Exponential Growth

Exponential Growth

Discrete Compounding Suppose that you were going to

invest $5000 in an IRA earning interest at an annual rate of 5.5%. How much interest would you earn during the 1st year? How much is in the account after 1 year?

Exponential Growth

Interest after 1 year:

Account value after 1 year:

What would happen during the 2nd year?

275$055.050001 rP

i

5275$055.15000055.050005000275$

1

1

iP

F

Exponential Growth Interest made during the 2nd year:

Value of account after 2nd year:

What about for the 3rd year?

13.290$055.052751

2 rF

i

5,565.13$055.15275055.05275527513.290$

2

21

iF

F

Exponential Growth

Interest made during 3rd year:

Value of the account after 3rd year:

08.306$055.0$5,565.132

3 rF

i

$5,871.21055.113.5565055.013.556513.556508.306$

3

32

iF

F

Exponential Growth

Summarizing our calculations:

$5,871.21055.15000055.113.5565

$5,565.13055.15000055.15275

275,5$055.15000

5000

3

055.15000

3

2

055.150002

1

2

F

F

F

P

Exponential Growth

From our calculations, a $5,000 investment into an account with an annual interest rate of 5.5% will have a value of F after t years according to the formula:

ttF 055.15000

Exponential Growth In general, P dollars invested at an annual

rate r, has a value of F dollars after t years according to:

Notice that the interest was paid on a yearly basis, while our money remained in the account. This is called compounding annually or one time per year.

tt rPF 1

Exponential Growth

What would happen if the interest was paid more times during the year?

Suppose interest is collected at the end of each quarter, (interest is paid four times each year). What would happen to our investment?

Exponential Growth

Since the annual interest rate is 5.5% this rate needs to be adjusted so that interest is paid on a quarterly basis. The quarterly rate is:

%375.14

5.541 n

rr

Exponential Growth

Interest made during 1st quarter:

Value of account after 1st quarter:

$68.754

055.05000

41

41

r

P

i

$5,068.754

055.015000

4

055.050005000

41

41

i

P

F

Exponential Growth

Interest made during the 2nd quarter:

Value of account after 2nd quarter:

$69.704

055.05068.75

41

41

42

r

F

i

$5,138.454

055.015068.75

4

055.05068.755068.75

$69.7042

41

42

i

F

F

Exponential Growth

Interest made during the 3rd quarter:

Account value after 3rd quarter:

70.65$4

055.05138.45

41

42

43

r

F

i

$5,209.104

055.015,138.45

4

055.05138.455138.45

$70.6543

42

43

i

F

F

Exponential Growth

Interest made during the 4th quarter:

Account value after 4th quarter:

$71.634

055.05209.10

41

43

44

r

F

i

$5,280.724

055.015209.10

4

055.05209.105209.10

$71.6344

43

44

i

F

F

Exponential Growth

Summarizing our results for 1 year:

$5,280.724

055.015000

4

055.015209.10

$5,209.104

055.015000

4

055.015138.45

$5,138.454

055.015000

4

055.0175.5068

$5,068.754

055.015000

5000

4

3

2

43

44

42

43

41

42

41

F

F

F

F

F

F

F

P

Exponential Growth

Notice that the exponent corresponds to the number of quarters in a year: So for 1 year there are 4 quarters So for 2 years there are 8 quarters So for 3 years there are 12 quarters So for 4 years there are 16 quarters So for t years there are 4t quarters

Exponential Growth

So the value of a $5,000 investment with an annual interest rate of 5.5% compounded quarterly after t years is given by:

t

tF4

4

055.015000

Exponential Growth

In general, P dollars invested at an annual rate r, compounded n times per year, has a value of F dollars after t years according to:

nt

n

rPF

1

Exponential Growth

From the last slide, we can also say:

In other words, we can find the present value (P) by knowing the future value (F).

nt

n

rFP

1

Exponential Growth

Notice for each of the 3 years the account that is compounded quarterly is worth more than the one compounded annually

n=1 n=4

t F F

1 $5,275.00 $5,280.72

2 $5,565.13 $5,577.21

3 $5,871.21 $5,890.34

Exponential Growth

It would seem the larger n is the more an investment is worth, but consider:

n=52 n=365

t F F

1 $5,282.55 $5,282.68

2 $5,581.07 $5,581.34

3 $5,896.45 $5,896.89

Exponential Growth

Notice value of the investment is leveling off when P, r, and t are fixed, but n is allowed to get really big.

This suggests that is leveling off to some special number

n

n

r

1

Exponential Growth

There is a clever technique that allows us to find this value. We let m = n/r, so that n = mr. For any value of r, m gets larger as n increases. We rewrite the expression:

rtmmrt

mP

rm

rPF

111

Exponential Growth

As m gets big,

10 2.59374246100 2.7048138291000 2.71692393210000 2.718145927100000 2.7182682371000000 2.71828046910000000 2.718281694100000000 2.718281786

m

m

11m

em

m

59057182818284.2

11

Exponential Growth So as m gets large,

This is for continuous compounding In Excel, use the function EXP(x)

rt

rtmmrt

Pe

mP

rm

rPF

111

Exponential Growth

So P dollars will grow to F dollars after t years compounded continuously at r % by the equation:

We can also find P by knowing F as follows:

rtPeF

rtFeP

Exponential Growth How do we compare investments with

different interest rates and different frequencies of compounding? Look at the values of P dollars at the end

of one year Compute annual rates that would

produce these amounts without compounding.

Annual rates represent the effective annual yield

Exponential Growth

In our current example when we compounded quarterly, after one year we had:

Notice we gained $280.72 on interest after a year. That interest represents a gain of 5.61% on $5000:

72.5280$4

055.015000

14

1

F

0561.05000

72.280y

Effective Annual Yield (y)

Exponential Growth

Effective annual yield (Discrete): find the difference between our money

after one year and our initial investment and divide by the initial investment.

Therefore, interest at an annual rate r, compounded n times per year has yield y:

111

n

n

n

r

P

Pnr

Py

Exponential Growth

You may need to find the annual rate that would produce a given yield.

Need to solve for r :

ryn

n

ry

n

ry

n

ry

n

ry

n

n

n

n

n

11

11

11

11

11

/1

/1

/1

This tells you the annual interest rate r that will produce a given yield when compounding n times a year. Note: This is only for Discrete Compounding

Exponential Growth

Effective Annual Yield (Continuous):

Annual interest rate:

1

rr

eP

PePy

1ln yr

Exponential Growth

Ex. Find the final amount if $10,000 is invested with interest calculated monthly at 4.7% for 6 years.

Soln. 49.250,13$

1000,10

1612

12047.0

nt

nrPF

Exponential Growth Ex. Find the annual yield on an

investment that computes interest at 4.7% compounded monthly.

Soln.

About 4.80%

0480.0

1048025794.1

11

1112

12047.0

n

nry

Exponential Growth

• Ex. Find the rate, compounded weekly, that has a yield of 9.1%

• Soln.

About 8.72%

087167685.0

1091.0152

1152/1

/1

nynr

Exponential Growth Examples that use the word

continuous to describe compounding period mean you use:

Ex. How much would you have after 3 years if an investment of $15,000 was placed into an account that earned 10.3% interest compounded continuously?

rtPeF

Exponential Growth

Soln.

94.430,20$

000,15 3103.0

e

PeF rt

Exponential Growth Ex. Find the annual rate of an investment

that has an annual yield of 9% when compounded continuously.

Soln.

Approx 8.62%

0862.0

109.0ln

1ln

yr

Exponential Growth

Where else can compound interest be used? Financing a home Financing a car Anything where you make monthly

payments (with interest) on money borrowed

Exponential Growth

The average cost of a home in Tucson is roughly around $225,000. Suppose you were planning to put down $25,000 now and finance the rest on a 30 year mortgage at 7% compounded monthly. How much would your monthly payments be?

Exponential Growth For a 30 year mortgage, you’ll be making

360 monthly payments.

At the end of the 360 months we want the present value (P) of all the monthly payments to add up to the amount you plan to finance, e.g. $200,000

The $200,000 is called the principal

Exponential Growth

Let’s say that Pk represents the present monthly value k months ago.

Then after 360 months, we want:

360321000,200 PPPP

Exponential Growth Since we’re borrowing money here,

each Pk can be expressed as

But where F represents the future value for Pk. In other words, F is your monthly payment.

k

k n

rFP

1

Exponential Growth

Remember we want:

So if we insert:

We have instead:

360321000,200 PPPP k

k n

rFP

1

36021

111000,200

n

rF

n

rF

n

rF

Exponential Growth

Now for a little algebra (factor out F):

Divide both sides by the stuff in [ ]

36021

111000,200n

r

n

r

n

rF

F

nr

nr

nr

36021

111

000,200

Exponential Growth The last result will tell us our monthly

payment F:

Notice that all we need to is figure out how to add up the numbers in the bottom. This is where we use Excel.

F

nr

nr

nr

36021

111

000,200

Exponential Growth

Since we’re compounding monthly at 7%, r = 0.07 and n = 12

So:

F

36021

1207.0

11207.0

11207.0

1

000,200

Exponential Growth

We’ll do the rest of our calculation in Excel

So our monthly payments F:

60.1330150.3076

000,200

1207.0

11207.0

11207.0

1

000,20036021

F

Exponential Growth

Now that we know what F is we can figure out what each Pk is.

Again, each Pk will tell us what F dollars was worth k months ago

We’ll again use Excel to answer this question.

Exponential Growth

In Excel:This number tells us that our monthly payment of $1330.60 was worth $1322.89 one month ago. Notice that as we descend down the table the values get smaller because we’re going farther back in time.

This number tells us how much of the monthly payment is for interest. Notice that as we descend the table the interest goes up. This tells us that in the beginning of a payment plan a lot of the monthly payment is toward interest and only a small portion is going toward principal while the reverse is true at the end.

End

Start

Exponential Growth

What your outstanding balance looks like with each monthly payment?

Balance

-500000

50000100000150000200000250000

0 100 200 300 400

Months

Bal

ance

$

Balance

Exponential Growth Things to notice:

After 360 months of payments of 1330.61, you’re really paying $479,019.60 on $200,000 borrowed.

The mortgage company has made 139% profit on your borrowing $200,000.