Chapter 11, Problem 3DC1

Summary Introduction

To determine:

The total nominal value of the portfolio as invested in common stock and in government bonds

Introduction:

Investment objective refers to the objective of the person for which the person is ready to invest money in order to get higher returns in order to meet the objective.

Explanation of Solution

Given,

Money invested is $2,000 per year.

Number of years is 40.

Tax rate is 15%

The average rate of common stock in table is 10.70%.

Formula to calculate after tax return,

$\text{After}\text{tax}\text{}\text{return}=\text{Rate}\text{of}\text{common}\text{stock}\times \text{after}\text{tax}\text{}\text{rate}$

Substitute 10.70% for rate of common stock and 0.85 for after tax rate.

$\begin{array}{c}\text{After}\text{tax}\text{}\text{return}=10.70\%\times 0.85\\ =9.095\%\end{array}$

After tax return rate is 9.095%.

Portfolio value as investment made in common stock

Formula to calculate the nominal value,

$\text{Nominal}\text{value}=\text{Payment}\times \frac{{\left(1+\text{rate}\right)}^{\text{number}\text{of}\text{years}}-1}{\text{rate}}$

Substitute $2,000 for the payment and 9.095% for rate and 40 for number of years.

$\begin{array}{c}\text{Nominal}\text{value}=\$2,000\times \frac{{\left(1+0.09095\right)}^{40}-1}{0.09095}\\ =\$2,000\times \frac{31.523}{0.09095}\\ =\$2,000\times 346.597\\ =\$693,194\end{array}$

Hence, the total nominal value is $693,194.

Average return on bonds is 7.24%.

Portfolio value as the investment is made in government bonds.

Formula to calculate after tax return,

$\text{After}\text{tax}\text{}\text{return}=\text{Rate}\text{of}\text{common}\text{stock}\times \text{After}\text{tax}\text{}\text{rate}$

Substitute 7.24% for rate of common stock and 0.85 for after tax rate.

$\begin{array}{c}\text{After}\text{tax}\text{}\text{return}=7.24\%\times 0.85\\ =6.154\%\end{array}$

After tax return rate is 6.154%.

Substitute $2,000 for the payment and 6.154% for rate and 40 for number of years.

$\begin{array}{c}\text{Nominal}\text{value}=\$2,000\times \frac{{\left(1+06154\right)}^{40}-1}{0.06154}\\ =\$2,000\times \frac{9.9007}{0.06154}\\ =\$2,000\times 160.882\\ =\$321,764.70\end{array}$

Hence, the total nominal value is $321,764.70.

Average return on treasury bills is 3.58%.

Portfolio value as the investment is made in treasury bills.

Formula to calculate after tax return,

$\text{After}\text{tax}\text{}\text{return}=\text{Rate}\text{of}\text{common}\text{stock}\times \text{After}\text{tax}\text{}\text{rate}$

Substitute 3.58% for rate of common stock and 0.85 for after tax rate.

$\begin{array}{c}\text{After}\text{tax}\text{}\text{return}=3.58\%\times 0.85\\ =3.043\%\end{array}$

After tax return rate is 3.043%.

Substitute $2,000 for the payment and 3.043% for rate and 40 for number of years.

$\begin{array}{c}\text{Nominal}\text{value}=\$2,000\times \frac{{\left(1+0.03043\right)}^{40}-1}{0.03043}\\ =\$2,000\times \frac{2.31695}{0.03043}\\ =\$2,000\times 76.14\\ =\$152,280\end{array}$

Hence, the total nominal value is $152,280.

Working notes:

Tax rate is 15%.

Formula to calculate post tax return,

$\begin{array}{c}\text{Post}\text{tax}\text{return}=1-\text{Tax}\text{rate}\\ =1-15\%\\ =1-0.55\\ =85\%\end{array}$

Post tax return is 85%.

Conclusion

The total nominal value of the portfolio is $693,194.