Discounted Cash Flow Valuation

1
Discounted Cash Flow Valuation

• Chapter 5

2
Key Concepts and Skills

• Be able to compute the future value of multiple
  cash flows
• Be able to compute the present value of multiple
  cash flows
• Be able to compute loan payments
• Be able to find the interest rate on a loan
• Understand how loans are amortized or paid off
• Understand how interest rates are quoted

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Chapter Outline

• Future and Present Values of Multiple Cash Flows
• Valuing Level Cash Flows Annuities and
  Perpetuities
• Comparing Rates The Effect of Compounding
  Periods
• Loan Types and Loan Amortization

4
Multiple Cash Flows FV Example 1

• Calculate the future value of the following
• Currently you have on your account 7000, and
  you will deposit each year 4000 for the next 3
  year. You can earn 8 annually on the account.
• How to calculate the FV?
• Find the value at year 3 of each cash flow and
  add them together.
• Today (year 0) FV 7000(1.08)3 8,817.984
• Year 1 FV 4,000(1.08)2
  4,665.600
• Year 2 FV 4,000(1.08)
  4,320.000
• Year 3 FV is same as PV
  4,000.000
• Total value in 3 years 8817.984 4666.6 4320
  4000 21,803.58
• There is an alternative calculation of this
  example on page 117. See Example 5.1.

5
Multiple Cash Flows FV Example 2

• Suppose you invest 500 in a mutual fund today
  and 600 in one year. If the fund pays 9
  annually, how much will you have in two years?
• FV 500(1.09)2 600(1.09) 1248.05
• How much will you have in 5 years if you make no
  further deposits after the first year ?
• First way FV 500(1.09)5 600(1.09)4
  1616.26
• Second way use the value at year 2 and keep for
  3 years FV 1248.05(1.09)3 1616.26
• NOTE There are two ways to calculate future
  values
• 1) Compound accumulated balances and forward one
  year at the time.
• 2) Calculate the FV of each cash flows and add
  them up.

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Multiple Cash Flows FV Example 3

• Suppose you plan to deposit 100 into an account
  in one year and 300 into the same account in
  three years. How much will be on the account in
  five years if the interest rate is 8?
• FV 100(1.08)4 300(1.08)2 136.049 349.92
  485.97

Similar example is 5.2. on page 118.
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Multiple Cash Flows PV Example 4

• An investment provides various cash flows in the
  future. You want to calculate the PV. You want to
  earn at least 12 return. How much should you pay
  for the investment where you will receive in 1
  year 200 in 2 year 400 in 3 year 600 an
  in 4 year 800.
• Calculate the PV? How?
• Find the PV of each cash flow using your
  required rate of return and sum them up
• Year 1 CF 200 / (1.12)1 178.571
• Year 2 CF 400 / (1.12)2 318.878
• Year 3 CF 600 / (1.12)3 427.068
• Year 4 CF 800 / (1.12)4 508.414
• The total PV 178.571 318.878 427.068
  508.414 1432.93

Example 5.3. from page 121.
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Timeline for PV calculation PV Example 4 contd

• How much should you pay for the investment where
  you will receive in
• 1 year 200 in 2 year 400 in 3 year
  600 an in 4 year 800.

Today
Year 1
Year 2
Year 3
Year 4
Time (in years)
200
400
600
800
178.571
318.878
427.068
508.414
1432.93
Example 5.3. from page 121.
9
Multiple Uneven Cash Flows Using Financial
Calculator

• Another way to calculate questions with multiple
  uneven cash flows is to use the financial
  calculator cash flow keys
• Cash flow keys of Texas Instruments BA-II Plus
• Press CF and enter the cash flows beginning with
  year 0.
• You have to press the Enter key for each cash
  flow
• Use the down arrow key to move to the next cash
  flow
• The F is the number of times a given cash flow
  occurs in consecutive years
• Use the NPV key to compute the present value by
  entering the interest rate for I, pressing the
  down arrow and then compute
• Clear the cash flow keys by pressing CF and then
  CLR Work

10
Decisions, Decisions Example 5

• Your broker calls you and tells you that he has
  this great investment opportunity. If you invest
  100 today, you will receive 40 in one year and
  75 in two years. If you require a 15 return on
  investments of this risk, should you take the
  investment?
• Calculate the PV of the 2 future cash flows and
  sum them
• PV 40PVfactor 75PVfactor
• 40/(1.15)75/(1.15)² 34.78256.7191.49
• Alternative calculation Use the calculator CF
  keys to compute PV
• CF CF00 C0140 F011 C0275 F021 NPV
  I15 CPT NPV91.49
• No the broker is charging more than you would
  be willing to pay. To be able to earn 15, you
  want to invest only 91.49, but the broker
  charges you initially 100.

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Decisions, Decisions Example 5 contd

• Recall Your broker calls you and tells you that
  he has this great investment opportunity. If you
  invest 100 today, you will receive 40 in one
  year and 75 in two years. If you require a 15
  return on investments of this risk, should you
  take the investment?
• Decision You accept an investment if that offers
  greater return than your required rate of return.
• Use the CF keys to compute the internal rate of
  return (IRR) of the investment and compare it
  with your required rate of return Use Cash Flow
  calculation as before.
• CF CF0 -100 C01 40 F01 1 C02 75 F02
  1
• NPV CPT IRR ?? IRR 8.88

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Annuities and Perpetuities Defined

• Annuity finite series of equal payments that
  occur at regular intervals
• If the first payment occurs at the end of the
  period, it is called an ordinary annuity. If the
  first payment occurs at the beginning of the
  period, it is called an annuity due (NOTE You
  can switch your calculator between the two types
  by using the 2nd BGN 2nd Set on the TI BA-II Plus
  )
• Examples of Annuity
• Sweepstakes receiving monthly the same amount of
  money
• Paying of loans where the monthly or annual
  payment is fixed/equal
• Saving up for retirement/children college funds/
  buying a car or house, where you save by putting
  in the investment the same amount of money each
  period.
• Perpetuity infinite series of equal payments

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Annuities and Perpetuities Basic Formulas

• Perpetuity PV C / r
• Annuities

Recall that the PV factor 1/ 1rt
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Annuity Finding PV Example 6

• Suppose you win the Publishers Clearinghouse 10
  million sweepstakes. The money is paid in equal
  annual installments of 333,333.33 over 30 years.
  If the appropriate discount rate is 5, how much
  is the sweepstakes actually worth today?
• Using annuity formula PV C annuity PV factor
• PV 333,333.33 1 1/1.0530 / .05
  5,124,150.29
• The PV of the sweepstakes is about 5,124,150.
• Using financial calculator
• PMT 333,333.33 N30 FV0 I/Y5 CPT PV
  5,124,150.29

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Annuity - Finding Payment Example 7

• Suppose you want to borrow 20,000 for a new car.
  You can borrow at 8 per year, compounded monthly
  (8/12 .66667 per month). If you take a 4 year
  loan (48 months), what is your monthly payment?
• Using the annuity formula from Slide 13
• 20,000 C1 1 / 1.00666748 / .0067 ? solve
  for C.
• C 20,000 / (1-1/1.00666748)/.0067
• C 488.26, thus the monthly payment is 488.26
• Using financial calculator
• N48 FV0 PV20,000 I/Y0.667 CPT PMT!!
• C with rounding to four digits is 488.63.

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Future Values for Annuities -Example 8

• Suppose you begin saving for your retirement by
  depositing 2000 per year in an IRA. If the
  interest rate is 7.5, how much will you have in
  40 years?
• FV of Annuity Cannuity FV factor
• With annuity formula
• FV 2000 (1.07540 1)/.075 454,513.04
• With financial calculator
• I/Y7.5 N40 PMT-2000 (PV0) CPT
  FV454,513.04

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Annuity Finding the Number of Payment -
Example 9

• Suppose you borrow 2000 at 5 and you are going
  to make annual payments of 734.42. How long
  before you pay off the loan?
• With annuity formula
• 2000 734.42(1 1/1.05t) / .05
• .1362 1 1/1.05t
• - . 8638 - 1/1.05t / cancel the negative
  sign both sides and take a reciprocal 1.1576
  1.05t solve for t ln(1.1576) / ln(1.05) 3
  years
• We do not take into account if you take on
  additional loan
• with financial calculator
• FV0 PMT734.42 PV2,000 I/Y5 CPT N3

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Annuity Finding the Rate - Example 10

• Suppose you borrow 10,000 from your parents to
  buy a car. You agree to pay 207.58 per month
  for 60 months. What is the monthly interest
  rate?
• Again use the annuity formula from slide 13
• 10,000 207.58 (1 1/(1r)60) / r
• 48.17 Annuity PV factor
• Use financial calculator Sign convention
  matters!!!
• N60 PV10,000 PMT -207.58 (FV0) CPT I/Y
  .75

All payments made / or cash outflows are
entered with a negative signs. Payments received
such as the mortgage loan entered with positive
sign.
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Annuity Finding the Rate Without a
Financial Calculator

• Trial and Error Process
• Choose an interest rate and compute the PV of the
  payments based on this rate
• Compare the computed PV with the actual loan
  amount
• If the computed PV loan amount, then the
  interest rate is too low
• If the computed PV interest rate is too high
• Adjust the rate and repeat the process until the
  computed PV and the loan amount are equal

This method is very time consuming!
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Perpetuity - Example 11

• Perpetuity is a special case of annuity, when the
  level stream of cash flow continues forever.
• For example, if an investment offer a cash flow
  of 500 forever, that is perpetuity and the PV
  C/ r, where r is the return you require on such
  an investment (for example 8).
• PV 500/.086,250
• This implies that you value such an investment
  for 6,250.

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Table 5.2
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EAR versus APR

• Effective Annual rate (EAR)
• This is the actual rate paid (or received) after
  accounting for compounding that occurs during the
  year
• If you want to compare two alternative
  investments with different compounding periods
  you need to compute the EAR and use that for
  comparison.
• Annual percentage rate (APR)
• This is the annual rate that is quoted by law
• By definition APR period rate times the number
  of periods per year
• Consequently, to get the period rate we rearrange
  the APR equation
• Period rate APR / number of periods per year

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EAR Formula and APR - Formula
Remember that the APR is the quoted rate, while
EAR is the rate you pay or earn.
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FV with Monthly Compounding

• Suppose you deposit 50 a month into an account
  that has an APR of 9, based on monthly
  compounding. How much will you have in the
  account in 35 years?
• Monthly rate .09 / 12 .0075
• Number of months 35(12) 420
• FV 501.0075420 1 / .0075 147,089.22
• With financial calculator
• PMT50 PV0 I/Y.75 N420 CPT 147,089.22

Again, we do not use EAR, because we make monthly
payments and we use the applicable monthly rate.
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Computing Payments with APRs

• Suppose you want to buy a new computer system and
  the store is willing to sell it to allow you to
  make monthly payments. The entire computer system
  costs 3500. The loan period is for 2 years and
  the APR is 16.9.
• What is your monthly payment?
• The monthly rate is .169 / 12 1.408 and
  investment period in number of months is 24.
• 3500 C1 1 / 1.0140824 / .01408
• C 172.88 The monthly payment using monthly
  rate 1.41 is 172.91
• With financial calculator
• N24 I/Y 1.408 (FV0) PV3500 CPT PMT
  172.87

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The different rates theory (Review)

• Discount rate used to calculate the PV of FV
• The discount factor is
• Internal Rate of Return (IRR) def. is on page
  236.
• IRR is the discount rate which makes the NPV (or
  Present Value) of the investment zero. Or
  alternately, the IRR on an investment is the
  required rate of return that results in zero PV,
  when used as discount rate.
• Rate of return or rate of interest
• The rate of interest, expressed as a proportion
  of the principal, at which interest is computed.
• Rate of return Calculated as the (value now
  minus value at time of purchase) divided by value
  at time of purchase.

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LOANS

• We briefly discuss different types of loans.
• This is section may be very useful for you. This
  knowledge can aid you in future loan negotiations
  and help you to make sound financial decisions.
• The focus is on amortized loans.

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Pure Discount Loans

• Treasury bills are excellent examples of pure
  discount loans. The principal amount is repaid
  at some future date, without any periodic
  interest payments.
• If a T-bill promises to repay 10,000 in 12
  months and the market interest rate is 7 percent,
  how much will the bill sell for in the market?
• PV FV Discount factor FV 1/(1r)
• FV1/1.07 10,000 / 1.07 9345.79

(similar example is Ex. 5.11 on page 138)
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Interest Only Loan

• Consider a 5-year, interest only loan with a 7
  interest rate. The principal amount is 10,000.
  Interest is paid annually.
• What would the stream of cash flows be?
• Years 1 4 Interest payments of .07(10,000)
  700
• Year 5 Interest principal 10,700
• This cash flow stream is similar to the cash
  flows on corporate bonds and we will talk about
  them in greater detail later.

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Amortized Loan with Fixed Payment

• Each payment covers the interest expense plus
  reduces principal (this is the general case for
  mortgage loans)
• Consider a 4 year loan with annual payments. The
  interest rate is 8 and the principal amount is
  5000.
• What is the annual payment using the annuity
  formula?
• 5000 C1 1 / 1.084 / .08
• C 1509.60
• What is the payment, using financial calculator?
• FV0 PV 5,000 I/Y8 N4 CPT PMT!! PMT
  -1,509.6
• In your answer say payment is 1,510 dollars
  (nonnegative).

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Amortization Table for Example 15
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Quick Quiz Different Loans

• What is a pure discount loan? What is a good
  example of a pure discount loan?
• What is an interest only loan? What is a good
  example of an interest only loan?
• What is an amortized loan? What is a good
  example of an amortized loan?