Chapter 28, Problem 4PS

a.

Summary Introduction

Adequate information:

Saving contribution = $5,000 pa

Retire in 30 year and expected to live after retirement 20

	Contribution	Withdrawal
R IRA	Tax	Tax Free
T IRA	Tax Free	Tax

To compute: 20-year retirement consumption if tax rate 30%, if choose traditional method.

Introduction: Annuity due is a case where payment is made at the beginning of each period. Interest is earned for each period.For example,Bank recurring deposit account.

Explanation of Solution

Here, future value of annuity is to be calculated because regular payments are made at the beginning of each period and tax is paid on the computed future value.

Formula used=FV = $p\left[\frac{\left[{(1+i)}^n-1\right]}{i}\right]$

  $\begin{array}{l}\text{Substitute the value:-}\hfill \\ \hfill \\ \text{FV}\left(\text{Future value }\right)\text{=20-year annuity consumption}\hfill \\ \text{P}\left(\text{annual annuity}\right)\text{=\$ 5000}\hfill \\ \text{I}\left(\text{int}\text{. rate}\right)\text{=5\%}\hfill \\ \text{N}\left(\text{period}\right)\text{=30}\hfill \\ \hfill \\ \text{Show calculations}\hfill \\ \hfill \\ \hfill \\ \text{ =5,000}\times \left[\text{66}\text{.43885}\right]\hfill \\ \text{ =332194}\text{.25}\hfill \\ \text{Before tax value is 332194}\text{.25}\hfill \\ \text{Tax rate 30\%}\hfill \\ \text{After tax value =332194}\text{.25}\times \left(\text{1-30\%}\right)\hfill \end{array}$

= 232,535.97

Final answer:-$232,535.97

Conclusion

Thus the 20-year annuity consumption value is $2,32,535.97.

b.

Summary Introduction

Adequate information:

Saving contribution = $5,000 pa

Retire in 30 year and expected to live after retirement 20

	Contribution	Withdrawal
R IRA	Tax	Tax Free
T IRA	Tax Free	Tax

To compute:Computation of 20-year retirement consumption if choose traditional method.

Introduction: Annuity due is a case where payment is made at the beginning of each period. Interest is earned for each period. For example: Bank recurring deposit account.

Explanation of Solution

Here, tax will be deducted from annual savings amount and later future value will be calculated using annuity formua. The calculations has been shown below:

  $\begin{array}{l}\text{Each period saving =5000}\\ \text{tax rate = 30\%}\\ \text{After tax saving amount =5000*(1-30\%)}\\ =3500\\ \text{Formula used:-}\\ \text{FV =p}\left[\frac{{\text{(1+i)}}^n-1}{\text{i}}\right]\\ \begin{array}{l}\text{Substitute the value:-}\hfill \\ \hfill \\ \text{FV}\left(\text{Future value }\right)\text{=20-year annuity consumption}\hfill \\ \text{P}\left(\text{annual annuity}\right)\text{= 3500}\hfill \\ \text{I}\left(\text{int}\text{. rate}\right)\text{=5\%}\hfill \\ \text{N}\left(\text{period}\right)\text{=30}\hfill \\ \hfill \\ \text{Show calculations}\hfill \\ \hfill \\ \hfill \\ \text{ =3500}\left[\text{66}\text{.43885}\right]\hfill \\ \text{ =2,32,535}\hfill \\ \text{final answer =2,32,535}\hfill \\ \hfill \\ \hfill \end{array}\end{array}$

Conclusion

20 year annuity consumption value under Roth IRA is 2,32,535.

c.

Summary Introduction

Adequate information:

Saving contribution = 5000 pa

Retire in 30 year and expected to live after retirement 20

	Contribution	Withdrawal
R IRA	Tax	Tax Free
T IRA	Tax Free	Tax

To discuss:The one that would provide better results if rate reduced 30% to 25% at retirement.

Introduction: Annuity due is a case where payment is made at the beginning of each period. Interest is earned for each period. For example: Bank recurring deposit account.

Explanation of Solution

Here, future value of both are compared using annuity formula and respective rate.

By traditional method

  $\begin{array}{l}\text{Each period saving =5000}\\ \text{Tax rate =25\%}\\ \text{Formula used:-}\\ \text{FV =p}\left[\frac{{\text{(1+i)}}^n-1}{\text{i}}\right]\\ \begin{array}{l}\text{Substitute the value:-}\hfill \\ \hfill \\ \text{FV}\left(\text{Future value }\right)\text{=20-year annuity consumption}\hfill \\ \text{P}\left(\text{annual annuity}\right)\text{= 5000}\hfill \\ \text{I}\left(\text{int}\text{. rate}\right)\text{=5\%}\hfill \\ \text{N}\left(\text{period}\right)\text{=30}\hfill \\ \hfill \\ \begin{array}{l}\text{Show calculations}\\ \left[\frac{{\text{(1+i)}}^n-1}{\text{i}}\right]\text{its value on given data is =66}\text{.43885}\end{array}\hfill \\ \hfill \\ \hfill \\ \text{ =5000}\left[\text{66}\text{.43885}\right]\hfill \\ \text{ =3,32,194}\text{.25}\hfill \\ \begin{array}{l}\text{after tax value is }\\ \text{ =3,32,194}\text{.25(1-25\%)}\\ \text{ =249145}\text{.68}\\ \\ \end{array}\hfill \\ \hfill \\ \hfill \end{array}\end{array}$

By Roth method

  $\begin{array}{l}\text{Each period saving =5000}\\ \text{tax rate = 30\%}\\ \text{After tax saving amount =5000*(1-30\%)}\\ =3500\\ \text{Formula used:-}\\ \text{FV =p}\left[\frac{{\text{(1+i)}}^n-1}{\text{i}}\right]\\ \begin{array}{l}\text{Substitute the value:-}\hfill \\ \hfill \\ \text{FV}\left(\text{Future value }\right)\text{=20-year annuity consumption}\hfill \\ \text{P}\left(\text{annual annuity}\right)\text{= 3500}\hfill \\ \text{I}\left(\text{int}\text{. rate}\right)\text{=5\%}\hfill \\ \text{N}\left(\text{period}\right)\text{=30}\hfill \\ \hfill \\ \begin{array}{l}\text{Show calculations}\\ \left[\frac{{\text{(1+i)}}^n-1}{\text{i}}\right]\text{its value on given data is =66}\text{.43885}\end{array}\hfill \\ \hfill \\ \hfill \\ \text{ =3500}\left[\text{66}\text{.43885}\right]\hfill \\ \text{ =2,32,535}\hfill \\ \text{after tax value is 2,32,535}\hfill \\ \hfill \\ \hfill \end{array}\\ \end{array}$

Conclusion

20 year annuity consumption value under traditional method is better when tax rate is reduce from 30% to 25% at retirement.