Chapter 5, Problem 14P

a.

Summary Introduction

To calculate: Future value of annuity a) $400 per year for 10 years at 10% b) $200 per year for 5 years at 5% c) $400 per year for 5 years at 0% d) Rework part a ,b and c as they are due.

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

Explanation of Solution

Solution:

The formula to calculate value of annuity is,

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

Here,

• FV is future value.
• C is monthly payment.
• i is interest rate.
• n is number of payments.

Substitute $400 for C, 10% for i and 10 years for n in equation (I).

$\begin{array}{c}FV=\$400\left(\frac{{\left(1+0.10\right)}^{10}-1}{0.10}\right)\\ =\$400\times \frac{1.60}{0.10}\\ =\$400\times 15.9374246\\ =\$6,374.96984\end{array}$

Conclusion

Hence, the future value of annuity is $6374.96984.

b.

Summary Introduction

To calculate: Future value of annuity a) $400 per year for 10 years at 10% b) $200 per year for 5 years at 5% c) $400 per year for 5 years at 0% d) Rework part a ,b and c as they are due.

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

Explanation of Solution

Solution:

The formula to calculate value of annuity is,

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

Here,

• FV is future value.
• C is monthly payment.
• i is interest rate.
• n is number of payments.

Substitute $200 for C, 5% for i and 5 years for n in equation (I)

$\begin{array}{c}FV=\$200\left(\frac{{\left(1+0.05\right)}^5-1}{0.05}\right)\\ =200\times \frac{0.27682156}{0.05}\\ =200\times 5.5256312\\ =1105.12624\end{array}$

Conclusion

The future value of annuity will be $1105.12624.

c.

Summary Introduction

To calculate: Future value of annuity a) $400 per year for 10 years at 10% b) $200 per year for 5 years at 5% c) $400 per year for 5 years at 0% d) Rework part a ,b and c as they are due.

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

Explanation of Solution

Solution:

The formula to calculate value of annuity is,

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

Here,

• FV is future value.
• C is monthly payment.
• i is interest rate.
• n is number of payments.

The formula to calculate future value is,

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

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

$\begin{array}{c}FV=\$700\times \left({\left(1+0.00\right)}^4+{\left(1+0.00\right)}^3+{\left(1+0.00\right)}^2+{\left(1+0.00\right)}^1+{\left(1+0.00\right)}^0\right)\\ =\$700\times 5\\ =\$3,500\end{array}$

Conclusion

The future value of annuity will be $3500.

d.a.

Summary Introduction

To calculate: Future value of annuity a) $400 per year for 10 years at 10% b) $200 per year for 5 years at 5% c) $400 per year for 5 years at 0% d) Rework part a ,b and c as they are due.

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

Explanation of Solution

Solution:

The formula to calculate value of annuity is,

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

Here,

• FV is future value.
• C is monthly payment.
• i is interest rate.
• n is number of payments.

The formula to calculate future value of annuity due is,

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

Here,

• FV is future value of annuity.
• C is the monthly payment.
• i is interest rate.
• n is number of payments.

Substitute $400 for C, 10% for i and 10 years for n in equation (II).

$\begin{array}{c}FV=\$400\times \frac{\left({\left(1+0.10\right)}^{10}-1\right)}{0.10}\times \left(1+0.10\right)\\ =\$400\times 17.5311671\\ =\$7,012.46684\end{array}$

Conclusion

The future value of annuity due is $7,012.46684.

b.

Summary Introduction

To calculate: Future value of annuity a) $400 per year for 10 years at 10% b) $200 per year for 5 years at 5% c) $400 per year for 5 years at 0% d) Rework part a ,b and c as they are due.

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

Explanation of Solution

Solution:

The formula to calculate value of annuity is,

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

Here,

• FV is future value.
• C is monthly payment.
• i is interest rate.
• n is number of payments.

Substitute $200 for C, 5% for i and 5 years for n in equation (II).

$\begin{array}{c}FV=\$200\frac{\left({\left(1+0.05\right)}^5-1\right)}{0.05}\times \left(1+0.05\right)\\ =\$200\times 5.80191276\\ =\$1,160.38255\end{array}$

Conclusion

The future value of annuity due is $1,160.38255.

c.

Summary Introduction

To calculate: Future value of annuity a) $400 per year for 10 years at 10% b) $200 per year for 5 years at 5% c) $400 per year for 5 years at 0% d) Rework part a ,b and c as they are due.

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

Explanation of Solution

Solution:

The formula to calculate value of annuity is,

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

Here,

• FV is future value.
• C is monthly payment.
• i is interest rate.
• n is number of payments.

The formula to calculate future value is,

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

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

$\begin{array}{c}FV=\left[\begin{array}{c}\$400\times \left({\left(1+0.00\right)}^4+{\left(1+0.00\right)}^3+{\left(1+0.00\right)}^2+{\left(1+0.00\right)}^1+{\left(1+0.00\right)}^0\right)\\ \times \left(1+0.05\right)\end{array}\right]\\ =\$400\times 5.25\\ =\$2,100\end{array}$

Conclusion

The future value of annuity due is $2,100.