Discounted Cash Flow Valuation

1
Chapter
5
Discounted Cash Flow Valuation
2
Key Concepts and Skills

• 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

3
Uneven Cash Flows

• You are considering an investment that will pay
  you 1000 in one year, 2000 in two years and
  3000 in three years. If you want to earn 10 on
  your money, how much would you be willing to pay?
• 1 N 10 I/Y 1000 FV PV? ? -909.09
• 2 N 10 I/Y 2000 FV PV? ? -1652.89
• 3 N 10 I/Y 3000 FV PV? ? -2253.94
• PV 909.09 1652.89 2253.94 4815.93

4
Multiple Uneven Cash Flows TI/LW

• Another way to use the financial calculator for
  uneven cash flows is to use the cash flow keys
• Texas Instruments BA-II Plus
• Clear the cash flow keys by pressing CF and then
  2nd CLR Work
• 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 computing NPV

5
Uneven Cash Flows

• Use CF button to enter CFs
• 2ND , CLR WORK clears the CF register
• Trick 1st CF CF0
• Enter CF 0 ?
• C01 1000 ENTER ? ?
• C02 2000 ENTER ? ?
• C03 3000 ENTER ? ?
• NPV, 10, ENTER, ? CPT
• NPV ? 4815.93

6
Decisions, Decisions

• Your broker has an 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, should you take the investment?
• Use the CFj keys to compute the value of the
  investment
• No ? Youre paying 100 its worth 91
• What return would you actually make?
• 2ND, CLR WORK
• CFo -100 ENTER ?, C01 40 ENTER ? ?, C02 75 ENTER,
  IRR, CPT

7
Quick Quiz Part 1

• Suppose you are looking at the following possible
  cash flows Year 1 CF 100 Years 2 and 3 CFs
  200 Years 4 and 5 CFs 300. The required
  discount rate is 7
• What is the value of the cash flows today?
• Heres where the Fs come in handy
• 2ND, CLR WORK
• CFo 0 ?, C01 ENTER ? ?, C02 200 ENTER ?, F02 2
  ENTER ?, C03 300 ENTER ?, F03 2 ENTER, NPV 7
  ENTER ? CPT

8
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
• Perpetuity infinite series of equal payments

9
Annuities and Perpetuities Basic Formulas

• Perpetuity PV CF / r
• Annuities

10
Annuities and the Calculator

• PMT key annuity
• Ordinary annuity versus annuity due
• You can switch your calculator between the two
  types by using the ltGOLDgt BEG/END
• If you see BEGIN in the display of your
  calculator, you have it set for an annuity due
• Most problems are ordinary annuities

11
Future Value of an Annuity

• The FV of annuity amount received the
  interest earned from time received until the
  future date

12
Future Value of an Annuity

• If you deposit 100 at the end of each year for
  three years in a savings account that pays 5
  interest per year, how much will you have at the
  end of three years?

100
100.00
100 (1.05)0
105.00 100 (1.05)1
110.25 100 (1.05)2
13
Future Value of an Annuity

• Financial calculator solution
• Inputs 3 N 5 I/Y -100 PMT
• CPT FV ? 315.25
• To solve the same problem, but for the present
  value instead of the future value, change the
  final input from FV to PV

14
Annuities Due

• If the three 100 payments at the beginning of
  each year, the annuity an annuity due.
• How will your answer be different?

15
Future Value of an Annuity Due

• 100 at the start of each year
• Each payment one year earlier ?earn interest
  for an additional year (period).

16
Future Value of an Annuity Due

• Financial calculator solution
• Switch to the beginning-of-period mode ? ltcolorgt
  BEG/END
• Inputs 3 N 5 I/Y -100 PMT
• FV? ? 331.0125

17
Present Value of an Annuity

• If you require a 5 return, how much would you
  pay today for a three-year annuity with payments
  of 100 at the end of each year?

18
Present Value of an Annuity
19
Present Value of an Annuity Due

• Payments at the beginning of each year
• Payments all come one year sooner
• Each payment would be discounted for one less
  year
• Present value of annuity due will exceed the
  value of the ordinary annuity by one years
  interest on the present value of the ordinary
  annuity

20
Present Value of Annuity Due

• 100
• 95.24
• 90.70

100
21
Present Value of Annuity Due

• Financial calculator solution
• Switch to the beginning-of-period mode ? 2ND BGN
  2ND SET
• Inputs Inputs 3 N 5 I/Y -100 PMT
• CPT PV ? 285.94
• Then switch back to the END mode
• 2ND BGN 2ND SET

22
More Annuities

• 1 million Lotto winner!!!!
• 40,000 per year for 25 years starting today
• 500,000 lump sum today
• Which is better? _at_8? _at_6?
• http//www.txlottery.org/faq/morequestions.cfm

23
Lotto

• 1 million Lotto winner!!!!
• 40,000 per year for 25 years starting today
• 500,000 lump sum today
• Pre-CVO, private firms would loan you the lump
  sum
• You repay it with your 40K check every year
• Rate on 300K loan?
• Rate on 500K loan?

24
Future Values for Annuities

• Suppose you begin saving for your retirement by
  depositing 2000 per year in an IRA. If the
  interest rate is 12.3, how much will you have in
  35 years?
• 35 N
• 12.3 I/Y
• -2000 PMT
• CPT FV? 926,533

25
Future Values for Annuities

• Suppose you begin saving for your retirement by
  depositing 2000 per year in an IRA. If the
  interest rate is 12.3, how much will you have in
  40 years?
• FV? ? 1,667,635
• Last slide, saving for 35 years ? FV 926,533
• 1,667,635 926,533 741,102 from 5 extra
  years of saving 2000 each year.

26
Annuity Due

• You are saving for a new house and you put
  10,000 per year in an account paying 8. The
  first payment is made today. How much will you
  have at the end of 3 years?
• 2ND BGN 2ND SET
• 3 N
• -10,000 PMT
• 8 I/Y
• CPT FV ? 35,061.12

27
Perpetuity

• Perpetuity formula PV C / r
• Preferred Stock is a perpetuity. Buyer of PS is
  promised a fixed cash dividend every period
  forever.
• If PS pays a 5 yearly dividend and investors
  require a 12 return, what is the price?

28
Finding the Payment

• Suppose you want to borrow 20,000 for a new car.
  You can borrow at 8 per year. If you take a 4
  year loan, what is your monthly payment?
• Whats different?
• Two methods

29
Finding the Payment

• Interest rate 8 per year, but monthly pmt
• Int rate 8/12 0.6667 per month
• 1 P/YR
• 48 N 20,000 PV 0.6667 I/Y
• CPT PMT ? 488.26
• Or

30
Finding the Payment

• Monthly PMT ? 2ND P/Y , 12 ENTER
• N of PMTs I/Y still annual rate
• 2ND QUIT
• 48 N 20,000 PV 8 I/Y
• CPT PMT ? -488.26

31
Credit Cards

• You realize that you have a 5000 balance on your
  credit card, which is being assessed 18 yearly
  interest. If you cut the credit card up and make
  100 payments every month on it, how long until
  youve paid it off?

32
Finding the Rate

• 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 interest rate?

33
Finding the Rate

• 2ND P/Y 1 ENTER, 2ND QUIT, 60 N, 10000 PV,
  -207.58 PMT
• CPT I/Y ? 0.75 per month
• 0.75 12 9.0 annually
• 2ND P/Y 1 ENTER, 2ND QUIT, 60 N, 10000 PV,
  -207.58 PMT
• CPT I/Y ? 9.0 per year

34
Future Values with Monthly Compounding

• Suppose you deposit 50 a month into an account
  that makes 9. How much will you have in the
  account in 35 years?

35
Future Values with Monthly Compounding

• P/Y 1 ? 420 N, 0.75 I/Y, 50 PMT, CPT FV
• --- or ---
• P/Y 12 ? 420 N, 9 I/Y, 50 PMT, CPT FV

36
Quick Quiz Part 2

• You want to have 1 million to use for retirement
  in 35 years. If you can earn 12 annually, how
  much do you need to deposit on a monthly basis if
  the first payment is made in one month?
• What if the first payment is made today?
• You are considering preferred stock that pays a
  yearly dividend of 6.00 and costs 75. What
  return does this imply?

37
Annual Percentage Rate

• This is the annual rate that is quoted by law
• By definition APR period rate (ie, 1 per
  month) times the number of periods per year (1
  12 12)
• Also called simple rate or nominal rate

38
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

39
Compounding Periods

• Which would you rather have
• 1. 100 compounded yearly at 10?
• 2. 100 compounded semiannually at 10?
• Both have 10 APR
• Assume 20-year investment, and find FV

40
Compounding Periods

• 1
• P/Y 1
• 20 N
• 10 I/Y
• -100 PV
• CPT FV
• 672.75

41
Compounding Periods

• 2
• 2ND P/Y 2 ENTER, 2ND QUIT
• 40 N ? N periods ? 40 6-month periods
• 10 I/YR (Still use yearly rate)
• -100 PV
• CPT FV
• 704.00
• Versus 672.75 for yearly compounding

42
Interest Rates

• APR Simple (Quoted) Interest Rate
• rate used to compute the interest payment paid
  per period
• Effective Annual Rate (EAR)
• annual rate of interest actually being earned,
  considering the compounding of interest

43
Interest Rates

• With annual compounding
• APR Effective Rate
• With semiannual/monthly compounding
• APR lt Effective Rate

44
Interest Rates

• Effective annual rate on TIBA/LW
• 2ND ICONV 10 ENTER ? NOM 10
• ? 2 ENTER ? 2 C/Y
• ? CPT ? EFF 10.25
• If compounded monthly, whats the effective
  annual rate on 10?

45
Decisions, Decisions II

• 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?

46
Decisions, Decisions II Continued

• To verify Suppose you invest 100 in each
  account. How much will you have in each account
  in one year?
• First Account
• 2ND P/Y 365 ENTER 2ND QUIT, 365 N 5.25 I/Y 100
  PV
• CPT FV ? 105.39
• Second Account
• 2ND P/Y 2 ENTER 2ND QUIT, 2 N 5.3 I/YR 100 PV
• CPT FV ? 105.37

47
APR - Example

• Suppose you want to earn an effective rate of 12
  and you are looking at an account that compounds
  on a monthly basis. What APR must they pay?

48
Amortized Loan --Home Mortgage

• 30-year, 100K loan _at_ 7 ? Compute PMT
• 2ND P/Y 12 ENTER 2ND QUIT
• 30 yrs 12 months 360 payments ? 360 N
• 7 I/Y
• 100,000 PV
• (0 FV)
• Hit CPT PMT to compute
• PMT 665.30

49
Home Mortgage

• Each payment 665.30 principal interest
• 1st payment
• 2ND AMORT P11 ENTER ? P21 ENTER ? ? ?
• 665.30 81.97 principal 583.33 interest
• New loan balance 100K - 81.97 99,918

50
Home Mortgage

• Each payment 665.30 principal interest
• 2nd payment
• 2ND AMORT P12 ENTER ? P22 ENTER ? ? ?
• 665.30 82.45 principal 582.85 interest
• New loan balance 99,918 - 82.45 99,836

51
Home Mortgage

• 1st year ? 12 payments 665.3012 7,984
• 12 payments
• 2ND AMORT P11 ENTER ? P212 ENTER ? ? ?
• 7,984 1,016 principal 6,968 interest
• New loan balance 100,000 - 1015.81 98,984

52
Home Mortgage

• 30-year 150,000 loan _at_ 7.5
• How long before loan is halfway paid off?
• If you pay 100/month extra, how long will it
  take to pay the loan completely off?

53
Amortized Loan

• 4 year loan with annual payments. The interest
  rate is 8 and the principal amount is 5000.
• What is the annual payment?
• Find the principal and interest paid in each year.

54
Amortization Table for Example