Chapter 5, Problem 15P

a.

Summary Introduction

To calculate: Present value of annuity at $400 per year for 10 years at 10%.

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 annuitization. The purpose of the annuity is not to the break the flow of income after retirement.

Explanation of Solution

Solution:

Formula to calculate present value of annuity is,

${\text{PV}}_{\text{Annuity}}=\text{C}\times \left[\frac{1}{\text{I}}-\frac{1}{\text{I}\times {\left(1+\text{I}\right)}^{\text{N}}}\right]$ (I)

Where,

• PV is present value.
• C is monthly payment made.
• I is interest rate.
• N is number of years.

Substitute $400 for C, 10% for I and N for 10 in equation (I).

  $\begin{array}{c}\text{PV}=\text{\$400}\times \left[\frac{1}{0.10}-\frac{1}{0.10\times {\left(1+0.10\right)}^{10}}\right]\\ =\$400\times \left(10-3.85542392\right)\\ =\$2457.8284\end{array}$

Conclusion

The amount of present value will be $2457.8284.

b.

Summary Introduction

To calculate: Present value of annuity at $200 per year for 5 years at 5%.

Explanation of Solution

Solution:

Substitute $200 for C, 5% for I and N for 5 in equation (I).

$\begin{array}{c}\text{PV}=\$200\times \left[\frac{1}{0.05}-\frac{1}{0.05\times {\left(1+0.05\right)}^5}\right]\\ =\$200\times \left(20-15.6705229\right)\\ =\$865.8954\end{array}$

Conclusion

The amount of present value will be $865.8954.

c.

Summary Introduction

To calculate: Present value of annuity at $400 per year for 5 years at 10%.

Explanation of Solution

Solution:

Formula to calculate present value of annuity is,

${\text{PV}}_{\text{Annuity}}=\text{C}\times {\displaystyle \sum \frac{1}{{\left(1+\text{I}\right)}^{\text{N}}}}$

Where,

• PV is present value.
• C is monthly payment made.
• I is interest rate.
• N for number of years.

Substitute $400 for C, 5% for I and 4, 3, 2, 1 and 0 for N.

$\begin{array}{c}\text{PV}=\$400\left(\frac{1}{{\left(1+0.00\right)}^4}+\frac{1}{{\left(1+0.00\right)}^3}+\frac{1}{{\left(1+0.00\right)}^2}+\frac{1}{{\left(1+0.00\right)}^1}+\frac{1}{{\left(1+0.00\right)}^0}\right)\\ =\$400\times 5\\ =\$2000\end{array}$

Conclusion

The value of annuity is $2000.

d.

Summary Introduction

To calculate: Annuity due.

Explanation of Solution

Solution:

Formula to calculate annuity due amount is,

${\text{PV}}_{\text{Annuity}}=\text{C}\times \left[\frac{1}{\text{I}}-\frac{1}{\text{I}\times {\left(1+\text{I}\right)}^{\text{N}}}\right]\times \left(1+\text{I}\right)$

Where,

• PV is present value.
• C is monthly payment made.
• I is interest rate.
• N is number of years.

Substitute $400 for C, 5% for I and 10 for N.(part (a))

$\begin{array}{c}\text{PV}=\text{\$400}\times \left(\frac{1}{0.10}-\frac{1}{0.10\times {\left(1+0.10\right)}^{10}}\right)\times \left(1+0.10\right)\\ =\$400\times \left(10-3.85542392\right)\times \left(1+0.10\right)\\ =\$2,703.61348\end{array}$

The amount will be $2,703.61348 for annuity due.

Substitute $200 for C, 5% for I and 5 for N. (part (b))

$\begin{array}{c}\text{PV}=\$200\times \left(\frac{1}{0.05}-\frac{1}{0.05\times {\left(1+0.05\right)}^5}\right)\times \left(1+0.05\right)\\ =\$200\times \left(20-15.6705229\right)\times \left(1+0.05\right)\\ =\$909.190\end{array}$

The amount will be $909.190 for annuity due.

Formula to calculate annuity due amount is, (part (c))

$\text{PV}=\text{C}\times {\displaystyle \sum \frac{1}{{\left(1+\text{I}\right)}^{\text{N}}}\times \left(1+\mathrm{I}\right)}$

Where,

• PV is present value.
• C is monthly payment made.
• I is interest rate.
• N for number of years.

Substitute $400 for C, 0% for I and 4, 3, 2, 1 and 0 for N.

$\begin{array}{c}\text{PV}=\left[\begin{array}{c}\$400\times \left(\frac{1}{{\left(1+0.00\right)}^4}+\frac{1}{{\left(1+0.00\right)}^3}+\frac{1}{{\left(1+0.00\right)}^2}+\frac{1}{{\left(1+0.00\right)}^1}+\frac{1}{{\left(1+0.00\right)}^0}\right)\\ \times \left(1+0\right)\end{array}\right]\\ =\$400\times 5\times 1\\ =\$2,000\end{array}$

Conclusion

The final amount of annuity due is for part a. is $2,703.61348, for part b. is $909.190 and for part c. is $2,000.