Time Value of Money

1
Time Value of Money
2
The Starting Point

• NPV analysis allows us to compare monetary
  amounts that differ in timing. We can also
  incorporate risk into the analysis, however we
  will not concern ourselves with this complication
  at this time.
• Two items need to be determined before you start
  the NPV analysis, future cash flows and interest
  rates. Forecasting these is often more an art
  than a science, however in many situations these
  are either known or can be estimated.

3
Items needed to solve these problems

• You will need to know all but one of the
  following
• interest rate i
• of periods n
• future value FV
• present value PV
• cash flow PMT

4
Methods to solve the problems

• A decent business calculator (e.g., HP10BII)
• A formula
• Tables
• A spreadsheet package (e.g., excel)

5
The following are useful formulas

• Future value of a single sum
• FV PV (1i)n
• Present value of a single sum
• PV FV 1/(1i)n

6
Simple versus compound interest

• Simple interest involves computing interest only
  on the original principal, not on any accrued
  interest. Compound interest involves calculating
  interest on interest.

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Future Value Simple Interest

• Example 1
• Invest 1 for 3 years _at_ 12 per annum.
• Period Beg. Amt. Interest End. Amt
• 1 1.00 0.12 1.12
• 2 1.00 0.12 1.24
• 3 1.00 0.12 1.36

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Future Value Compound Interest

• Example 2
• Period Beg. Amt. Interest End. Amt Formula
• 1 1.0000 0.12 1.1200 1.121
• 2 1.1200 0.13 1.2544 1.122
• 3 1.2544 0.15 1.4049 1.123
• n 3, i 12, PV 1, FV ?

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Future Value

• Example 3
• Invest 5 at the end of each year for 4 years _at_
  12. What is the FV?
• 5 5 5 5
• now 1 2 3
  4
• 5 x 1.00 5.00
• 5 x 1.12 5.60
• 5 x 1.2544 6.2720
• 5 x 1.4049 7.0245
• 4.779 23.897 This is the same as
  the future value of an ordinary annuity
• n 5, i 12, Pmt 5, FV ?

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Present Value

• In each of the cases so far we wished to
  determine what a dollar would be worth in the
  future. We can also go the other direction.
  Often we wish to know what future sums are worth
  today. This is called present value (PV)

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Present Value

• Example 4
• What is the PV of a 10 dollars received 1 year
  from today assuming 12 interest?
• ? 10
• Now 1
• Note that 8.93 grows to 10 in 1 year _at_ 12
• 8.93 x 1.12 10
• n 1, i 12, V FV, PV ?

12
Present Value

• Example 5
• What is the PV of 4 received 3 years from today
  and 4 received 2 and 1 year from today at 5
  interest?
• 4 4 4
• Now 1 2 3
• 4 x .9524 3.810
• 4 x .9070 3.628
• 4 x .8638 3.455
• 2.7232 10.893
• n 3, i 5, PMT 4, PV ?

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Non- Annual Periods

• So far we have computed FV of a single sum and an
  annuity and also PV of a single sum and an
  annuity. Each are basically the reverse of the
  other. Each has been computed with one
  compounding period per year. Often the
  compounding period is shorter.

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Future values with non-annual deposits

• Example 6
• What is the FV of a 75,000 deposit made every 6
  months for 3 years using an annual rate of 10?
• 0 1 2 3 4 5
  6 7 8 9 10
• 75 75 75 75
  75 75
• ((1.056)-1)/.05 x 75,000
• 6.80191 x 75,000 510,143
• n6, i 10, pmt 75,000, FV ?
• Note Be sure to set your calculator to 2
  payments per year.

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Other Items to Solve For

• N how long will it take a sum to grow to a
  certain FV at a given interest rate
• i what interest rate is required to grow a
  certain sum to a given FV in a given length of
  time
• PMT what payment is required to pay off a loan
  at a given interest rate in a set amount of time

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Solving for n

• Example 7
• How many periods does it take for 130 to grow to
  261.48 _at_ 15 per annum?
• n ?, i 15, PV 130, FV -261.48

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Solving for i

• Example 8
• At what annual interest rate will 175 grow to
  377.81 in ten years?
• n 10, i ?, PV 175, FV -377.81

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Find the required payment

• Example 9
• Compute the required semi-annual payment in order
  to have 14,000 at the end of 5 years _at_ 8
• 14,000
• 0 1 2 3 4 5
  6 7 8 9 10
• x x x x x x x x x
  x
• n10, i8, PMT ?, FV -14,000

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Car payments

• Example 10
• What would be your monthly car payment on a
  15,000 4 year loan _at_ 10. Payments are made at
  the end of each month.
• PV 15,000 n 48 i 10 pmt

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Car payment

• Example 10 (continued)
• Instead of a 4 year loan, compute the payment for
  a 5 year (60 payment) loan.
• PV 15,000 n 60 i 10 pmt

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Car payment

• Example 10 (continued)
• Leave the loan at 5 years, but lower the interest
  rate to 8. Compute the payment.
• PV 15,000 n 60 i 8 pmt

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Car payment

• Example 10 (continued)
• With the 5 year, 8 loan, assume the maximum
  payment you can afford is 275. How much of a
  loan can you afford?
• n 60 i 8 pmt 275 PV

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Car payment

• Example 10 (continued)
• Go back to the 15,000, 5 year, 10 loan.
• How much of the 12th payment applies toward
  principal? Interest? What is the remaining
  balance?
• Do the same for the 36th payment?
• Do the same for the 13th 24th payments combined?

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Present Value of an Annuity

• Example 11
• You win a 4,000,000 lottery that pays 200,000
  per year for 20 years. What is the present value
  of the lottery assuming a rate of 10?
• n 20, i 10, PMT 200,000 PV

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Uneven cash flows

• Up to this point we have assumed cash flows are
  the same each period. This is common for
  mortgage and lease payments. Things are not
  nearly as tidy when you need to determine if a
  project makes financial sense. Typically you
  will experience cash flows from revenues and
  expenses that vary each period.

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Uneven cash flows

• This is the situation firms face when attempting
  to decide if a new location makes economic sense.
  Luckily this situation can still be handled with
  your financial calculator. You will be using a
  few new keys
• CFj, Nj, IRR/YR, and NPV

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Internal rate of return

• IRR/YR is used to compute the internal rate of
  return. This represents the interest rate that
  the project is earning over its life. This is
  similar to solving for i in the previous problems.

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Net present value

• Sometimes you may know that you need a minimum
  return (internal rate of return) to take on a new
  location. You can then use this interest rate in
  the calculation and then compute the present
  value of all the combined cash flows. The
  summary number is the net present value of the
  project. If the project is earning a return
  greater the the required IRR, the NPV will be
  positive, otherwise it will be negative.

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IRR and NPV example

• Example 16
• You wish to determine the IRR and NPV of a
  project with the following projected cash flows
• At inception -10,000
• End of year 1 3,000
• End of year 2 1,000
• End of year 3 3,000
• End of year 4 8,000
• Determine the IRR and then the NPV if the
  required return is 15

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Tips

• 1. Draw time lines
• 2. Put in all the knowns
• 3. Be sure to use the period interest rate
• 4. Make sure the answer passes the smell test
  (e.g., is the present value lt the future value?)