Chapter 4, Problem 72QP

Summary Introduction

To calculate: The return of the investment offerings.

Introduction:

The total sum of interest that is due for a particular time is known as the interest rate. The interest rate is due for a period either as a proportion of the sum borrowed or deposited, or as the proportion of the sum lent.

Answer to Problem 72QP

The return of the investment offering is 7.21%.

Explanation of Solution

Given information:

A college savings program is offered by a financial planning service. The plan states that an individual must make six annual payments of $15,000 each with a first payment from today on the 12^th birthday of Person X’s son. Starting from the 18^th birthday of Person X’s son, the plan provides $32,000 in a year for 4 years.

Note: Here, it is essential to determine the rate of interest that makes the present value of annuity for the college cost equivalent to the future value of annuity for the savings.

Formula to calculate the present value annuity is as follows:

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

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

Compute the present value of annuity is as follows:

$\begin{array}{c}\text{Present value annuity}=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ =\$32,000\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^4}\right)\right]}{r}\right\}\end{array}$

Formula to calculate the future value annuity is as follows:

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

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

Compute the future value of annuity is as follows:

$\begin{array}{c}\text{Future value annuity}=C\left\{\frac{\left[{\left(1+r\right)}^t-1\right]}{r}\right\}\\ =\$15,0000\left\{\frac{\left[{\left(1+r\right)}^t-1\right]}{r}\right\}\end{array}$

The equation of the present value of annuity and the equation of the future value of annuity are set equal to each other:

$32,000\left[\frac{\left\{1-\left[\frac{1}{{\left(1+r\right)}^4}\right]\right\}}{r}\right]=15,000\left\{\frac{\left[{\left(1+r\right)}^6-1\right]}{r}\right\}$

Cancelling the r term in the denominator and 1,000 in the numerator on both the sides we get:

$32\left[\left\{\left[\frac{{\left(1+r\right)}^4-1}{{\left(1+r\right)}^4}\right]\right\}\right]=15\left\{\left[{\left(1+r\right)}^6-1\right]\right\}$

$\begin{array}{l}\text{Take}{\left(1+r\right)}^4\text{from}\text{denominator}\text{of}\text{LHS}\text{to}\text{numerator}\text{of}\text{RHS}\\ 32\left[{\left(1+r\right)}^4-1\right]=15\left\{\left[{\left(1+r\right)}^6-1\right]\right\}{\left(1+r\right)}^4\end{array}$

Note: LHS is the left hand side and RHS is the right hand side.

$\begin{array}{l}\text{Simplify}\text{further}\\ 32{\left(1+r\right)}^4-32=15\left\{\left[{\left(1+r\right)}^{10}-{\left(1+r\right)}^4\right]\right\}\\ 32{\left(1+r\right)}^4-32=15{\left(1+r\right)}^{10}-15{\left(1+r\right)}^4\end{array}$

$\begin{array}{l}\text{Bring}\text{all}\text{terms}\text{to}\text{LHS}\\ 15{\left(1+r\right)}^{10}-32{\left(1+r\right)}^4-15{\left(1+r\right)}^4+32=0\\ 15{\left(1+r\right)}^{10}-47{\left(1+r\right)}^4+32=0\end{array}$

Note: The return on the investment offering is calculated by using a spreadsheet.

Compute the return on the investment with the spreadsheet:

Step 1:

• Type the simplified formulae in H6 of the spreadsheet and consider the r value as H7.

Step 2:

• Assume the r value as 0.10%.

Step 3:

• In the spreadsheet, go to data and select the what-if analysis.
• In what-if analysis, select the goal seek.
• In the set cell, select H6 (the formulae).
• The ‘To value’ is considered as 0(the value of the present value of annuity).
• The H7 cell is selected by changing the cell.

Step 4:

• Following the previous step click OK in the goal seek. The goal seek status appears with the r value.

Step 5:

• The r value appears to be 7.21211963873623%.

Hence, the r value is 7.21%.