Time Value of Money Discounted Cash Flow Analysis

1
Time Value of MoneyDiscounted Cash Flow Analysis
2
Time Value of Money

• A dollar received today is not worth the same
  amount as a dollar to be received in the future
  WHY?
• You should receive Interest on the dollar
  received today if it is invested.

3
A Simple Example

• You deposit 100 today in an account that earns
  5 interest annually for one year.
• How much will you have in one year?
• Value in one year Current Value Interest
  Earned
• 100 100(.05)
• 100(1.05) 105
• The 100 today has a Future Value of 105
• or
• The 105 next year has a Present Value of 100

4
Using a Time Line

• An easy way to represent this is on a time line
• Time 0 1 year
• 5
• 100 105

Beginning of First Year
End of First year
5
What would the 100 be worth in 2 years?

• You would receive interest on the interest you
  received in the first year (the interest
  compounds)
• Value in 2 years Value in 1 year interest
• 105 105(.05) 105(1.05) 110.25
• Or substituting 100(1.05) for 105
• 100(1.05)(1.05)
• 100(1.05)2 110.25

6
On the time line

• Time 0 1 2
• Cash -100 105 110.25
• Flow

Beginning of year 1
End of Year 1 Beginning of Year 2
End of Year 2
7
Generalizing the Formula

• 110.25 (100)(1.05)2
• This can be written more generally
• Let t The number of periods 2
• i The interest rate per period .05
• PV The Present Value 100
• FV The Future Value 110.25
• FV PV(1i)n
• (110.25) (100)(1 0.05)2
• This works for any combination of n, i, and PV

8
Future Value Interest Factor

• FV PV(1i)n (1i)n is called the
• Future Value Interest Factor (FVIFi,n)
• FVIFs can be found in tables or calculated
• Interest Rate 4.0 4.5 5.0 5.5
• Periods
• 1
• 2
• 3

1.1025
OR (1.05)2 1.1025 Either way original
equation can be rewritten FV PV(1i)n
PV(FVIFi,n)
9
Calculation MethodsFV PV(1i)n

• Tables using the Future Value Interest Factor
  (FVIF)
• Regular Calculator
• Financial Calculator
• Spreadsheet

10
Using the tables

• FVIF5,2 1.1025
• Plugging it into our equation
• FV PV(FVIFi,n)
• FV 100(1.1025) 110.25

11
Using a Regular Calculator

• Calculate the FVIF using the yx key
• (1.05)21.1025
• Proceed as Before
• Plugging it into our equation
• FV PV(FVIFi,n)
• FV 100(1.1025) 110.25

12
Financial Calculator

• Financial Calculators have 5 TVM keys
• N Number of Periods 2
• I interest rate per period 5
• PV Present Value -100
• FV Future Value ?
• PMT Payment per period 0
• After entering the portions of the problem you
  know, the calculator will provide the answer

13
Financial Calculator Example

• On an HP-10B calculator you would enter
• 2 N 5 I -100 PV 0 PMT FV
• and the screen shows 110.25

14
Spreadsheet Example

• Excel has a FV command
• Excel command FV(rate,nper,pmt,pv,type)
• FV(0.05,2,0,100,0)
• 110.25
• notes
• The inputs needed are basically the same as on
  the financial calculator
• Type refers to whether the payment is at the
  beginning (type 1) or end (type0) of the year

15
Practice Problem

• If you deposit 3,000 today into a CD that pays
  4 annually for a period of five years, what will
  it be worth at the end of the five years?

FV PV(1i)n PV(FVIFi,n)
FVIF0.04,5 (10.04)5 1.216652
FV 3,000(10.04)53,000(1.216652) FV
3,649.9587
16
Compounding Interest

• Assume that 100 years ago your ancestors invested
  5 at 6. In the first year there would have
  been 0.30 in interest.
• If you took out the interest each year you would
  have received a total of 0.30(100) or 30 in
  interest
• How much would the 5 be worth if the interest
  reinvests?

5(1.06)100 1,696.51
17
Compounding Interest

• Leaving the interest in the account allows you to
  earn interest upon the interest. The impact of
  the interest compounds or increases over time.
• The more periods interest is allowed to
  accumulate, the greater the impact of the
  compounding will be.

18
Compounding at Different Rates of Interest
4,338.58 7
1,696.51 6
657.51 5
19
Calculating Present Value

• We just showed that FVPV(1i)n
• This can be rearranged to find PV given FV, i and
  n.
• Divide both sides by (1i)n

which leaves PV FV/(1i)n
20
Example

• If you wanted to have 110.25 at the end of two
  years and could earn 5 interest on any deposits,
  how much would you need to deposit today?
• PV FV/(1i)n

110.25
PV
100
(10.05)2
21
Present Value Interest Factor

• PV FV/(1i)n 1/(1i)nis called the
• Present Value Interest Factor (PVIFi,n)
• PVIFs can be found in tables or calculated
• Interest Rate 4.0 4.5 5.0 5.5
• Periods 0
• 1
• 2
• 3

0.907029
OR 1/(1.05)2 0.907029 Either way original
equation can be rewritten PV FV/(1i)n
FV(PVIFi,n)
22
Calculation MethodsPV FV/(1i)n

• Tables using the Present Value Interest Factor
  (PVIF)
• Regular Calculator
• Financial Calculator
• Spreadsheet

23
Using the tables

• PVIF5,2 .9070
• Plugging it into our equation
• PV FV(PVIFi,n)
• PV 110.25(0.9070) 100.00

24
Using a Regular Calculator

• Calculate the PVIF using the yx key
• (1/(1.05))2.9070
• Make sure to divide first then square
• Proceed as Before
• Plugging it into our equation
• PV FV(PVIFi,n)
• PV 110.25(0.9070) 100.00

25
Financial Calculator

• Financial Calculators have 5 TVM keys
• N Number of Periods 2
• I interest rate per period 5
• PV Present Value ?
• FV Future Value 110.25
• PMT Payment per period 0
• After entering the portions of the problem you
  know, the calculator will provide the answer

26
Financial Calculator Example

• On an HP-10B calculator you would enter
• 2 N 5 I 110.25 FV 0 PMT PV
• and the screen shows -100.00

27
Spreadsheet Example

• Excel has a PV command
• Excel command PV(rate,nper,pmt,fv,type)
• FV(0.05,2,0,110.25,0)
• -100.00
• notes
• The inputs needed are basically the same as on
  the financial calculator
• Type refers to whether the payment is at the
  beginning (type 1) or end (type0) of the year

28
Example

• Assume you want to have 1,000,000 saved for
  retirement when you are 65 and you believe that
  you can earn 10 each year.
• How much would you need in the bank today if you
  were 25?

29
Put the problem on a time line

• Age 25 35 45 55 65
• Years 0 10 20 30 40

1,000,000
PV?
PVIF40,10 1/(1.1)40 0.02209493

• PV 1,000,000/(1.10)401,000,000(.02209493)
• PV 22,094.93

30
What if you are currently 35? Or 45?

• If you are 35 you would need
• PV 1,000,000/(1.10)30 57,308.55
• If you are 45 you would need
• PV 1,000,000/(1.10)20 148,643.63
• This process is called discounting (it is the
  opposite of compounding)

31
Example 2

• You decide to attend law school after completing
  your MBA. You believe that you will need
  100,000 when you start Law School in three
  years. How much would you need in the bank today
  at 7 to have enough for tuition?

100,000/(1.07)3 81,629.7878
PVIF7,3 .8163 100,000(.8163) 81,630
32
PV and FV Practice Problem

• You hope to buy a new car when you graduate in
  two years, you believe the car will cost 25,000.
  If you can earn 9 each year, how much would you
  need to put in the bank today to be able to buy
  the car in two years?

Interest Rate?
FV or PV?
Number of Periods?
PV 25,000/1.092
PV 21,041.99
33
Solving for the interest rate

• PV FVt/(1i)n or PV(1i)nFV
• Rearrange the above equation
• FV/PV (1i)tn
• (FV/PV)1/n 1i
• (FV/PV)1/n-1 i

34
An Example

• What interest rate would you need to double your
  investment of 1,000 over the next five years?
• 2,000 1,000(1i)5
• 2,000/1,000 2 (1i)5
• 2(1/5) (1i)5(1/5)1i
• 1.1468 1.1468

35
Rule of 72 A shortcut

• How long does it take for a sum of money to
  double in value from compounding at a given rate?
• If the interest rate is between 5 and 20 then
  the sum will double in approximately 72/r
• If you are earning 8 interest your money would
  double in approximately 72/8 9 years

36
An Introduction to determining the Correct
Interest Rate

• So far we have just assumed a level of interest
  rate for our problems.
• How should the correct interest rate be
  determined?
• Interest rates are also linked to the level of
  risk (we will see this in detail later in the
  semester). Generally, greater risk results in
  greater return.

37
Opportunity Cost

• An opportunity cost represents the cost of the
  best foregone alternative.
• When calculating Time Value problems the correct
  rate combines the idea of risk and return and
  opportunity cost.
• Opportunity Cost Rate the rate of return on the
  best available alternative investment of equal
  risk.

38
Solving for the number of periods

• FV PV(1i)n
• Rearrange FV/PV (1i)n
• Take the natural log of both sides
• ln(FV/PV) n(ln(1i))
• n ln(FV/PV)/(ln(1i))

39
Questions

• What happens to the PV of a future sum as the
  level of interest rate (discount rate) increases
  (or decreases)?
• What happens to the FV as the interest rate
  increases (or decreases)?
• What happens to the PV of a future sum if the
  number of periods increases (or decreases)?
• What happens to the FV of a current sum if the
  number of periods increases (or decreases)?

40
Annuities

• Annuity A series of equal payments made over a
  fixed amount of time. An ordinary annuity makes
  a payment at the end of each period.
• Example A 4 year annuity that makes 100 payments
  at the end of each year.
• Time 0 1 2 3 4
• CFs 100 100 100 100

41
Future Value of an Annuity

• The FV of the annuity is the sum of the FV of
  each of its payments. Assume 6 a year
• Time 0 1 2 3 4
• 100 100 100 100 FV of CF

100(1.06)0100.00
100(1.06)1106.00
100(1.06)2112.36
100(1.06)3119.10
FV 437.4616
42
FV of An Annuity

• This could also be written
• FV100(1.06)0 100(1.06)1 100(1.06)2
  100(1.06)3
• FV100(1.06)0 (1.06)1 (1.06)2(1.06)3
• or for any n, i, payment, and t

43
FVIF of an Annuity (FVIFAr,t)

• Just like for the FV of a single sum there is a
  future value interest factor of an annuity
• This is the FVIFAi,n
• FVannuityPMT(FVIFAi,n)

44
FVIFA

• The FVIFA can be approximated by
• FVIFA (1i)n-1/iFVIFi,n-1/i

45
Calculation Methods

• Tables - Look up the FVIFA
• FVIFA6,4 4.374616 FV 100(4.374616)
  437.4616
• Regular calculator -Approximate FVIFA
• FVIFA (1i)t-1/i FV 100(4.374616)
  437.4616
• Financial Calculator
• 4 N 6 I 0 PV -100 PMT FV 437.4616
• Spreadsheet
• Excel command FV(rate,nper,pmt,pv,type)
• Excel command FV(.06,4,100,0,0)437.4616

46
Practice Problem

• Your employer has agreed to make yearly
  contributions of 2,000 to your Roth IRA.
  Assuming that you have 30 years until you retire,
  and that your IRA will earn 8 each year, how
  much will you have in the account when you retire?

47
Put the problem on a time line

• Age 35 36 37 64 65
• Years 0 1 2 29 30
• 2,000 2,000 2,000 2,000

FVIFA30,8 (10.08)30-1/0.08 113.28
48
Alternative Solution Methods

• Financial Calculator
• 30 N 8 I 0 PV -2000 PMT
• FV 226,533.42
• Spreadsheet
• Excel command FV(rate,nper,pmt,pv,type)
• Excel command
• FV(.08,30,0,-2000,0)226,566.42

49
Practice Problem 2

• Assume you want to have 1,000,000 for retirement
  at age 65. If you deposit the same amount each
  year and are 20 years old today how much will you
  need to deposit each year if you earn 9?
• 1,000,000 PMT(FVIFA45,9)
• 1,000,000 PMT(525.8587345)
• 1,901.6514
• What if you wait until you are 30 to start
  saving?
• 1,000,000 PMT(FVIFA35,9)
• PMT 4,635.83

50
Present Value of an Annuity

• The PV of the annuity is the sum of the PV of
  each of its payments
• Time 0 1 2 3 4
• 100 100 100 100

100/(1.06)194.3396
100/(1.06)288.9996
100/(1.06)383.9619
100/(1.06)479.2094
PV 346.5105
51
PV of An Annuity

• This could also be written
• PV100/(1.06)1100/(1.06)2100/(1.06)3100/(1.
  06)4
• PV1001/(1.06)11/(1.06)21/(1.06)31/(1.06)4
  
• or for any i, payment, and t

52
PVIF of an Annuity PVIFAr,t

• Just like for the PV of a single sum there is a
  present value interest factor of an annuity

This is the PVIFAi,n
PVannuityPMT(PVIFAi,n)
53
PVIFA

• The PVIFA can be approximated by

54
Calculation Methods

• Tables - Look up the PVIFA
• PVIFA6,4 3.465105 FV 100(3.465105)
  346.5105
• Regular calculator -Approximate FVIFA
• PVIFA (1/i)-1/i(1i)n FV 100(3.465105)
  346.5105
• Financial Calculator
• 4 N 6 I 0 FV -100 PMT PV 346.5105
• Spreadsheet
• Excel command PV(rate,nper,pmt,fv,type)
• Excel command PV(.06,4,100,0,0)346.5105

55
Example Solving for the Required Annuity Payment

• Your grandfather has retired, he currently has
  2,000,000 saved to finance his retirement. How
  much could he spend each of the next 20 years if
  his deposits earn 7 annually?
• 2,000,000 PMT(PVIFA20,7)
• 2,000,000 PMT(10.594)
• 188,785.85

56
Annuity Due

• The payment comes at the beginning of the period
  instead of the end of the period.
• Time 0 1 2 3 4
• CFs Annuity 100 100 100 100
• CFs Annuity Due 100 100 100 100
• How does this change the calculation methods?

57
Future Value of an Annuity Due

• The FV of the annuity is the sum of the FV of
  each of its payments. Assume 6 a year
• Time 0 1 2 3 4
• 100 100 100 100 FV of CF

100(1.06)1106.00
100(1.06)2112.36
100(1.06)3119.10
100(1.06)4126.25
FV 463.7093
58
FV of Annuity Due

• Compare the annuity due to a regular annuity with
  the same number of payments and interest rate.
• There is one more period of compounding for each
  payment, Therefore
• FVAnnuity Due FVAnnuity(1i)

59
Present Value of an Annuity Due

• The PV of the annuity due is the sum of the PV of
  each of its payments
• Time 0 1 2 3 4
• 100 100 100 100

100/(1.06)0100
100/(1.06)194.3396
100/(1.06)288.9996
100/(1.06)383.9619
PV 367.3011
60
PV of Annuity Due

• PVAnnuity Due There is one less period of
  discounting for each payment, Therefore
• PVAnnuity Due PVAnnuity(1i)

61
Which would you Choose?

• On December 31, 2003 Norman and DeAnna Shue of
  Columbia, South Carolina had reason to celebrate
  the coming new year after winning the Powerball
  Lottery. They had 2 options.

110 Million Paid in 30 yearly payments
of 3,666,666
60 Million
62
So what option should the Shue Family choose?

• Lets assume their local banker tells them they
  can earn 3 interest each year on a savings
  account. Using that as the interest rate what is
  the PV of the 30 payments?

63
Present Value of an Annuity Due

• The PV of the annuity is the sum of the PV of
  each of its payments
• Time 0 1 2
  3 29
• 3.6M 3.6M 3.6M 3.6M 3.6M

3.6M/(1.03)03.6M
3.6M/(1.03)13.559M
3.6M/(1.03)23.456M
3.6M/(1.03)33.355M
3.6M/(1.03)291.555M
PV 74,024,333
64
Wrong Choice?

• It would cost 74,024,333 to generate the same
  annuity payments each year, the Shues took the
  60 Million instead of the 30 payments, did they
  made a mistake?
• Not necessarily, it depends upon the interest
  rate used to find the PV.
• The rate should be based upon the risk associated
  with the investment. What if we used 6 instead?

65
Present Value of an Annuity

• The PV of the annuity is the sum of the PV of
  each of its payments
• Time 0 1 2
  3 29
• 3.6M 3.6M 3.6M 3.6M 3.6M

3.6M/(1.06)03.6M
3.6M/(1.06)13.459M
3.6M/(1.06)23.263M
3.6M/(1.06)33.078M
3.6M/(1.06)29676,708
PV 53,499,310
66
What is the right rate?

• Remember the correct rate is based upon the
  opportunity cost.
• The Lottery invests the cash payout (the amount
  of cash they actually have) in US Treasury
  securities to generate the annuity since they are
  assumed to be free of default.
• In this case a rate of 4.87 would make the
  present value of the securities equal to 60
  Million (20 year Treasury bonds at the time of
  the winnings yielded 5.02)

67
Intuition

• Over the last 50 years the SP 500 stock index as
  averaged over 9 each year, the PV of the 30
  payments at 9 is 41,060,370
• If you can guarantee a 9 return you could buy an
  annuity that made 30 equal payments of
  3.6Million for 41,060,370 and used the rest of
  the 60 million for something else.

68
Perpetuity

• A perpetuity is a constant cash flow that is
  received forever.
• The PV of a perpetuity would be

69
Perpetuity

• However the formula can be simplified

70
Amortization of a Loan

• You want to borrow 1,000 and pay it off over
  three years. Assume that you are charged 6 each
  year. How much will your payment be?
• 1,000 PV PMT ????
• 1,000 PMT (PVIFA6,3)
• 1,000 PMT(2.67)
• PMT 374.11

71
Amortization

• You pay a total of 374.11(3) 1,122.33
• A portion of each payment represents interest
  charges, the portion of the payment that is
  interest changes with each payment
• You can find the amount of interest by
  multiplying the balance at the beginning of the
  period by the interest rate.
• At the beginning of the loan, the balance is
  1,000 so there is 1,000(.06) 60 in interest.

72
Amortization

• The remainder of the payment pays off principal.
• 374.11 - 60314.11
• The remaining principal at the end of the period
  will then be
• 1,000 314.11 685.89
• The process then repeats itself every period
  until the original balance of the loan is paid
  off.

73
Amortization

• Beginning
  Ending
• Year Balance Payment Interest Principal
  Balance

1,000
374.11
60.00
314.11
685.89
1
685.89
352.93
374.11
41.15
332.96
2
374.11
21.18
0.00
3
352.93
352.93
74
Credit Card Debt

• Assume that you currently have a 5,183.66
  balance on your credit card, and it charges you
  18 interest every year (1.5 in interest each
  month).
• The Credit Card company require you to make a
  minimum monthly payment of 80 each month, how
  long do you think it would take to pay off the
  balance?

75
Credit Card Problem

• Your PV is 5,183.66
• You pay 80 each month and have a monthly
  interest rate of 1.5.
• You are solving for the number of periods it
  would take to pay off the debt, (in other words
  how many months of paying 80 each month has a PV
  of 5,183.66

76
Amortization Credit Card Debt

• Beginning
  Ending
• Month Balance Payment Interest Principal
  Balance

1
5,183.66
5,181.41
80
77.75
2.25
2
5,181.41
80
77.72
2.28
5,179.13
78.81
240
78.86
80
1.19
0.05
77
Uneven Cash Flow Streams

• What if you receive a stream of payments that are
  not constant? For example
• Time 0 1 2 3 4
• 100 100 200 200 FV of CF
• 200(1.06)0200.00
  200(1.06)1212.00 100(1.06)2112.36 1
  00(1.06)3119.10 FV 643.4616

78
FV of An Uneven CF Stream

• The FV is calculated the same way as we did for
  an annuity, however we cannot factor out the
  payment since it differs for each period.

79
PV of an Uneven CF Streams

• Similar to the FV of a series of uneven cash
  flows, the PV is the sum of the PV of each cash
  flow. Again this is the same as the first step
  in calculating the PV of an annuity the final
  formula is therefore

80
A Second Example

• Ivan Pudge Rodriquez signed a contract reported
  to be worth 40 Million to play baseball over the
  next four years for the Detroit Tigers.
• The contract pays Pudge 7M this year, 8 M next
  year, 11M in each of the following years plus
  3M extra the last year if the team does not
  retain him for another year. What is the PV of
  his contract?

81
PV of Playing Baseball

• Given an interest rate of 5 Pudges contract is
  only worth 34.9 Million
• With an interest rate of 10 Pudges contract is
  only worth 30 Million
• Which is the best way to value the contract?

82
Quick Review

• FV of a Single Sum FV PV(1i)n
• PV of a Single Sum PV FV/(1i)n
• FV and PV of annuities and uneven cash flows are
  just repeated applications of the above two
  equations

83
Semiannual Compounding

• Often interest compounds at a different rate than
  the periodic rate.
• For example
• 6 yearly compounded semiannual
• This implies that you receive 3 interest each
  six months
• This increases the FV compared to just 6 yearly

84
Semiannual CompoundingAn Example

• You deposit 100 in an account that pays a 6
  annual rate (the periodic rate) and interest
  compounds semiannually
• Time 0 1/2 1 3 3
• -100 106.09
• FV100(1.03)(1.03)100(1.03)2106.09

85
Effective Annual Rate

• The effective Annual Rate is the annual rate that
  would provide the same annual return as the more
  often compounding
• EAR (1inom/m)m-1
• m of times compounding per period
• Our example
• EAR (1.06/2)2-11.032-1.0609

86
Inflation

• We have ignored the impact of inflation
• It is possible to adjust the interest rate for
  the impact of inflation
• Assume you have 100 today and after investing it
  for one year you have 116.60.
• What return did you receive?
• 16.6

87
Inflation

• Assume that inflation was 6 over the same time
  as your investment,
• How much did your purchasing power increase?
• (1r)(106) 116.6
• r .10 10

88
Real Interest Rate

• Since your purchasing power did not change your
  real return was zero (therefore the real rate of
  interest is zero)

89
Purchasing Power Example

• Jared eats a 5 subway sandwich for lunch every
  day, he has budgeted 100 each month (100/5 20
  sandwiches).
• If he puts 100 away to spend in one year in an
  account earning 16.6 and the price of sandwiches
  increases by 5, how many sandwiches can he buy
  each month in one year?
• 116.6/5.25 22.21 vs.
• 116.6 /5 23.32 without the price increase

90
Generally

91
The Fisher Effect

• Let R The nominal rate of interest
• r the real rate of interest
• h the inflation rate
• The Fisher Effect States
• 1R (1r)(1h)
• Or
• R r h (rh)
• Which interest rate is more important to
  investors?

92
For Next Time

• Try the practice problems let me know if you
  would like to see any of them in class.