FIN 504: Financial Management

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FIN 504 Financial Management

• Lecture 4 Time Value of Money II

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Time Value of Money II

• Time Lines
• Annuities
• Perpetuities

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Time Lines

• The Use of Time Lines
• Temporal Indices
• t
• T
• 1, 2, 3,

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C0
C1
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CT
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Annuities
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Annuities

• Annuities
• A finite series of constant cash flows, e.g.,
• 100 per year for 5 years
• 10 per month for 7 months
• Variables
• Cash Flow Amount
• The Date of the First Payment
• The Period (weekly, quarterly, annually)
• The Length (or Number) of Payments

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Annuities

• A 5 year annual, annuity of 500 beginning in
  year 1
• TECHNICAL NOTE When we use the term annuity we
  mean an annuity in arrears, i.e., an annuity
  whose first payment begins next (not this)
  period. An annuity that begins this period is
  called an annuity due and will be considered
  toward the end of the lecture.

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Annuities

• Annuities can always be valued as a series of one
  time cash flow (r 7)

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Annuities

• But there is a general present value formula that
  is often more convenient
• NOTE The formula assumes that the first payment
  begins next period!

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Annuities

• Thus

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Annuities

• And there is a general future value formula
• NOTE The formula assumes that the first payment
  begins next period and it give the future value
  in year T!

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Annuities

• Thus

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Annuities

• We can check our calculations to make sure that
  the future value of the annuity (properly
  discounted) gives its present value

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Delayed Annuities
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Delayed Annuities

• Annuities may not begin in the next period. What
  if the annuity begins in two or three years, not
  next year. I call these delayed annuities.
• If the cash flow begins in year 2, the annuity
  formula will calculate the value in year 1. So to
  get the present value, we need to discount
  everything one more year.

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Delayed Annuities

• In general, for year the annuity is delayed, we
  need to discount the result by one additional
  year.
• If s is the years delayed, then we can adapt
  our formula.

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Delayed Annuities

• EXAMPLE
• Find the present value of a 10 year of 30.00
  annuity that begins in 8 years (r 6).

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Cash Flow, Time and Interest Rates
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Cash Flow, Time and Interest Rates

• As in our previous lecture, we may need to
  calculate for than just the present or future
  value. Annuities have three additional variables
• Cash Flow
• Interest Rate
• Time

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Cash Flow Problems

• Present value cash flow problems are especially
  important since they are the equivalent to the
  period payments on a loan.
• For example, if I borrow at 1,000.00 at 11 for
  5 years, what are my monthly payments? That is a
  question of calculating the cash flows of an
  annuity.

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Cash Flow Problems

• We can easily solve the present value formula for
  C

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Cash Flow Problems

• If I borrow at 1,000.00 at 11 for 5 years, what
  are my annual payments?

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Cash Flow Problems

• Future value cash flow problems are the
  savings problems.
• For example, if I want to have 100,000.00 in 5
  years and the rate of interest is 11, how much
  do I have to save per year?

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Cash Flow Problems

• We can easily solve the future value formula for
  C

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Cash Flow Problems

• If I want to have 100,000.00 in 5 years and the
  rate of interest is 11, how much do I have to
  save per year?

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Interest Rate Problems

• If you examine the formula for either the present
  or future value of an annuity, you may note two
  things
• The variable r occurs more than once, and
• In one case the variable r has an exponent.
• In such situations, it is not possible to find a
  closed form solution for r.
• Interest rate problems can, however, easily be
  solved with a spreadsheet.

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Time Problems

• The are Two Versions of Time Problems
• Present Value How long will it take to repay a
  loan?
• Future Value How long will it take to save a
  certain amount?

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Time Problems-Present Value

• We can solve the present value formula for t

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Time Problems-Present Value

• If I borrow at 1,000.00 at 11, and I want to
  make annual payments of 350.00, how long will it
  take me to repay the loan?

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Time Problems-Future Value

• We can solve the future value formula for t

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Time Problems-Future Value

• If I put 1,000 in my account annually at 7, how
  long will it take for my balance to reach
  10,000?

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Non-Annual Annuities
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Non-Annual Annuities

• Unfortunately, not all annuities have annual cash
  flows bonds pay semi-annual coupons, loans often
  have monthly payments, and we can put money in a
  bank quarterly, weekly, daily or even hourly.
• This mean that we need a mechanism for adapting
  all of our previous formulae for non-annual
  periods.

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Non-Annual Annuities

• The good new is that this is not at all
  difficult. Four simple rules
• Let m stand for the number of periods in a
  year. E.g., m 4 for quarterly cash flows, m
  52 for weekly cash flows, etc.
• For t in a formula substitute tm.
• For r in a formula substitute r/m.
• If C is the annual cash flow, substitute C/m,
  but if C is the period cash flow do not change it.

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Non-Annual Annuities

• EXAMPLE
• What is the present value of an annuity that pays
  10 per week and lasts for 3 year (r 15)?
• Note that since the problem has the period
  (weekly) cash flow, I do not divide C by m!

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Annuities Due
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Annuities Due

• An annuity due is an annuity that begins this
  period, not next.
• Four Year Annuity of 100 per Year
• Four Year Due Annuity of 100 per Year

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Annuities Due

• This means that an annuity due of y for t
  periods is exactly the same as an annuity of y
  for t-1 periods plus y.

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Annuities Due

• What is the value of a 5 year annual, annuity due
  of 500 (r 4)? Note it is the same as 500
  plus the value of a 4 year annual, annuity of
  500 (r 4).

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Perpetuities
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Perpetuities

• Perpetuity
• An infinite series of constant cash flows, e.g.,
• 100 per year forever
• 10 per month forever
• Variables
• Cash Flow Amount
• The Date of the First Payment
• The Period (weekly, quarterly, annually)

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Perpetuities

• Valuing a Perpetuity
• Two Notes
• Since perpetuities are infinite, they cannot have
  a future value.
• For non-annual perpetuities, follow the same
  rules that apply to non-annual annuities.

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Growing Perpetuities
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Growing Perpetuities

• A Growing Perpetuity
• An infinite series of changing cash flows, e.g.,
• If g 5, then
• 100.00, 105.00, 110.25, 115.76,
• Variables
• Cash Flow Amount
• The Date of the First Payment
• The Period (weekly, quarterly, annually)
• Growth Rate

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Growing Perpetuities

• Valuing a Growing Perpetuity
• Three Notes
• The growth can be positive or negative, e.g., the
  cash flow can either increase or decline at x
  per period.
• C1 refers to next periods cash flow.
• There are economic reasons why g is never greater
  than r.

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Growing Perpetuities

• EXAMPLE
• What is the present value of 1,000 per year
  growing at 3 per year (r 10)

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Cash Flow and Interest Rate Problems Non-Annual
Perpetuities and, Delayed Perpetuities
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Cash Flow and Interest Rate Problems

• Given the simplicity of the perpetuity formula,
  it is easy to solve it for the cash flow and the
  interest rate.

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Non-Annual Perpetuities

• These work just like a non-annual annuities
• Remember that C is the period cash flow.

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Delayed Perpetuities

• Finally, delayed perpetuities are adjusted in the
  same way as delayed annuities