NPV and Alternative Investment Rules
November 27, 2001
Objectives of lecture on captital budgeting rules
In previous lectures we have already argued that the appropriate goal from shareholders
point of view is to maximize NPV. But in practice other method are used as well. ⇒
Comparison of diﬀerent capital budgeting techniques.
• Payback period rule
• Discounted payback period rule
• Average accounting return (AAR)
• Internal rate of return (IRR)
• Proﬁtability index
Then will discuss in more detail problems using NPV in capital budgeting.
Why should investment rule be based on NPV?
When we discussed the separation theorem, we have shown that management which invests
in projects with positive NPV ⇔ increases shareholder value.
NPV is appropriate rule since:
• NPV uses cash ﬂows not earnings: cash ﬂows really what is available to pursue cor-
porate investment plans.
• NPV uses all cash ﬂows: Cash ﬂows exhaustively describe an investment opportunity
from the point of view of valuation. Other methods ignore cash ﬂows after a certain
point in time.
• NPV discounts the cash ﬂows properly: Captures the time value of money as revealed
by the markets.
Why are other methods used in practice?
• Historical reasons and convenience.
• Alternative methods less complicated.
• Lack of appropriate estimates for cash ﬂows and risk adjusted discount rates.
Pay Back Period Rule
Idea: Many investment projects start with a cash outﬂow followed by cash inﬂows in the
future. ⇒ How many years does it take until initial cash outﬂow has been paid back ?
Answer: Pay back period .
Application: Management decides about what it considers as acceptable (upper limit)
time until initial investment is paid back.
⇒ Rule: Undertake all investment projects with a pay back period less than say 5 years.
⇒ Ranking Criteria: Undertake investment project with shortest pay back period.
Pros of Payback Period
• Upper limit on cut oﬀ time limits risk exposure of investment projects.
• Favors liquidity of ﬁrm.
• Respects short term investment horizon.
Cons of Payback Perido
Projects considered as identical with respect to payback period can be extremely diﬀerent:
• Diﬀerent cash ﬂows after cut oﬀ period are ignored.
• Timing of cash ﬂows before cut oﬀ period is not taken into account: Two projects
with same initial cash outﬂow of $100 followed by cash inﬂows of $10 over 10 year for
one project and no cash inﬂows for 9 year but $ 100 in the 10th year have the same
pay back period! (Should prefer the ﬁrst project since intermediate cash inﬂows can
be reinvested.) ⇒ Payback period method ignores time value of money.
• Biased against long-term projects and therefore against shareholder interests.
• Criteria may not exist and/or acceptance criteria is arbitrary.
Discounted Pay Back Period
Idea: Correct payback period method such that time value of money is taken into account.
Pros: Same as pay back period. But in addition reﬂect opportunity costs of money as
valued by the ﬁnancial markets.
Cons: As pay back period. But in addition get more complicated since have to discount
as for NPV rule.
Averge Accounting Return (AAR)
Idea: It is often easier to determine projects earning rather than cash ﬂows. Calculate rate
of return of projects earning by dividing its earnings by the project’s book value. Since
both vary oﬀer time take time averages ⇒ Average Accounting Return.
Average net income
Application:Proceed in two steps
1. Determine average net income:
Average Net Income = (Revenuen − Expensesn
−Depreciationn − Taxesn )
2. Determine average investment:
Average Investment = (Value of investmentn
− Depreciationn )
Decison Rule for ARR
⇒ Rule: Management deﬁnes hurdle rate. Undertake investment when AAR is higher
than this hurdle rate.
⇒ Ranking criteria: Invest in project with highest hurdle rate.
Pros of ARR
• Accounting information available.
• Easy to calculate.
• Does always exist.
Cons of ARR
• Ignores the time value of money: average net income can be same but income per
period can be very diﬀerent.
• Benchmark (hurdle rate) arbitrary.
• Based on book values: Do not reﬂect means available for investment project as cash
ﬂows to and book values do not reﬂect valuation by the market.
Internal Rate of Return (IRR)
Idea: Given the cash ﬂow structure of a given investment project at which interest rate is
its NPV equal to zero ?
Application: Management ﬁxes target rate of return.
IRR solves the following equation:
(1 + IRR)n
IRR corresponds to the “yield” of the investment project.
Investing and Financing
1. Investing: Undertake investment if IRR is higher than a speciﬁed hurdle rate which
may correspond to discount rate from ﬁnancial markets. (Opportunity cost of capital).
2. Financing: Accept ﬁnancing project if IRR is lower than a speciﬁed hurdle rate
which may correspond to the discount rate from ﬁnancial markets (Opportunity cost
Ranking criteria: Undertake investment project with highest IRR and accept ﬁnancing
project with lowest IRR.
NPV and IRR
Remark: IRR is very close to NPV:
• Both are based on all cash ﬂows and capture time value of money.
• Both evaluate investment project by a simple number
But IRR does not always exist or is not necessarily unique.
Pros of IRR
• Uses all cash ﬂows.
• Not based on accounting information.
• Easy to understand.
• Easy to communicate.
Cons of IRR
• Does not capture diﬀerent scale of project.
• Does not always exist (“is not always real”) and may not be unique:
Example: Consider cash ﬂows C0 , C1 , C2 . Then IRR is given by solution
C0 + + =0
1 + IRR (1 + IRR)2
then deﬁne X = 1+IRR and try to ﬁnd X.
The equation deﬁning IRR is quadratic and you may remember that the general
solution for X is then
−C1 + C1 − 4C0 C2
−C1 − 2
C1 − 4C0 C2
and consequently we will have two solutions for IRR if C1 − 4C0 C2 is not equal to
2 − 4C C < 0 we have one IRR which is not a real number !
zero. Furthermore if C1 0 2
Remark : As you can see this does not occur if the initial investment is negative and
all other cash ﬂows are positive (Investment) or vice versa all but the last cash ﬂows
are positive (Financing).
Cons IRR continued
• Does not distinguish between investing and ﬁnancing. Example: If we multiply all
cash ﬂows by the same number, the IRR does not change ! Consequently if we compare
an investment project with a ﬁnancing project we obtain the same IRR!.
• Re-investment rate assumption: IRR rule does not discount at opportunity costs
of capital, it implicitly assumes that the time value of money is the IRR. Implicit
assumption: ⇒ Re-investment rate assumption.
Therefore the IRR assumes that investor can reinvest their money at the IRR which
is diﬀerent from the market-determined opportunity costs. It is logically not coherent
that the re-investment rate should be diﬀerent if they are in the same risk class.
• Value Additivity Principle: The value of the ﬁrm should be equal to the sum of the
values of the individual projects of the ﬁrm.
Illustration: Violation of Value Additivity Principle
Therefore if we have choice between two mutually exclusive projects and an indepen-
dent project, it should not occur that project one is preferred to project two but if
mixed with the independent project, project two is preferred. But this can happen
– Project 1: -100,0,550 ⇒ IRR=134.4%
– Project 2: -100,225,0 ⇒ IRR=125.0%
– Project 3: -100,450,0 ⇒ IRR=350 %
– Project 1+3: -200,450,550 ⇒ IRR=212.8%
– Project 2+3: -200,675,0 ⇒ IRR=237.5%
Idea: IRR is attractive since it is not dependent on scale of investment. It is a “per dollar”
quantity. The proﬁtability index does the same thing for NPV: Amount of NPV generated
for each investment dollar.
PV of cash inﬂows
initial cash outﬂow
PV of cash outﬂows
PIF in =
initial cash inﬂow
Application: Management deﬁnes benchmark for index
⇒ Undertake investment if proﬁtability index is higher than benchmark.
⇒ Undertake investment with highest proﬁtability index.
Pros and Cons for Proﬁtability Index
• respects limited availability of investment funds.
• easy to understand and communicate
• same decision as NPV for independent projects
Cons: As IRR non-additive ⇒ Problems when evaluate mutually exclusive projects.
Conclusion: Valuation principles
Only NPV satisﬁes all of the following requirements:
• Considers all cash ﬂows.
• Cash ﬂows are discounted at the true opportunity cost.
• Selects from mutually exclusive projects the one that maximizes shareholder value.
• Respects value-additivity principle.
Conﬂict among Criteria
Exercise: Consider the following projects:
• A: -1000,100,900,100,-100,-400
• B: -1000,0,0,300,700,1300
• C: -1000,100,200,300,400,1250
• D: -1000,200,300,500,500,600
Show that payback rule chooses A, ARR rule B, NPV with discount rate 10% C and IRR
Diﬃculties Using NPV Rule
In practice several issues make the use of NPV rule diﬃcult:
• Measurement of cash ﬂows
• Incremental cash ﬂows not accounting income
• Measurement of opportunity costs of capital
• Investments with unequal lives
Measurement of cash ﬂows
• Incremental cash ﬂow: What part of the cash ﬂow is exclusive consequence of invest-
ment project ? ⇒ Ignore Sunk costs ⇒ Include Opportunity costs ⇒ Correct for Side
• Projection of future cash ﬂows.
• Real versus nominal cash ﬂows.
• Capital Cost Allowance (CCA). .
If we assume straight-line depreciation then the net cash ﬂow (NCF) can be obtained in
two equivalent ways:
1. Total project cash ﬂow approach
2. Tax Shield Approach
Remark: In practice we have to use Declining Balance CCA
Total project cash ﬂow approach
N CF = (N 0I − CCA) × (1 − tc ) + CCA
• tc : Corporate tax rate
• NOI: Net operating income
• NOI-CCA: Taxable income
• (N OI − CCA) × (1 − tc ): Net income
Tax shield approach
N CF = N OI × (1 − tc ) + tc × CCA
• N OI × (1 − tc ): After tax net operating income.
• tc × CCA: Tax shield (tax savings).
Measurement of opportunity costs of capital
• Real versus nominal interest rate (The Fisher relation). ⇒ Right discount rate de-
pends on whether cash ﬂows are real or nominal.
• Correction of discount factor for risk.
Investments with unequal lives
• Matching cycles: ⇒ repeat projects until timing matches.
Example Cost project A −500, 120, 120, 120 project B −600, 100, 100, 100, 100. To
compare consider smallest common multiple 12 and compare PV of 4 repetitions of
projection A with 3 repetitions of project B.
• Equivalent annual costs: Consider example as before.
1 1 1
EAC A = P V A [ + + ]−1
1 + r (1 + r)2 (1 + r)3
1 1 1 1
EAC B = P V B [ + 2
1 + r (1 + r) (1 + r) (1 + r)4