299X159 Lecture 3

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299X-159 Lecture 3

• Finance Applications with Excel Annuities an
  Amortization

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Example 1

• Suppose we invest 500 at the end of each year
  for 5 years, at an interest rate of 4,
  compounded annually.
• How much money will we have at the end of year 5?

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Example 1 (cont.)
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Example 1 (cont.)

• At the end of year 5, we will have a total
  investment S as follows
• S 500 500(1.04)1 500(1.04)2 500(1.04)3
  500(1.04)4 dollars
• To add up this sum, we can use the following idea!

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Example 2

• Find a formula for the following sum for non-zero
  real numbers a and r
• Sn a ar ar2 arn-1
• Solution Write down the equation for Sn, then
  multiply both sides of the equation by r to get
• Sn a ar ar2 arn-1
• rSn ar ar2 arn-1 arn
• Subtract the first equation from the second and
  solve for Sn
• rSn Sn arn a
• (r-1)Sn arn - a
• Sn a(rn 1)/(r-1)

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Sequences and Series

• Well see ideas like those used in Example 2
  appear in many mathematical settings!
• Here is some useful terminology!
• A sequence is a function whose domain is the
  positive integers.
• We can think of a sequence as a list of numbers
  written in a definite order a1, a2, a3,
• For example, the Fibonacci sequence is
• 1, 1, 2, 3, 5, 8, 13, 21,
• Another type of sequence is a geometric sequence,
  such as
• 1/2, 1/4, 1/8, 1/16,

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Sequences and Series (cont.)

• An idea related to sequences is series.
• A series is an infinite sum constructed from a
  sequence.
• Given the sequence a1, a2, a3,
• a corresponding series is
• a1 a2 a3

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Sequences and Series (cont.)

• We study series by looking at sequences of
  partial sums
• S1 a1
• S2 a1 a2
• Sn a1 a2 an
• In Example 2, we were finding a closed-form
  expression for the nth partial sum of a geometric
  series.

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Example 1 (cont.)

• Using the result from Example 2, with a 500, r
  1.04, and n 5, we get
• S 500 500(1.04) 500(1.04)2 500(1.04)3
  500(1.04)4
• 500(1.045 1)/(1.04 1)
• 2708.16 dollars
• An investment of this type is called an annuity!

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Annuities

• An annuity is a sequence of equal payments made
  at equal periods of time.
• An ordinary annuity is an annuity in which
  payments are made at the end of each time period,
  with interest compounded each time period.
• We call the time between payments the payment
  period and the time from the beginning of the
  first payment period to the end of the last
  period the term of the annuity.
• The future value of the annuity is the final sum
  after all of the payments have been made it is
  the sum of the all the compound amounts of all
  payments, compounded to the end of the term.

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Annuities (cont.)

• Example 1 is an example of an ordinary annuity,
  with payments of 500, payment period of 1 year,
  and term of 5 years.
• In general, the future value of an annuity of n
  payments of R dollars each at the end of each
  consecutive interest period, with interest
  compounded at a rate I per period, is

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Annuities Due

• Annuities due are annuities in which payment is
  made at the beginning of a payment period.
• One can show that the future value of an annuity
  due is

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Sinking Funds

• One application of annuities is to make periodic
  payments to guarantee a certain amount of money
  is available at some future date.
• Such an investment is known as a sinking fund.

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Example 3

• A 45-year-old teacher puts 1000 in a retirement
  account at the end of each quarter until the
  teacher reaches the age of 60 and makes no
  further deposits.
• If the account pays 8 interest compounded
  quarterly, how much will be in the account when
  the teacher retires at age 65?

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Example 3 (cont.)

• Solution This investment is a combination of an
  ordinary annuity with a fixed principal earning
  compound interest!
• The ordinary annuitys future value S after 15
  years is found with
• R 1000
• i 0.08/4
• n 154
• S R((1i)n-1)/i
• This is followed by investing principal S for 5
  more years, earning compound interest with
• P S
• i 0.08/4
• n 54
• A P(1i)n
• We can use the built-in Excel function FV to help
  with the calculations (see next page)!
• We find that the investment yields 169,474.59.

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Example 3 (cont.)
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Example 4

• How much money would I have to invest at the end
  of each month, compounded monthly, at annual
  interest rate of 9 to have 1,000,000 in twenty
  years?
• Repeat with investment terms of 30 years and 40
  years.

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Example 4 (cont.)

• Solution This is an example of a sinking fund
  with
• S 1,000,000
• i 0.09/12
• n 2012
• Using Excels PMT function, we find that monthly
  payments of R 1497.26 will yield 1,000,000 in
  20 years.
• For 30 years, R 546.23 and for 40 years, R
  213.61.
• Moral of Example 4 Invest as soon as possible!

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Present Value of an Annuity

• Suppose instead of investing money in an ordinary
  annuity, by making n payments of R dollars each
  at an interest rate i per period, we want to
  invest P dollars once at the beginning of term,
  at the same compound interest rate, to get the
  same amount as the annuity would yield.
• To find P, all we need to do is to set the
  formulas for the future values of compound
  interest and an ordinary annuity equal to each
  other!

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Present Value of an Annuity (cont.)

• Solving for P and simplifying (check!) we find
  that

• We call P the present value of the annuity.
• One of the biggest uses of the present value of
  an annuity formula is figure out periodic
  payments on a loan!

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Example 5

• Our offer to purchase a house for 130,000 has
  been accepted!
• We can afford a down payment of 20 of the sale
  price and want to take out a mortgage to cover
  the balance.
• A local bank is offering a fixed 30-year loan at
  an annual interest rate of 6.618.
• What would our monthly payment be for this loan?
• How much interest would we pay over the lifetime
  of the loan?

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Example 5 (cont.)

• Solution Our loan amount is 130,000 -
  0.20130,000 104,000 dollars.
• Paying a lump sum of 104,000 at the beginning of
  the loan would pay off the loan, so if we take
  the loan amount to be the present value of an
  ordinary annuity, we can figure out our monthly
  payment!

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Example 5 (cont.)

• With
• P 104,000
• i 0.06618/12
• n 3012 360
• we can find the monthly payment R from the
  present value of an annuity formula by solving
  for R
• This calculation can be done in Excel with the
  PMT function!
• Our payment is 665.44 per month.
• For this loan, well end up paying 665.44360 -
  104,000 135,558.40 dollars in interest!

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Amortization

• A loan is amortized if both the principal and
  interest are paid by a sequence of equal
  payments.
• Thus, the loan in Example 5 is amortized by
  making 360 monthly payments of 665.44.
• A useful way to work with a loan is via an
  amortization schedule, which is a table that
  keeps track of how much of each payment is used
  to pay for interest and how much is used to pay
  off the principal loan amount.
• Remark When loans for property such as a house
  are made, usually the home is used as a security
  against the borrower being unable to make loan
  payments. Such an arrangement is called a
  mortgage.

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Example 6

• Make an Amortization Schedule for the house loan
  in Example 5.
• Use this schedule to determine how long it will
  take to pay off half of the loan (i.e. to owe
  only 52,000).

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Example 6 (cont.)
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Example 6 (cont.)

• The 258th payment (21.5 years) is the one at we
  will have paid off half of the house loan!
• Technically, we still owe 51,805.91 at this
  point.
• Notice that for each payment, the amount going
  towards interest decreases and the amount going
  towards the principal increases!

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Homework

• As a prize in a contest, you are offered 1000
  now or 1210 in 5 years. If money can be
  invested at 6 compounded annually, which is
  larger?
• A bank pays 5 interest compounded monthly. Find
  the compound amount for an investment of 1000
  for one year. At what rate would the money have
  to be invested to earn the same amount with an
  investment paying simple interest at the same
  rate (this rate is called the effective rate of
  the stated rate of 5).
• Construct an amortization schedule for the house
  loan in Example 6 with loans for 20 years, 15
  years, and 10 years.
• Print out a copy of the 10 year amortization
  schedule.
• How much interest do you pay over the lifetime of
  each of these mortgages?
• At what point has half of the principal been paid
  off for each of these mortgages?

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References

• Finite Mathematics and Calculus with Applications
  (4th edition), by Margaret Lial, Charles Miller,
  and Raymond Greenwell.