Write

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Write

• Describe in words how we compute a growth factor
  if we are given a percentage increase.
• Describe in words how we compute a decay factor
  if we are given a percentage decrease.
• Is it possible to have a percentage decrease of
  more than 100? Why or why not?

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Lesson 26Compound Interest

• Thursday 5 January 2006

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What is interest?

• Interest is payment for the use of borrowed
  money.
• Some interest is paid to banks or other
  institutions that provide us with loans.
• Some interest is paid by banks to their
  depositors for the use of their money.
• What? Are you telling me that when I put my
  money into a bank, it doesnt stay there?
• Thats right. The bank loans it out to other
  people. Thats how they make money.

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How is interest computed?

• Well, simple interest is just an exponential
  function, of course.
• For example, if you earn 6 a year, and deposit
  1000, the amount of money you will have after x
  years is equal to
• 1000(1.06)x

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What other kind of interest is there?

• Well, the deal is this there are a lot of ways
  to think about that 6 youre earning.
• We refer to interest being broken down and
  applied in several chunks as compound interest.

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The Compound Interest Formula

• First, some vocabulary
• Annually Once a year
• Semi-annually Twice a year
• Quarterly Four times a year
• Monthly 12 times a year
• Weekly 52 times a year
• Daily 365 times a year
• Hourly 8760 times a year

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The Compound Interest Formula

• p principal (the amount you invest or borrow)
• r rate of interest, expressed as a decimal
• n number of installments per year
• x number of years

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STOP. WAIT.

• This really isnt difficult. Its just a little
  complicated. Well address this.
• Dont panic.
• OK. Now, lets look at this again.

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The Compound Interest Formula

• p principal (the amount you invest or borrow)
• r rate of interest, expressed as a decimal
• n number of installments per year
• x number of years

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The Compound Interest Formula

• So, lets work with this.
• Suppose you invest 1000 for 5 years at a rate of
  10 a year.
• To see the difference that the compounding of
  this interest makes, lets compute the total
  amount for several different compounding periods.

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The Compound Interest Formula

• Can you believe this?
• The rate is constant at 10 for the entire
  table.
• The only difference is how frequently this
  interest is computed but this makes a real
  difference!

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So for example

• You borrow 10,000 to buy a car at 12 interest,
  compounded quarterly. How much will you owe in 4
  years if you pay none back?
• Credit card interest is computed daily. If you
  borrow 850 for 1 year at 17 , how much will you
  owe?

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Practice

• 400 at 8 compounded monthly
• 1,000 at 4 compounded quarterly

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Why does compounding change things?

• Well, heres the deal when you compound the
  interest, you start earning interest on the
  interest!
• For example, 10 compounded once on 100 will
  give you 100 10 110.
• But if you look at that 10 as 2.5 four times
• First quarter add 2.5 of 100 102.50
• Second quarter add 2.5 of 102.50!!!!!
• Thats the trick when you compound, you earn
  interest on your interest, and the more often you
  compound, the greater the effect.

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The Effect of Compounding Converges

• The amount keeps going up, but it goes up by less
  and less and less.
• We call this type of behavior convergence.
• Can we figure out what the limit of this
  convergence is? What the amount would be if we
  compounded continuously?

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Continuously Compounded Interest

• Yep. And it involves e.
• Whats e, you ask? Well, its about 2.7, and
  its a big freaking deal.

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Ladies and Gentlemen e

• The mathematical constant e (occasionally called
  Euler's number after the Swiss mathematician
  Leonhard Euler) is the base of the natural
  logarithm function.
• e ? 2.71828 18284 59045
• Alongside the number ? and the imaginary unit i,
  e is one of the most important mathematical
  constants.

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Continuous Interest

• p principal (the amount you invest or borrow)
• e about 2.7
• r rate of interest, expressed as a decimal
• t number of years

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How do we use this?

• Well, its pretty simple.
• Suppose we want to know how much money youd owe
  if you borrowed 5,000 at 3 compounded
  continuously after five years.

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For Example

• I have 100 to put into the bank for 5 years.
  Im offered 10 interest compounded quarterly,
  or 8 compounded continuously. Which should I
  take? Why?

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Practice

• If you invest 750 at 11 compounded
  continuously, how much will you have after 10
  years?

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Summary

• Interest is of two types
• Simple where you just compute it once a year
• Compound where it is computed in pieces several
  times during the period in question.
• Compound interest requires a slight alteration to
  the exponential formula.
• Further, compound interest is of two types
• Regular compounded
• A finite number of compound periods.
• Continuously compounded
• Interest is compounded infinitely, all the time.