Chapter 11, Problem 35QP

a.

Summary Introduction

To determine: The Expected Return and Standard Deviation for each Security.

Introduction:  Expected Return is a process of estimating the profits and losses an investor earns through the expected rate of returns. Standard deviation is apportioned of distribution of a collection of figures from its mean.

Answer to Problem 35QP

The Expected Return is 12.50%, Variance is 0.002125 and Standard Deviation for Asset 1 is 4.61%. The Expected Return is 12.50%, Variance is 0.002125 and Standard Deviation for Asset 2 is 4.61%. The Expected Return is 12.50%, Variance is 0.002125 and Standard Deviation for Asset 3 is 4.61%.

Explanation of Solution

Determine the Expected Return, Variance and Standard Deviation for Asset 1

Using excel spreadsheet we calculate the expected return, variance and standard deviation as,

Excel Spreadsheet:

Excel Workings:

Therefore the Expected Return is 12.50%, Variance is 0.002125 and Standard Deviation for Asset 1 is 4.61%

Determine the Expected Return, Variance and Standard Deviation for Asset 2

Using excel spreadsheet we calculate the expected return, variance and standard deviation as,

Excel Spreadsheet:

Excel Workings:

Therefore the Expected Return is 12.50%, Variance is 0.002125 and Standard Deviation for Asset 2 is 4.61%

Determine the Expected Return, Variance and Standard Deviation for Asset 3

Using excel spreadsheet we calculate the expected return, variance and standard deviation as,

Excel Spreadsheet:

Excel Workings:

Therefore the Expected Return is 12.50%, Variance is 0.002125 and Standard Deviation for Asset 3 is 4.61%

b.

Summary Introduction

To determine: The Covariance and Correlation for each pair of security.

Answer to Problem 35QP

Solution: The Covariance is 0.00125 and Correlation is 0.5882 for Security 1 and 2. The Covariance is -0.002125 and Correlation is -1 for Security 1 and 3. The Covariance is -0.00125 and Correlation is -0.5882 for Security 2 and 3.

Explanation of Solution

Determine the Covariance for Security 1 and 2

Using excel we find the covariance and correlation for security 1 and 2 as,

Excel Spreadsheet:

Excel Workings:

Therefore the Covariance for Security 1 and 2 is 0.00125

Determine the Correlation for Security 1 and 2

$\begin{array}{c}Correlation{\left(\rho \right)}_{\left(1,2\right)}=\left[\frac{Co\mathrm{var}ianc{e}_{\left(1,2\right)}}{{\sigma }_{Security1}\times {\sigma }_{Security2}}\right]\\ =\left[\frac{0.00125}{4.61\%\times 4.61\%}\right]\\ =\left[\frac{0.00125}{0.002125}\right]\\ =0.588177or0.5882\end{array}$

Therefore the Correlation for Security 1 and 2 is 0.5882

Determine the Covariance for Security 1 and 3

Using excel we find the covariance and correlation for security 1 and 3 as,

Excel Spreadsheet:

Excel Workings:

Therefore the Covariance for Security 1 and 3 is -0.00125

Determine the Correlation for Security 1 and 3

$\begin{array}{c}Correlation{\left(\rho \right)}_{\left(1,3\right)}=\left[\frac{Co\mathrm{var}ianc{e}_{\left(1,3\right)}}{{\sigma }_{Security1}\times {\sigma }_{Security3}}\right]\\ =\left[\frac{-0.002125}{4.61\%\times 4.61\%}\right]\\ =\left[\frac{-0.002125}{0.002125}\right]\\ =-0.9999or-1\end{array}$

Therefore the Correlation for Security 1 and 3 is -1

Determine the Covariance for Security 2 and 3

Using excel we find the covariance and correlation for security 2 and 3 as,

Excel Spreadsheet:

Excel Workings:

Therefore the Covariance for Security 2 and 3 is -0.00125

Determine the Correlation for Security 2 and 3

$\begin{array}{c}Correlation{\left(\rho \right)}_{\left(2,3\right)}=\left[\frac{Co\mathrm{var}ianc{e}_{\left(2,3\right)}}{{\sigma }_{Security2}\times {\sigma }_{Security3}}\right]\\ =\left[\frac{-0.00125}{4.61\%\times 4.61\%}\right]\\ =\left[\frac{-0.00125}{0.002125}\right]\\ =-0.588177or-0.5882\end{array}$

Therefore the Correlation for Security 2 and 3 is -0.5882.

c.

Summary Introduction

To determine: The Expected Return and Standard Deviation of Portfolio of Security 1 and Security 2.

Answer to Problem 35QP

Solution: The Expected Return is 12.50% and Standard Deviation of Portfolio of Security 1 and Security 2 is 4.11%.

Explanation of Solution

Determine the Expected Return on Portfolio of Security 1 and 2

$\begin{array}{c}Expected\mathrm{Re}turn\left(E_{rp}\right)=\left[\begin{array}{l}\left(Weigh{t}_{Security1}\times Expected\mathrm{Re}turn{\left(E_r\right)}_{Security1}\right)+\\ \left(Weigh{t}_{Security2}\times Expected\mathrm{Re}turn{\left(E_r\right)}_{Security2}\right)\end{array}\right]\\ =\left[\left(50\%\times 12.50\%\right)+\left(50\%\times 12.50\%\right)\right]\\ =\left[\text{0}\text{.0625}+\text{0}\text{.0625}\right]\\ =12.50\%\end{array}$

Therefore the Expected Return on Portfolio of Security 1 and 2 is 12.50%

Determine the Variance on Portfolio of Security 1 and 2

$\begin{array}{c}Variance\left({\sigma }_P^2\right)=\left[\begin{array}{l}{\left(Weigh{t}_{Asset1}\right)}^2\times {\left({\sigma }_{Asset1}\right)}^2+{\left(Weigh{t}_{Asset2}\right)}^2\times {\left({\sigma }_{Asset2}\right)}^2+\\ \left(2\times Weigh{t}_{Asset1}\times {\sigma }_{Asset1}\times Weigh{t}_{Asset2}\times {\sigma }_{Asset2}\times {\rho }_{\left(1,2\right)}\right)\end{array}\right]\\ =\left[\begin{array}{l}{\left(0.50\right)}^2\times {\left(0.0461\right)}^2+{\left(0.50\right)}^2\times {\left(0.0461\right)}^2+\\ \left(2\times 0.50\times 0.0461\times 0.50\times 0.0461\times 0.5882\right)\end{array}\right]\\ =\left[\left(0.25\right)\times \left(0.002125\right)+\left(0.25\right)\times \left(0.002125\right)+0.000625\right]\\ =\left[0.000531+0.000531+0.000625\right]\\ =0.001687\end{array}$

Therefore the Variance on Portfolio of Security 1 and 2 is 0.001687

Determine the Standard Deviation on Portfolio of Security 1 and 2

$\begin{array}{c}S\tan dardDeviation\left({\sigma }_P\right)=\sqrt{Variance\left({\sigma }_P^2\right)}\\ =\sqrt{0.001687}\\ =0.04108or4.11\%\end{array}$

Therefore the Standard Deviation on Portfolio of Security 1 and 2 is 4.11%.

d.

Summary Introduction

To determine: The Expected Return and Standard Deviation of Portfolio of Security 1 and Security 3.

Answer to Problem 35QP

The Expected Return is 12.50% and Standard Deviation of Portfolio of Security 1 and Security 3 is 0%.

Explanation of Solution

Determine the Expected Return on Portfolio of Security 1 and 3

$\begin{array}{c}Expected\mathrm{Re}turn\left(E_{rp}\right)=\left[\begin{array}{l}\left(Weigh{t}_{Security1}\times Expected\mathrm{Re}turn{\left(E_r\right)}_{Security1}\right)+\\ \left(Weigh{t}_{Security3}\times Expected\mathrm{Re}turn{\left(E_r\right)}_{Security3}\right)\end{array}\right]\\ =\left[\left(50\%\times 12.50\%\right)+\left(50\%\times 12.50\%\right)\right]\\ =\left[\text{0}\text{.0625}+\text{0}\text{.0625}\right]\\ =12.50\%\end{array}$

Therefore the Expected Return on Portfolio of Security 1 and 3 is 12.50%

Determine the Variance on Portfolio of Security 1 and 3

$\begin{array}{c}Variance\left({\sigma }_P^2\right)=\left[\begin{array}{l}{\left(Weigh{t}_{Asset1}\right)}^2\times {\left({\sigma }_{Asset1}\right)}^2+{\left(Weigh{t}_{Asset3}\right)}^2\times {\left({\sigma }_{Asset3}\right)}^2+\\ \left(2\times Weigh{t}_{Asset1}\times {\sigma }_{Asset1}\times Weigh{t}_{Asset3}\times {\sigma }_{Asset3}\times {\rho }_{\left(1,3\right)}\right)\end{array}\right]\\ =\left[\begin{array}{l}{\left(0.50\right)}^2\times {\left(0.0461\right)}^2+{\left(0.50\right)}^2\times {\left(0.0461\right)}^2+\\ \left(2\times 0.50\times 0.0461\times 0.50\times 0.0461\times \left(-1\right)\right)\end{array}\right]\\ =\left[\left(0.25\right)\times \left(0.002125\right)+\left(0.25\right)\times \left(0.002125\right)+\left(-0.001062\right)\right]\\ =\left[0.000531+0.000531+\left(-0.001062\right)\right]\\ =0\end{array}$

Therefore the Variance on Portfolio of Security 1 and 3 is 0

Determine the Standard Deviation on Portfolio of Security 1 and 3

$\begin{array}{c}S\tan dardDeviation\left({\sigma }_P\right)=\sqrt{Variance\left({\sigma }_P^2\right)}\\ =\sqrt{0}\\ =0\%\end{array}$

Therefore the Standard Deviation is 0%.

e.

Summary Introduction

To determine: The Expected Return and Standard Deviation of Portfolio of Security 2 and Security 3.

Answer to Problem 35QP

The Expected Return is 12.50% and Standard Deviation of Portfolio of Security 2 and Security 3 is 2.09%.

Explanation of Solution

Determine the Expected Return on Portfolio of Security 2 and 3

$\begin{array}{c}Expected\mathrm{Re}turn\left(E_{rp}\right)=\left[\begin{array}{l}\left(Weigh{t}_{Security2}\times Expected\mathrm{Re}turn{\left(E_r\right)}_{Security2}\right)+\\ \left(Weigh{t}_{Security3}\times Expected\mathrm{Re}turn{\left(E_r\right)}_{Security3}\right)\end{array}\right]\\ =\left[\left(50\%\times 12.50\%\right)+\left(50\%\times 12.50\%\right)\right]\\ =\left[\text{0}\text{.0625}+\text{0}\text{.0625}\right]\\ =12.50\%\end{array}$

Therefore the Expected Return on Portfolio of Security 2 and 3 is 12.50%

Determine the Variance on Portfolio of Security 2 and 3

$\begin{array}{c}Variance\left({\sigma }_P^2\right)=\left[\begin{array}{l}{\left(Weigh{t}_{Asset2}\right)}^2\times {\left({\sigma }_{Asset2}\right)}^2+{\left(Weigh{t}_{Asset3}\right)}^2\times {\left({\sigma }_{Asset3}\right)}^2+\\ \left(2\times Weigh{t}_{Asset2}\times {\sigma }_{Asset2}\times Weigh{t}_{Asset3}\times {\sigma }_{Asset3}\times {\rho }_{\left(2,3\right)}\right)\end{array}\right]\\ =\left[\begin{array}{l}{\left(0.50\right)}^2\times {\left(0.0461\right)}^2+{\left(0.50\right)}^2\times {\left(0.0461\right)}^2+\\ \left(2\times 0.50\times 0.0461\times 0.50\times 0.0461\times \left(-0.5882\right)\right)\end{array}\right]\\ =\left[\left(0.25\right)\times \left(0.002125\right)+\left(0.25\right)\times \left(0.002125\right)+\left(-0.00063\right)\right]\\ =\left[0.000531+0.000531+\left(-0.00063\right)\right]\\ =0.000438\end{array}$

Therefore the Variance on Portfolio of Security 2 and 3 is 0.000438

Determine the Standard Deviation on Portfolio of Security 2 and 3 is 0

$\begin{array}{c}S\tan dardDeviation\left({\sigma }_P\right)=\sqrt{Variance\left({\sigma }_P^2\right)}\\ =\sqrt{0.000438}\\ =0.020918or2.09\%\end{array}$

Therefore the Standard Deviation is 2.09%.

f.

Summary Introduction

To determine: The Results of parts (a), (c), (d) and (e).

Explanation of Solution

• The correlation between's the profits on two securities is below 1, there is an advantage to diversification.
• A portfolio that comprise of negatively correlated portfolios can accomplish higher risk diminishment than a positively correlated portfolio and the expected return of the stock being even.
• Applying appropriate weights on perfectly negatively correlated securities can lessen portfolio change to 0.