Principles of Finance - Time Value of Money.pdf

Principles of Finance - Time Value of Money.pdf

1. Time Value of Money
Bodie, Chapters4 & 5,8 & 9
Mishkin, Chapter4
Cornett, Chapters4 & 5
Time Value of Money 1

2. Content
The Time Value of Money
1. Calculate future values and understand compounding
2. Calculate present values and understanding discounting
3. Apply the time value of money equation using formula, calculator
and spreadsheet.
4. Determine the future value of multiple cash flows
5. Determine the future value of an annuity
6. Build and analyze amortization schedules
7. Compute the effective annual rate on a loan and investment
2Time Value of Money

3. Financial Decisions
•Costs and benefits being spread out over time
•The values of sums of money at different dates
•The same amounts of money at different dates
have different values.
•Virtually every decision involves TIME and
UNCERTAINTY
3Time Value of Money

4. Time Value of Money
•The time value of money refers to a dollar in
hand today being worth more than a dollar
received in the future, because you can invest
today’s dollar in an interest-bearing account
that grows in value over time.
Time Value of Money 4

5. Time Value of Money
•Two forms of time value of money calculations are
commonly used in finance for security valuation
purposes:
1.Value of a lump sum
•A lump sum payment is a single cash payment received at the
beginning or end of some investment horizon
2.Value of annuity payments
•Annuity payments are a series of equal cash flows received at
fixed intervals over the entire investment horizon
© 2022 McGraw-Hill Education.

6. Value of a lump sum
Future Value
•We refer to translating a value today into a
value in the future as compounding, whereas
discounting is translating a future value into the
present.
•Future value = Present value + Interest
Time Value of Money 6

7. Moving money through time
Future
Value
Present
Value
Time Value of Money 7
Compounding
Discounting
→ finding the equivalent value to money at different points in time

8. Lump Sum Valuation
•Present Value of a Lump Sum
•Future Value of a Lump Sump
© 2022 McGraw-Hill Education.

9. i, r, k: What is the difference?
•From now on, if you look at other books and
our documents, you will see other notations for
the interest rate, including r for return or the
required rate of return, and k for the cost of
capital.
•These really represent the same concept: the
time value of money.
Time Value of Money 9

10. Valuing a lump sum
Future value
•Suppose you deposit $100 in an account that
earns 5% interest per year. If you do not make
any withdrawals, how much will you have in the
account at the end of 2 years, 10 years and 30
years?
Time Value of Money 10

11. Methods of Solving Future Value
•Method 1: The equation
•FV = 1000 x1+0.05
30
•Method 2: The spreadsheet
=
11Time Value of Money

12. Using computer
Investing $ today
• Invest (Expense) $
today in present to
earn greater return in
the future.
• Earn interest
(revenue), plus
principal
•PV = (-)
•FV = +
Borrowing $ today
•Borrow (Inflow) $ today
in present to use now,
then repay with interest
in the future.
•Pay interest (expense),
plus principal
• PV = +
•FV = (-)
12Time Value of Money

13. Loan Payment
•If you borrow $20,000 and the interest on the
loan is 9% per year, all payable at the end of
the loan, what is the amount that you must
repay if the loan is for four years?
13Time Value of Money

14. Future value and Inflation
•You suppose to buy a house five years from now and
want to invest enough money now to pay for it. You
have in mind a house now costs $100,000 and the
interest rate you can earn for your money is 8% per
year (as deposit IR)
•How much money should you invest now? If the
inflation rate is 5% per year.
Time Value of Money 14

15. Annual percentage rate (APR)
•When an interest rate is stated in terms of a
rate per year, but interest is compounded more
frequently than once per year, the stated
annual rate is referred to as the annual
percentage rate (APR), but the actual
calculation requires using the rate per
compound period and the number of
compound periods.
Time Value of Money 15

16. Savings deposit
•A customer wants to deposit one billion VND
in a TCB saving deposit with _____
1.an APR of 5.5%, monthly compounding. How
much is his total savings account after three
years?
2.an APR of 5.5%, quarterly compounding.
How much is his total savings account after
three years?
Time Value of Money 16

17. Valuing a lump sum
What is the effective annual rate of a mortgage rate
that is advertised at 8% (APR) over the next twenty
years and paid with quarterly payment?
a.8.3%
b.8.29%
c.1.006%
d.8.24%
e.None of the above
Time Value of Money 17

18. Compounding a savings deposit
•Suppose you have a saving deposit
of $1000, paying 8.4%, quarterly
compounding. How much will you
have in the account in 5 years.
Time Value of Money 18

19. Revenue estimation
•Consider the Walt Disney Company. At the end
of fiscal year 2023, analysts expected Disney’s
earnings to grow at a rate of 13.9% per year, in
the long-term.
• If Disney’s earnings for fiscal year 2018 were
$2,750 per share and if we concur with the
analysts, we can estimate the earnings per
share for fiscal years into the future.
•How much is the estimated earnings per share
for 2030?
Time Value of Money 19

20. Savings
Suppose you receive $1000 in an account
paying 6.1% annually compounding
interest for 5 years. How much is the
present value of the deposit?
Time Value of Money 20

21. Don’t Discount Discounting
•We refer to translating a value back in time as
discounting, which requires determining what a
future amount or cash flow is worth today.
•Discounting is used in valuation because we
often want to determine the value today of
some future value or cash flow (e.g., what a
bond is worth today if it promised interest and
principal repayment in the future).
Time Value of Money 21

22. Calculate the Interest
Rate/Yield/Discount rate/Growth rate
•What is the discount rate on my future cash?
•At what rate is my money growing over time?
• ?????? =(
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????????????
)
1/�
-1
22Time Value of Money

23. Interest rate
•If you deposit $35000 in the bank today and
five years will get back $40000, what is your
growth rate? IR=?
23Time Value of Money

24. Calculation of the time period
•The time period is the waiting time for a
present value to mature into a desired value.
•n =
ln(
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????????????
)
ln(1+??????)
24Time Value of Money

25. •Suppose you have a choice of saving $1
million with the following terms, with interest
paid at the end of the deposit:
•5% APR, quarterly interest
•5.5% APR, semi-annual interest
•6% APR, annual interest
Under which deposit terms would you have the
largest payment at the end of three years?
Time Value of Money 25
EAR and Frequency

26. VALUING A
STREAM OF CASH
FLOWS
26Time Value of Money

27. Present Value of Multiple
Payment Streams
•You invest $5000 in bank with APR = 12%,
compounding monthly, beginning two years
from now and $4000 beginning three years
from now, $3000 beginning four years from
now.
• Compute the present value of the amount you
receive after ten years from now.
Time Value of Money 27

28. Future Value of Multiple
Payment Streams
•Suppose you plan to put away some years to
build up a nest egg to use as a down payment
on a house. You start off by putting away
$2,000 today, and over the next three years
you are able to put away $3,000 at the end of
the first year, $4,000 at the end of the second
year, and $5,000 at the end of the third year.
•How much will you have saved by the end of
the 4th year if your investment rate is 5% per
year?
28Time Value of Money

29. Valuation of a Stream of Cash Flow
•You have just graduated and need money to
buy a new car. Your rich Uncle Henry will lend
you the money so long as you agree to pay
him back within four years, and you offer to
pay him the rate of interest that he would
otherwise get by putting his money in a savings
account. Based on your earnings and living
expenses, you think you will be able to pay him
$5000 in one year, and then $8000 each year
for the next three years.
•If Uncle Henry would otherwise earn 6% per
year on his savings, how much can you borrow
from him?
29Time Value of Money

30. Three Rules of Time Travel
30Time Value of Money

31. Annuities and
Perpetuities
31Time Value of Money

32. Annuity
•An annuity is a series of even cash flows.
Because the cash flows are the same amount,
the math is simpler.
•An annuity is a stream of N equal cash flows
paid at regular intervals. Most car loans,
mortgages, and some bonds are annuities.
Time Value of Money 32

33. Annuity Valuation
•Present Value of an Annuity
•Future Value of an Annuity
33© 2022 McGraw-Hill Eduation.

34. Future Value of an annuity stream
•Say you decide to put away $100 at the end of
every month for the next five year. If you can
earn 4.8% on the account, what is the value of
the account at the end of the five year?
• Unlike the previous problem, you do not put
any money away today.
• The first deposit is at the end of the first year.
34Time Value of Money

35. Back and Forth with Annuities 1
•Kitty and Red put $1,500 into a college fund
every year for their son, Eric, on his birthday,
with the first deposit one year from his birth (at
his very first birthday). The college fund has a
guaranteed annual growth or interest rate of
7%. At his eighteenth birthday, they will pay the
last $1,500 into the fund. How much will be in
the college fund for Eric immediately following
this last payment?
35Time Value of Money

36. Back and Forth with annuities
•Ben and Donna determine that upon
retirement, they will need to withdraw $50,000
annually at the end of each year for the next
thirty years. They know that they can earn 4%
each year on their investment.
•What is the present value of this annuity? In
the other words, how much will Ben and Donna
need in their retirement account (at the
beginning of their retirement) to generate this
future cash flow?
36Time Value of Money

37. Payment (PMT)
•Your biotech firm plans to buy a new DNA
sequencer for $100,000. The seller requires
that you pay 25% of the purchase price as a
down payment, but is willing to finance the
remainder by offering a 48-month loan with
equal monthly payments and an interest rate of
6%.
•What is the monthly loan payment?
37Time Value of Money

38. Personal finance
Three Loan Payment Methods
1. Amortized Loan (Interest and Principal as You
Go)
→ Amortization Schedules
2. Discount Loan (Interest and Principal at the
Maturity of Loan
3. Interest Only Loan
→Interest as You Go
→ Consols (Multiple Interest) ;
→ Coupon Bond (Multiple Interest + Principal at
Maturity of Loan)
38Time Value of Money

39. Amortized Loan
•You take a loan of 2 billions VND for 20 years
with annual payment and the interest rate of
9%.
•How much you annual payment?
•Draw the amortization schedule.
Time Value of Money 39

40. 40
Time Value of Money
Amortization schedule
You take a loan of 2 billions VND for 20
years with the interest rate of 9%.

41. Interest Only Loan
•Borrower will pay multiple equal interest during
the years and principal at maturity.
For instance, a borrowing of 5 billion VND for 5
years, IR= 15%, interest only loan will have five
cashflows which pay annually 750 million VND
at the end of each year and a lump sum of 5
billion at the end of the fifth year.
•If the interest rate for bank savings deposit is
12%, how much is the monthly payment which
is enough to pay for the above debt?
Time Value of Money 41

42. Perpetuities
•Let’s look at still another example. Suppose
you are evaluating an investment that promises
$10 every year forever. This type of cash flow
stream is referred to as a perpetuity.
Time Value of Money 42

43. Perpetuities
•A perpetuity is a stream of equal cash flows
that occur at regular intervals and last forever.
•One example is the British government bond
called a consol (or perpetual bond). Consol
bonds promise the owner a fixed cash flow
every year, forever.
43Time Value of Money

44. Present Value of A Perpetuity
•A perpetuity with payment C and interest rate r:
44Time Value of Money

45. Perpetuities
•You want to endow an annual MBA graduation
party at your alma mater. You want the event to
be a memorable one, so you budget $30,000
per year forever for the party. If the university
earns 8% per year on its investments, and if
the first party is in one year’s time, how much
will you need to donate to endow the party?
45Time Value of Money

46. Financial Math: Consols
•The government of the United Kingdom has
had bonds outstanding that never mature.
These bonds are referred to as consolidated
stock, or consols, that are part of the
government debt.
•The bonds currently pay interest at a rate of
2.5% per year, paid four times a year. What is
the value of one Consolidated Stock if the
appropriate discount rate is 4%?
•Face value = 100
Time Value of Money 46

47. Application: Growing Cash Flows
➢Growing Perpetuity
➢Growing Annuity
47Time Value of Money

48. Growing Perpetuity
•A growing perpetuity is a stream of cash
flows that occur at regular intervals and grow at
a constant rate forever.
48Time Value of Money

49. Present Value of Growing Perpetuity
(Suppose g < r)
Cash flow C,
growing at rate g every period until period n
49Time Value of Money

50. Growing Perpetuity
•You planned to donate money to your alma mater
to fund an annual $30,000 MBA graduation party.
Given an interest rate of 8% per year, the required
donation was the present value of
•………………………………………………today
•Before accepting the money, however, the MBA
student association has asked that you increase
the donation to account for the effect of inflation on
the cost of the party in future years.
•Although $30,000 is adequate for next year’s party,
the students estimate that the party’s cost will rise
by 4% per year thereafter. To satisfy their request,
how much do you need to donate now?
50Time Value of Money

51. Growing Annuity
•A growing annuity is a stream of N growing
cash flows, paid at regular intervals. It is a
growing perpetuity that eventually comes to an
end.
51Time Value of Money

52. Growing Annuity
•Ellen considered saving $10,000 per year for
her retirement. Although $10,000 is the most
she can save in the first year, she expects
her salary to increase each year so that she
will be able to increase her savings by 5%
per year. With this plan, if she earns 10% per
year on her savings, how much will Ellen
have saved at age 65?
52Time Value of Money

53. Interest Rate
53Time Value of Money

54. Annual and Periodic Interest Rate
•Let’s assume that you purchase a CD for
$500 with a promised annual percentage
rate (APR) of 5%.
•The annual percentage rate (APR) is
the yearly rate that you earn by investing
or charge for borrowing.
54Time Value of Money

55. Annual and Periodic Interest Rate
•However the financial institution quotes 5% interest
rate on an annual basis, these institutions in fact often
pay interest quarterly, monthly or even daily.
•The period in which the financial institution applies
interest or the frequency of times at which it adds
interest to an account each year is the compounding
period or compounding periods per year (C/Y)
55Time Value of Money

56. Effective annual rate (EAR)
•EAR = (1+
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56Time Value of Money

57. Financial Math: Financial Leasing
57Time Value of Money

58. PRESENT VALUE
AND THE NPV
DECISION RULE
58Time Value of Money

59. Evaluating long-term cash flows
•One of the most important tools in business
and investing is evaluating an investment that
provides cash flows over a long time period.
•We refer to these types of decisions as capital
budgeting because we use capital—that is,
long-term sources of funds—and are
evaluating what we can spend now to get
these future benefits—hence, the budgeting
part
Time Value of Money 59

60. Different investment:
Thing One and Thing Two
•Using a 10% cost of
capital, we can apply
the time value of
money skills to give
us a better idea of
which is better:
Time Value of Money 60

61. Net Present Value (NPV)
•When we compute the value of cash today, we
refer to it as the present value (PV).
•Similarly, we define the net present value
(NPV) of a project or investment as the
difference between the present value of its
cash inflows and the present value of its cash
outflows:
•NPV = PV(Cash inflows) - PV(Cash outflows)
61Time Value of Money

62. Net Present Value Decision Rule
Time Value of Money 62

63. Net Present Value (NPV)
•As long as the NPV is positive, the
decision increases the value of the
firm and is a good decision
regardless of your current cash
needs or preferences regarding
when to spend the money.
63Time Value of Money

64. The NPV Decision Rule
•When making an investment decision, take the
alternative with the highest NPV.
•Choosing this alternative is equivalent to
receiving its NPV in cash today.
64Time Value of Money

65. NPV
•Problem
Suppose you started a Web site hosting business and then decided to return to
school. Now that you are back in school, you are considering selling the business
within the next year. An investor has offered to buy the business for $200,000
whenever you are ready. If the interest rate is 10%, which of the following three
alternatives is the best choice?
1. Sell the business now.
2. Scale back the business and continue running it while you are in school for one
more year,
and then sell the business (requiring you to spend $30,000 on expenses now, but
generating
$50,000 in profit at the end of the year).
3. Hire someone to manage the business while you are in school for one more year,
and then sell the business (requiring you to spend $50,000 on expenses now, but
generating $100,000 in profit at the end of the year).
65Time Value of Money

66. Choosing among alternative Plans
66Time Value of Money

67. NPV when n → ∞
Time Value of Money 67

68. Internal Rate of Return
•This interest rate is called the
internal rate of return (IRR),
defined as the interest rate that sets
the net present value of the cash
flows equal to zero.
68Time Value of Money

69. Internal Rate of Return
•Suppose you have an investment opportunity
that requires you to put up $50,000 and has
expected cash inflows of $28,809.52 after one
year and $28,809.52 after two years.
•IRR =10%
Time Value of Money 69

70. Internal Rate of Return Decision Rule
•The internal rate of return is a yield—what we
earn, on average, per year.
•How do we use it to decide which investment, if
any, to choose?
•Let’s revisit investments Thing One and Thing
Two and the IRRs that we just calculated for
each. If, for similar risk investments, owners
earn 10% per year, then both Thing One and
Thing Two are attractive investments.
Time Value of Money 70

71. •The decision rule for the internal rate of return
is to invest in an investment if it provides a
return greater than the cost of capital. The cost
of capital, in the context of the IRR, is a hurdle
rate—the minimum acceptable rate of return.
Time Value of Money 71
Internal Rate of Return Decision Rule

72. Time Value of Money 72
Internal Rate of Return Decision Rule

73. •When evaluating mutually exclusive
investments, the one with the highest IRR may
not be the one with the best NPV.
•The IRR may give a different decision than
NPV when evaluating mutually exclusive
investments because of the reinvestment
assumption:
• NPV assumes cash flows reinvested at the
cost of capital.
• IRR assumes cash flows reinvested at the
internal rate of return.
Time Value of Money 73
IRR and Mutually Exclusive Investment

74. IRR and Mutually Exclusive Investment
•What if we were forced to choose between Thing
One and Thing Two because they are mutually
exclusive or there is a limit on how much we can
invest? Thing Two has a higher IRR than Thing
One—so at first glance we might want to accept
Thing Two.
•What about the NPV of these investments? What
does the NPV tell us to do? If we use the higher
IRR, it tells us to go with Thing Two.
•Choosing the investment with the higher net
present value is consistent with maximizing wealth.
Why? Because if the cost of capital is 10%, we
would calculate different NPVs and come to a
different conclusion.
Time Value of Money 74

75. This reinvestment assumption may cause
different decisions in choosing among mutually
exclusive investments when one or more of the
following apply:
• The timing of the cash flows is different among
the investments.
• There are scale differences (that is, very
different cash flow amounts).
• The investments have different useful lives.
Time Value of Money 75
IRR and Mutually Exclusive Investment

76. •With respect to the role of the timing of cash
flows in choosing between two investments:
Thing Two’s cash flows are received sooner
than Thing One’s. Part of the return on either is
from the reinvestment of its cash inflows. And
in the case of Thing Two, there is more return
from the reinvestment of cash inflows.
•The question is “What do you do with the cash
inflows when you get them?” We generally
assume that if you receive cash inflows, you’ll
reinvest those cash flows in other assets.
Time Value of Money 76
IRR and Mutually Exclusive Investment
The timing of the cash flows

77. IRR
•A major publisher has offered to pay Star $1
million upfront if he agrees to write a book
about his experiences. He estimates that it will
take him three years to write the book. The
time that he spends writing will cause him to
forgo alternative sources of income amounting
to $500,000 per year.
•IRR=?
Time Value of Money 77

78. IRR
•Star has informed the publisher that it needs to
sweeten the deal before he will accept it. In
response, the publisher offers to give him a
royalty payment when the book is published in
exchange for taking a smaller upfront payment.
Specifically, Star will receive $1 million when
the book is published and sold four years from
now, together with an upfront payment of
$550,000. Should he accept or reject the new
offer?
Time Value of Money 78

79. Exercise
•You have just taken out a five-year loan from a
bank to buy an engagement ring. The ring costs
$6000. You plan to put down $2000 and borrow
$4000. You will need to make annual payments
of $1250 at the end of each year. Show the
timeline of the loan from your perspective. How
would the timeline differ if you created it from
the bank’s perspective?
Time Value of Money 79

80. Exercise
•You are the lucky winner of the $30 million
state lottery. You can take your prize money
either as
•(a) 30 payments of $1 million per year (starting
today), or
•(b) $14 million paid today.
•If the interest rate is 8%, which option should
you take?
Time Value of Money 80

81. Exercise
Suppose you invest $3500 today and receive
$9500 in five years.
•a. What is the IRR of this opportunity?
•b. Suppose another investment opportunity
also requires $3500 upfront, but pays an equal
amount at the end of each year for the next
five years. If this investment has the same IRR
as the first one, what is the amount you will
receive each year?
Time Value of Money 81