2time value of money.pptx

PowerPoint
to accompany
Chapter 4
NPV and the Time
Value of Money

Copyright © 2011 Pearson Australia (a division of Pearson Australia Group Ltd) –
9781442502000 / Berk/DeMarzo/Harford / Fundamentals of Corporate Finance / 1st edition
Question
 You have $1000 today
 Competitive market annual interest rate
is 10% for the next 20 years.
 How much will you have in 2034 if you
invest $1000 today?

4.1 The Timeline
 A series of cash flows lasting several periods is
defined as a stream of cash flows, which can
be presented on a timeline.
 Date 0 represents the present, date 1 is one year
later.
3

Example 4.1
Constructing a Timeline (pp.88-9)
Problem:
 Suppose you must pay tuition fees of $10,000
per year for the next four years.
 Your tuition payments must be made in equal
instalments of $5,000 each every six months.
 What is the timeline of your tuition payments?
4

Solution:
 Assuming today is the start of the first semester,
your first payment occurs at date 0 (today).
 The remaining payments occur at six-month
intervals. Using one-half year (six months) as the
period length, we can construct a timeline as
follows:
5
Example 4.1
Constructing a Timeline (pp.88-9)

4.2 Valuing Cash Flows at
Different Points in Time
 Three important rules
1. Comparing and combining values
2. Compounding
3. Discounting
6

Rule 1: Comparing and combining values
 A dollar today and a dollar in one year are not
equivalent.
 Having money now is more valuable than having
money in the future—if you have the money today,
you can earn interest on it.
7
You can only compare or combine
values at the same point in time.

Rule 2: Compounding
 The process of moving forward along the timeline to
determine a cash flow’s value in the future (future
value) is known as compounding.
 Suppose we have $1,000 today and we wish to
know the equivalent amount in one year’s time.
 We use the current market interest rate of r = 10%
and multiply your original investment by (1 + r),
which is what you have at the end of the period:
8
($1,000 today) x
(1.10% $ in one year)
= $1,100 in one year
$ today

Rule 2: Compounding
 We can apply this rule repeatedly.
 Suppose we want to know how much the $1,000
is worth in two years’ time.
 If the interest rate for year 2 is also 10%, then we
convert:
9
($1,100 in one year) x
(1.10% $ in two years)
= $1,210 in two years
$ in one year

Rule 2: Compounding
 This effect of earning interest on both the original
principal plus the accumulated interest, ‘interest
on interest’, is known as compound interest.
 Compounding the cash flow a third time,
assuming the competitive market interest rate is
fixed at 10%, we get:
10
To calculate cash flow’s future value, you
must compound it.

(1 ) (1 ) (1 ) (1 )
times
n
n
FV C r r r C r
n
          
Future Value of a Cash Flow
(Eq. 4.1)
11
Future value of a cash flow
 To calculate a cash flow C’s value n periods into
the future, we must compound it by n intervening
interest rates factors:
FORMULA!

Figure 4.1 The Composition of
Interest over Time
12

Rule 3: Discounting
 The process of finding the equivalent value today
of a future cash flow is known as discounting.
Present Value of cash flow
13
To calculate the value of a future cash flow at
an earlier point in time, we must discount it.
(Eq. 4.2)
PV = C / (1+r)n =
C
(1 + r) n
FORMULA!

Table 4.1 The Three Rules of
Valuing Cash Flows
14

Example 4.2 Present Value of a Single
Future Cash Flow (p.93)
Problem:
 You are considering investing in a savings bond
that will pay $15,000 in ten years.
 If the competitive market interest rate is fixed at
6% per year, what is the bond worth today?
15

Solution:
Plan:
 First set up your timeline. The cash flows for this
bond are represented by the following timeline:
 Thus, the bond is worth $15,000 in ten years.
 To determine the value today, we calculate the
present value using Eq. 4.2 and our interest rate
of 6%.
16

Execute & Evaluate:
17

Applying the rules of valuing cash flows
 Suppose we plan to save $1,000 today, and
$1,000 at the end of each of the next two years.
 If we earn a fixed 10% interest rate on our
savings, how much will we have three years from
today?
18

Applying the rules of valuing cash flows
 Another approach to the problem is to calculate the
future value in year 3 of each cash flow separately.
 Once all three amounts are in year-3 dollars, we
can then combine them.
19

Example 4.3
Calculating the Future Value (p.96)
Problem:
 Let’s revisit the savings plan we considered
earlier: we plan to save $1,000 today and at the
end of each of the next two years.
 At a fixed 10% interest rate, how much will we
have in the bank three years from today?
20

Solution:
Plan:
 We’ll start with the timeline for this savings plan:
 First, we’ll calculate the present value of the cash
flows. Then, we’ll calculate its value three years
later (its future value).
21
Example 4.3

Execute:
 Here, we treat each cash flow separately, then
combine the present values and calculate the
future value in year 3.
22
Example 4.3

Evaluate:
 This answer of $3,641 is precisely the same
result we found earlier.
 As long as we apply the three rules of valuing
cash flows, we will always get the correct
answer.
23
Example 4.3

4.3 Valuing a Stream of Cash
Flows
 Most investment opportunities have multiple cash
flows at different points in time.
 The timeline represents the general formula for
the present value of a cash flow stream:
Present value of cash flows:
24
PV = C0 +
C1
+
C2
+ … +
C n
( 1 + r ) ( 1 + r )2 ( 1 + r )n
FORMULA!
(Eq. 4.3)

Example 4.4 Present Value of a Stream of
Cash Flows (pp.97-8)
Problem:
 You borrow money from your uncle to buy a car and
agree to pay him back within four years, offering him
the same rate of interest that his savings account
would give him.
 You think you would be able to pay him $5,000 in one
year, and then $8,000 for the next three years.
 If your uncle would otherwise earn 6% per year on his
savings, how much can you borrow from him?
25
1 2 3 4
Promised Cash Flow: $5,000 $8,000 $8,000 $8,000

Execute:
 We can calculate the PV of this stream of CFs as
follows:
 If your uncle deposits your payments in the bank
each year, how much will he have four years
from now?
26
PV =
5,000
+
8,000
+
8,000
+
8,000
( 1.06) ( 1.06)2 ( 1.06)3 ( 1.06)4
= 4,716.98 + 7,119.97 +6,716.95 +6,336.75 =
$24,890.65

Execute (cont’d):
 We need to calculate the FV of the annual deposits, by
calculating the bank balance each year:
 To verify, suppose your uncle left $24,890.65 in the bank
earning 6% interest. In four years, he would have:
FV = $24,890.65 x (1.06)4 = $31,423.87 in 4 years
27

Evaluate:
 We get the same answer (within one cent of rounding)
for both ways.
 Thus, your uncle should lend you $24,890.65 in
exchange for the promised payments.
 This amount is less than the total you are willing to pay
him ($5,000 + $8,000 + $8,000 + $8,000 = $29,000) due
to the time value of money.
28

4.4 The Net Present Value of a
Stream of Cash Flows
 Most investment opportunities have multiple cash
flows at different points in time.
 We can represent any investment decision on a
timeline as a cash flow stream where the cash
outflows (investments) are negative cash flows
and the inflows are positive cash flows.
 Thus, the NPV of an investment opportunity is
also the present value of the stream of cash
flows of the opportunity:
29
NPV = PV (benefits) – PV (costs) = PV (benefits – costs)
FORMULA!

Example 4.5 Net Present Value of an
Investment Opportunity (p.99)
Problem:
 You have been offered the following investment
opportunity: if you invest $1,000 today, you will
receive $500 at the end of the next two years,
followed by $550 at the end of the third year.
 If you could otherwise earn 10% per year on your
money, should you undertake the investment
opportunity?
30

0 1 2 3
-$1,000 $500 $500 $550
Solution:
Plan:
 To decide whether you should accept this
opportunity, we’ll need to calculate the NPV by
calculating the present value of the cash flow
stream.
31

Execute:
Evaluate:
32

4.5 Perpetuities, Annuities and
other Special Cases
 In this section we consider two types of
cash flow streams:
 Perpetuities
 Annuities
33

PV (C in perpetuity) =
C
r
other Special Cases
 Perpetuities
 A perpetuity is a stream of equal cash flows
that occur at regular intervals and last forever.
 Note that the first cash flow does not occur
immediately—it arrives at the end of the first
period.
Present value of a perpetuity:
34
(Eq. 4.4)
FORMULA!

other Special Cases
 To illustrate, if market interest rate is 5%, by
investing $100 in the bank today, you can, in
effect, create a perpetuity paying $5 per year:
35

Example 4.6
Endowing a Perpetuity (p.102)
Problem:
 You want to endow an annual graduation party at
your university.
 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?
36

Solution:
Plan:
 The timeline of the cash flows you want to provide
is:
37
Example 4.6

38
Example 4.6

other Special Cases
 Annuities
 An annuity is a stream of n equal cash flows
paid at regular intervals, which ends after some
fixed number of payments.
Present value of an annuity:
39
FORMULA!

other Special Cases
 With an initial $51.90 investment at 5% interest,
you can create a 15-year annuity of $5 per year.
40

Example 4.7 Present Value of a Lottery
Prize Annuity (pp.104-5)
Problem:
 You are the lucky winner of a lottery.
 You can take your prize money either as:
 (a) 30 payments of $1 million per year
(starting today), or
 (b) $15 million paid today.
 If the interest rate is 8%, which option should you
take?
41

Solution:
Plan:
 Option (a) provides $30 million in prize money
paid over time, which has to be converted in to
present value.
 The first payment of $1 million at date 0 is
already stated in present value terms, but we
need to calculate the present value of the
remaining payments—a 29-year annuity of $1
million per year.
42

43

Evaluate:
 The reason for the difference is the time value of
money. If you have the $15 million today, you
can use $1 million immediately and invest the
remaining $14 million at an 8% interest rate.
 This strategy will give you $14 million  8% =
$1.12 million per year in perpetuity!
44

(annuity) (1
1
1 (1 )
(1 )
1
((1 ) 1)
N
N
N
N
FV PV r)
C
r
r r
C r
r
  
 
   
 
 

 
   
(Eq. 4.6)
45
 Future value of an annuity
other Special Cases
FORMULA!

Example 4.8 Retirement Savings Plan
Annuity (pp.106-7)
Problem:
 Ellen is 35 years old and has decided it is time to
plan seriously for her retirement.
 At the end of each year until she is 65, she will
save $10,000 in a retirement account.
 If the account earns 10% per year, how much will
Ellen have saved at age 65?
46

Solution:
Plan:
 As always, we begin with a timeline to keep track
of both the dates and Ellen’s age:
 Ellen’s savings plan looks like an annuity of
$10,000 per year for 30 years.
 To determine the amount Ellen will have in the
bank at age 65, we’ll need to calculate the future
value of this annuity.
47
Annuity (pp.106-7)

48
Annuity (pp.106-7)

other Special Cases
 Growing cash flows
 A growing perpetuity is a stream of cash
flows that occur at regular intervals and grow
at a constant rate forever.
 For example, a growing perpetuity with a first
payment of $100 that grows at a rate of 3%
has the following timeline:
49

(Eq. 4.7)
50
 Present value of a growing perpetuity
PV( growing perpetuity) = C
r – g
FORMULA!
other Special Cases

Example 4.9
Endowing a Growing Perpetuity (p.108)
Problem:
 In Example 4.6, you planned to donate money to
your university to fund an annual $30,000
graduation party.
 Given an interest rate of 8% per year, the required
donation was the present value of $375,000.
 However, the student association has asked that
you increase the donation to account for inflation.
 The students estimate that the party’s cost will rise
by 4% per year.
 To satisfy their request, how much do you need to
donate now?

Solution:
Plan:
 The cost of the party next year is $30,000, and the
cost then increases 4% per year forever.
 From the timeline, we recognise the form of a
growing perpetuity and can value it that way.
52
Example 4.9

53
Example 4.9

(Eq. 4.8)
54
4.6 Solving for Variables other than
Present Value or Future Value
 Solving for cash flows
 Let’s consider the example of a loan—you
know how much you want to borrow (PV) and
you know the interest rate, but you do not
know how much you need to repay each year.
 If the payments are an annuity, we can invert
the annuity formula to solve for the payment:
Loan payment:
C =
P
1
1 -
1
r (1 + r )n
FORMULA!

Example 4.10
Calculating a Loan Payment (p.110)
Problem:
 Your firm plans to buy a warehouse for
$100,000.
 The bank offers you a 30-year loan with equal
annual payments and an interest rate of 8% per
year.
 The bank requires that your firm pay 20% of the
purchase price as a down payment, so you can
borrow only $80,000.
 What is the annual loan payment?
55

Solution:
Plan:
 We start with the timeline (from the bank’s
perspective):
 Using Eq. 4.8, we can solve for the loan payment,
C, given N = 30, r = 8% (0.08) and P = $80,000.
56
Example 4.10

57
Example 4.10

2time value of money.pptx

Recommended

Recommended

More Related Content

Similar to 2time value of money.pptx

Similar to 2time value of money.pptx (20)

More from NgoMinh32

More from NgoMinh32 (12)

Recently uploaded

Recently uploaded (20)

2time value of money.pptx