Exponential Growth

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Exponential Growth
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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?

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Exponential Growth

• Interest after 1 year
• Account value after 1 year
• What would happen during the 2nd year?

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Exponential Growth

• Interest made during the 2nd year
• Value of account after 2nd year
• What about for the 3rd year?

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Exponential Growth

• Interest made during 3rd year
• Value of the account after 3rd year

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• Summarizing our calculations

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• 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

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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.

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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?

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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

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Exponential Growth

• Interest made during 1st quarter
• Value of account after 1st quarter

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Exponential Growth

• Interest made during the 2nd quarter
• Value of account after 2nd quarter

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• Interest made during the 3rd quarter
• Account value after 3rd quarter

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• Interest made during the 4th quarter
• Account value after 4th quarter

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• Summarizing our results for 1 year

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• 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

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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

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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

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• From the last slide, we can also say
• In other words, we can find the present value (P)
  by knowing the future value (F).

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Exponential Growth

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

n1 n4
t F F
1 5,275.00 5,280.72
2 5,565.13 5,577.21
3 5,871.21 5,890.34
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Exponential Growth

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

n52 n365
t F F
1 5,282.55 5,282.68
2 5,581.07 5,581.34
3 5,896.45 5,896.89
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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

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Exponential Growth

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

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• As m gets big,

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• So as m gets large,
• This is for continuous compounding
• In Excel, use the function EXP(x)

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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

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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

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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

Effective Annual Yield (y)
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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

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Exponential Growth

• You may need to find the annual rate that would
  produce a given yield.
• Need to solve for r

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
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Exponential Growth

• Effective Annual Yield (Continuous)
• Annual interest rate

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

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• Ex. Find the annual yield on an investment that
  computes interest at 4.7 compounded monthly.
• Soln.
• About 4.80

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• Ex. Find the rate, compounded weekly, that has a
  yield of 9.1
• Soln.
• About 8.72

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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?

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Exponential Growth

• Soln.

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• Ex. Find the annual rate of an investment that
  has an annual yield of 9 when compounded
  continuously.
• Soln.
• Approx 8.62

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• Where else can compound interest be used?
• Financing a home
• Financing a car
• Anything where you make monthly payments (with
  interest) on money borrowed

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• 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?

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Exponential Growth

• For a 30 year mortgage, youll 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

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Exponential Growth

• Lets say that Pk represents the present monthly
  value k months ago.
• Then after 360 months, we want

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• Since were 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.

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• Remember we want
• So if we insert
• We have instead

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• Now for a little algebra (factor out F)
• Divide both sides by the stuff in

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• 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.

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• Since were compounding monthly at 7, r 0.07
  and n 12
• So

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• Well do the rest of our calculation in Excel
• So our monthly payments F

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• 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
• Well again use Excel to answer this question.

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End

• 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 were 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.
Start
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Exponential Growth

• What your outstanding balance looks like with
  each monthly payment?

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Exponential Growth

• Things to notice
• After 360 months of payments of 1330.61, youre
  really paying 479,019.60 on 200,000 borrowed.
• The mortgage company has made 139 profit on your
  borrowing 200,000.