3' Time Value of Money 1

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3. Time Value of Money (1)

• 3.1 Future Values Compound Interest
• 3.2 Present values
• 3.3 Multiple Cash Flows
• 3.4 Perpetuities and Annuities

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3.1 Future values and compound interest rate

• Deposit100, interest rate10
• Interest earning100x0.110
• Future value of 100 after 1 year10010110100
  x(10.1)
• Future value after 2 years100x(10.1)x(10.1)
  121
• Future value after 3 years100x(1.1)x(1.1)x(1.1)
  
• Future value after 4 years100x
  (1.1)x(1.1)x(1.1)x(1.1)
• Future value after n years100x(1r)n

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3.2 Present values

• A dollar today is worth more than a dollar
  tomorrow. Why?
• Future value of 100 after 1 year100x(1.1)110.
• So if Ill get 110 next year, I can sacrifice
  100. i.e.
• y110/1.1100 (present value of 110).
• How about present value of 121 which will be
  paid 2 years from now?
• y121/(1.1)x(1.1)100
• Present valuefuture value after t periods/(1r)t

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Present values (cont.d)

• Example 1Veronica wants to buy a gift for her
  grandmother. After an extensive investigation, it
  turns out that grandma likes to go to grand
  canyon and to have a wild ride. The cost of the
  trip is 2000. annual interest rate is 8. How
  much money should Veronica put aside to this year
  to purchase the trip ticket next year?
• PV2000/(10.08)1851.852
• How much money needed to purchase the ticket 3
  years from now?
• PV2000 / (10.08)3 1587.664
• If Veronica puts 1600 in a bank, then she can
  buy the trip next year. What is the interest
  rate?
• 16002000/(1r)
• 1r2000/16001.25
• r0.25 equivalently 25

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Example (Veronica Grandma) cont.d

• If Veronica puts 1400 in a bank, she can buy the
  ticket 2 years from now. Whats the interest
  rate?
• 14002000/(1r)2
• (1r)2 2000/14001.428571,
• (1r)1.195229
• r 0.195229
• If Veronica puts 1400 in a bank, she can buy the
  ticket to Hawaii 2 years from now. Whats the
  price of the ticket if interest rate is 20?

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Example (Veronica Grandma) cont.d

• Let the interest rate be 8. If Veronica can
  only put 1500 this year, how many years later
  can she purchase the ticket?
• 1,500 2000/ (10.08)t then
• (10.08)t1.333
• Hence tlog(1.333)/log(10.08)3.78 can be
  rounded to 4 years.

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Examples (Present values)

• 2. Sharon is crazy for a house in Far West.
  Annual interest rate of banks is 10.There are
  two payment offers for the house.
• (i) 5,000 down payment another 5,000 payment
  2 years from now.
• (ii) 9,900 if it is paid in full.
• (iii) pay nothing now, your grandson will pay
  500,000 for you 40 years later.
• Rank the options.
• PV1 50005000/(10.1)2 9132.231
• PV2 9900
• PV3 500,000/(10.1)40 11047.46
• Does your ranking change if annual rate change?
  If annual interest rate drops/increases?

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Examples (Present values)

• 3. Rex is willing to pay 100 to buy IOU which
  will payoff 120 next year.Whats the interest
  rate?
• PVfuture value next year/ (1r)1
• 100120/(1r)
• 1r120/1001.2. Hence, r0.2 or equivalently
  20.
• Whats the interest rate if Rex pays 100 now
  it pays off 120, 2 years from now?
• 100120/(1r)2
• (1r)2120/1001.2
• (1r)sqrt(1.2)1.095445
• R0.095445 or 9.5445.

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3.3 Multiple Cash Flows

• In real world, there are multiple cash flows,
  stream of cash flows. Instead paying or getting
  paid as lump sum at the end period, securities
  yield a stream a cash flows over a period of
  time.
• Example 1 Steven wants to buy a new car. There
  are two options
• Option 1 pay 15,500.
• Option 2 pay 8,000 as an installment and pay
  4,000 in each of the next two years.
• Which option is better for Steven?

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Examples (Multiple Cash Flows )

• Example 2 Assume a bond pays off 100 at t0
  (current), 1100 at t1 and nothing at t2. If the
  interest rate is 5, whats future value at t2?
  Whats the present value of cash flows?
• Example 3 Joseph wants to buy a gift for his
  grandfather. His grandfather wants to see the
  soccer match between Turkey and Brasil. The cost
  of the ticket is 1600 (including travel and
  accommodation expenses) . Let annual interest
  rate be 10. If Joseph can put 500 this year
  500 next year how many years does he need?

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3.4 Perpetuities and Annuities

• Annuity Equally spaced level stream of cash
  flows.
• Perpetuity stream of cash flows that lasts
  forever.
• How is it possible that perpetuity has a finite
  value with infinite payments??
• Example Teri buys a perpetuity for 100 which
  pays 20 per year. Interest rate earned each
  year20/1000.20, i.e. 20
• r C/PVP
• ACAO implies
• PVPC/r
• where r Interest rate on perpetuity,
• C Cash payment,
• PVP Present value of perpetuity
• What is the difference between present value of a
  single cash payment and PVP?

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Examples ( Perpetuities and Annuities)

• Example Eric is a wealthy philanthropist and
  wants to endow a chair in finance department. The
  chair will provide 100,000 forever with 10
  interest rate. What is the present value of
  perpetuity?
• PVP1C/r100,000/0.11,000,000
• What is the PVP if he plans to start it four
  years from now?
• PVP after three yearsC/r100,000/0.11,000,000
• current PVP1,000,000 / (10.1)3751,315

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3.5 Growing perpetuity

• Perpetuity grows at a constant rate. g
• PV of growing perpetuity
• Ex Catherine will rent her apartment to Sean for
  1000 per year and expect to increase the rent by
  5 at the end of every year. If we assume that
  these people live forever, how much does Kelsey
  offer to buy the apartment right now? (interest
  rate is 10)

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3.4 Annuities

• Present value of t year Annuity (PVA)
• Example An auto dealer offers you a deal. For
  the new Ford Mustang, all you need to do is to
  pay 4,000 a year at the end of the next 3 years.
  What is the present value of this payment?
• Present Value4,000x 9,947.41

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Annuity Due

• Annuity Due Level stream of cash flows starting
  immediately.
• PV annuity due 1PV ordinary annuity of t-1
  payments

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Example (Annuity Due)

• Example Edgar is planning to buy a shopping mall
  net worth of 125 million. He pays a down payment
  of 25 million. Rest is financed by a bank at
  r10. If the bank sets up a mortgage payments
  over the next 360 years. Whats the fair annual
  mortgage payment?
• PVannual payment x 100
• annual payment
• 1.028 million

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Future Value of Annuity

• Example John saves 3,000 at the end of every
  year for the next four years. How much will they
  be worth at the end of 4 years? (interest rate is
  8)
• PV 3,000 x 4-year Annuity factor
• 3,000 x
• 9,936
• Future value of annuity 9,936 x (1.08)413,518
• Future value of annuity of 1 a yearPV annuity x
  (1r)t
• x (1r)t