Annuities, Part 3

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Annuities, Part 3

• Section 5.3

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Annuity Due

• We have looked at ordinary annuities, when the
  payment is made at the end of the payment period.
  
• These are common as a means of funding future
  regular income, such as retirement accounts or
  college tuition.
• An annuity due has payments at the beginning of
  the payment period.
• These may arise in the case of loans or other
  regular bills.

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Present Value of an Annuity Due

• Auto insurance premiums are 40 per month, due at
  the beginning of the month. If the interest rate
  is 8 compounded monthly, how much would it cost
  to pay for one year in advance?
• We want to find the present value of all the
  payments.
• Note that the present value of the first payment
  is still 40.
• The present value of the second payment is 40(1
  .08/12)-1.
• The present value of the last payment is 40(1
  .08/12)-11.
• So we want to add 40 40(1 .08/12)-1 40(1
  .08/12)-2 . . . 40(1 .08/12)-11.
• From the formula for a geometric sum we get

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Present Value of an Annuity Due

• But notice that we had 40 (because the first
  payment is due now) 40(1 .08/12)-1 40(1
  .08/12)-2 . . . 40(1 .08/12)-11 (or the
  present value of an 11 month ordinary annuity).
• In general, for an annuity due with n payments of
  R with periodic rate r, the present value A is
  given by
• where is the
  present value of an annuity of 1 for n - 1
  periods at periodic rate r.

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Amount of an Annuity

• The amount of an annuity is the future value of
  all the payments at the end of the term of the
  annuity.
• To find the amount of an annuity we need to find
  the future value of all the payments.
• For an ordinary annuity, the future value of the
  first payment R is R(1 r)n-1,
• the second payment R(1 r)n-2,
• the third payment R(1 r)n-3,
• and so on, until the next to last payment has
  future value R(1 r)1, and the last has value R.
  

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• So we want to add
• R(1 r)n-1 R(1 r)n-2 R(1 r)n-3 . . .
• . . . R(1 r)3 R(1 r)2 R(1 r)1 R
• From the geometric sum formula we get that
• We let the expression
• and then we can write

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Amount of an Annuity, example

• Find the amount (future value) of an annuity with
  payments of 500 made at the end of each month
  for 5 years at 7.2 compounded monthly.
• R 500, r .006, n 60
• So

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Amount of an Annuity Due

• For an annuity due our formula changes somewhat.
  The first payment earns interest for all n
  periods, and the last payment earns interest for
  one period.
• S R(1 r)n R(1 r)n-1 R(1 r)n-2 . .
  .
• . . . R(1 r)3 R(1 r)2 R(1 r)1
• But this is the same as an ordinary annuity for n
  1 periods minus the last payment.
• So the future value of an annuity due is given
  by
• where

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Sinking Fund

• A sinking fund is a fund into which periodic
  payments are made to have funds on hand for a
  future financial obligation.
• Examples are funds set up by businesses to
  finance future purchases of equipment, future
  building expenses, scheduled maintenance costs,
  etc.

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Sinking Fund

• After paying off her car, Wiley Wilma decides to
  set up a sinking fund to replace her car when she
  needs to. She expects her present car will last
  for another 5 years and wants to have 25,000 to
  buy her new car. If she can trade in her car for
  2000 and get 4.2 on her money compounded
  monthly, what should her payments be?
• This amounts to setting up an annuity for the
  difference between the cost and the trade-in.
• S 23,000, r .0035, and n 60

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Key Suggested Problems

• Sec. 5.3 27, 29, 31, 33, 35, 39, 41