Chapter 5, Problem 14P

a.

Summary Introduction

To calculate: Future value of annuity of $500 for a year for 8 years at 14%

Annuity:

It is an agreement under which person pays the lump sum payment or number of small payments and in return gets the amount at later date or upon annuitization. The purpose of the annuity is   not to break the flow of income after retirement.

Explanation of Solution

Given,

The annuity is $500 per year.

The interest rate is 14% or 0.14.

The numbers of years are 8 years.

The formula to calculate value of annuity is equation (I).

    $F{V}_{\text{Annuity}}=\text{C}\times \left(\frac{{\left(\text{1+i}\right)}^{\text{n}}\text{-1}}{\text{i}}\right)$

Here,

• FV stands for future value.
• C is for monthly payment.
• I is interest rate.
• n stands for no of payments.

Substitute $500 for C, 0.14 , n for 8 years in equation (I)

    $\begin{array}{c}\text{FV}=\text{\$500}\left(\frac{{\left(\text{1}+\text{0}\text{.14}\right)}^8-\text{1}}{0.14}\right)\\ =\$5\text{00}\times \frac{\text{1}\text{.85}}{\text{0}\text{.14}}\\ =\$5\text{00}\times \text{13}\text{.21}\\ =\$6,605\end{array}$

The annuity is $6,605.

Conclusion

Hence, the future value of annuity is $6,605.

(b)

Summary Introduction

To calculate: Future value of annuity of $250 for a year for 4 years at 7%

Annuity:

It is an agreement under which person pays the lump sum payment or number of small payments and in return gets the amount at later date or upon annuitization. The purpose of the annuity is   not to break the flow of income after retirement.

Explanation of Solution

Given,

The annuity is $250 per year.

The interest rate is 7%.

The numbers of years are 4 years.

Substitute $250 for C, 0.07 , n for 4 years in equation (I)

    $\begin{array}{c}\text{FV}=\text{\$250}\left(\frac{{\left(\text{1}+\text{0}\text{.07}\right)}^4-\text{1}}{0.07}\right)\\ =\$\text{250}\times \frac{0.311}{\text{0}\text{.07}}\\ =\$\text{250}\times 4.44\\ =\$1,110.\end{array}$

Conclusion

The future value of annuity will be $1,110.

(c)

Summary Introduction

To calculate: Future value of annuity of $700 for a year for 4 years at 0%

Annuity:

It is an agreement under which person pays the lump sum payment or number of small payments and in return gets the amount at later date or upon annuitization. The purpose of the annuity is   not to break the flow of income after retirement.

Explanation of Solution

Given,

The annuity is $700 per year.

The interest rate is 0%.

The numbers of years are 4 years.

The formula to calculate the future value of an annuity when interest rate is 0,

    $FV=C\times {\displaystyle \sum {\left(1+i\right)}^n}$

Substitute $700 for C, 0.00 for r, n for 4 years.

    $\begin{array}{c}\text{FV}=\$700\times \left({\left(1+0.00\right)}^4+{\left(1+0.00\right)}^3+{\left(1+0.00\right)}^2+{\left(1+0.00\right)}^1+{\left(1+0.00\right)}^0\right)\\ =700\times 5\\ =3500\end{array}$

Conclusion

The future value of annuity will be $3500

(d)

Summary Introduction

To rework: Part a, b and c as they are due.

Annuity:

It is an agreement under which person pays the lump sum payment or number of small payments and in return gets the amount at later date or upon annuitization. The purpose of the annuity is   not to break the flow of income after retirement.

Explanation of Solution

The formula to calculate future value of annuity due,

    $F{V}_{\text{Annuit}y}=C\left(\frac{{(1+i)}^n-1}{i}\right)\times \left(1+i\right)$

Were,

• FV stands for future value of annuity.
• C symbolizes the monthly payment.
• I is for interest rate.
• N is for number of payments.

d.a.

Summary Introduction

To calculate: Future value of annuity of $500 for a year for 8 years at 14%

Annuity:

It is an agreement under which person pays the lump sum payment or number of small payments and in return gets the amount at later date or upon annuitization. The purpose of the annuity is   not to break the flow of income after retirement.

Explanation of Solution

Given,

The annuity is $500 per year.

The interest rate is 14% or 0.14.

The numbers of years are 8 years.

Substitute C for $500, i for 14%, n for 8 years in equation {(d) I}

    $\begin{array}{c}\text{FV}=\$500\times \frac{\left({\left(1+0.14\right)}^8-1\right)}{0.14}\times \left(1+0.14\right)\\ =\$500\times 15.09\\ =\$7,545\end{array}$

Conclusion

The future value of annuity due is $7,545.

d.b

Summary Introduction

To calculate: Future value of annuity of $250 for a year for 4 years at 7%

Annuity:

It is an agreement under which person pays the lump sum payment or number of small payments and in return gets the amount at later date or upon annuitization. The purpose of the annuity is   not to break the flow of income after retirement.

Explanation of Solution

Given,

The annuity is $250 per year.

The interest rate is 7%.

The numbers of years are 4 years.

Substitute C for $250, I for 7%, n for 4 years.

    $\begin{array}{c}\text{FV}=\$250\frac{\left({\left(1+0.07\right)}^4-1\right)}{0.07}\times \left(1+0.07\right)\\ =\$250\times 4.75\\ =\$1,187.68\end{array}$

Conclusion

The future value of annuity due is $1,187.68.

d.c

Summary Introduction

To calculate: Future value of annuity of $700 for a year for 4 years at 0%

Annuity:

It is an agreement under which person pays the lump sum payment or number of small payments and in return gets the amount at later date or upon annuitization. The purpose of the annuity is   not to break the flow of income after retirement.

Explanation of Solution

Given,

The annuity is $700 per year.

The interest rate is 0%.

The numbers of years are 4 years.

The following formula will be used to solve.

    $F{V}_{Annuity}=C\times {\displaystyle \sum {\left(1+i\right)}^n\times \left(1+i\right)}$

Substitute C for $700, i for 0%, n for 4 years in equation.

    $\begin{array}{c}\text{FV}=\$700\times \left({\left(1+0.00\right)}^3+{\left(1+0.00\right)}^2+{\left(1+0.00\right)}^1+{\left(1+0.00\right)}^0\right)\\ \times \left(1+0.00\right)\\ =700\times 5\\ =3,500\end{array}$

Conclusion

The future value of annuity due is $3,500.