# DCF (NPV) approach

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```					  DCF (NPV) approach                                            Last week

Argued that the appropriate objective of the
firm was to maximize shareholder value
Finance decision involves playing ( exchanging)
Valuing certain cash flows                                     cash to get more money
We can exchange cash now v. 1. cash later 2.
TIP                       cash uncertain 3. cash in another currency
If you do not understand
Investment decisions are undertaken only if
something,
they raise value.
ask me!                   Valuation is key to finance.
Today: Study of valuation under certainty.

2

The plan of the lecture                                         Financial choices

Review the concept of the time value of money                   Which would you rather receive today?
present value (PV)
discount rate (r)                                                TRL 1,000,000,000 ( one billion Turkish
discount factor (DF)                                            lira )
net present value (NPV)
USD 652.72 ( U.S. dollars )
Review of the formula for calculating
perpetuity                                                    Both payments are absolutely guaranteed.
annuity
What do we do?
Interest compounding
How to use a financial calculator

3                                                      4

Financial choices with
Financial choices                                               time

We need to compare “apples to apples” -                         Which would you rather receive?
this means we need to get the TRL:USD                              \$1000 today
exchange rate                                                      \$1200 in one year
From www.bloomberg.com we can see:                              Both payments have no risk, that is,
USD 1 = TRL 1,637,600                                          there is 100% probability that you will be
Therefore TRL 1bn = USD 610.64                                    paid
there is 0% probability that you won’t be
paid

5                                                      6
Financial choices with
time                                                        Present value
Why is it hard to compare ?                                 In order to have an “apple to apple”
\$1000 today                                              comparison, we convert future payments to
\$1200 in one year
the present values
This is not an “apples to apples” comparison. They
have different units                                             this is like converting money in TRL to money in
\$1000 today is different from \$1000 in one year                 USD
Why?                                                            Certainly, we can also convert the present value to
A cash flow is time-dated money                               the future value.
⌧It has a money unit such as USD or TRL                      Although these two ways are theoretically the same,
⌧It has a date indicating when to receive money              but the present value way is more important and has
more applications, as to be shown in stock and bond
valuations.
7                                                                      8

Present value for the cash
Some terms                                                  flow at end of period 1

Finding the present value of some cash
C1
flows is called discounting.                                    PV =               = DF1 × C1
Finding the future value of some cash                                       1 + r1
flows is called compounding.
DF1 =         1
(1+ r1 )1
C1 is the cash flow at the end of period 1
PV is the present value (value now) of the cash flow in period 1
DF1 is called discount (present value) factor for the cash flow
r1 is the discount rate, or interest rate.
9                                                                      10

Present value for the cash
flow at end of period t
Example 1
What is the present value of \$100 received
in one year (next year) if the interest rate                                     Ct
is 7%?                                                     PV =                                  = DFt × Ct
PV=100/(1.07)1 =?
\$100
(1 + rt )        t

PV=?    Year one

\$93.46                                                     Replacing “1” with “t” allows the
formula to be used for cash flows at any
point in time
11                                                                      12
Example 3
Example 2
You just bought a new computer. The payment terms are
What is the present value of \$100 received                                      \$3000 at the end of 2 years. If you can earn 8% on your
in year five if the discount rate is 7%?                                        money (may be thru an online bank), how much money
PV=100/(1.07)5 =?                                                           should you set aside today in order to make the payment
when due in two years?

PV =                        = \$2,572.02
\$100
\$71.30                                                                                               3000
PV=?               Year 5
(1.08 ) 2

13                                                                 14

Explanation of the
discount factor                                                                Example 4

Discount (present value) Factor                                                Given two dollars, one received a year from
now and the other dollar two years from now.

DF t =                                 1                                 Assume r1 = 5% and r2 = 7%. What is the
present value for each dollar received?
(1 + rt ) t
DF1=1.00/(1+0.05)=0.95
The present value of one US dollar received in the                             DF2=1.00/(1+0.07)2=0.87
future; or, you can think of it as the price of \$1 to
be received at the end of period t.
15                                                                 16

Present value of multiple
cash flows                                                                     Example 5

For a cash flow received in multiple years                                     John is given the following set of cash flows
PVs can be added together to evaluate                                          and discount (interest) rates. What is the PV?
multiple cash flows.                                                             C1 = 100    r1 = 0.1
\$100    \$200    \$50

C1                C2                        CN                    C 2 = 200 r2 = 0.09
PV =                  +                  + .... +                                                         PV=?                   Yr 3
r3 = 0.07
Yr 1   Yr 2
(1+ r1 )   1
(1+ r2 )   2
(1 + rN )    N         C3 = 50
N
= ∑ C i × DFi                                                             PV=100/(1.1)1 + 200/(1.09)2 + 50/(1.07)3 =?
i =1
\$300.06
17                                                                 18
Projects                                                                         Examples of projects
A “project” is a term that is used to describe the                               An entrepreneur starts a company:
following activity:                                                                 initial investment is negative cash outflow.
spend some money today                                                           future net profit is cash inflow .
receive cash flows in the future                                              An investor buys a share of IBM stock
A stylized way to draw project cash flows is as follows:                            cost is cash outflow; dividends are future cash inflows.
A lottery ticket:
cash flows               cash flows            investment cost: cash outflow of \$1
in year one (probably positive)   in year two
jackpot: cash inflow of \$20,000,000 (with some very small
probability…)
Initial investment                                                          Thus projects can range from real investments, to
(negative cash flows)
financial investments, to gambles (the lottery ticket).

19                                                                    20

Firms or companies                                                               Net present value
A firm or company can be regarded as a                                           A net present value NPV) is the sum of
set of projects.                                                                 the initial investment (usually made at
capital budgeting is about choosing the best                                  time zero) and the PV of expected future
projects in real asset investments.                                           cash flows.
How do we know whether one project is                                                NPV = C0 + PV (C1 LCT )
worth taking?                                                                                            T                     T
Ct
= C0 + ∑                     = ∑ Ct × DFt
t =1 (1 + rt )
t
t =0

21                                                                    22

NPV rule                                                                         Example 7

If the NPV of a project is positive, the firm                                    Given the data for the following project,
should go ahead to take this project.                                            what is the NPV?        -\$50
\$50     \$10

C0 = −50
C1 = 50      r1 = 7.5%
C 2 = 10     r2 = 8.0%                     Yr 1      Yr 2

Yr 0

NPV=-50+50/(1.075)+10/(1.08)2 =

23                                                                    24
Example 8                                                      Present Values
Discount                      Cash                 Present
Assume that the cash flows                                     Period
from the construction and sale                                           Factor                         Flow                Value
of an office building is as                                      0         1 .0                      − 150,000             − 150,000
follows. Given a 7% interest                                     1         1
1.07      = .935            − 100,000             − 93,500
(discount) rate, find the net                                    2         1
= .873           + 300,000             + 261,900
(1.07 )2
present value.                                                                                 NPV = Total =               \$18,400
Year 0    Year 1   Year 2
− 150,000 − 100,000 + 300,000
25                                                                                     26

Now let s study some
interesting patterns of cash
flows…
Perpetuities
We are going to look at the PV of a perpetuity starting one year
Perpetuity                                                    from now.
Definition: if a project makes a level, periodic payment into
Annuity                                                       perpetuity, it is called a perpetuity.
Let’s suppose your friend promises to pay you \$1 every year,
starting in one year. His future family will continue to pay you and
your future family forever. The discount rate is assumed to be
constant at 8.5%. How much is this promise worth?
\$1    \$1 \$1   \$1     \$1                \$1
PV
???

Yr1    Yr2   Yr3 Yr4   Yr5         Time=infinity

27                                                                                     28

Calculate the PV of a
perpetuity
Perpetuities (continue)
Calculating the PV of the perpetuity is easy (ignore          Consider the perpetuity of one dollar
the process if you hate math)
every period your friend promises to pay
PV =
C
+
C
+L+
C               you. The interest rate or discount rate is
(1 + r )1 (1 + r ) 2     (1 + r ) ∞         8.5%.
∞                      ∞
= C ⋅∑
1
= C.∑ α i =
C          Then PV =1/0.085=\$11.765, not a big gift.
i =1   (1 + r ) i
i =1    r

29                                                                                     30
Value the annuity
Annuities
Well, a friend might not pay you forever.                           If you’d like to know the derivation of the
Instead, consider a friend that promises to pay                    following formulae, come see me after
you \$C every year, for the next “T” years. This                    class.
is called an annuity.
Can you think of examples of annuities in the
real world?                                                                   ⎛1      1      ⎞
\$1 \$1 \$1 \$1 \$1         \$1
PV = C ⎜ −         T ⎟
⎝ r (1 + r ) r ⎠
PV
???

Yr1   Yr2   Yr3 Yr4   Yr5   Time=T

31                                                   32

Example for annuities                                              My solution

you win the \$1million dollar lottery! but                          Using the formula for the annuity
wait, you will actually get paid \$50,000
⎛ 1          1        ⎞
per year for the next 20 years if the                                 PV = 50,000 * ⎜     −               ⎟
discount rate is a constant 7% and the first                                        ⎝ 0.07 1.07 20 * 0.07 ⎠
payment will be in one year, how much                                   = \$529,700.71
have you actually won (in PV-terms) ?

33                                                   34

Example                                                            Solution

You agree to lease a car for 4 years at \$300
per month. You are not required to pay any
money up front or at the end of your
agreement. If your discount rate is 0.5% per
month, what is the cost of the lease?

⎡ 1           1         ⎤
Lease Cost = 300 × ⎢     −
⎣ .005 .005(1 + .005)48 ⎥
⎦
Cost = \$12,774.10

35                                                   36
Perpetuities with a                                            More detailed proof…(ignore this if
growth rate                                                    you hate math)

What is the PV of the perpetuity with a cash flow of C
in the next period and then growing at a rate of g at                    C      C(1+ g)     C(1+ g)∞
PV =           +         +L+
very period in the future?                                             (1+ r)1 (1+ r)2       (1+ r)∞

C (1 + g ) ∞
C ∞ (1+ g)i                     1+ g
C            C (1 + g )                            =       ⋅∑           Let' s call α ≡      , where 0 < α < 1 then:
PV =                +                +L+                          1+ g i=1 (1+ r)i                 1+ r

(1 + r )1       (1 + r ) 2          (1 + r ) ∞        PV =
C ∞ i         C                              C 1−α ∞+1    C     1
∑α = 1+ g (1+α +α 2 +......+α∞ −1) = 1+ g ( 1−α −1) = 1+ g (1−α −1)
1+ g i=1
∞ (1 + g )i −1            C                       =
C          1
−1) =
C 1+ r − (r − g) C 1+ g
=           )=
C
=C⋅ ∑                      =
(                                             (
1+ g (r − g) /(1+ r)       1+ g   r−g         1+ g r − g r − g
i =1 (1 + r )
i         r−g
37                                                                            38

Future value (i=interest rate
here)                                                           Manhattan Island Sale

The formula for converting the present value to                         Peter Minuit bought Manhattan Island for \$24 in 1630.
future value:
To answer, determine \$24 is worth in the year 2004,
FVt =i = PVt = 0 × (1 + rt =i )i                                 compounded at 8%.

PVt =0 = present value at time zero                                                           FV = \$ 24 × (1 + .08 ) 374
FVt =i
rt =i
= future value in year i                                                                     = \$ 75 .979 trillion
= discount rate during the i years

Ct =i                                                                                                   FYI - The value of Manhattan Island land is
figure.
well below this figure.
39                                                                            40

Simple interest v.
compounded interest rate                                        Compound Interest

In simple interest rate, FV=P(1+rt).                                             18
16            10% Simple
In compounded interest rate, FV=p(1+r)^t                                         14
12            10% Compound
I finance, we almost always mean
FV of \$1

10
compounded interest rate.                                                         8
6
4
2
0
0

3

6

9
12

15

18

21

24

27

30

Number of Years
41                                                                            42
Interest compounding                                     Compound Interest

The interest rate is often quoted as the simple          Example
interest rate, which is often called as APR, the           Suppose you are offered an automobile loan at an APR of
annual percentage rate, or the Stated Interest             6% per year. What does that mean, and what is the true
Rate.                                                      rate of interest, given monthly payments?

If the interest rate is compounded m times in
each year and the APR is r, the effective
annual interest rate is
m
⎛1 + r ⎞ − 1
⎜       ⎟
⎝     m⎠
43                                                                           44

The effect of compounding
Compound Interest                                        frequency
i        ii         iii       iv                     v
Periods Interest              Value                  Annually
per     per        APR        after                  compounded
year    period     (i x ii)   one year               interest rate
1         6%        6%        1.06                            6.000%

1.032
Effective interest rate = (1.005)12 − 1
2         3          6                    = 1.0609            6.090

4         1.5        6        1.0154 = 1.06136                6.136

= 6.1678%                       12         .5        6        1.00512 = 1.06168               6.168
52      .1154        6        1.00115452 = 1.06180            6.180

365     .0164        6        1.000164365 = 1.06183           6.183
45                                                                           46

What is the future value (FV) of an
Now let’s learn how to use                               initial \$100 after 3 years, if interest
calculators                                              rate (i) = 10%?

Have you bought one yet?                                  Finding the FV of a cash flow or series of cash
flows when compound interest is applied is
called compounding.
FV can be solved by using the arithmetic,
methods.
0               1                 2                   3
10%

100                                                   FV = ?
47                                                                           48
Solving for FV:                                                  Solving for FV:
The arithmetic method                                            The calculator method
After 1 year:                                           Solves the general FV equation.
FV1 = PV ( 1 + i ) = \$100 (1.10)
= \$110.00                                         Requires 4 inputs into calculator, and will
After 2 years:                                          solve for the fifth. (Set to P/YR = 1 and END
FV2 = PV (1+i)(1+i) = \$100 (1.10)2                   mode.)(P/YR=periods per year, END=cashflow happens
=\$121.00                                          end of periods)

After 3 years:
INPUTS        3        10     -100       0
FV3 = PV ( 1 + i )3 = \$100 (1.10)3
=\$133.10                                                         N       I/YR     PV      PMT      FV
After n years (general case):                            OUTPUT                                         133.10
FVn = PV ( 1 + i )n
49                                                           50

What is the present value (PV) of \$100
Solving for PV:
due in 3 years, if I/YR = 10%?                                   The arithmetic method

Finding the PV of a cash flow or series of
cash flows when compound interest is                           PV = FVn / ( 1 + i )n
applied is called discounting (the reverse of
compounding).                                                  PV = FV3 / ( 1 + i )3
The PV shows the value of cash flows in                          = \$100 / ( 1.10 )3
terms of today’s worth.
= \$75.13
0                   1          2                3
10%

PV = ?                                         100
51                                                           52

Solving for FV:
Solving for PV:                                                  3-year ordinary annuity of \$100 at
The calculator method                                            10%

Exactly like solving for FV, except we                           \$100 payments occur at the end of each
have different input information and                             period, but there is no PV.
are solving for a different variable.

INPUTS         3       10               0          100          INPUTS        3        10       0      -100
N        I/YR    PV      PMT         FV                         N       I/YR     PV      PMT      FV
OUTPUT                         -75.13                            OUTPUT                                          331

53                                                           54
Solving for PV:
3-year ordinary annuity of \$100 at                      What is the PV of this uneven
10%                                                     cash flow stream?

\$100 payments still occur at the end of
each period, but now there is no FV.                       0           1         2          3            4
10%

100          300           300         -50
90.91
247.93
INPUTS        3     10              100    0
225.39
N    I/YR     PV      PMT    FV
-34.15
OUTPUT                    -248.69                     530.08 = PV
55                                                          56

Solving for PV:                                         Solving for I:
What interest rate would cause \$100 to
Uneven cash flow stream                                 grow to \$125.97 in 3 years?

Input cash flows in the calculator’s “CFLO”            Solves the general FV equation for I.
register:
CF0 = 0
CF1 = 100
CF2 = 300
CF3 = 300
INPUTS         3             -100       0     125.97
CF4 = -50
N    I/YR     PV      PMT       FV
Enter I/YR = 10, press NPV button to get
NPV = \$530.09. (Here NPV = PV.)                         OUTPUT               8

57                                                          58

The Power of Compound                                   Solving for FV:
Interest                                                Savings problem

A 20-year-old student wants to start saving for          If she begins saving today, and sticks to
retirement. She plans to save \$3 a day. Every day,       her plan, she will have \$1,487,261.89 when
she puts \$3 in her drawer. At the end of the year,       she is 65.
she invests the accumulated savings
(\$1,095=\$3*365) in an online stock account. The
stock account has an expected annual return of
12%.
INPUTS         45   12        0      -1095

How much money will she have when she is 65                              N    I/YR     PV      PMT       FV
years old?                                                OUTPUT                                       1,487,262

59                                                          60
Solving for FV:
Savings problem, if you wait until
you are 40 years old to start
If a 40-year-old investor begins saving
today, and sticks to the plan, he or she will
have \$146,000.59 at age 65. This is \$1.3
million less than if starting at age 20.
Lesson: It pays to start saving early.

INPUTS     25      12       0     -1095
N      I/YR     PV     PMT       FV
OUTPUT                                    146,001

61

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