Handout 1

1
Handout 1

• (Chapter 4)

2
Time Value of Money

• When you examine the cost and benefit of an
  investment project, you will need to compare the
  money you pay today and money you would receive
  in the future. For example, you would need to
  compare the money you would spend today to build
  a factory, and money that will be generated by
  this investment in the future.
• First, we will learn the method of comparing the
  value of money at different points in time.
• See the next slide

3
Example

• Suppose you sell a piece of raw land. You
  received offers from two potential buyers
  Buyer-A offers to pay 10,000 today. Buyer-B
  offers to pay 11,424 a year from today.
• Which offer has more value to you? There are
  two methods we can use to answer to this
  question. (1) Future Value and (2) Present Value.

4
Example (Contd)

• First, we use the concept of future value to
  compare the values of the two offers.
• Future value gives you the method to convert
  the value of the money you have today into the
  value of the money in the future. (For example,
  you can convert the value of the money today into
  the value of the money one year from today. )
• Before seeing the computational method, it is
  informative to think why the value of money you
  have today should be different from the money you
  will receive in the future. ( See
  next slide)

5
Example (Contd)

• The value of the money you have today is
  different from the value of money you will
  receive in the future (for example one year from
  today) because the money you have today can be
  deposited in the bank account and you can earn
  additional cash in the form of interest.
• Therefore, the value of the money today, and the
  value of the money in the future is different by
  the amount equal to the interest.

6
Example (contd)

• Suppose that interest rate is 12. If you deposit
  10,000 in the bank account today, you will
  receive the interest of the amount equal to
  10,0000.121,200.
• This means that, 10,000 today will grow in one
  year to
• 10,000(10,0000.12)10,0001.1211,200
• 11,200 is the one-year future value of 10,000
  you have today. In other words, 10,000 you will
  receive today, and 11,200 that you will receive
  a year from today have the same value. This is
  the basic idea behind the concept of future
  value. Using the concept of future value of
  money, we can compare the value of two offers.

7
Example (Answer 1)

• Buyer A offers to pay 10,000 today, and Buyer B
  offers to pay 11,424 a year from today. Suppose
  that interest rate you can earn is 12.
• Then the one-year future value of the buyer As
  offer is 10,0001.1211,200. This is smaller
  than the buyer Bs offer. Therefore, the value of
  buyer Bs offer is greater than the value of
  Buyer As offer.
• Next slide summarizes the computation of future
  value.

8
Future value One period Case

• Suppose that you have C0 today. The interest
  rate is r. Then, (one period) future value of C0
  is given by
• FV C0(1 r)
• Note The subscript for C0 means that this is
  the money you have at time zero (today).

9
Example (contd)

• The same question can be answered by using the
  concept of Present Value. The present value
  method gives you the tool to convert the money
  you will receive in the future into the value of
  todays money.
• Remember, the Buyer B offers to pay you 11,424
  one year from today. How can we convert this
  amount into todays value? ( see next slide)

10
Example (Contd)

• We can convert 11,424 that you will receive a
  year from now to todays value by asking the
  following question.
• How much money would you have to put in the
  account so that you will receive 11,424 a year
  from today?
• To answer this question, let PV be the value you
  have to put in the account. Then PV can be
  computed by solving the following equation.
  See next slide.

11

• Assuming that the interest rate you can earn is
  12, the equation you will solve is given by
• PV1.12 11,424
• Therefore
• PV11,424/1.1210,200
• This means that 11,424 you will receive a year
  from today is equivalent to 10,200 you receive
  today.
• Since 10,200 is still greater than the Buyer As
  offer (10,000 today), the value of Buyer Bs
  offer is greater than Buyer As offer.
• Next slide summarizes the computation of present
  value.

12
Present Value One period Case

• Suppose you will receive C1 one year from today.
  Let r be the interest rate that you can earn.
  Then the present value of C1 is given by

Note The subscript for C1 refers to the fact
that you will receive the amount one year from
today.
13
Discount rate

• The interest rate, r, that you use for the
  computation of present value is called the
  discount rate.
• The interest rate r is the amount you can receive
  by depositing (or lending) the money. If you see
  the interest rate from the borrowers side,
  interest rate is the cost of borrowing the money.
  In this sense, discount rate can be thought of as
  the Cost of Capital.

14
Discount rate (contd)

• As can be seen from the formula for the present
  value, discount rate greatly influence the
  computation of the present value.
• Appropriate choice of the discount rate will be
  discussed in chapter 12 of our textbook. Till
  then, for simplicity, we will assume that
  discount rate is given.

15
Future value-Multiperiod Case-

• Previously, we talked about the computation of
  future value for one period only. In the
  following slides, we consider the computation of
  future value for multiperiod case.

16
Future value-Multiperiod Case- Example

• Suppose that you have 100. What will be the
  value of this amount evaluated at two years from
  today? Suppose that the annual interest rate is
  9. See nest slide

17
Future value-Multiperiod Case- Example

• First, notice that you will receive the first
  interest payment one year from now.
• You re-deposit the initial 100 plus the interest
  earned during the first year.
• Therefore, two years from today, the initial 100
  will grow to the amount given by the circled
  amount in the table below. This is the two period
  future value of 100.

(1001.09)1.09 118.81
1001.09109
Future value evaluated at two periods from today
18
Future value-Multiperiod Case- Example

• Therefore, if you have 100 today and if the
  interest rate is 9, the future value of this
  amount evaluated at 2 years from today is given
  by
• FV1001.091.09118.81
• We will see the formula for computing future
  value for longer periods. However, before doing
  so, we will take a closer look at the example.

19
Future Value-Compounding-

• In the previous example, the initial amount of
  money 100 was left in the account for 2 years.
  More specifically, the initial money was left in
  the account for one year, then in the second
  year, initial money and the interest earned
  during the first year was re-deposited in the
  account.
• Such a process of leaving the money in the
  capital market for several periods is called
  Compounding

20
Simple interest and interest on interest

• When a lender keep the money for several periods,
  interest will earn interest.
• To see this, consider you deposit 1 today. Two
  years from now, this amount will grow to
• 1(1r)(1r) 1 2r r2
• Continues to next slide

21
Simple interest and interest on interest (Contd)

• 1(1r)(1r) 1 2r r2

Simple Interest This is the amount you would
earn by investing 1 this year, and 1 next year.
Interest on Interest This is the amount you
will earn in the second period by re-investing
the interest earning that you have received at
the end of first year (r).
22
Simple interest and interest on interest (Contd)

• Therefore, when money is invest at compound
  interest, each interest payment is re-invested.
• In other words, money makes money and the money
  that the money makes makes more money.

23
Future Value Multiperiod Case-Formula for
computation-

• We discussed how to compute future value for two
  period case. This can be generalized for longer
  periods.
• Suppose you have C0 today. Let r be the interest
  rate. The future value of the money evaluated at
  T years from now is given by
• FVC0(1r)T

24
Exercise 1

• Suh-Pyng Ku puts 500 in a saving account. The
  account earns 7 percent, compounded annually. How
  much will Ms Ku have at the end of three year?

Answer 500(10.07)3612.52
25
Power of compounding

• Notice that in the Suh-Pyng Ku example, the
  amount she receives in 3 years (612.52), is
  considerably higher than the amount she would
  receive if she had not re-invested the interest.
• 612.52 gt 500 35000.07 605

Amount she earned by investing at compound
interest.
Amount she would receive if she had not
re-invested interest. (This is the amount you
would receive if she only earned the simple
interest
26
Power of compounding (Contd)

• For two period case, the power of compounding
  seems to be trivial. However, compounding will
  have a significant effect for longer periods.
• Exercise Open the Excel Sheet Compound
  interest rate and simple interest rate. Use
  Suh-Pyng Ku example, compute the future values
  for each period up to 30 years from today. Also
  compute the value you would have if you do not
  re-invest the interest (simple interest case).
  Plot both case.

27
Answer
For longer period, compounding has a significant
effect. As can be seen from the graph future
value (which is compounded) have much greater
values than the simple interest case.
28
Exercise 2

• The following exercise see the future value from
  a slightly different point of view.
• Suppose you won 10,000 in the lottery today.
  You want to buy a car in 5 years. You estimate
  that the car will cost 16,105 at that time. What
  interest rate must you earn in order to afford
  this car?

29
Present value-Multiperiod Case-

• We have talked about the present value of cash
  that you receive one year from today only.
• We can compute the present value for longer
  periods.
• Suppose you will receive CT T years from today.
  What is the present value of CT?

30
Present value-Multiperiod Case- (Contd)

• We can answer the question by asking the
  following question.
• How much money you have to put in the
  account so that you receive CT T years from
  today
• This question can be answered by solving the
  following equation.
• See next slide

31
Present value-Multiperiod Case- (Contd)

• PV(1r)TCT
• Therefore
• PVCT/(1r)T

32
Present Value Multiperiod Case-Formula for
Computation-

• Suppose that you will receive CT T years from
  today. If the discount rate is r, the present
  value of this cash is given by

33
Exercise 3

• Bernard Dumas will receive 10,000 three years
  from now. Bernard can earn 8 on his investment,
  and so the appropriate discount rate is 8. What
  is the present value of the cash flow?

34
Present Value Multiple Cash flow

• Suppose you won a lottery of 5000 in which you
  will receive the equal installment over the 5
  years (1000 each for five years). The first
  payment starts one year from today. What is the
  present value of the lottery? Assume that
  discount rate is 7. Use the Excel Sheet Present
  Value of Lottery for computation.

35
Net Present Value of an investment

• Net present value of an investment is the sum of
  the present values of all of the cash flows minus
  the present value of the cost.
• See the example in the next slide

36
Net present value of an investment. Example

• Royal Petroleum is considering investing in a
  project to develop an oil well in Africa. Since
  it takes some time to build the oil well, if they
  decide to invest in this project, the first cash
  flow will start one year from today. Once the
  well is build, the company expect to have a
  constant cash flow of 10 billion dollars every
  year for 20 years. At the end of 20th year, the
  company will sell the well, which is expected to
  generate 20 billion cash. The initial cost of
  building the oil well is 100 billion. If the
  company decide to take in the project, the
  company has to pay the cost up front.
• What is the Net Present Value of this investment
  Project? Assume the discount rate of 10. Use the
  excel sheet Royal Petroleum to answer the
  question.

37
A Note on the Computation of NPV using Excel

• The Excel Function NPV(discount rate, )
  will automatically compute the Net Present Value.
• A note should be made. Excel Function start
  discounting from the first cell. Therefore, the
  if the first cell is (?C0), it will discount (?
  C0) as well.
• To avoid discounting the initial cost, you should
  compute the NPV in the following way.
• NPVB8 NPV(0.08, B9B28)

38
Algebraic Formula of net present value of an
investment

• Suppose you invest C0 in a project today. C0
  represent the initial cost. This investment
  generate cash flow of C1 one year from now, C2
  two years from now,, and CT T years from now.
  Then the discounted cash flow of this investment
  project can be written in the table below.

CT/(1r)T
? C0
C1/(1r)
C2/(1r)2
39
Algebraic Formula of net present value of an
investment

• Therefore, the Net Present value of an investment
  is given by

40
Compounding periods

• So far we have assumed that compounding occurs
  yearly. Sometimes, compounding may occur more
  frequently.
• Suppose a bank pays 10 interest rate compounded
  semiannually.
• This means (1) a 1000 worth deposit in the
  bank would be worth 10001.051,050 after 6
  months. And (2) this amount will be reinvested
  for the second half of the year, and at the end
  of the year, you will have 10001.051.051,102.5
  0
• Therefore, you end up earning more tha 10 at the
  end of the year.
• The 10 interest rate in this example is called
  the Stated Annual Interest Rate.

41
Example

• If a bank pays 10 of interest compounded
  quarterly, how much a 1000 deposit worth one
  year from now? This means that you would earn
  2.5 interest for each quarter. So the future
  value will look like in the following table.

• Clearly, the value of the 1000 at the end of the
  year (i.e., end of the forth quarter) is given by
  formula below

42
Compounding m times a year

• We have considered the examples of the semiannual
  compounding and quarterly compounding. We can
  generalize this idea for the case where
  compounding occurs m times a year.
• Suppose that you invest C0 in an account that
  compounds m times a year with the stated annual
  interest rate of r.
• What would be the value of C0 at the end of the
  year?
• See next slide

43
Compounding m times a year (Contd)

• For this problem, you can think of one year as
  having m separate periods, each period having an
  interest rate of r/m.
• Then the problem is the same as computing the
  future value of C0 deposited over m periods with
  interest equal to r/m
• Therefore, the value of C0 at the end of the year
  is given by the formula in the next slide

44
Compounding m times a year-Algebraic form-

• Let r be the stated annual interest rate. Let
  C0 be the initial deposit (investment). Then,
  compounding the investment m times a year
  provides end-of-year wealth of

45
Exercise

• What is the future value of 1000, a year from
  now, in an account with a stated annual rate of 8
  percent if
• a) Compounded annually
• b) Compounded semiannually
• c) Compounded monthly
• d) Compounded Daily

46
Stated annual interest rate and effective annual
interest rate.

• Consider a bank account with stated annual
  interest rate of 10 which is compounded
  semiannually. If you deposit 100, this will grow
  at the end of the year to

• Notice that a different bank account with an
  interest rate of 10.25 (compounded annually)
  would give you the same end-of-the-year wealth.
  Therefore, 10.25 is called the effective annual
  interest rate. We can generalize this idea to the
  case where compound occurs m times a year.
  See next slide

47
Formula for effective annual interest rate

• Suppose a bank account with the stated annual
  interest of r, compounded m times a year. Then,
  the effective annual interest rate is given by
• Effective annual interest rate (1r/m)m -1
• This means that, a bank account with an
  interest rate equal to (1r/m)m-1 compounded
  annually would give you exactly the same
  end-of-the-year wealth as the bank account with
  stated annual interest r compounded m times a
  year.

48
Compounding periods -Multiyear case-

• Previously, we have calculated the future value,
  compounded m-times a year over one year.
• What would be the future value of an deposit in
  an bank account that compounds m times a year if
  you keep the deposit for more than one year (for
  example, T years)?

49
Compounding periods -Multiyear case- (Contd)

• To think of this problem, you can again consider
  one year as having m separate sub-periods, each
  having interest rate of r/m. Therefore, if you
  keep the deposit for T years, there will be a
  total of MT sub-periods. This is shown in the
  table below.

• Therefore, the future value of the initial
  deposit C0 evaluated at T years from now is given
  by the formula in the next slide.

50
Compounding Periods Multiyear case-
Computational formula -

• Let C0 be the initial amount of deposit
  (investment), and r be the stated annual interest
  rate. Compounding the investment m times a year
  for T years provides for future value of wealth

51
Exercise

• EX1
• Suppose you deposit 50 in a bank account that
  compounds semiannually. The stated annual
  interest rate is 0.12. If you leave the amount in
  the account for 3 years, what value the initial
  deposit grows to?
• EX2
• Open Excel Sheet Compounding Period Exercise.
  Compute the future value of the initial deposit
  of 50 evaluated 3 years from now for different
  number of compoundings (i.e., different number of
  m). Graph the relationship between the future
  value and number of compounding.

52
Answer Ex2
53
Continuous compounding (Formula)

• What if we ask the bank to compound continuously
  (i.e., to increase the number of compounding as
  much as possible)?
• It is known that C0 (1r/m)rT C0erT as m
  increases to infinity. (The number e 2.718.)
• Let r be the stated annual interest rate. If the
  interest rate is compounded continuously, the
  future value of C0 at the end of T years is
  given by
• FV C0erT
• where 2.718.

54
Exercise

• Compute the future value of 50 continuously
  compounded for 3 years at a stated annual
  interest rate of 12.

55
Present values of common types of cash flow
streams

• We consider the following four classes of
  cash flow streams
• Perpetuity
• A constant stream of cash flows that lasts
  forever.
• Growing perpetuity
• A stream of cash flows that grows at a constant
  rate forever.
• Annuity
• A level stream of cash flow that lasts for a
  fixed number of periods.
• Growing annuity
• A stream of cash flows that grows at a constant
  rate for a fixed number of periods.

56
Perpetuity

• A constant stream of cash flows that lasts
  forever.

The formula for the present value of a perpetuity
is
57
Perpetuity Example

• What is the value of a British consol that
  promises to pay 15 each year forever. The
  interest rate is 10-percent.

58
Growing perpetuity

• Imagine an apartment building where cash flow to
  the landlord next year is 1000. Suppose that the
  cash flow rises at 5 percent per year. If the
  rise will continue indefinitely, how can we
  compute the present value of this cash flow? The
  appropriate interest rate is 0.08. To answer this
  question, we will use the growing perpetuity
  formula.

59
Growing perpetuity (Contd)

• Let us generalize the previous example. Suppose
  you have an asset that gives you the first cash
  flow of C a year from now. After the first year,
  the cash flow grows at a constant rate g.
• Let the appropriate interest be r.
• This cash flow is illustrated in the figure in
  the next slide.

60
Growing Perpetuity

The formula for the present value of a growing
perpetuity is
61
Growing Perpetuity Example

• Consider a security that pays dividend forever
  starting from next year. The first dividend
  payment is 1.30 and dividends are expected to
  grow at 5 forever.
• If the discount rate is 10, what is the value of
  this promised dividend stream?

62
Annuity

• Annuity is a level stream of regular payments
  that lasts for a fixed number of periods.
• Examples Mortgage payments
• Let T be the number of period. Then annuity can
  be represented in the figure in the next slide.

63
Annuity Formula
The formula for the present value of an annuity
is
Annuity Factor ATr
64
Example

• Should you buy an asset that will generate income
  of 1,200 at the end of each year for eight
  years? The price of the asset is 6,200 and the
  annual interest rate is 10.

65
Delayed Annuity

• When we work with annuity or perpetuities, we
  have to get the timing right. Consider the
  following example.
• Exercise
• Example Danielle Caravello will receive a
  four-year annuity of 500, beginning at date 6.
  If the interest rate is 10, what is the present
  value of her annuity?
• See nest slide

66
Delayed annuity Exercise (Contd)

• We can answer to the question in the previous
  slide either by (1) inputting the cash flow in
  the excel, or by (2) using the annuity formula in
  a smart way. We would like to be able to use both
  method (1) and (2).
• First, find the present value by using method
  (1). Open Excel Sheet Delayed annuity exercise.
  Find the present value of the annuity.
• Steps for the method (2) are described in the
  next slide.

67
Delayed annuity Exercise (contd)

• We can solve the exercise by using annuity
  formula. There are two steps to do this.
• Step one
• Compute the present discount value of annuity
  using the PV formula for annuity
• The resulting value will give you the present
  value of the annuity evaluated at date 5. Now,
  compute this value.
• Note It is easy to be mistaken that this
  value is the PV evaluated at date 6. However, we
  can see that this is the PV evaluated at period 5
  by remember that annuity formula for annuity
  starting from period 1 gives the present value at
  date 0.

68
Delayed annuity Exercise (contd)

• Step two
• Since the value you obtained in the Step one
  gives the present value of the annuity evaluated
  at date 5, discount the present value of the
  annuity back to date 0.
• Now, compute the present value of the delayed
  annuity. Check to see if your result coincide
  with the result you obtained using Excel.

69
Annuity in advance

• The annuity formula in slide 37 assume that the
  first payment begins a full period hence. What
  happens if the annuity begins today?
• Example
• Suppose you won a lottery that pays you
  50,000 a year for 20 years. If the first payment
  occurs immediately, what will be the PV of this
  lottery? Interest rate is 8.

70
Exercise-Equating PV of two annuities-

• Consider the following example.
• Harold and Helen Nash are saving for the
  college education of their newborn daughter
  Susan. The Nashes estimate that college expenses
  will run 30,000 per year when their daughter
  reaches college in 18 years. The interest rate in
  the next few decades will be 14. How much money
  must they deposit in the bank each year so that
  their daughter will be fully supported through
  four years of education.
• To simplify the computation, assume that the
  baby is born today, and the parents make the
  first annual tuition payment on her 18th
  birthday. The parents will make equal bank
  deposits on each of her first 17th birthdays but
  no deposit at date 0.

71
Growing annuity

• In the previous slides regarding annuity, we
  considered the case where you receive constant
  amount over time.
• Often cash flows increase over time. For example,
  rent of an apartment often increases.
• Growing annuity is the annuity that grows at a
  constant rate over time.

72
Growing annuity (contd)

• Consider the following setting. You receive the
  annuity starting from the date 1. The initial
  amount is C. Afterward, the cash flow will
  increase at the rate of g. The cash flow
  continues for T years.
• This situation is described in the figure in the
  next slide.

73
Growing Annuity Formula
The formula for the present value of a growing
annuity
74
PV of Growing Annuity
You are evaluating an income property that is
providing increasing rents. Net rent is received
at the end of each year. The first year's rent is
expected to be 8,500 and rent is expected to
increase 7 each year. Each payment occur at the
end of the year. What is the present value of the
estimated income stream over the first 5 years if
the discount rate is 12?
34,706.26