Chapter 15, Problem 4PA

Summary Introduction

(a)

To determine:

The annual needs of A & Y after retirement.

Introduction:

Annual needs means the amounts needs to spent on current expenses of the individual. An individual not only think for the present need but also consider the future needs and that is the reason they invest amount in retirement plans.

Explanation of Solution

Given,

The amount of current expenditure is $67,000.

The formula to calculate annual need is,

$\text{Annual needs}=\text{Current expenditure}\times 70\%+\$5,000$

Substitute $67,000 for current expenditure in the above formula.

$\begin{array}{c}\text{Annual needs}=\$67,000\times 70\%+\$5,000\\ =\$46,900+\$5,000\\ =\$51,900\end{array}$

Conclusion

Thus, the annual needs of A & Y after retirement are $51,900.

Summary Introduction

(b)

To determine:

The annual requirement of A &Y in today’s dollars with the effect of taxes

Explanation of Solution

Given,

Annual need is $51,900 as calculated in part a.

Average tax rate is 12% at the time of retirement.

The formula to calculate the annual requirement is,

$\text{Annual requirement}=\frac{\text{Annual needs}}{\left(1-\text{Tax rate}\right)}$

Substitute $51,900 for annual needs and tax rate is 12% in the above formula.

$\begin{array}{c}\text{Annual requirement}=\frac{\$51,900}{\left(1-0.12\right)}\\ =\frac{\$51,900}{0.88}\\ =\$58,977\end{array}$

Conclusion

Thus, the annual requirement after adjusting for tax is $58,977.

Summary Introduction

(c)

To determine:

The projected annual income shortfall in today’s dollars.

Explanation of Solution

Given,

The annual requirement after adjusting for tax is $58,977 as calculated in part b.

The current expenditure excluding the amount of cruise is $57,000.

The formula to calculate annual income shortfall is,

$\text{Annual income shortfall}=\text{Annual requirement}-\text{Current expenses}$

Substitute $58,977 for annual requirement and $57,000 for current expenses in the above formula.

$\begin{array}{c}\text{Annual income shortfall}=\$58,977-\$57,000\\ =\$1,977\end{array}$

Conclusion

Thus, the projected annual income shortfall is $1,977.

Summary Introduction

(d)

To determine:

The future value of shortfall of A & Y.

Explanation of Solution

Given,

Annual income shortfall is $1,977 as calculated in part c.

The retirement occurs after 30 years.

Inflation rate is 5%

The formula to calculate the future value of annual shortfall is,

$\text{shortfall}=\text{Present value of shortfall}\times {\left(1+\text{Inflation rate}\right)}^{\text{n}}$

Substitute $1,977 for present value of shortfall, 5% for inflation rate and 30years for n in the above formula.

$\begin{array}{c}\text{Shortfall}=\$1,977\times {\left(1+0.05\right)}^{30}\\ =\$1,977\times {\left(1.05\right)}^{30}\\ =\$1,977\times 4.322\\ =\$8,544.48\end{array}$

Conclusion

Thus, future value of shortfall is $8,544.48.

Summary Introduction

(e)

To determine:

The annual investment need to fulfill the retirement goals.

Introduction:

Annual investment is the amount kept aside by the individual that is used in the future as retirement benefit. As after the retirement the source of income is not the salary so the individual has to save annually for the future.

Explanation of Solution

Given,

Future value of shortfall is $8,544.48 as calculated in part d.

The retirement occurs after 25 years.

The rate of return is 8%.

The formula to calculate annual investment is,

$\text{Annual investemnt}=\frac{\text{Future value of shortfall}}{{\left(1+\text{Rate of return}\right)}^{\text{n}}}$

Substitute $8,544.48 for future value of shortfall, rate of return is 8% and 25 years for n in the above formula.

$\begin{array}{c}\text{Annual investemnt}=\frac{\$8,544.48}{{\left(1+0.08\right)}^{25}}\\ =\frac{\$8,544.48}{{\left(1.08\right)}^{25}}\\ =\frac{\$8,544.48}{6.85}\\ =\$1,247\end{array}$

Conclusion

Thus, the annual investment need to fulfill the retirement goals is $1,247.