Time Value of Money II

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Chapter 5

• Time Value of Money II

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Multiple Payments Uneven Cash Flow Streams

• Suppose you wanted to calculate the PV of several
  cash payments. For example 500 after 1 year,
  1000 after 2 years, and 3000 after 3 years.
  This can always be done by calculating the PV of
  each payment and adding them together. (Assume
  the appropriate interest rate of 10).

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• You could do the same thing for future value
  calculations.

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Annuities

• Although every problem involving multiple
  payments can be solved as above there are short
  cuts.
• Simple AnnuityA set of n-equal payments with the
  first coming at the end of the first period and
  the last coming at the end of the n-th period.
• Examples

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• Annuity Due A set of n-equal payments with the
  first coming at the beginning of the first period
  and the last coming at the beginning of the n-th
  period.
• Examples

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Present Values of Annuities

• Simple Annuities
• What is the present value of an annuity of 25,
  first payment coming in one year and the last
  coming at the end of 5 years? (Assume a 10
  interest rate)

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• For This Example

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• Using a Financial Calculator

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Changing Interest Rates

• What is the present value of a simple annuity of
  100, for 5 years at 5, 10, and 15? What is
  the relation between interest and PVs?

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• At 5

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• At 10
• At 15

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Changing Time Periods

• What is the present value of a simple annuity of
  100, for 5, 10 and 15 years at 10?

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• For 10 years
• For 15 years

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Annuities Due

• Annuities Due
• What is the present value of an annuity of 25,
  first payment due immediately and the last coming
  at the beginning of the 5th year? (Assume a 10
  interest rate)

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• For a ordinary simple annuity
• Why is an annuity due more valuable?

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• Annuities Due on a Financial Calculator
• Your display should show the word BEGIN on the
  bottom

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Future Values of Annuities

• Future Values
• Simple Annuities
• What is the future value of an annuity of 25,
  first payment due in one year and the last coming
  at the end of 5 years? Assume an interest rate
  of 10.

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• For this example

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• Using a financial calculator

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More Examples

• Your great grandfather spent 1000 on his and her
  model Ts in 1935. Over the next 65 years the
  average return on small cap stocks was 14. If
  your great grandfather had instead invested 100
  a year in stocks for you for the next 60 years,
  how rich would you be today?

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• Annuity Due
• What is the future value of an annuity of 25,
  first payment due immediately and the last coming
  at the beginning of 5 years? Assume an interest
  rate of 10.

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Solving for Unknown Payments

• You are developing a plan for your future
  retirement. You believe that you could
  comfortably retire in 40 years if you have saved
  1,000,000. From your investigations you
  determined that you can earn an average 10
  return on your investment. How much will you
  need to save each year. (Assume your payments
  will come at the end of the year).

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• Using a financial calculator
• You can do the same thing with the present value
  of an annuity formula.
• To calculate mortgage and car payments etc.

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Solving for i

• You are developing a plan for your future
  retirement. You believe that you could
  comfortably retire in 40 years if you have saved
  1,000,000. You can afford to save 2,000 a
  year. What interest rate would you need to earn?

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• How can you solve?
• Trial and error
• Tables
• Financial calculator

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Perpetuities

• Perpetuities Typically annuities call for
  payments to be made over some finite period of
  timefor example, 100 a year for 3 years. When
  a set of payments are set to go on indefinitely
  the annuities are called perpetuities.
• Examples
• Preferred Stock
• Perpetual Bonds
• Common Stock

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• A preferred stock pays an annual dividend of 5
  per year with no set maturity date. Assume the
  appropriate interest rate is 8 and that the
  payments come at the end of the year.

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• if you paid 62.50 what would be your average
  return?
• Value is determined by the return that investors
  want on their investments.

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Effective Rates of Return

• Suppose the annual rate of interest is 12
  (sometimes called the annual rate) and you put 1
  into a savings account. How much will you have
  after one year if your bank advertises.
• Annual Compounding
• The effective annual rate (EAR) is the interest
  rate expressed as if it were compounded annually.
  The true interest earned.

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• Semi Annual Compounding

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• Quarterly Compounding

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• Monthly Compounding

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• Daily Compounding

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• Continuous compounding you need only understand
  that there is a limit to the effects of
  compounding.

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A General Formula for EAR

• We have a general formula for calculating
  effective interest rates.

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• Example
• Your credit card advertises an 18 APR compounded
  monthly. What is your actual cost?

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Adjusting PV and FV Formulas

• Adjusting earlier formulas for non-annual
  compounding
• Adjust the interest rate (i/m)
• Where m is the number of compounding periods
• Adjust the number of periods (n?m)
• For example a 5 year car loan with monthly
  payments requires 60 payments.

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• PV and FV formulas

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• Present value of an annuity (PVA)

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• Future value of an annuity (FVA)

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Amortizing a Loan

• Example
• You are applying for a five year 100,000
  mortgage loan. You will make monthly payments at
  the end of each month and the annual percentage
  rate is 12. What is your monthly payment? Fill
  out the following Amortization table to show how
  much of each payment goes toward interest
  payments and what part pays down the principal of
  the loan.

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• An Amortization Table

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Hints on Approaching TVM Problems

• Do you have an annuity
• What is the compounding period?
• Are you looking for a PV or is a PV given?
• Draw a timeline
• Are you looking for a FV or is a FV given?
• If an annuity are you looking for a payment? Or
  is a payment given?
• What is the interest rate?

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Examples

• Problem 22 Beginning 3 months from now, you
  want to be able to withdraw 2,000 each quarter
  from your bank account to cover college expenses
  over the next 4 years. If the account pays 1
  percent interest quarterly, how much do you need
  to have in your bank account today to meet your
  expense needs over the next 4 years?

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• What is the relationship between the value of an
  annuity and the level of interest rates? Suppose
  you just bought an eight year annuity of 10,000
  per year when interest rates are 10 percent.
  What happens to the value of the investment rates
  suddenly drop to 6 percent? What if they rise to
  14 percent?

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• You and your twin brother both plan to retire in
  40 years. You being the miserly one plan to
  start saving 2500 a year (starting at the end of
  this year) for ten years. After that you will
  stop adding and just earn interest on the
  balance. Your brother the prodigal wont start
  saving until the end of the 11th year and will
  save an equal amount each year for the next 30
  years. How much must he save each year if he
  wants to retire with the same amount as you?
• You

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• First how much will you have?

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• Your Brother

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• A student once asked me this question because it
  was relevant to a business dealing You sold a
  property on land-contract two years ago. The
  terms were 8 amortized over 30 years (assume
  monthly payments), but the contract called for a
  balloon payment after 5 years. The amount of the
  loan was 100,000. The owner of the contract
  wants to now sell it but the potential buyer
  wants to earn 10 on the contract. At what price
  should the contract be sold?

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• Whats the payment?

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• Whats the Balloon payment?

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• What do you get if you buy this contract two
  years after it was issued?
• Investors now want to earn 10 on the contract

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The End