beta for new portfolio would be weighted average of beta of individual securities in the portfolio. So, risk free securities (security with standard deviation as 0) would decrease the weighted average.

Thus, the systematic risk for new portfolio with government securities would be lower than that of the original portfolio.

d.

Summary Introduction

To comment: If exchanging ABC stock with XYZ stock makes no difference on the portfolio, keeping expected return and standard deviation same for both the stocks.

Introduction: An investor may invest in various stocks to reduce the risk of losses. Such a theory is called correlation theory. It is believed that an investor takes a lot of risk to achieve higher returns on their investment portfolio.

Explanation of Solution

Given,

  $\begin{array}{l}\text{E}\left({\text{r}}_{\text{ABC}}\right)\text{ = E}\left({\text{r}}_{\text{XYZ}}\right)\hfill \\ {\text{σ}}_{\text{ABC}}{\text{ = σ}}_{\text{XYZ}}\hfill \end{array}$

As per the given information, standard deviation and expected return of stock ABC and XYZ are same. The covariance of each stock with the original portfolio is unknown. Thus, it would be difficult to draw conclusion on the same.

If covariance of each stock with the original portfolio would have been provided then the stock that would lead to lower standard deviation for the entire portfolio would have been selected, keeping all other factors constant.

e.

Summary Introduction

To discuss: (i) One limitation of standard deviation being used as a measure of risk when investor is concerned about loss than high returns, and (ii) Any other measure of risk in such a situation

Introduction: An investor may invest in various stocks to reduce the risk of losses. Such a theory is called correlation theory. It is believed that an investor takes a lot of risk to achieve higher returns on their investment portfolio.

Explanation of Solution

(i) If the investor is more concerned about the loss due to risk of investment than higher returns, then the investor should avoid using standard deviation of return as a measure of risk.

One limitation of using standard deviation of return measure risk is that it does not differentiate between positive and negative movement in the prices.

(ii) Besides variance, range of returns could be used to measure the risk.

Range of returns: It considers the highest and lowest expected return in the future. Higher range indicates greater variability and risk whereas lower range indicates less variability and risk.