NPV as Investment Criterion

Corporate Finance 2-0
© Professor Ho-Mou Wu
Capital Investment Decisions
2.1 Net Present Value
2.2 Project Valuation in a Riskless World
Fisher’s Principle
2.3 Present Value and Compounding
2.4 Present Value with Special Cash Flows
(RWJ Ch 3, 4)

Investment Decision
Example 1:
Suppose an investment that promises to pay $10,000
in one year is offered for sale for $9,500. Your
interest rate is 5%. Should you buy?
• If you were to be promised $10,000 due in one year
when interest rates are at 5-percent, your investment
be worth $9,523.81 in today’s dollars.

2.1 Net Present Value : FV and PV
• The amount that a borrower would need to set aside
today to to able to meet the promised payment of
$10,000 in one year is call the Present Value (PV) of
$10,000.
Note that $10,000 = $9,523.81×(1.05).
• If you were to invest $10,000 at 5-percent interest
for one year, your investment would grow to
$10,500 : $10,500 = $10,000×(1.05).
The total amount due at the end of the investment is
call the Future Value (FV).

Net Present Value
• The Net Present Value (NPV) of an investment is the
present value of the expected cash flows, less the cost
of the investment.
: So you should Invest.
Back to Example 1:

Net Present Value as the Investment Criterion
In the one-period case, the formula for NPV can be
written as:
If we had not undertaken the positive NPV project
considered on the last slide, and instead invested our
$9,500 elsewhere at 5-percent, our FV would be less
than the $10,000 that investment promised and we
would be unambiguously worse off in FV terms as
well:
$9,500×(1.05) = $9,975 < $10,000.
,where is cash flow at date 1

2.2 Project Evaluation in a Riskless World
C0
C0
C1
C1
Y1=1.2m
Y0=1m
Saver (lending)
B
Spender (borrowing)
A
Y
)
r
1
(
Y
Y
)
Y
(
PV 1
0



1
1+r
slope = -(1+r)
Why do we use NPV as the investment criterion ?
Assume Perfect Capital Market and Two Period

(I) Saving (Financing) Decision

Use PV to Check Feasibility of Consumption plan
Example 2:
Is the consumption plan C0＝0.9m and C1＝1.325m feasible?
Use the PV formula to evaluate it.
If r＝10%, 0.9＋＝2.105＝PV(C)＞1＋＝2.091＝PV(Y)
: not feasible
If r＝20%, 0.9＋＝2.004＝PV(C)＞1＋＝2.000＝PV(Y)
: not feasible
If r＝30%, 0.9＋＝1.919＝PV(C)＜1＋＝1.923＝PV(Y)
: feasible!

(Ⅱ) Investment Opportunities

Corporate Investment Decision-Making
Consumption
at
t+1
Positive NPV projects shift the
shareholder’s opportunity set out,
which is unambiguously good.
All shareholders agree on their
preference for positive NPV
projects, whether they are
borrowers or lenders.

(Ⅲ) Investment Opportunities with Financial Markets
Financial markets present
saving/borrowing
opportunities, as represented
by the dotted straight line.
Suppose the company (farm)
chooses D, its owners can
then use financial markets for
saving or borrowing. Both
investors are happier than in
(Ⅱ), but D is not the optimal
investment plan yet. C0
B
A
D
E
C1
slope = -(1+r)
PV(D)

Project Valuation in a Riskless World
C0
A’
D
E
slope = -(1+r)
PV(Y)
C1
B’
Y*
Y1
Y0
Y* is the optimal
investment plan, which is
the one that maximizes
NPV(Y)＝PV(Y)－E or
PV(Y).
Perfect capital market
(borrowing rate＝lending
rate) is assumed.

Corporate Investment Decision-Making
• In reality, shareholders do not vote on every
investment decision faced by a firm and the
managers of firms need decision rules to operate by.
• All shareholders of a firm will be made better off if
managers follow the NPV rule—undertake positive
NPV projects and reject negative NPV projects.

Optimal Investment Plan
Net Present Value NPV＝
＝PV(Y)－E
Therefore, the best investment plan is the one that maximizes NPV(Y);
and the best investment plan is independent of investors’ preferences.
PV＝
NPV＝＝

Fisher’s Separation Principle
Given perfect capital market and certainty, the optimal
investment plan is the one that maximizes the net
present value of available production plans, without
regard to the individuals’ subjective preferences that
enter into their consumption/saving decisions. (Irving
Fisher)
This is the basis for using the present value as the
evaluation criterion.
Separation of investment and financing decisions
Separation of ownership and management.

Why do we use NPV or PV as investment criterion

NPV as Investment Criterion

2.3 Present Value and Compounding
• How much would an investor have to set aside today
in order to have $20,000 five years from now if the
current rate is 15%?
0 1 2 3 4 5
$20,000
PV

How Long is the Wait?
Example 5: If we deposit $5,000 today in an account
paying 10%, how long does it take to grow
to $10,000?

Example 6: Assume the total cost of a college education
will be $50,000 when your child enters college
in 12 years. You have $5,000 to invest today.
What rate of interest must you earn on your
investment to cover the cost of your child’s
education?
What Rate Is Enough?

2.4 PV with Special Cash Flows
• 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 stream of constant cash flows 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.

Perpetuity
A constant stream of cash flows that lasts forever.
0
…
1
C
2
C
3
C
The formula for the present value of a perpetuity is:

Growing Perpetuity
A growing stream of cash flows that lasts forever.
0
…
1
C
2
C×(1+g)
3
C ×(1+g)2
The formula for the present value of a growing perpetuity is:

Annuity
A constant stream of cash flows with a fixed maturity.
0 1
C
2
C
3
C
The formula for the present value of an annuity is:
T
C


Growing Annuity
A growing stream of cash flows with a fixed maturity.
0 1
C
The formula for the present value of a growing annuity:

2
C×(1+g)
3
C ×(1+g)2
T
C×(1+g)T-1

What Is a Firm Worth?
• Conceptually, a firm should be worth the present
value of the firm’s cash flows.
• The tricky part is determining the size, timing and
“risk” of those cash flows : we will probe further in
later class..

NPV as Investment Criterion

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