Chapter 5, Problem 46QP

a)

Summary Introduction

To calculate: The present value of the payments at an ordinary annuity and the present value of annuity due.

Introduction:

The present value of the cash flows in the future with a particular discount rate is the present value of annuity. The repeating payment that is made at the starting of every period is the annuity due.

Answer to Problem 46QP

• The present value of annuity is $54,619.45.
• The present value annuity due is $58,715.90.

Explanation of Solution

Given information:

Person X is going to receive $13,500 per year for 5 years. The correct interest rate is 7.5%

Suppose the payments are in the form of an ordinary annuity, the present value is computed with the following formula:

Formula to calculate the present value annuity for sixty months:

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

Note: C denotes the annual cash flow, r denotes the rate of exchange, and t denotes the time period.

Timeline of the present value of annuity:

Compute the present value annuity:

$\begin{array}{c}\text{Present value annuity}=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ =\$13,500\left\{\frac{\left[1-\left(\frac{1}{{\left(1+0.075\right)}^5}\right)\right]}{0.075}\right\}\\ =\$13,500\left\{\frac{\left[1-\left(\frac{1}{{\left(1.075\right)}^5}\right)\right]}{0.075}\right\}\\ =\$13,500\left\{\frac{\left[1-\left(\frac{1}{1.435629326}\right)\right]}{0.075}\right\}\end{array}$

$\begin{array}{c}=\$13,500\left\{\frac{\left[1-0.696558632\right]}{0.075}\right\}\\ =\$13,500\left\{\frac{0.303441367}{0.075}\right\}\\ =\$54,619.45\end{array}$

Hence, the present value of annuity is $54,619.45.

Timeline of the present value of annuity due:

Formula to compute the present value of annuity due:

${\text{PVA}}_{\text{due}}=\left(1+r\right)\text{PVA}$

Note: The PVA is the present value of annuity and r is the rate of interest.

Compute the present value of annuity due:

$\begin{array}{c}{\text{PVA}}_{\text{due}}=\left(1+r\right)\text{PVA}\\ \text{=}\left(1+0.075\right)\$54,619.45\\ =\left(1.075\right)\$54,619.45\\ =\$58,715.90\end{array}$

Hence, the present value of annuity due is $58,715.90.

b)

Summary Introduction

To calculate: The future value of annuity and the future value of annuity due,

Introduction:

The value of a group of recurring payments at a particular date in the future is the future vale of annuity.

Answer to Problem 46QP

• The future value of annuity is $78,413.28.
• The future value of annuity due is $84,294.27.

Explanation of Solution

Given information:

Person X is going to receive $13,500 per year for 5 years. The correct interest rate is 7.5%

The future value of the annuity can be calculated as follows:

Timeline of the future value of annuity:

Formula to calculate the future value annuity:

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

Note: C denotes the annual cash flow or annuity payment, r denotes the rate of interest, and t denotes the number of payments.

Compute the future value annuity:

$\begin{array}{c}\text{Future value annuity}=C\left\{\frac{\left[{\left(1+r\right)}^t-1\right]}{r}\right\}\\ =\$13,500\left\{\frac{\left[{\left(1+0.075\right)}^5-1\right]}{0.075}\right\}\\ =\$13,500\left\{\frac{\left[{\left(1.075\right)}^5-1\right]}{0.075}\right\}\\ =\$13,500\left\{\frac{\left[1.435629326-1\right]}{0.075}\right\}\end{array}$

$=\$78,413.28$

Hence, the future value of annuity is $78,413.28.

Timeline of the future value of annuity due:

Formula to compute the future value of annuity due:

${\text{FVA}}_{\text{due}}=\left(1+r\right)\text{FVA}$

Note: The FVA is the future value of annuity and r is the rate of interest.

Compute the present value of annuity due:

$\begin{array}{c}{\text{FVA}}_{\text{due}}=\left(1+r\right)\text{FVA}\\ \text{=}\left(1+0.075\right)\$78,413.28\\ =\left(1.075\right)\$78,413.28\\ =\$84,294.27\end{array}$

Hence, the future value of annuity due is $84,294.27.

c)

Summary Introduction

To find: The highest present value, the ordinary annuity due or the ordinary annuity, and the highest future value.

Introduction:

The present value of the cash flows in the future with a particular discount rate is the present value of annuity. The repeating payment that is made at the starting of every period is the annuity due.

Answer to Problem 46QP

The present value of annuity due has the highest present value and the future value of annuity due has the highest future value

Explanation of Solution

If the rate of interest is positive, then the present value of the annuity due will be the highest when compared to the present value of annuity. Every cash flow in an annuity due has got a year before; that means there is one period less to discount in every cash flow.

If the rate of interest is positive, then the future value of the annuity due will be the highest when compared to the future value of annuity. Every cash flow is made a year before; therefore, every cash flow gets an extra period of compounding.

Hence, the present value of annuity due has the highest present value and the future value of annuity due has the highest future value.