LESSON 9 – SUMMARY

Methods of evaluating Investments

Comparison between the approaches
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The Net Present Value (NPV) approach
 Internal Rate of Return (IRR)
 Profitability Index (PI)
 Discounted Payback Period (DPP)

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EVALUATING INVESTMENT PROJECTS

(1/2)
After computing the Cash Flows the next step is the decision-
making process for accepting or rejecting the project, based on

The cash flows represent the money which one must invest if
undertaking the project and the money obtained by it.

Obviously the return has to be enough to allow the recovery of
the capital invested, but also to generate a profitability to that
capital.

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those Cash Flows.
Thus, the initial question focuses on the magnitude of that
return, i.e., what should be its size to consider that the
corresponding project is a good one.
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EVALUATING INVESTMENT PROJECTS

(2/2)
This return can be divided in two components:

Risk-free interest rate: if we have a certain capital available to invest, we could,
without any risk, apply it in a secure investment (German Government bonds)

Risk premium: the investment in a particular project (business) has of course a
certain degree of risk (we are not sure that the estimated values will end up
happening in the future, in other words, we are not sure that the cash flows of
the project will materialize by the forecasted values).

Thus, the profitability associated with the project should cover these
two components. The greatest difficulty is in defining the risk
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and we would get the interest rate associated with this type of investment;
premium, since every project has different risk levels (for example, a
project of replacement of an equipment will certainly have a lower
risk than the expansion of the productive capacity or the creation of a
new business area).
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EVALUATION APPROACHES: NET PRESENT
VALUE
(1/3)



Thus, the invested capital will not only have to be recovered
but also has to generate an annual return of 10%.
On the other hand we know that the cash flows of the project
are distributed over time. To compare or work with them, we
have to use them in the same point in time. Usually we
discount them all to zero (we have a much better perception of
representing 1,000 € today than 1.000 € in five years).
The discount rate should naturally be the 10% that we
identified above, i.e., should be the minimum annual return
that we demand for our investment, since this rate represents
the financial value of time considering jointly the project risk.
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
Let's imagine that in face of the risk of the project we believe
that, if the same generates a return of 10%, it represents a
good investment, i.e., we wish to undertake it.
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EVALUATION APPROACHES: NET PRESENT
VALUE
(2/3)

Generalizing for any project, the NPV (VAL, in portuguese)
is calculated as follows:

Where,
I0
 CFn
 n
 r


CF3
CFn
CF1
CF2



....

(1  r )1 (1  r ) 2 (1  r )3
(1  r ) n
Investment undertook today
Cash Flow of period n
Number of years in project lifetime
Discount rate (demanded return)
Decision Rule:
If NPV > 0
 If NPV = 0
 If NPV < 0

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VAL / NPV   I 0 
Undertake the Investment
Undertake or Re-evaluate the project
Do not undertake the Investment
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EVALUATION APPROACHES : NET PRESENT
VALUE
(3/3)
EXAMPLE
project:
Cash Flow
0
1
2
3
4
(650)
265
330
365
425
Assuming an annual discount rate of 10%:
1)
Compute the NPV.
2)
Analyze the outcome and make the decision about undertaking or not the
project.
3)
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Consider that a firm forecasted the following cash flows for an investment
What is the impact on the NPV of a change in the discount rate?
In Excel use the function NPV() .
06
EVALUATION APPROACHES : INTERNAL
RATE OF RETURN
(1/3)

The NPV, despite having a clear decision rule, it represents an
absolute value does not tell us much about the degree of
attractiveness of the project. All of us are more sensitive and better
understand the outcomes in rates when we are talking about
profitability.

This concept of rate is presented in the second approach: the Internal
Rate of Return (IRR). This method provides a single number (rate)
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absolute number. Thus, it has no appealing reading, since the
that is able to summarize the merits of a project. In practice the IRR
approach is very close to the NPV one, without being NPV.
07
EVALUATION APPROACHES : INTERNAL
RATE OF RETURN
(2/3)

Generalizing for any project, IRR (TIR, in portuguese) can
be calculated as follows:

Where,
I0
 CFn
 n


CF3
CFn
CF1
CF2


 .... 
1
2
3
(1  TIR) (1  TIR)
(1  TIR)
(1  TIR) n
Investment undertook today
Cash Flow of period n
Number of years of the project lifetime
Decision Rule:
If IRR > Demanded Return
 If IRR = Demanded Return
project
 If IRR < Demanded Return

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0  I0 
Undertake the investment
Undertake or Re-evaluate the
Do not undertake
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EVALUATION APPROACHES : INTERNAL
RATE OF RETURN
(3/3)
EXAMPLE
1)
Compute the IRR.
2)
Analyze the outcome and make the decision about undertaking or not the
project.
3)
Illustrate graphically the relationship between the NPV and the discount
rate, identifying the IRR.
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Consider the information of the previous example and:
In Excel use the function IRR() .
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EVALUATION APPROACHES:
PROFITABILITY INDEX
(1/3)
The Profitability Index (PI) is essentially a different way of presenting the
information already conveyed by the NPV. Represents the absolute
return, reported to the present moment, per unit of capital invested.

It is a ratio of the present value of the future cash flows after CAPEX over
the CAPEX.

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
The reading of the outcome is of: for each money unit invested, it is
possible to get PI units of return. This relationship is established at the
time zero, i.e., at the beginning of the project.
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EVALUATION APPROACHES :
PROFITABILITY INDEX
(2/3)

Generalizing for any project, the PI (IRP, in portuguese)
n
can be calculated as follows:
CFt  I t

Where,
It
 CFt
 n
 r


t 0
n
t
It

t
t  0 (1  r )
Investment undertook today
Cash Flow of period t
Number of years of the lifetime of the project
Demanded Return
Decision Rule:
If IRP > 1
 If IRP = 1
 If IRP < 1

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IRP 
 (1  r )
Undertake the investment
Undertake or Re-evaluate the project
Do not undertake
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EVALUATION APPROACHES:
PROFITABILITY INDEX
(3/3)
EXAMPLE
1)
Compute the PI.
2)
Analyze the outcome and make the decision about undertaking or not the
project.
3)
Assuming that on year 3 the firm acquires an equipment by the price of
25, recalculate the PI.
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Still using the same example:
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EVALUATION APPROACHES: DISCOUNTED
PAYBACK PERIOD
(1/3)

The previous approaches value the profitability of the project. However, in
some circumstances, as much or more important as profitability is the
out in a context of political, economic or social instability. In these
situations it is important to know how long it takes to recover the
investment.

The DPP is fundamentally a risk indicator and not a profitability
measure.

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pace of recovery of the investment. Imagine a project that will be carried
The procedure to be followed in its calculation is derived from the sum of
the discounted cash flows (opposite to the DPP, the simple PP does not
take into account the time value of money) up to a value of zero (breakeven).
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EVALUATION APPROACHES: DISCOUNTED
PAYBACK PERIOD
(2/3)

Generalizing for any project, the DPP (PRI, in portuguese)
can be calculated as follows:
0
t 0

Where,
CFt
 r


CFt
(1  r )t
Cash Flow of period t
Demanded Return
Decision Rule:
If PRI < Lifetime of the project*
 If PRI = Lifetime of the project*
the project
 If PRI < Lifetime of the project*

* or any particular cut-off date
Undertake the investment
Undertake or Re-evaluate
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PRI
Do not undertake
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EVALUATION APPROACHES: DISCOUNTED
PAYBACK PERIOD
(3/3)
EXAMPLE
Still using the same example:
Compute the DPP.
2)
Analyze the outcome and make the decision about undertaking or not the
project.
3)
Discuss what is the relevance of the cash flow of the year 4 for the
calculation and conclusion about the DPP.
Period
Cash Flow
Discounted
Cash Flow
Accumulated
Discounted
Cash Flow
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1)
0
1
...
n
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COMPARING THE METHODS – A SINGLE
PROJECT
(1/3)
The NPV is a Cash approach, reflecting the
increase in value for the shareholder. Assumes
reinvestment of intermediate cash flows at a
minimum required rate.
 IRR is an annual rate and represents the
internal rate of profitability, assuming the
reinvestment of project cash flows at the IRR
itself.
 The PI represents a weight factor for each euro
invested in the project.
 The DPP is a measure of time and indicates the
length of time required for recovery of the
investment.

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COMPARING THE METHODS – A SINGLE
PROJECT

(2/3)
Accept the project:
NPV > 0 , IRR > r , PI > 1 , DPP < n

Accept/Re-evaluate the project:
NPV = 0 , IRR = r , PI = 1 , DPP = n

Reject the project:
NPV < 0 , IRR < r , PI < 1 , DPP > n
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In the presence of a project where there is an
initial outflow and later inflows, all methods
reach the same conclusion:

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COMPARING THE METHODS – A SINGLE
PROJECT

In the presence of a project where the cash flows
are "abnormal" the method to adopt is of the
NPV.
The IRR approach has problems:



Does not distinguish between financing or investing
operations.
May deliver more than one outcome (Multiple IRR).
In calculating the DPP it is necessary to be
cautious about the fact that the accumulated
value of the discounted cash flows may change its
sign more than once.
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
(3/3)
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