Chapter 4: Discounted cash flow valuation

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Chapter 4 Discounted cash flow valuation

• Corporate Finance
• Ross, Westerfield, and Jaffe

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Outline

• 4.1 Future value
• 4.2 Present value
• 4.3 Other parameters
• 4.4 Multiple cash flows
• 4.5 Comparing rates
• 4.6 Loan types

3
Definitions

• Present value (PV) earlier money on a time line.
• Future value (FV) later money on a time line.
• Interest rate (i), e.g., discount rate, required
  rate, cost of capital exchange rate between
  earlier money and later money.
• The number of time periods on a time line (N).
• PV ? FV time value of money via the exchange
  rate, i.e., interest rate, i.

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End-of-period cash flows

• By default, in this class cash flows occur at the
  end of each period.
• If cash flows occur at the beginning of each
  period, it will be explicitly specified.

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One equation one solution

• In general, we have one equation 0 f (PV, FV,
  i, N).
• Since we have only one equation, we can only
  allow for one unknown parameter (variable). That
  is, if wed like to calculate the value of a
  parameter, say FV, the values of the remaining
  parameters, i.e., PV, i, and N, need to be known.

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FV example I

• Suppose that we buy a 12-month CD at 12 annual
  interest rate for 10,000.
• FV PV ? (1 i)N 10,000 ? (1 12)1
  11,200.

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Do not compare apples with oranges

• Why N 1 while the CD matures in 12 months? The
  key is that
• The time frequency of i and N must be the same.
• If we use annual interest rate, then we need to
  measure the investment period using the unit of
  year. In this case, 12 months equal a year so N
  1.
• What is the value of N if the example provided us
  monthly interest rate, say 0.96 per month?
• Any volunteer?

8
Compounding

• Of course, the previous formula, FV PV ? (1
  i)N, is based on the notion of compounding.
• Compounding the process of accumulating interest
  on an investment over time to earn more interest.
• Earn interest on interest.
• Reinvest the interest.
• A popular method.

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FV example II

• Deposit 50,000 in a bank account paying 5. How
  much will you have in 6 years?
• Formula FV PV ? (1 i)N 50,000 ? (1 5)6
  67,000.
• Financial table (Table A.3) FV 50,000 ?
  1.3401 67,000.
• Financial calculator 6 N 5 I/Y 50000 PV CPT
  FV. The answer is FV -67,004.7820. Ignore the
  negative sign.

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Texas Instruments BAII Plus (keys)

• FV future value.
• PV present value.
• I/Y period interest rate.
• Interest is entered as a percent.
• N number of time periods.
• Clear the registers (CLR TVM, i.e., 2nd FV) after
  each calculation otherwise, your next
  calculation may come up with a wrong answer.

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FV example, III

• Jacob invested 1,000 in the stock of IBM. IBM
  pays a current dividend of 2 per share, which is
  expected to grow by 20 per year for the next 2
  years. What will the dividend of IBM be after 2
  years?
• Formula FV PV ? (1 i)N 2 ? (1 20)2
  2.88.
• Table A.3 FV 2 ? 1.4400 2.88.
• Calculator 2 PV 20 I/Y 2 N CPT FV. The
  answer is -2.8800.

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Discounting

• Discounting the process of calculating the
  present value of future cash flows.
• We call i the discount rate when we try to solve
  for present value. Depending on the question,
  this rate can be interest rate, cost of capital,
  or opportunity cost.

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PV example, I

• Suppose that you need 4,000 to pay your tuition.
  1-year CD interest rate is 7. How much do you
  need to put up today?
• Formula PV FV / (1 i)N 4,000 / (1 7)1
  3,738.3.
• Table A.1 PV 4,000 ? 0.9346 3,738.4.
• Calculator 4000 FV 7 I/Y 1 N CPT PV. The
  answer is -3,738.3178.

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PV example, II

• Suppose that you are 21 years old. Your annual
  discount (return) rate is 10. How much do you
  need to invest today in order to reach 1 million
  by the time you reach 65?
• Formula PV FV / (1 i)N 1,000,000 / (1
  10)44 15,091.
• Table A.1 does not have the present value factor
  for N 44. This is the limitation of using a
  financial table. Thus, we will focus on the
  other 2 methods in the following discussions.
• Calculator 1000000 FV 10 I/Y 44 N CPT PV.
  The answer is -15,091.1332.

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PV relationship, I

• Holding interest rate constant the longer the
  time period, the lower the PV.
• What is the present value of 5,000 to be
  received in 5 years? 10 years? The discount rate
  is 8
• 5 years 5 N 8 I/Y 5000 FV CPT PV. The answer
  is PV -3,402.9160.
• 10 years 10 N 8 I/Y 5000 FV CPT PV. The
  answer is PV -2,315.9674.

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PV relationship, II

• Holding time period constant the higher the
  interest rate, the smaller the PV.
• What is the present value of 5,000 received in 5
  years if the interest rate is 10? 15?
• 10 10 I/Y 5 N 5000 FV CPT PV. The answer is
  PV -3,104.6066.
• 15 15 I/Y 5 N 5000 FV CPT PV. The answer is
  PV -2,485.8837.

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The other parameters

• Recall that 0 f (PV, FV, i, N).
• We can find the value of i or N as long as we
  know about the values of the other parameters.
• The easiest way is to use a financial calculator.
• They are formulas, i.e., analytical solutions,
  for i and N as well. But these are not the focus
  of the course.

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Interest rate example

• Suppose that you deposit 5,000 today in a bank
  account paying interest rate i per year. If you
  reach 10,000 in 10 years, what rate of return
  are you being offered?
• Calculator 5000 PV -10000 FV 10 N CPT I/Y.
  The answer is I/Y 7.1773.
• Note that for entering -10000 FV, this is the
  sequence 10000 / FV.

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Time period example

• Suppose that you have 10,000 today. You want to
  retire as a millionaire. The annual rate of
  return that you can earn on the market is 10.
  In how many years can you retire?
• Calculator 10000 PV -1000000 FV 10 I/Y CPT N.
  The answer is N 48.3177.

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Multiple cash flows

• When there are multiple cash flows need to be
  discounted or compounded, the PV or FV of
  multiple cash flows are simply the sum of
  individual PVs or FVs, respectively.

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Multiple cash flow example

• Dennis has won the Kentucky State Lottery and
  will receive 2,000 (cash flow 1)in a year and
  5,000 (cash flow 2) in 2 years. Dennis can earn
  6 in his money market account, so the
  appropriate discount rate is 6.
• PV PV1 PV2 2,000 / (1 6)1 5,000 / (1
  6)2 6,337.
• That is, Dennis is equally inclined toward
  receiving 6,337 today and receiving 2,000 and
  5,000 over the next 2 years.

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Multiple cash flow example, Excel
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Annuity

• (Ordinary) Annuity a level of stream of cash
  flows for a fixed period of time (multiple, equal
  cash flows).
• Same dollar amount per period, making calculation
  much easier.
• FV C ? (1 i)N 1 / i .
• PV C ? 1 1 / (1 i)N / i .
• C is the fixed periodical payment.

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Annuity PV example

• Suppose that you want to buy a car. You can
  afford to pay 632 per month for the next 48
  months. You borrow at 1 per month for 48
  months. How much can you borrow?
• Formula PV C ? 1 1 / (1 i)N / i
  632 ? 1 1 / (1 1)48 / 1 24,000.
• Calculator 632 PMT 1 I/Y 48 N CPT PV. The
  answer is PV -23,999.5424.

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Annuity FV example

• Suppose that you put 3,000 per year into a Roth
  IRA. The account pays 6 per year. How much
  will you have when you retire in 30 years?
• Formula FV C ? (1 i)N 1 / i
  3,000 ? (1 6)30 1 / 6
  237,174.56.
• Calculator 3000 PMT 6 I/Y 30 N CPT FV. The
  answer is FV -237,174.5586.

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Other parameters for annuity

• An insurance company offers to pay you 10,000
  per year for 10 years if you will pay 67,100 up
  front. What is the rate of return?
• Calculator -67100 PV 10000 PMT 10 N CPT I/Y.
  The answer is I/Y 8.0003.

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Annuity due

• Annuity due an annuity for which the cash flows
  occur at the beginning of the period.
• For calculating PV and FV of an annuity due, we
  can use the following formula Annuity due value
  ordinary annuity value ? (1 i).

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Annuity due example

• You are going to rent an apartment for a year.
  You have 2 choices (1) pay the monthly rent,
  500, at the beginning of the month, or (2) pay
  the entire years rent, 5,000, today. Suppose
  that you can earn 1 every month. Which is the
  better choice?
• Ordinary PV 500 PMT 1 I/Y 12 N CPT PV. The
  answer is PV -5,627.5387.
• Annuity due PV ordinary PV ? (1 i)
  5,627.5387 ? 1.01 5,683.8141.
• You would want to pay 5,000 today if you can.

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Growing annuity

• Growing annuity a finite number of growing cash
  flows, where the constant growth rate is g.
• PV C ? 1 ((1 g) / (1 i))N / (i g)
  .

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Growing annuity example

• Emily has just been offered a job at 80,000 a
  year. She anticipates her salary increasing by
  9 a year until her retirement in 40 years.
  Given an interest rate of 20, what is the
  present value of her lifetime salary?
• PV C ? 1 ((1 g) / (1 i))N / (i g)
  80,000 ? 1 ((1 9) / (1 20))40
  / (20 9) 711,730.71.

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Perpetuity

• Perpetuity a constant stream of cash flows
  without end.
• PV C / i.

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Perpetuity example

• Preferred stock promises the buyer a fixed cash
  dividend every period (usually every quarter)
  forever. Suppose that VTinsurance Inc. wants to
  sell preferred stock. The quarterly dividend is
  1 per share. The required rate of return for
  this issue is 2.5 per quarter. What is the fair
  value of this issue?
• PV C / i 1 / 2.5 40 (per share).

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Growing perpetuity

• Growing perpetuity an infinite cash flow stream
  that grows at a constant rate, g.
• PV C1 / (i g), C1 is the cash flow at time 1.

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Growing perpetuity example

• Toyota is expected to pay a dividend (annual
  dividend) of 3 per share in a year. Investors
  also anticipate that the annual dividend will
  rise by 6 per year forever. The applicable
  discount rate is 11. What is the present value
  of future dividends?
• PV C1 / (i g) 3 / (11 6) 60 per
  share.

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Comparing rates, I

• Rates are quoted in many different ways.
• Tradition.
• Legislation.
• Effective annual rate (EAR) the actual rate paid
  (or received) after accounting for compounding
  that occurs during the year.
• When comparing two alternative investments with
  different compounding frequencies, one needs to
  compute the EARs and use them for reaching a
  decision.

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Comparing rates, II

• Annual percentage rate (APR) or stated annual
  interest rate the annual rate without
  consideration of compounding.
• APR period rate ? the number of periods per
  year, m.
• EAR 1 (APR / m)m 1.

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Rate example, I

• You went to a bank to borrow 10,000. You were
  told that the rate is quoted as 8 compounded
  semiannually. What is the amount of debt after
  a year?
• FV PV ? (1 i)N 10,000 ? (1 4)2
  10,816.
• EAR 1 (APR / m)m 1 1 (8 / 2)2 1
  8.16.

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Rate example, II

• What is the APR if the monthly rate is 1?
• APR 1 ? 12 12.
• What is the monthly (period) rate if the APR is
  6 with monthly compounding?
• Period (monthly) rate 6 / 12 0.5.

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Continuously compounding

• FV PV eAPRthe number of years , where e has
  the value of 2.718.
• Suppose that you invest 1,000 at a continuously
  compounded rate of 10 for a year.
• FV PV eAPRthe number of years 1,000
  e101 1,105.20. So, EAR 10.52.

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APR vs. EAR in real life

• By Trust-in-saving law, banks need to disclose
  EAR ( or called annual percentage yield (APY), or
  effective annual yield (EAY)). So you get the
  correct rate when you save.
• By Trust-in-lending law, banks need to disclose
  APR, the stated (quoted) rate. So you get a
  seemingly low rate when you borrow.
• Lesson the Congress is a good friend of the
  banking industry?

41
Pure discount loans

• Pure discount loans the borrower receives money
  today and repays a single lump sum at some time
  in the future.
• Treasury bills U.S. government borrows money and
  promises to repay a fixed amount at some time
  less than one year. Suppose that the maturity is
  12 months. The face value is 10,000. The
  market discount rate is 7. How much do you need
  to pay for the T-bill?
• PV FV / (1 i)N 10,000 / (1 7)1
  9,345.79.

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Amortized loans

• Amortized loans the loans that are paid off by
  making regular principal reductions.
• Payment per period interest a portion of
  principal.
• The most common type of amortized loans require
  borrowers make a single, fixed payment every
  period, i.e., annuity.

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Buying a house, I

• You are ready to buy a house and you have 20,000
  for a down payment and closing costs. Closing
  costs are estimated to be 5,500. The interest
  rate on the loan is 6 per year with monthly
  compounding (.5 per month) for a 30-year fixed
  rate loan. You are able to buy the house at
  154,500. What is the monthly payment? Suppose
  that you have an annual salary of 50,000. What
  is the ratio of the mortgage payment to your
  monthly income?

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Buying a house, II

• Down payment 20,000 5,500 14,500.
• Loan 154,500 14,500 140,000.
• Calculator 140000 PV 0.5 I/Y 360 N CPT PMT.
  The answer is PMT -839.3707.
• PMT/income 839.3707 / (50,000 / 12) 20.14.
• Banks usually do not want to see this ratio to be
  higher than 25.

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Interest-only loans

• Interest-only loans borrower pays interest each
  period and repay the entire original principal at
  some time in the future.
• Example bonds.
• This serves as a launch point for next topic
  Chapter 5 How to value bonds and stocks.

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Review let us work on this one

• Q11, P. 120 Conoly Co. Has identified an
  investment project with the following cash flows.
  If the discount rate is 10, what is the PV?
• Year 1 1,200. Year 2 600. Year 3 855. Year
  4 1,480.

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Review let us work on this one

• Concept 3, p. 118. Suppose that two athletes
  sign 10-year contracts for 80 million. In one
  case, we are told that the 80 million will be
  paid in 10 equal installments. In the other
  case, we are told that the 80 million will be
  paid in 10 installments, but the installments
  will decrease by 5 per year. Who got the better
  deal? Why?

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Assignment

• Please submit your work on problem 40 (p. 123),
  50 (p. 123) and 56 (p. 124) in 1 week.