Section 5.5

Exponential Functions and

Investing

Copyright ©2013 Pearson Education, Inc.

Objectives

Find future value of investments when interest is compounded k times per year

Find future value of investments when interest is compounded continuously

Find the present value of an investmentUse graphing utilities to model investment data

Example

To numerically and graphically view how $1000 grows at 6% interest, compounded annually, do the following:a. Construct a table that gives the future values S = (1 + 0.06)t for t = 0, 1, 2, 3, 4, and 5 years.b. Graph the function for the given values in a.c. Graph the function as a continues function.d. Is there another graph that more accurately represents the amount that would be in the account at any time during the first 5 years that the money is invested?

Example (cont)

Solutiona. b.

Example (cont)

Solutionc. d.

Future Value of an Investment with Annual CompoundingIf P dollars are invested at an interest rate r per year, compounded annually, the future value S at the end of t years is S = P(1 + r)t.

Future Value of an Investment with Periodic CompoundingIf P dollars are invested for t years at the annual interest rate r, where the interest is compounded k times per year, then the interest rate per period is r/k, the number of compounding periods is kt, and the future value that results is given by

1 dollars

ktrS Pk

Example

a. Write the equation that gives the future value of $1000 invested for t years at 8%, compounded annually.

b. Write the equation that gives the future value of $1000 invested for t years at 8%, compounded daily.

c. Graph the equation from parts (a) and (b) on the same axes, with t between 0 and 30.

d. What is the additional amount of interest earned in 30 years from compounding daily rather than annually?

Example (continued)

Solutiona. P = 1000 and r = 0.08

b. P = 1000, r = 0.08, and k = 365

1000(1 0.08)

1000(1.08)

t

t

SorS

3650.081000 1365

t

S

Example (continued)

Solutionc.

d. Amount compounded annually = $10,062.66 Amount compounded continuously = $11,020.28 Interest = $11,020.28 $10,062.66 = $957.62

Future Value of an Investment with Continuous CompoundingIf P dollars are invested for t years at an annual interest rate r, compounded continuously, then the future value S is given by

S = Pert dollars

Example

a. What is the future value of $3500 invested for 10 years at 9%, compounded continuously?

b. How much interest will be earned on this investment?

Solutiona.

b. Interest = $8608.61 – $3500 = $5108.61

0.09(10)35008608.61

S e

Present ValueThe lump sum that will give future value S in n compounding periods at rate i per period is the present value

or, equivalently, (1 )(1 )

nn

SP P S ii

Example

What lump sum must be invested at 10%, compounded semiannually, for the investment to grow to $15,000 in 7 years?

SolutionS = 15,000i = 0.10/2 = 0.05n = 2(7) = 14

14

15,000 7576.02(1 0.05)

P

Example

The data in Table give the annual return on an investment of $1.00 made on March 31, 1990. The values reflect reinvestment of all distributions and changes in net asset value but exclude sales charges.a. Use an exponential function to model these data.b. Use the model to find what the fund would amount to on March 31, 2001 if $10,000 was invested March 31, 1990 and the fund continued to follow this model after 2000.c. Is it likely that this fund continued to grow at this rate in 2002?

Example (cont)

Solutiona. Use technology to create an

exponential model for the data.

b. March 31, 2001 is 11 years after March 31, 1990.Use the equation from part (a) with x = 11.

c. No. Most stocks and mutual funds decreased in value in 2002 because of terrorist attacks in 2001.

( ) 1.1609(1.2004)xA x

11( ) $10,000[1.1609(1.2004) ]86,572.64

A x