Time value of money calculations quantify an intuitive idea:

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Time Value of Money

• Time value of money calculations quantify an
  intuitive idea
• It is better to receive a sum money of money now
  than the same sum of money later.
• It is better to pay a sum of money later than the
  same sum of money now.

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Time Value of Money

• The mathematics
• A present amount (P) growing at r per period for
  n periods will grow to a future amount (F) given
  by the following formula
• F P (1 r)n

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Time Value of Money

• The notation
• The term in brackets is the future value factor
  and is available in tables, financial
  calculators, and Excel functions.
• FV, n, r (1 r)n

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Time Value of Money

• Suppose you have 10,000 to invest for five years
  and will earn 10 per period.
• The initial amount is known as the present value
• The amount the investment will accumulate to is
  known as the future value

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Time Value of Money

• When the present value is known, we can calculate
  the future value.
• Example how much will the investment be worth in
  five years?
• FV 10,000 FV, n5, r10
• 10,000 1.6105 16,105

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Time Value of Money
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Time Value of Money

• Periodic payments of a constant amount are known
  as annuities
• Example suppose you invest 10,000 per year for
  three years to earn 10 per year.

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Time Value of Money

• The future value of the investment will depend on
  whether the payments are made at the beginning or
  end of the year.
• An annuity with end of period payments is known
  as an annuity in arrears (or an ordinary
  annuity).
• An annuity with beginning of period payments is
  known as an annuity in advance.

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Time Value of Money

• For an ordinary annuity
• F 10,000 FV, n2, r10
• 10,000 FV, n1, r10
• 10,000 FV, n0, r10
• (10,000 1.2100) (10,000 1.1000)
  (10,000 1.0000) 33,100

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Time Value of Money

• Future value factors are available for annuities.
  The problem can be expressed as
• F 10,000 FVA, n3, r10
• 10,000 3.3100
• 33,100

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Time Value of Money

• End of period payments

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Time Value of Money

• For an annuity in advance
• F 10,000 FV, n3, r10
• 10,000 FV, n2, r10
• 10,000 FV, n1, r10
• (10,000 1.3310) (10,000 1.2100)
  (10,000 1.1000) 36,410

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Time Value of Money

• To find the future value of an annuity in advance
  using tables for annuities in arrears
• Look up the factor for (n1) periods and subtract
  1. For the previous example
• F 10,000 (FVA, n4, r10) 1
• 10,000 4.6410 1
• 36,410

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Time Value of Money

• Beginning of period payments

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Time Value of Money

• When the future value is known, we can solve for
  the present value.
• The mathematics
• P F / (1r)n
• The notation
• P F PV, n, r

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Time Value of Money

• Example how much do we have to invest today at
  10 in order to have 16,105 in 5 years?
• PV 16,105 PV, n5, r10
• 16,105 .6209 10,000

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Time Value of Money

• We often need to calculate the present value of
  an annuity
• Example you will need to spend 10,000 per year
  for three years. How much must you deposit into a
  fund earning interest at 10 to provide for these
  future expenditures?

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Time Value of Money

• For an ordinary annuity (end of period payments
• P 10,000 PV, n1, r10
• 10,000 PV, n2, r10
• 10,000 PV, n3, r10
• (10,000 .9091) (10,000 .8264)
  (10,000 .7513) 24,869

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Time Value of Money

• End of period payments would require an initial
  deposit of 24,869.

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Time Value of Money

• For an annuity in advance
• P 10,000 PV, n0, r10
• 10,000 PV, n1, r10
• 10,000 PV, n2, r10
• (10,000 1.0000) (10,000 .9091)
  (10,000 .8264) 27,355

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Time Value of Money

• Beginning of period payments would require an
  initial deposit of 27,355

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Time Value of Money

• To find the present value of an annuity in
  advance using tables for annuities in arrears
• Look up the factor for (n-1) periods and add 1.
  For the previous example
• P 10,000 (PVA, n2, r10) 1
• 10,000 1.7355 1
• 27,355

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Time Value of Money

• Time value of money calculations are based on the
  mathematics of compound growth.
• The present value and future value are just the
  values of the cash flows at the beginning and
  end of the period of growth, respectively.

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Time Value of Money

• Solving time value of money problems
• Identify timing and amount of cash inflows and
  outflows, e.g., investment and returns, borrowing
  and repayment.
• Select a convenient point in time and set the
  value of the inflows equal to the value of the
  outflows.
• Solve for the unknown.

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Exercise 1a

• The estimated current annual cost of attending
  TCU as an undergraduate student is, according to
  the TCU website, 37,380. Suppose you are the
  proud parent of a newborn daughter who will
  attend TCU in 18 years and the annual cost of
  attending TCU increases by 8 per year.

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Exercise 1a

• What will the annual cost amount to when your
  daughter is ready to start her freshman year at
  TCU?

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Exercise 1a

• F 37,380 FV, n18, r8
• 37,380 3.9960 149,371
• Answer it will cost 149,371 for your daughters
  freshman year at TCU if the cost increases 8 per
  year.

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Exercise 1b

• Suppose you can write a check today and deposit
  it into a savings account earning 6 per year to
  fully pay for your daughters freshman year of
  college. What is the amount of the check you need
  to write?

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Exercise 1b

• P 149,371 PV, n18, r6
• 149,371 .3503 52,331
• Answer 52,331 deposited in an account earning
  6 per year will pay for the expected cost of
  your daughters freshman year at TCU.

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Exercise 1c

• Suppose instead that you will make equal annual
  deposits each year on your daughters birthday
  into the savings account. You want the balance in
  the account to equal the expected cost of your
  daughters freshman year at TCU. If the fund
  earns 6 per year, what is the amount of the
  equal annual deposits?

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Exercise 1c

• Solved as a PV problem
• 52,331 A PVA, n18, r6
• A 10.8276
• A 52,324/10.8276 4,833

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Exercise 1c

• Solved as a FV problem
• 149,371 A FVA, n18, r6
• A 30.9057
• A 149,371/30.9057 4,833
• Answer 4,833 per year deposited into an account
  earning 6 will pay for your daughters freshman
  year at TCU.

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Exercise 1c
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Exercise 2

• You have your eye on a new BMW coupe that costs
  50,000. Suppose you put 10,000 down and finance
  the rest at an annual rate of 12 (1 per month).
  Calculate the monthly payment on a 4-year and
  5-year loan.

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Exercise 2

• 48 month loan
• 40,000 A PVA, n48, r1
• A 37.9740
• A 40,000/37.9740 1,053.35
• On a 48 month loan, the monthly payment will be
  1,053.35.

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Exercise 2

• 60 month loan
• 40,000 A PVA, n60, r1
• A 44.9550
• A 40,000/44.9550 889.78
• On a 60 month loan, the monthly payment will be
  889.78.

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Exercise 3a

• You have the opportunity to invest in a security
  that will pay the following future cash flows
  2,500 every six months for 5 years and 100,000
  at the end of 5 years. You are willing to invest
  provided that you will earn a 6 annual return,
  compounded semi-annually (3 every six months).

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Exercise 3a

• What is the maximum amount you would be willing
  to pay for this security?

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Exercise 3a

• P 2,500 PVA, n10, r3 100,000 PV,
  n10, r3
• P (2,500 8.5302) (100,000 .7441)
• 95,734.90
• Answer investing 95,734.90 in this security
  would produce a return of 6.

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Exercise 3b

• Suppose instead that you are willing to invest
  provided that you will earn a 4 annual return,
  compounded semi-annually (2 every six months).
  What is the maximum amount you would be willing
  to pay for this security?

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Exercise 3b

• P 2,500 PVA, n10, r2 100,000 PV,
  n10, r2
• P (2,500 8.9826) (100,000 .8203)
• 104,491.29
• Investing 104,491.29 in this security would
  produce a 4 annual return.

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Exercise 4

• An investment opportunity offers the following
  future cash flows 5,000 at the end of year 1,
  12,000 at the end of year 2, and 8,000 at the
  end of year 3. The cost of this opportunity,
  payable today is 20,000. Your company has more
  than 20,000 to invest but management insists on
  earning at least an 8 return on its investment.
  Should you take this opportunity?

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Exercise 4

• If PV _at_ 8 is greater than 20,000, this project
  will earn more than an 8 return.
• P 5,000 PV, n1, r8
• 12,000 PV, n2, r8
• 8,000 PV, n3, r8
• P (5,000 .9259) (12,000 .8573) (8,000
  .7938) 21,268
• Since the PV of the future cash flows at 8 will
  exceed the cost of the investment, this
  investment will earn more than the required 8
  return.

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Exercise 5

• TXU management estimates that it will cost 1
  billion to decommission the Comanche Peak nuclear
  power plant at the end of its useful life (36
  years). The Nuclear Regulatory Commission
  requires that TXU establish a fund to pay for the
  decommissioning. What is the required payment if
  TXU deposits an equal annual amount into a fund
  earning 4 at the end of each year?

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Exercise 5

• 1,000 million A FVA, n36 r4
• A 77.5983
• A 1,000 million/77.5983 12.9 million
• Answer an annual deposit of 12.9 million into
  the decommissioning fund will pay for the
  expected cost of decommissioning Comanche Peak.