Copyright © Cengage Learning. All rights reserved. Compound Interest and Continuous Growth SECTION 6.3

Copyright © Cengage Learning. All rights reserved.

Compound Interest andContinuous Growth

SECTION 6.3

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Learning Objectives

1 Use the compound interest formula to calculate the future value of an investment

2 Construct and use continuous growth models

3 Use exponential models to predict and interpret

unknown results

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Compound Interest

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Compound Interest

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Example 1 – Compound Interest

Consider the following situation. In April 2007, Heritage Bank offered a 1-year certificate of deposit (CD) with a minimum investment of $1000 at a nominal interest rate of 4.42% compounded quarterly. (Nominal means “in name only.”) What is the value of the CD after 1 year?

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Compound Interest

We can now find the future value of the CD after 1 year by using a quarterly growth factor of 1.01105 (an increase of 1.105%) as shown in Table 6.17.

Table 6.17

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Compound Interest

Note that the interest payments continue to increase throughout the year.

This is due to compound interest, which occurs when interest is paid on the initial amount invested as well as all previously earned interest.

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The Compound Interest Formula

Table 6.19 shows compounding frequencies and associated formulas for compound interest.

Table 6.19

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In March 2007, MetLife Bank advertised a 5-year CD with a 4.64% nominal interest rate compounded daily.

Assuming someone could continue to invest their money at this rate and compounding frequency up until retirement at age 65, compare the future value of a $5000 investment of a 25-year-old to the future value of a $10,000 investment for a 45-year-old.

Assume no additional deposits or withdrawals are made.

Example 2 – Comparing Future Values

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Example 1 – Solution

The 25-year-old will be accumulating interest 365 days a year for 40 years (the time until she retires), while the 45-year-old will be accumulating interest 365 days a year for 20 years (the time until he retires). Using the compound interest formula, we have

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Deducting the amount of the initial investment, we see that the 25-year-old earned $26,986.69 in interest while the 45-year-old earned $15,292.96 in interest.

cont’d

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Compound Interest as an Exponential Function

Since compound interest causes an account to grow by a fixed percent each year, we may categorize the compound interest formula as an exponential function.

First, consider the following property of exponents.

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From this property we can rewrite the compound interest formula as follows.

If we then let can rewrite the compound interest

formula in the form y = abx.

A = Pbt

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The annual percentage yield (APY) is the actual amount of interest earned during the year and takes into account not only the nominal rate but also the compounding frequency.

The Truth in Savings Act of 1991 requires banks to publish the APY for all savings accounts and CDs so consumers can more easily compare offers from different banks and make informed financial decisions.

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Example 3 – Modeling Compound Interest with Exponential Functions

An investor makes a $3000 initial deposit in a CD with a 3.68% interest rate compounded weekly.

a. Use the compound interest formula to find an exponential function to model the future value of the investment.

b. Determine the annual percentage yield for the investment.

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Example 3(a) – Solution

So models the future value of the CD.

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Example 3(b) – Solution

From the exponential model, we can see the annual growth factor is 1.0375 and the corresponding annual percentage yield is 3.75% (1.0375 – 1).

cont’d

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Continuous Growth Factor

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Example 4 – Finding the Continuous Growth Factor

Using the compound interest formula, examine what happens to $1 invested in an account with a 100% interest rate if interest is compounded an increasing number of times per year. Discuss what you discover.

Solution:

Based on the examples we have seen thus far, it appears that the value of the account will increase as we increase the compounding frequency.

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To see whether compounding every hour, minute, or second will dramatically increase the account value, consider Table 6.20.

Table 6.20

cont’d

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The table reveals an interesting result. It appears the account will never be worth more than $2.72 after 1 year, and the maximum annual growth (APY) will never be more than 171.8% (a growth factor of approximately 2.718).

This number ( 2.718) is so important in mathematics

that it has its own name: e.

cont’d

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We represent the notion of infinitely large n by using the term continuous compounding.

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Find the future value after 4 years of $2000 invested in an account with a 5.3% nominal interest rate compounded continuously.

Solution:

Using the formula for continuous compound interest, we have

Example 5 – Continuous Compound Interest

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After 4 years, the account will be worth $2472.30.

cont’d

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e: The Natural Number

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Converting from General Form to Using e

Consider the following situation. The total annual health-related costs in the United States in billions of dollars can be modeled by the function

where t is the number of years since 1960.

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The function H is similar to continuous compound interest, so we could use an exponential function involving the number e to model the annual health-related expenditures.

Since we are not dealing with an interest-bearing account, we will use the more generalized form of f (x) = aekx instead of the more specific A = Pert.

By finding the value of k such that ek = b, we can rewrite the exponential function using the number e.

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We see that k = 0.09649, so e0.09649 = 1.1013. Thus, we rewrite H as

The functions and are equivalent, provided we disregard round-off error.

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Example 6 – Converting f (x) = aekx to f (x) = abx

Rewrite the function f (x) = 1000e0.07x in the form f (x) = abx.

Solution:

Since b = ek, we can find the growth factor b by evaluating e0.07.

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Continuous Growth (Decay) Rate

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We know that functions of the form f (x) = abx exhibit exponential growth when b > 1 and exponential decay when b < 1. Since b = ek, what effect does k have on the value of b?

Let’s consider the results when k = 0.1 and when k = –0.1.

It appears that f (x) = aekx has exponential growth when k > 0 and exponential decay when k < 0.

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Copyright © Cengage Learning. All rights reserved. Compound Interest and Continuous Growth SECTION 6.3