Discounted Cash Flow Valuation

1
Chapter 5

• Discounted Cash Flow Valuation

2
Key Concepts and Skills

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

3
Chapter Outline

• Future and Present Values of Multiple Cash Flows
  
  
• Annuities and Perpetuities
• Comparing interest rates The Effect of
  Compounding
  
• Loan Types and Loan Amortization

4
Multiple Cash Flows FV Example 1

• 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?
  
• -500 PV 9i 2N Compute FV 594.05
• -600 PV 9i 1N Compute FV 654.00
• Add 1248.05

5
Multiple Cash Flows FV Example 2

• You deposit 100 into an account in one year and
  300 into the account in 3 years earning 8
  interest. How much will you have in five years?
  
• -100PV 8i 4N Compute FV 136.05
• -300PV 8i 2N Compute FV 349.92
• Add 485.97

6
Example 2 Timeline
0
1
2
3
4
5
100
300
136.05
349.92
485.97
7
Multiple Cash Flows - PV Example 1

• An investment will pay you 200 in one year, 400
  in two years, 600 the next year, and 800 at the
  end of the next year.
  
• You can earn 12 on similar investments. How
  much would you pay for this one?
  
• 200FV 12i 1N Compute PV
• Repeat for years 2, 3, and 4 adjusting the N
  number for the number of years
  
• Answer 1,432.93

8
Example 1 Timeline
9
Multiple Cash Flows PV Example 3

• You are considering an investment that will pay
  you 1,000 in one year, 2,000 in two years and
  3,000 in three years. If you want to earn 10
  on your money, how much would you be willing to
  pay?
• 1000 FV 10i 1N Compute PV 909.09
• 2000 FV 10i 2N Compute PV 1,652.89
• 3000 FV 10i 3N Compute PV
  2,253.94
  
• PV 909.09 1,652.89 2,253.94 4,815.92

10
Caveat Emptor!

• A stockbroker calls you and tells you that he has
  a 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?
• How do we solve this?

11
Bad Broker Advice!

• 40FV 1N 15i Compute PV
• 75FV 2N 15i Compute PV
• 34.78 56.71 91.49
• You do not make the investment because in
  Management 133 you learned how to evaluate an
  investment!
  
• Broker

12
Annuities and Perpetuities

• Annuity a pattern of equal payments that occur
  at regular intervals
  
• Ordinary Annuity when the first payment occurs
  at the end of the period
  
• Annuity Due when the first payment occurs at
  the beginning of the period
  
• Remember your ABCs
• A B C D Annuity Due occurs at the Beginning
• Perpetuity infinite series of equal payments

13
Annuities and the Calculator

• You can use the PMT key on the calculator for the
  equal payment
  
• The sign convention still holds
• Ordinary annuity versus annuity due
• You can switch your calculator between the two
  types by using the 2nd BGN 2nd Set on the TI
  BA-II Plus
  
• If you see BGN or Begin in the display of
  your calculator, you have it set for an annuity
  due
  
• Most problems are ordinary annuities

14
Annuity Lottery Example

• Congratulations! You won 10 million in the
  lottery. The money is paid in equal annual
  installments of 333,333.33 over 30 years. If
  the discount rate is 5, how much is the
  sweepstakes actually worth today?
• PV 333,333.331 1/1.0530 / .05
  5,124,150.29
  
• 333,333.33PMT 5i 30N Compute PV

15
Annuity vs. Annuity Due

• Suppose an annuity due has five payments of 400
  each with a 10 discount rate. Compute the PV of
  an ordinary annuity and the annuity due.
  
• Ordinary
• 400PMT 4N 10i Compute PV 1,267.95
• Annuity Due
• 400PMT 5N 10i Compute PV 1,667.95
• Hint Be sure to adjust calculator for an
  annuity due (begin)

16
Calculating a payment

• You want to borrow 20,000 for a new car. You
  qualify for a four-year loan at 8 per year,
  compounded monthly. What is your car payment?
  
• Try it!

17
Calculating a payment

• -20,000PV
• .66667i (8/12)
• 48N (4 years x 12)
• Compute PMT
• Answer 488.26

18
Finding the Number of Payments Credit Card Debt

• You charged 1000 on your credit card for spring
  break. You can only afford to make the minimum
  payment of 20/month. The interest rate is
  1.5/month.
• How long will it take to pay for spring break?
• Try it!

19
Credit Card Debt Solution

• 1000 PV
• -20 PMT
• 1.5 i
• Compute N
• Answer 93.11
• 93.11 what? How does that translate to years?

20
Finding the Number of Payments For a Personal Loan

• You borrow 2000 from a friend at 5 interest
  rate. You agree to make annual payments of
  734.42.
  
• How long will it take you to pay off the loan?
• Try it!

21
Finding the Number of Payments Another Example

• The hard way to solve this problem!
• 2000 734.42(1 1/1.05t) / .05
• .136161869 1 1/1.05t
• 1/1.05t .863838131
• 1.157624287 1.05t
• t ln(1.157624287) / ln(1.05)

22
The Easy Solution Using Your Financial Calculator!

• 2000PV
• -734.42 PMT
• 5i
• Compute N
• Answer Three years

23
Finding the Rate

• Suppose you borrow 10,000 from your rich uncle
  for a trip to Hawaii! You agree to pay 207.58
  per month for 60 months. What is the monthly
  interest rate?
• Sign convention matters!!!
• 60 N
• 10,000 PV
• -207.58 PMT
• CPT I/Y
• Answer .75 per month

24
Future Values for Annuities

• You decide you want to retire at 40, so you begin
  saving for your retirement by depositing 2,000
  per year in an IRA.
  
• If the interest rate is 7.5, how much will you
  have in 40 years?
  
• Which annuity will have produce the most amount
  of money for retirement, an Ordinary Annuity or
  an Annuity Due?
  

25
Annuity Solution

• FV(Ordinary) 454,513.04
• FV(Due) 488,601.52
• 2000 PMT 7.5i 40N Compute FV
• Change calculator to BEGIN mode for Annuity Due
• All things being equal, the annuity due will
  always have the higher dollar amount because the
  money has a longer time to compound.
  
• Remember, the greatest law in the universe is the
  law of compound interest!
  

26
Perpetuity

• A perpetuity is a annuity with an infinite life,
  making continual annual payments
  
• Perpetuity formula PV C/r
• C Cash flow
• r return
• A perpetual cash flow of 500 with an 8 return
• would be computed as
• PV C/r 500/.08 6,250

27
Effective Annual Rate (EAR)

• This is the actual or true interest rate paid or
  earned (received).
  
• The effective rate reflects the impact of
  compounding frequency.
  
• If you want to compare two alternative
  investments with different compounding periods
  you must compute the EAR and use that for
  comparison.

28
Annual Percentage Rate

• This is the annual (nominal) rate that must be
  disclosed to consumers on credit cards and on
  other loans as a result of truth in lending
  laws.
• 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

29
Computing APRs

• What is the APR if the monthly rate is .5?
• .5(12) 6
• What is the APR if the semiannual rate is .5?
• .5(2) 1
• What is the monthly rate if the APR is 12 with
  monthly compounding?
  
• 12 / 12 1
• Can you divide the above APR by 2 to get the
  semiannual rate? NO!!! You need an APR based on
  semiannual compounding to find the semiannual
  rate.

30
Things to Remember

• You ALWAYS need to make sure that the interest
  rate and the time period match.
  
• If you are looking at annual periods, you need an
  annual rate.
  
• If you are looking at monthly periods, you need a
  monthly rate.
  
• If you have an APR based on monthly compounding,
  you have to use monthly periods for lump sums, or
  adjust the interest rate appropriately if you
  have payments other than monthly

31
Computing EARs - Example

• Suppose you can earn 1 per month on 1 invested
  today.
  
• What is the APR? 1(12) 12
• How much are you effectively earning?
• FV -1PV 1i 12N Compute FV 1.1268
• Rate 12.68
• Suppose if you put it in another account, you
  earn 3 per quarter.
  
• What is the APR? 3(4) 12
• How much are you effectively earning?
• -1PV 3i 4N Compute FV 1.1255
• Rate 12.55

32
Compounding Comparison

• You are looking at two savings accounts. One pays
  5.25, with daily compounding. The other pays
  5.3 with semiannual compounding. Which account
  should you use?
• First account calculator sequence
• 5.25 shift NOM, 365 shift P/YR, shift EFF
  5.3899
  
• Second account calculator sequence
• 5.3 shift NOM, 2 shift P/YR, shift EFF
  5.3702
  
• Which account should you choose and why?

33
Computing Payments with APRs

• Suppose you want to buy Plasma TV that costs
  3500 and the store is willing to allow you to
  make monthly payments. The loan period is for 2
  years and the interest rate is 16.9 with monthly
  compounding. What is your monthly payment?
• Monthly rate 16.9 / 12 i
• Number of months 2(12) 24 N
• -3500 PV
• Compute pmt 172.88

34
Future Values with Monthly Compounding

• Suppose you deposit 50 per 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 9/12 i
• Number of months 35(12) 420N
• -50 PMT
• Compute FV 147,089.22

35
Present Value with Daily Compounding

• You need 15,000 in 3 years for a new car. If
  you can deposit money into an account that pays
  an APR of 5.5 based on daily compounding, how
  much would you need to deposit?
• Number of days 3(365) 1095N
• Daily rate 5.5 / 365i
• 15,000FV
• Compute PV 12,718.56

36
Quick Quiz Part 5

• What is the definition of an APR?
• What is the effective annual rate?
• Which rate should you use to compare alternative
  investments or loans?
  
• Which rate do you need to use in the time value
  of money calculations?

37
Discount Loans Example

• Treasury bills are 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?
  
• 10,000FV 7i 1N
• Compute PV 9,345.79

38
Interest-Only Loan - Example

• 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

39
Amortized Loan with Fixed Payment - Example

• Each payment covers the interest expense plus
  reduces principal
  
• Consider a 4-year loan with annual payments. The
  interest rate is 8 and the principal amount is
  5000.
  
• What is the annual payment?
• -5000 PV 4N 8i
• COMPUTE PMT 1,509.60

40
Quick Quiz

• 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?

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End of Chapter 5!