Chapter 5, Problem 6QP

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

a) The present value for the annuity payment of $2,100 which is given for 7 years at an interest rate of 8% is $10,933.38.

b) The present value for the annuity payment of $1,095 which is given for 9 years at an interest rate of 7% is $7,134.18.

c) The present value for the annuity payment of $11,000 which is given for 18 years at an interest rate of 9% is $96,311.88.

d) The present value for the annuity payment of $30,000 which is given for 28 years at an interest rate of 11% is $258,048.65.

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 8%. The timeline is as follows:

The annuity payment of $1,095 is given for 9 years at a rate of 7%. The timeline is as follows:

The payment of $11,000 is given for 18 years at a rate of 9%. The timeline is as follows:

The payment of $30,000 is given for 28 years at a rate of 11%. 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 8%:

$\begin{array}{c}\text{Present value annuity}=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ =\$2,100\left\{\frac{\left[1-\left(\frac{1}{{\left(1+0.08\right)}^7}\right)\right]}{0.08}\right\}\\ =\$2,100\left\{\frac{\left[1-\left(\frac{1}{{1.08}^7}\right)\right]}{0.08}\right\}\\ =\$2,100\left\{\frac{\left[1-\left(\frac{1}{1.713824269}\right)\right]}{0.08}\right\}\end{array}$

$\begin{array}{c}=\$2,100\left\{\frac{1-0.583490395}{0.08}\right\}\\ =\$2,100\left\{\frac{0.416509604}{0.08}\right\}\\ =\$2,100\times 5.206370059\\ =\$10,933.38\end{array}$

Hence, the present value annuity is $10,933.38.

Compute the present value annuity for $1,095 at 7%:

$\begin{array}{c}\text{Present value annuity}=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ =\$1,095\left\{\frac{\left[1-\left(\frac{1}{{\left(1+0.07\right)}^9}\right)\right]}{0.07}\right\}\\ =\$1,095\left\{\frac{\left[1-\left(\frac{1}{{1.07}^9}\right)\right]}{0.07}\right\}\end{array}$.

$\begin{array}{c}=\$1,095\left\{\frac{\left[1-\left(\frac{1}{1.838459212}\right)\right]}{0.07}\right\}\\ =\$1,095\times 6.515232249\\ =\$7,134.18\end{array}$

Hence, the present value annuity is $7,134.18.

Compute the present value annuity for $11,000 at 9%:

$\begin{array}{c}\text{Present value annuity}=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ =\$11,000\left\{\frac{\left[1-\left(\frac{1}{{\left(1+0.09\right)}^{18}}\right)\right]}{0.09}\right\}\\ =\$11,000\left\{\frac{\left[1-\left(\frac{1}{{1.09}^9}\right)\right]}{0.09}\right\}\end{array}$

$\begin{array}{c}=\$11,000\left\{\frac{\left[1-\left(\frac{1}{4.717120417}\right)\right]}{0.09}\right\}\\ =\$11,000\times 8.755625109\\ =\$96,311.88\end{array}$

Hence, the present value annuity is $96,311.88.

Compute the present value annuity for $30,000 at 11%:

$\begin{array}{c}\text{Present value annuity}=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ =\$30,000\left\{\frac{\left[1-\left(\frac{1}{{\left(1+0.11\right)}^{28}}\right)\right]}{0.11}\right\}\\ =\$30,000\left\{\frac{\left[1-\left(\frac{1}{{1.11}^{28}}\right)\right]}{0.11}\right\}\end{array}$

$\begin{array}{c}=\$30,000\left\{\frac{\left[1-\left(\frac{1}{18.57990145}\right)\right]}{0.11}\right\}\\ =\$30,000\times 8.60162183\\ =\$258,048.65\end{array}$

Hence, the present value annuity is $258,048.65.