Chapter 5, Problem 6QP

Calculating Annuity Values. For each of the following annuities, calculate the present value.

Annuity Payment	Years	Interest Rate
$ 2,100	7	5%
1,095	9	10
11,000	18	8
30,000	28	14

Summary Introduction

To calculate: The present value.

Introduction:

An annuity is a level stream of flow of cash for a particular period. They often appear in financial arrangements and it is a shortcut used for finding the values.

Answer to Problem 6QP

• The present value for the annuity payment of $2,100 which is given for 7 years at an interest rate of 5% is $12,151.38.
• The present value for the annuity payment of $1,095 which is given for 9 years at an interest rate of 10% is $6,306.13.
• The present value for the annuity payment of $11,000 which is given for 18 years at an interest rate of 8% is $103,090.76.
• The present value for the annuity payment of $30,000 which is given for 28 years at an interest rate of 14% is $208,819.87.

Explanation of Solution

Given information:

The details of annuity payment, number of years, and interest rate are provided. The annuity payment of $2,100 is given for 7 years at an interest rate of 5%. The timeline is as follows:

The annuity payment of $1,095 is given for 9 years at a rate of 10%. The timeline is as follows:

The payment of $11,000 is given for 18 years at a rate of 8%. The timeline is as follows:

The payment of $30,000 is given for 28 years at a rate of 14%. The timeline is as follows:

Formula to calculate the present value annuity:

$\text{Present value annuity}=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}$

Note: C denotes the annuity payment or an annual cash flow, r denotes the rate of exchange, and t denotes the period.

Compute the present value annuity for $2,100 at 5%:

$\begin{array}{c}\text{Present value annuity}=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ =\$2,100\left\{\frac{\left[1-\left(\frac{1}{{\left(1+0.05\right)}^7}\right)\right]}{0.05}\right\}\\ =\$2,100\left\{\frac{\left[1-\left(\frac{1}{{1.05}^7}\right)\right]}{0.05}\right\}\\ =\$2,100\left\{\frac{\left[1-\left(\frac{1}{1.407100423}\right)\right]}{0.05}\right\}\end{array}$

$\begin{array}{c}=\$2,100\left\{\frac{1-0.71068133}{0.05}\right\}\\ =\$2,100\left\{\frac{0.289318669}{0.05}\right\}\\ =\$2,100\times 5.786373397\\ =\$12,151.38\end{array}$

Hence, the present value annuity is $12,151.38.

Compute the present value annuity for $1,095 at 10%:

$\begin{array}{c}\text{Present value annuity}=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ =\$1,095\left\{\frac{\left[1-\left(\frac{1}{{\left(1+0.10\right)}^9}\right)\right]}{0.10}\right\}\\ =\$1,095\left\{\frac{\left[1-\left(\frac{1}{{1.10}^9}\right)\right]}{0.10}\right\}\end{array}$.

$\begin{array}{c}=\$1,095\left\{\frac{\left[1-\left(\frac{1}{2.357947691}\right)\right]}{0.10}\right\}\\ =\$1,095\times 5.759023816\\ =\$6,306.13\end{array}$

Hence, the present value annuity is $6,306.13.

Compute the present value annuity for $11,000 at 8%:

$\begin{array}{c}\text{Present value annuity}=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ =\$11,000\left\{\frac{\left[1-\left(\frac{1}{{\left(1+0.08\right)}^{18}}\right)\right]}{0.08}\right\}\\ =\$11,000\left\{\frac{\left[1-\left(\frac{1}{{1.18}^9}\right)\right]}{0.08}\right\}\end{array}$

$\begin{array}{c}=\$11,000\left\{\frac{\left[1-\left(\frac{1}{3.996019499}\right)\right]}{0.08}\right\}\\ =\$11,000\times 9.371887136\\ =\$103,090.76\end{array}$

Hence, the present value annuity is $103,090.76.

Compute the present value annuity for $30,000 at 14%:

$\begin{array}{c}\text{Present value annuity}=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ =\$30,000\left\{\frac{\left[1-\left(\frac{1}{{\left(1+0.14\right)}^{28}}\right)\right]}{0.14}\right\}\\ =\$30,000\left\{\frac{\left[1-\left(\frac{1}{{1.14}^{28}}\right)\right]}{0.14}\right\}\end{array}$

$\begin{array}{c}=\$30,000\left\{\frac{\left[1-\left(\frac{1}{39.2044926}\right)\right]}{0.14}\right\}\\ =\$30,000\times 6.96066228\\ =\$208,819.87\end{array}$

Hence, the present value annuity is $208,819.87.