Time Value of Money

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Time Value of Money

• FIL 404
• Prepared by Keldon Bauer

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Cash Flow Time Lines

• You win a contest, and you have the option of
  taking 1.4 million now or 250,000 per year for
  five years.
• Which should you take?
• The answer comes through taking into
  consideration the time value of money.

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Cash Flow Time Lines

• The first step is visualizing the cash flows by
  drawing a cash flow time line.
• Time lines show when cash flows occur.
• Time 0 is now.

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Cash Flow Time Lines

• Outflows are listed as negatives.
• Inflows are positive.
• State the appropriate interest rate, which
  represents your opportunity costs

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Future Value

• Future value is higher than today, because if I
  had the money I would put it to work, it would
  earn interest.
• The interest could then earn interest.
• Compounding allowing interest to earn interest
  on itself.

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Future Value - Example

• If you invest 1,000 today at 8 interest per
  year, how much should you have in five years (in
  thousands).

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Future Value

• For one year, the future value can be defined as

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Future Value

• The second year, the future value can be stated
  as follows

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Future Value

• Therefore, the general solution to the future
  value problem is
• The Excel formula is
• FV(Interest, Term, Payments, Present Value,
  Type)

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Future Value - Excel
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Future Value

• Interest can be seen as the opportunity growth
  rate of money.

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Present Value

• Present value is the value in todays dollars of
  a future cash flow.
• If we are interested in the present value of 500
  delivered in 5 years

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Present Value

• The general solution to this problem follows from
  the solution to the future value problem

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Present Value - Excel

• The Excel formula is
• PV(Interest, Term, Payments, Future Value,
  Type)

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Present Value - Excel
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Present Value

• Since the discount rate is the opportunity cost,
  the present value represents what I would have to
  give up now to get the future value specified.

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Interest Rates

• If we know the amount we need at time n and the
  amount we can invest at time zero, then we must
  only solve for the interest rate.

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Interest Rates

• Solving for interest rates algebraically

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Interest Rates - Example
In Excel RATE(Term, Payment, Present Value,
Future Value, Type, Guess)
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Interest Rate - Excel
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Time Periods

• If the present value, future value and interest
  rate are known, but the number of time periods is
  not. Then n can be found algebraically

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Time Periods - Example

• If we use the last example of investing 100, we
  want 500 in future, and the current market
  interest is 8, n can be found

In Excel NPER(Interest, Payment, Present
Value, Future Value, Type)
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Time Periods - Excel
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Annuities

• Definition A series of equal payments at a
  fixed interval.
• Two types
• Ordinary annuity Payments occur at the end of
  each period. (Default in Excel)
• Annuity due Payments occur at the beginning of
  each period. (Set the type 1 in Excel)
• In Excel, use the same formulas introduced so
  far, just specify payment and type.

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Ordinary Annuity

• Example is a regular payment of 100 for five
  years earning 8 interest.

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Ordinary Annuity Future Value

• The future value of an ordinary annuity can be
  found as follows

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Ordinary Annuity - Example
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Ordinary Annuity - Example
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Annuity Due

• Example is a regular payment of 100 for five
  years earning 8 interest.

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Annuity Due Future Value

• The future value of an annuity due can be found
  by noticing that the annuity due is the same as
  an ordinary annuity, with one more compounding
  period

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Annuity Due - Example
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Annuity Due - Excel
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Ordinary Annuity - Present Value

• Example is a regular payment of 100 for five
  years earning 8 interest.

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Ordinary Annuity - Present Value

• The present value of an ordinary annuity can be
  found as follows

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Ordinary Annuity - Example
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Ordinary Annuity - Excel
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Annuity Due - Present Value

• Example is a regular payment of 100 for five
  years earning 8 interest.

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Annuity Due - Present Value

• The future value of an annuity due can be found
  as follows

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Annuity Due - Example
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Annuity Due - Excel
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Annuities - Finding Interest Rate

• Interest rates cannot be solved directly.
• Calculators and computers search for the correct
  answer (there is only one correct answer).
• It guesses and then iteratively goes higher or
  lower.

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Perpetuities

• What would you have to pay to be paid 2,000 per
  year forever (given a market rate of 8)?

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Uneven Cash Flow Streams

• If payments are irregular or come at irregular
  intervals, we can still find the PV (or FV).
• Take the present value (or future value) of
  individual payments and sum them together.

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Uneven Cash Flows - Example
1,136.51 Present Value
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Uneven Cash Flows - Excel

• Excel can do this in one argument
• NPV(Interest, Array of Payments starting with
  payment for time 1).
• If you want to include a payment in time zero,
  add it to the above argument separately.

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Uneven Cash Flows - Excel
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Uneven Cash Flow - Example
Future Value 1,669.91
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Uneven Cash Flows - Excel
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Finding Interest Rate

• As with annuities, interest rates for uneven cash
  flow streams cannot be solved directly.
• Calculators search for the correct answer, called
  an IRR (there may be more than one correct
  answer).
• It guesses and then iteratively goes higher or
  lower.

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Compounding

• The more often one compounds interest, the faster
  it grows.

Annual
Semi-Annual
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Compounding

• Why is there a difference in future value?
• Because interest is earned on itself faster!
• How would you adjust to compound monthly?
• How would you make an adjustment in annuities?

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Effective Annual Rate

• To convert the other compounding periods to an
  effective annual compounding rate (EAR) use the
  following formula

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Effective Annual Rate - Example

• 8 monthly compounding loan is equal to what in
  effective annual rate?

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Fractional Time Periods

• If you invest 100 for nine months at an EAR of
  8, how does one calculate the future value?
• The same way one did before.

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Amortized Loans

• A loan with equal payments over the life of the
  loan is called an amortized loan.
• Loan mathematics are the same as an annuity.
• Loan amounts are the present value.
• Periodic loan payments are the payments.

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Amortized Loans

• The present value of a monthly loan uses the
  annuity formula adjusted for monthly payments

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Amortized Loans

• Payments on a given loan can be found by solving
  for PMT in the previous equation

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Amortized Loans - Example

• What is the payment on a 30 year loan of 150,000?

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Amortized Loans - Excel
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Amortization Schedules

• Amortization schedules show how much of each
  payment goes toward principal and how much toward
  interest.
• The easiest way of calculating one by hand is by
  calculating the outstanding loan balance
  month-by-month, and then taking the difference in
  loan balance from month to month as the principal
  portion of the payment.

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Amortization Schedules

• The portion in the amortized loan formula that
  says nm can be interpreted as months remaining.
• So to find the part of the first 1,100.65 that
  is paid toward the principal one would realize
  that at the beginning one had all 150,000
  outstanding.
• I have posted an Excel example on the web.

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Amortization Schedules

• After the first month, one has 359 payments left.
  Therefore the loan principal outstanding is

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Amortization Schedules

• The difference in principal outstanding is the
  part of the payment that went toward principal.
  In this instance, 150,000-149,899.35100.65
• The rest of the payment went toward interest. In
  this instance that would be 1,100.65-100.651,000
  .

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Amortization Schedules
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Different Types of Interest

• Simple Interest (i or isimple) - The rate we have
  used thus far to calculate interest.
• Periodic Interest (i or iperiodic) - The interest
  paid over a certain period.

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Different Types of Interest

• Effective Annual Rate (EAR) Described earlier
  as the rate that would be charged to get the same
  compounded annual rate.
• Annual Percentage Rate (APR)