TABLE OF CONTENTS

2.1        Exponential Growth:  Compound Interest

 

Goals:  

  • Understand the nature of exponential growth
  • Write exponential model given initial value and growth factor or growth rate
  • Understand the difference between growth factor and growth rate.
  • Recognize data that has either a linear or exponential growth pattern

 

Terms to know:  

·         compound interest

·         effective annual yield

·         interest

·         nominal interest rate

·         simple interest

 

Concept Prep:  Real world linear vs. exponential  

 

Dynagraph Lab Activity

 

Let’s take a closer look at the first scenario in the concept prep exercises. 

 

a)  After the first year, your new salary

 

                                                =  Original salary  +  3% of original salary

 

                                                =   40,000 + .03(40,000)  

                                                =   40,000 +  1200    

                                                =   41,200

 

Note that your raise would be $1200.

                                                                                                                             

After the second year, your new salary would be

 

                                                =   Former salary  +  3% of former salary

                                                =   41,200 + .03(41,200)  

                                                =   41,200 +  1236    

                                                =   42,436

 

Note that your raise this time would be $1236.

 

After the third year, your new salary would be

 

                                                =  Former salary  +  3% of former salary

                                                =   42,436 + .03(42,436)  

                                                =   42,436 + 1273.08   

                                                =   43,709.08

 

Note that your raise this time would be $1273.08.

 

Notice also that because we are taking 3% of a larger number each successive year, the size of the raise was greater than that of the previous year.  A very different situation occurs when the raise is the same each year, as in the situation below.

 

b)  With a starting salary of $40,000 and a $1200 raise each year, at the end of the third   

      year your salary would be 

 

                                       =  Original salary + 3  1200

                                          =  40,000 + 3600

                                          =  43,600

 

 

                                         *     *     *

 

 

Year

Salary with 3% annual raise

 

Year

Salary with $1200 annual raise

 

 

 

 

 

0

$40,000

 

0

$40,000

1

$41,200

 

1

$41,200

2

$42,436

 

2

$42,400

3

$43,709

 

3

$43,600

4

$45,020

 

4

$44,800

5

$46,371

 

5

$46,000

 

                                                   TABLE 1.  A  Comparison of salaries

 

 

Suppose we wanted to generalize the first scenario above, i.e. we are earning 3% annual raises.

 

After the first year, your salary is

 

Factor out the GCF, 40,000:                                          

 

 

 

 

 

 

After the second year, your salary is

 

Factor out the GCF, 40,000(1.03):

 

 

 

 

 

 

After the third year, your salary is

 

Factor out the GCF, 40,000(1.03)2:

 

 

 

 

 

 

 

Are you noticing a pattern?  What would the salary be after the 10th year?

 

If we let t = number of years since the beginning of the contract, and continue in the above fashion, we find that the amount of money, A, depends upon t:

 

Notice that the variable, t, is in the exponent. 

 

 

 

 

A function of the form A(t) = Ca t where a > 0  and a  1 is an exponential function. 

 

The number C gives the initial value of the function (when t = 0) and the number a is the growth (or decay) factor. 

 

Note that if the growth rate is r, the growth factor is 1 + r.

 

 

 

 

Example 1.  Write a formula for an exponential function with initial value of 3,000 and a growth factor of 1.06.

 

SOLUTION.   The general exponential function is of the form A(t) = Ca t , with C as the initial value and a as the growth factor, so the function is given by 

 

.

                                                                                   

                                               

 

Example 2.  Write a formula for an exponential function with initial value of 10 and growing 3.5% every time period.

 

SOLUTION.   Since the initial value is growing at a rate of 3.5%, the growth factor is

1 +. 035 or 1.035.  The function is given by

 

 

                                                                       

 

 

Example 3.   How much money would be in an account after 10 years if you deposited $2500 in a mutual fund which compounds interests 4% annually?

 

SOLUTION.   Since the initial value is growing at a rate of 4%, the growth factor is

1 +. 04 or 1.04.  The function is given by

 

 

 

 

After 10 years, we would have

 

 

 

Not bad!  Notice that you added nothing to the initial amount. The increase is due solely to interest earned on the original amount.

 

 

 

 

Example 4.   Okay, suppose now you plan to stash away this $2500 for 40 years (i.e. you’re saving for retirement already  smart!).  Because you don’t need the money, you can invest in a riskier stock and it has been earning 12% per year, compounded once every year.  Assuming that it continues to earn 12% compounded annually, how much money would you have at the end of 40 years?

 

SOLUTION.   You’re not going to believe this.  Take a guess at how much you would have, before reading on.

 

 

 

 

 

 

 

 

When figuring compound interest, we use an exponential function, usually written:

 

A(t) = P(1 + r)t

 

Where r is the interest rate, P is the principle, and t is the time in years. Note that this formula is just a slight adaptation of the general exponential formula.

 

     Checkpoint Exponential Functions 1

 

 

More on Compound Interest

 

In practice, most banks, savings and loans, stocks don’t figure compound interest annually.  Find an advertisement either in the newspaper or online and report to the class at least 3 different compounding periods that you found.

 

Some investment firms compound interest quarterly.   Let’s take that $2500 again and invest it in a CD that advertises 12% interest compounded quarterly.

 

Think about it.  The bank is not going to give you 12% four times a year, because the annual rate would then be much higher than 12%.  They will take that 12% and divide it by 4 and give you that rate 4 times a year.

 

 So, our formula now becomes:

 

 

 

   

 

How many times will interest be compounded ?  

 

If compounding occurs 4 times a year, then after 1 year, there would be 4 compoundings, after 2 years there would be 4  2 = 8  compoundings after 3 years 4  3 = 12, etc.  In general, there would  be 4t  compoundings, where t is the number of years of the investment.

 

After 40 years, then, our $2500 would earn    

 

                                                           

about $50,000 more than the final amount with annual compounding!

 

 

 

 

It’s quite interesting to note the effect of different interest rates on the amount earned. 

 

Example 5.   Let’s take that $2500 and figure the amount you would have after 40 years at each of the following interest rates.   Write each answer as a dollar amount rounded to the nearest cent.

 

a)    10%                                                                 b) 8%

c)     6%                                                            d) 4%

 

          SOLUTION

        a)   

 

        b)   

 

        c)   

 

        d )   

 

 

 

Example 6.   Now, keep the interest rate steady at 8% and figure how much you would have if interest were compounded:

 

a)    annually

b)   semi-annually

c)    quarterly

d)   monthly

e)    daily

 

        SOLUTION

        a)             

 

        b)     

          

        c)       

           

        d)         

 

 

        e)             

 

 

MORAL OF THE STORY:   SAVE YOUR  MONEY!!!

 

 

 

 

 

 

 

 

 

When figuring compound interest that is compounded more than once a year, we use the function given by:

 

A= P (1 +  )nt    where

P  is the principle or present value

r  is the advertised interest rate,

n  is the number of compoundings per year, and

t is the time in years.

 

 

 

        Checkpoint Exponential Functions 2

 

 

Effective vs Nominal Interest Rate

 

Let’s take another look at the situation where $2500 is invested at 8% interest

compounded quarterly.   At the end of 1 year we would have

 

,

which simplifies to

 

 

 

Notice that if we take  we get 1.0824, which is equivalent to having 8.24% interest

figured only once.  So when we take 8% compounded quarterly, we are actually earning

8.24% a year!   This higher rate is called the effective annual rate or effective annual yield

in the business world.  The original or advertised rate of 8% is called the nominal rate,

because it is the rate “in name” only.

 

 

 

Example 7.  Find the effective annual yield for an account that gives 5.5% nominal interest compounded monthly.

 

SOLUTION.

 

Use the compound interest formula for t = 1 and P = 1:                                                                                                                                        

 

Use your calculator to round the answer to4 decimal places:       

 

 

Subtract 1 and change the decimal to a percent:                             5.64%                                     

 

The effective annual yield is 5.64 %

 

 

More worked examples

 

Homework problems

 

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