Introduction to Corporate Finance

PART 1: INTRODUCTION

3-4 ADDITIONAL APPLICATIONS

INVOLVING LUMP SUMS

Equations 3.1 and 3.2 are just two ways of writing a mathematical relationship linking the present

and future value of cash flows to each other and to the interest rate and investment horizon. In many

situations, the objective is not to find the present or future value of a cash payment, but instead to

answer a question about the interest rate or the investment horizon. The series of examples that follows

illustrates other kinds of problems that can be solved by using some form of Equation 3.1.

You saved $1,000, and you plan to put it into an

investment earning 8% interest. How many years

will it take you to triple your money? To solve this

problem, start with Equation 3.1:

FV = PV × (1 + r)n

In this case, we know PV = $1,000 and r = 0.08.

We also know that FV = $3,000, because the goal is

to triple the initial $1,000 investment. The unknown

quantity is n, the number of years needed for

$1,000 to grow to $3,000 if the interest rate is 8%.

Therefore, we have

$3,000 = $1,000 (1 + 0.08)n

Dividing both sides by $1,000 leaves a simplified

equation

3.0 = (1.08)n

To solve this last expression algebraically, we

take the natural logarithm of each side and then

simplify terms as follows

ln(3.0) = ln(1.08)n

ln(3.0) = ln(1.08) x n

ln(3.0) ÷ ln(1.08) = n

1.0986 ÷ 0.0770 = n

14.3 years = n

Figure 3.5 illustrates the calculator and

spreadsheet solutions for this problem.

example

FIGURE 3.5 USING A CALCULATOR TO FIND INVESTMENT TIME PERIOD

Formula B4: =NPER(B2,0,B1,B3,0)

Number of years 14.3

Future value –$3,000

8%

$1,000

Interest rate

Present value

Input

1,000

8

–3,000

PV

I

FV

CPT

N

Solution 14.3

Function

Row

Column

Calculator Spreadsheet

1

2

3

4

5

A B

In the preceding example, we solved for the number of periods required for a lump sum to grow to a

particular future value at a particular interest rate. Here, again, Excel offers a quick way to find the answer.

You can use the ‘number of periods’ function in Excel, which has the syntax =nper(rate,pmt,pv,fv,type).

In the previous problem, $1,000 is the present value, $3,000 is the future value, and 8% is the rate (the