Chapter 4, Problem 44QP

Summary Introduction

To calculate: The present value of annuity.

Introduction:

The current worth of the future value of money is termed as present value. The cash flow’s present value in future with a particular rate of discount is the present value of annuity.

Answer to Problem 44QP

Solution:

The present value of annuity for the first 7 years is $99,134.79 and the present value of annuity for the remaining 8 years is $133,166.63.

Explanation of Solution

Given information:

The payments for a fifteen-year annuity are made at the end of each month and it pays $1,750 per month. The rate of interest is compounded monthly and the rate is twelve percent for the first seven years and six percent for the rest eight years and this rate is also compounded monthly.

Timeline of the sales:

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 annual cash flow, r denotes the rate of exchange, and t denotes the period. From the above information, it is necessary to compute the present value of annuity; but the rate of interest varies at the life of annuity. At first, it is essential to determine the last eight years’ present value of annuity.

Compute the present value of annuity for the last eight years:

$\begin{array}{c}{\text{Present value annuity}}_2=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ \text{Present value annuity}=\$1,750\left\{\frac{\left[1-\left(\frac{1}{{\left(1+\frac{0.06}{12}\right)}^{96}}\right)\right]}{\frac{0.06}{12}}\right\}\\ =\$1,750\left\{\frac{\left[1-\left(\frac{1}{{\left(1+0.005\right)}^{96}}\right)\right]}{0.005}\right\}\\ =\$1,750\left\{\frac{\left[1-\left(\frac{1}{1.614142708}\right)\right]}{0.005}\right\}\end{array}$

$\begin{array}{c}=\$1,750\left\{\frac{\left[1-0.619523908\right]}{0.005}\right\}\\ =\$1,750\left\{\frac{0.380476092}{0.005}\right\}\\ =\$1,750\times 76.0952184\\ =\$133,166.63\end{array}$

Hence, the present value of annuity for the last eight years is $133,166.63.

Formula to calculate the present value:

$\text{Present value}=\frac{\text{Cash flow}}{{\left(1+\text{r}\right)}^{\text{t}}}$

Note: r denotes rate of discount and t denotes the number of years. The value of this cash flow is calculated. The lump sum is discounted for today.

Compute the present value:

$\begin{array}{c}\text{Present value }=\frac{\text{Cash flow}}{{\left(1+\text{r}\right)}^{\text{t}}}\\ =\frac{\$133,166.63}{{\left(1+\frac{0.12}{12}\right)}^{84}}\\ =\frac{\$133,166.63}{{\left(1.01\right)}^{84}}\\ =\frac{\$133,166.63}{2.306722744}\end{array}$

$=\$57,729.79451$

Note: This is the present value of cash flow for today.

Hence, the present value of cash flow for today is $57,729.79451.

Compute the present value of annuity for the first seven years:

$\begin{array}{c}{\text{Present value annuity}}_1=C\left\{\frac{\left[1-\left(\frac{1}{1+r^t}\right)\right]}{r}\right\}\\ =\$1,750\left\{\frac{\left[1-\left(\frac{1}{{\left(1+\frac{0.12}{12}\right)}^{84}}\right)\right]}{\frac{0.102}{12}}\right\}\\ =\$1,750\left\{\frac{\left[1-\left(\frac{1}{{\left(1+0.01\right)}^{84}}\right)\right]}{0.01}\right\}\\ =\$1,750\left\{\frac{0.566484527}{0.01}\right\}\end{array}$

$=\$99,134.79223$

Hence, the present value of annuity for the first seven years is $99,134.79223.

Formula to calculate the present value of today’s cash flow:

$\text{Present value of cash flow today}={\text{PVA}}_1+{\text{PVA}}_2$

Note: The PVA refers to present value of annuity.

Compute the present value of the cash flow today:

$\begin{array}{c}\text{Present value of cash flow today}={\text{PVA}}_1+{\text{PVA}}_2\\ =\$99,134.79223+\$57,729.79451\\ =\$156,864.59\end{array}$

Hence, the present value of the cash flow today is $156,864.59.