Accepted Manuscript

Investment decisions and sensitivity analysis: NPV-consistency of

rates of return

Andrea Marchioni, Carlo Alberto Magni

PII: S0377-2217(18)30007-9
DOI: 10.1016/j.ejor.2018.01.007
Reference: EOR 14908

To appear in: European Journal of Operational Research

Received date: 20 August 2016

Revised date: 1 January 2018
Accepted date: 3 January 2018

Please cite this article as: Andrea Marchioni, Carlo Alberto Magni, Investment decisions and sensitivity
analysis: NPV-consistency of rates of return, European Journal of Operational Research (2018), doi:
10.1016/j.ejor.2018.01.007

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 ACCEPTED MANUSCRIPT

Highlights
• We introduce a definition of strong NPV-consistency for relative measures of worth

• We measure strong NPV-consistency in terms of the projects value drivers

• We use Sensitivity Analysis for measuring strong NPV-consistency

• Average Return On Investment is defined as a capital-weighted mean of period rates

• Average Return On Investment is strongly NPV-consistent

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Investment decisions and sensitivity analysis:

NPV-consistency of rates of return
Andrea Marchioni∗
Department of Economics “Marco Biagi”

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University of Modena and Reggio Emilia

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andrea.marchioni@unimore.it

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Carlo Alberto Magni†
Department of Economics “Marco Biagi”

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CEFIN − Center for Research in Banking and Finance
University of Modena and Reggio Emilia
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magni@unimo.it
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Abstract
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Investment decisions may be evaluated via several different metrics/criteria, which are func-
tions of a vector of value drivers. The economic significance and the reliability of a metric depend
on its compatibility with the Net Present Value (NPV). Traditionally, a metric is said to be
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NPV-consistent if it is coherent with NPV in signalling value creation. This paper makes use
of Sensitivity Analysis (SA) for measuring coherence between rates of return and NPV. In par-
ticular, it introduces a new, stronger definition of NPV-consistency that takes into account the
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influence of value drivers on the metric output. A metric is strongly NPV-consistent if it signals
value creation and the ranking of the value drivers in terms of impact on the output is the same
as that provided by the NPV. The degree of (in)coherence is calculated with Spearman’s (1904)
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correlation coefficient and Iman and Conover’s (1987) top-down coefficient. We focus on the class
of AIRRs (Magni 2010, 2013) and show that the average Return On Investment (ROI) enjoys
strong NPV-consistency under several (possibly all) methods of Sensitivity Analysis.

Keywords. Finance, sensitivity analysis, investment decisions, NPV, Return On Investment,

consistency, average, AIRR.

JEL codes. D81, M41, G31, G12, D92, Q28.

∗
Viale Berengario 51, 41121 Modena, Italy.
†
Viale Berengario 51, 41121 Modena, Italy. Corresponding author

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1 Introduction
In capital budgeting many different criteria are used for evaluating a project, measuring
economic efficiency, and making decisions. Net Present Value (NPV) is considered the
most theoretically reliable tool, since it correctly measures shareholder value creation
(Brealey and Myers 2000, Ross, Westerfield and Jordan 2011). However, in practice, many
other metrics are used; in particular, relative measures of worth such as internal rate of
return (IRR), profitability index (PI), modified internal rate of return (MIRR), Return
On Investment (ROI), etc. Recently, a more general notion of rate of return, labeled
AIRR (Average Internal Rate of Return) has been developed by Magni (2010, 2013),

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based on a capital-weighted mean of holding period rates. The AIRR approach consists
in associating the capital amounts invested in each period with the corresponding period

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returns by means of a weighted arithmetic mean. Magni (2010, 2013) showed that any
AIRR is NPV-consistent: decisions made by an investor who adopts NPV are the same as

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those made by an investor who adopts AIRR. Magni (2013) showed that many traditional
metrics can be viewed as belonging to the class of AIRRs, including IRR, PI, MIRR. As a
special case, this approach makes use of the Return On Investment (ROI) to get an average

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ROI, which is the ratio of the total project return to the total invested capital. Whatever
the depreciation pattern, the average ROI exists and is unique, it has the unambiguous
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nature of investment rate, independent of the value drivers, and decomposes the economic
value created into economic efficiency (the difference between average ROI and cost of
capital) and investment scale (the sum of the committed amounts).
However, while traditional NPV-consistency is important, under uncertainty, an NPV
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or a rate of return are not the only factors that drive a decision. The investigation of the
risk factors that mainly influence the value of the objective function is no less important.
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Sensitivity analysis (SA) investigates the variation of an objective function under

changes in the key inputs of a model, so aiming at identifying the most important risk
factors affecting the output (and, therefore, the decision) and ranking them. There are
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many different SA techniques (see Pianosi et al. 2016, Borgonovo and Plischke 2016) and,
given a technique, different objective functions may or may not lead to different results.
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This paper positions itself in the interfaces of operational research (OR) and finance.
The strict connections between operations management and finance were recognized long
since (e.g., Small 1956, Weingartner 1963, Adelson 1965, Hespos and Strassman 1965,
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Teichroew et al. 1965a,b, Rivett 1974, Ignizio 1976) and scholarly contributions in the
field have grown dramatically in the last decades (e.g., Rosenblatt and Sinuany-Stern
1989, Grubbström and Ashcroft 1991, Murthi et al. 1997, Meier et al. 2001, Gondzio and
Kouwenberg 2001, Baesens et al. 2003, Steuer and Na 2003, Xu and Birge 2008, Koç et
al. 2009, Fabozzi et al. 2010, Thomas 2010, Seifert et al. 2013).
The relation between OR and finance is bidirectional. On one side, finance provides
a rich toolkit of theories, criteria, and methodologies which enable operational managers
to better understand the impact of their decisions so as to maximize the shareholders’
wealth: “In order to make decisions managers need criteria of goodness, decision tools,
and an understanding of the environment in which they operate . . . The main elements of

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this are that the right criterion of goodness is the maximisation of shareholder wealth and
that firms operate in something close to a perfect capital market.” (Ashford et al. 1988).
On the other side, operational research sets the aims and scope of financial modeling
for managerial purposes: As opposed to finance theory which uses financial modeling
for describing the behavior of the “average” investor and deriving the pricing process of
financial assets, operational managers use financial modeling from the point of view of
an individual decision maker with specific needs, constraints and preferences (Spronk and
Hallerbach 1997). Further, operations research itself provides techniques and tools that
may be applied to several finance problems (Board et al. 2003).
This paper is in line with the bidirectional relation between operations and finance.

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Specifically, it recognizes the fundamental roles of economic and financial measures of

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worth such as the NPV and the ROI for decision-making and, at the same time, applies an
OR technique (SA) to such financial measures in order to investigate their compatibility.

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As such, it falls within that strand of the OR literature which makes use of various
economic efficiency measures for managerial purposes, including the NPV (e.g., Yang
et al. 1993, Baroum and Patterson 1996, Herroelen et al. 1997, Cigola and Peccati

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2005, Borgonovo and Peccati 2006a, Wiesemann et al. 2010, Leyman and Vanhoucke
2017), the IRR (Rapp 1980, Nauss 1988, Hazen 2003, 2009, Hartman and Shafrick 2004,
Dhavale and Sarkis 2018), the ROI (e.g., Danaher and Rust 1996, Myung et al. 1997,
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Brimberg and ReVelle 2000, Brimberg et al. 2008, Li et al. 2008, Menezes et al. 2015,
Magni 2016) and the return to outlay (Kumbhakar 2011). This work is strictly linked
with some recent methodological papers within this field which evaluate rationality and
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robustness of various efficiency measures and/or their sensitivity to changes in the key
parameters. Specifically, Magni (2015) showed that the average ROI (labeled average
ROA) is reliable for measuring economic efficiency in industrial applications; Mørch et al.
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(2017) used the average ROI as the objective function in a problem of renewal of shippings,
and compared the results with those obtained from the traditional NPV maximization.
Borgonovo and Peccati (2004, 2006b) studied the impact of the key drivers of an industrial
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project on NPV, IRR, and value at any time. Borgonovo et al. (2010) applied SA in a
project financing transaction to assess the degree of coherence between NPV and debt
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service coverage ratio. Talavera et al. (2010) applied SA to the IRR of photovoltaic
grid-connected systems. Percoco and Borgonovo (2012) applied SA to IRR and NPV and
studied the coherence between the two metrics in terms of importance of key drivers.
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We investigate the coherence of average ROI and NPV and give a new, more stringent,
definition of NPV-consistency (strong coherence), according to which a metric is strongly
NPV-consistent under a given SA technique if it is NPV-consistent in the traditional sense
and, in addition, the ranking of the project’s value drivers (in terms of influence on the
output) is the same. If a metric is not NPV-consistent, the degree of inconsistency may be
measured by two alternative indices: Spearman’s (1904) coefficient or Iman and Conover’s
(1987) top-down coefficient.
We find that the average ROI is strongly NPV-consistent under many techniques,
even in a strict sense (the relevances of the parameters are the same). As a result, the
average ROI is a reliable measure of worth which can coherently be associated with NPV

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in investment evaluation, assessment of economic efficiency, and decision-making.

The remaining part of the paper is structured as follows. Section 2 presents the average
ROI and the notion of NPV-consistency. Section 3 briefly describes some known SA
methods and Section 4 introduces the notion of pairwise coherence according to which any
two functions are strongly coherent if the ranking of the model parameters coincides. This
section shows that, under many SA techniques, a function f and an affine transformation
of it share the same (ranking and) relevances of parameters, so they are strongly coherent
in a strict sense. Section 5 shows that the average ROI is strongly NPV-consistent in
a strict sense under many SA techniques. Some numerical examples are illustrated in
section 6. Some concluding remarks end the paper. (An Appendix is devoted to some

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other AIRRs, including non-strongly consistent ones such as IRR, MIRR and EAIRR.)

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2 AIRR, average ROI, and NPV consistency

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Let P be a project and let F = (F0 , F1 , . . . , Fp ) 6= 0 its estimated stream of free cash
flows (FCFs), where F0 < 0 is the investment cost and p is the lifetime of the project.

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Let τ be the tax rate, Rt be the revenues, Ot be the operating costs, and let Dept denote
depreciation, t = 1, 2, . . . , p. Then,
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operating profit
z }| {
Ft = (Rt − Ot − Dept )(1 − τ ) +Dept (1)
= (Rt − Ot )(1 − τ ) + τ · Dept .
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Revenues and costs are often estimated in terms of some key inputs such as prices, quantity
produced and sold, unit costs, growth rates, etc. There may be several types of costs,
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such as energy, material, labor, selling, general, and administrative expenses, etc. For
example,
 s
X 
Ft = q · p0 (1 + gp ) − t
O0j (1 + gOj )t (1 − τ ) + τ · Dept (2)
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j=1

where p0 denotes the initial price, q denotes the annual quantity sold, O0j denotes the
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initial amount of the j-th item of cost, gp and gOj are the growth rates, and s is the
number of cost items involved in the project under consideration. Let k be the (assumed
constant) cost of capital (COC). We assume that the COC is exogenously fixed by the
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decision-maker/analyst. It is well-known that net present value (NPV) measures the

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economic value created: NPV = pt=0 Ft (1 + k)−t . Therefore, the NPV decision criterion
may be stated as follows:

Definition 1. (NPV criterion) A project creates value (i.e., it is worth undertaking) if

and only if the project NPV, computed at the discount rate k, is positive: NPV(k) > 0.

Let C = (C0 , C1 , . . . , Cn ) be any vector representing some notion of capital, such that
C0 = −F0 and Cn = 0 and let It = Ft + Ct − Ct−1 be the associated return. An AIRR,
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denoted as ı̄, is defined as the ratio of the overall return I = pt=1 It (1 + k)−(t−1) earned

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Pp
by the investor to the overall capital committed C = t=1 Ct−1 (1 + k)−(t−1) :

I
ı̄ = (3)
C

or, equivalently, as the weighted mean of period rates associated with the capital stream
C: Pp
it Ct−1 dt−1
ı̄ = Pt=1
p
t=1 Ct−1 dt−1

where dt = (1+k)−t is the discounting factor and it = It /Ct−1 is the period rate of return,
t = 1, 2, . . . , p (see Magni 2010, 2013).
Magni (2010, 2013) defined a project a net investment if C > 0 and a net financing

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if C < 0. In such a way, the financial nature of any project (and its associated rate of

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return) can be identified as an investment project or a financing project (respectively, an
investment rate or a financing rate).

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Traditionally, it is widely accepted that a metric/criterion ϕ is said to be NPV-
consistent if and only if a decision maker adopting ϕ makes the same decision suggested
by the NPV criterion. We can formalize this standard notion as follows.

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Definition 2. (NPV-consistency) A metric/criterion ϕ is NPV-consistent if, given a
cutoff rate k, the following statements are true:
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(i) an investment project creates value if and only if ϕ > k
(ii) a financing project creates value if and only if ϕ < k.
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Magni (2010, 2013) showed that, given a cash-flow stream F, if ϕ = ı̄, then the metric
is NPV-consistent, since, for any vector C, the following product structure holds:
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NPV(1 + k) = C(ı̄ − k). (4)

The above definition and eq. (4) are particularly interesting because they show that
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the AIRR approach enables reframing the NPV in terms of product of a capital base C
and an excess return ı̄ − k. This means that the economic value created is determined by
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two factors: The project scale (C) and the project’s economic efficiency, ı̄ − k. The same
NPV can be created either by investing a large capital amount at a small rate or investing
a small capital at a high rate. Furthermore, the general definition stated above enables
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the analyst to understand whether value is created because capital is invested at a rate of
return which is higher than the COC or because capital is borrowed at a financing rate
which is smaller than the COC (see also Magni 2015).
We now consider the special case of AIRR where Ct = Bt is the capital which remains
invested in the project at time t: Bt = Bt−1 − Dept and B0 = −F0 , so that It is the
operating profit: It = Ft + Bt − Bt−1 = Ft − Dept = (Rt − Ot − Dept )(1 − τ ). The
associated period return rate is the Return on Investment (ROI):

Operating profit (Rt − Ot − Dept )(1 − τ )

ROIt = = . (5)
Invested capital Bt−1

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Thus, the AIRR becomes

Total Return I
ı̄(B) = = (6)
Total Invested Capital B
Pp

where I = t=1(Rt − Ot − Dept )(1 − τ ) · dt−1 is the overall operating profit generated
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by the project and B = pt=1 Bt−1 dt−1 is the overall invested capital, expressing the size
of the investment. As seen above, ı̄(B) may be viewed as a weighted average of ROIs:

ı̄(B) = α1 ROI1 + α2 ROI2 + . . . + αp ROIp (7)

where αt = Bt−1 dt−1 /B. We call ı̄(B) average ROI.1 As (4) holds for any C (and, there-

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fore, for B = (B0 , B1 , . . . , Bn ) as well), NPV(1 + k) = B · (ı̄(B) − k) so the average ROI is

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NPV-consistent (see also Magni 2015). It is also worth noting that the average ROI has
the compelling property of existence and uniqueness for any project. Also, its financial

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nature does not depend on the value drivers nor the cost of capital: It is unambiguously
determined as an investment rate, since B0 = −F0 > 0, which implies B > 0. This makes
it a good candidate as a reliable measure of worth.

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Owing to (1) and (2), the NPV is a function of several variables (the prospective
revenues and costs). Practically, the analyst selects depreciation for every period, Dep1 ,
Dep2 , . . . , Depp , then estimates the amount of sales, the initial price(s), the costs for
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labor, material, maintenance, energy, the growth rates, the tax rate, etc. These variables
are risk factors, also known as value drivers, for they affect the FCFs. Hence, given the
project COC, the project NPV is computed. For example, using (2),
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p Ps j

X q · p0 (1 + gp )t − j=1 O0 (1 + gOj )t (1 − τ ) + τ · Dept
NPV = F0 + . (8)
(1 + k)t
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t=1

It is evident that the average ROI depends on these same value drivers, given that ROIt
depends on them. From (5),
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Ps j

q · p0 (1 + gp )t − j=1 O0 (1 + gOj )t − Dept (1 − τ )
ROIt = .
Bt−1
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Exploiting (4), one can describe the AIRR as a function of the overall capital C:
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NPV
ı̄ = ı̄(C) = k + (1 + k). (9)
C

Figure 1 graphically describes the AIRR function ı̄(C) for a value-creating project; each
pair (C, ı̄(C)) represents an NPV-consistent rate of return; among the infinitely many
AIRRs, we highlight the average ROI, which is the AIRR associated with the capital
stream B.
The project’s aim is to check whether the coherence of average ROI and NPV, which is
guaranteed in a traditional sense, remains valid if changes in value drivers are considered.
The analysis of change in a model’s inputs and the impact on the model output is the
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Magni (2015) used the expression average ROA for this measure.

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𝑖̅(𝐶)

Average
ROI

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B C

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Figure 1: Graph of the AIRR function for a positive-NPV project

purpose of Sensitivity Analysis (SA).2 US

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3 Sensitivity analysis
In the definition of Saltelli et al. (2004, p. 45), sensitivity analysis (SA) is the “study of
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how the uncertainty in the output of a model (numerical or otherwise) can be apportioned
to different sources of uncertainty in the model input.”
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Given a model and a set of inputs (parameters), the SA investigates the relevance of
parameters in terms of variability of the model output. In the literature there exist many
SA techniques (see Borgonovo and Plischke 2016, Pianosi et al. 2016, for review of SA
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methods). A model can be described as consisting of an objective function f defined on

the parameter space A, which maps vector of inputs onto a model output y:
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f : A ⊂ Rn → R, y = f (α), α = (α1 , α2 , . . . , αn ) . (10)

The vector α = (α1 , α2 , . . . , αn ) ∈ A ⊂ Rn is the vector of inputs or parameters or key

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drivers and y(α) is the output of the model. Let α0 = α10 , α20 , . . . , αn0 ∈ A be the base-
case, a representative value (e.g., mean value, most probable value, etc.). The relevance
of a parameter αi (also known
 as importancemeasure) quantifies the impact of αi on the
output variation. Let R = R1f , R2f , . . . , Rnf be the vector of the relevances. The latter
f

determines the ranking of the parameters in the following way. Input αi is defined to be
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Ekern (1981) and Foster and Mitra (2003) provide conditions under which a project’s NPV is greater than
a second project’s NPV irrespective of the COC. Assuming that the second project is the null alternative, those
conditions identify those projects which are robust under changes in the COC. Those conditions hold for any
AIRR as well, given that any AIRR is NPV-consistent in the traditional sense. In this paper, we measure the
robustness of the project with respect to the estimates of revenues and costs and focus on their impact on NPV
and rate of return.

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more relevant than αj if and only if |Rif | > |Rjf |. The parameters are equally relevant
for f if |Rif | = |Rjf |. The rank of αi , denoted as rif , depends on the importance measure:
αi has a higher rank (it has a greater  impact on the output) than αj if it has greater
f f f f
relevance. Let r = r1 , r2 , . . . , rn be the vector of ranks.
Pn n·(n+1)
f i n+1
The average rank is rM = i=1
n = 2
n = 2 . The high parameters (or top
f
parameters) are those whose rank is higher than the average rank rM ; the low parameters
f
are those parameters whose rank is smaller than rM .
Following we briefly describe some well-known (global and local) SA techniques.

(i) Standardized regression coefficient (global SA)

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Let V denote variance and σ denote standard deviation. Consider the linear regression
with dependent variable f and explanatory variables αi , ∀i = 1, . . . , n, estimated with

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OLS method: f = β0f + ni=1 βif · αi + u. The standardized regression coefficient SRCif
measures the importance of αi (Saltelli and Marivoet 1990, Bring 1994, Saltelli et al.

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2008):
βif · σ(αi )
SRCif = . (11)
σ(f )

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(ii) Sensitivity indices in variance-based decomposition methods (global SA)
In variance-based methods, the importance of a parameter is generally represented
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through the First Order Sensitivity Index (FOSI) and the Total Order Sensitivity Index
(TOSI) (Saltelli et al. 2008). The FOSI, here denoted as SIi1,f , measures the individual
effect of the parameter on the output variance:
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V (E(f |αi ))
SIi1,f = , (12)
V (f )
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where V (E(f |αi )) is the variance of the expectation of f upon a fixed value of αi .3
The TOSI, here denoted as SIiT,f , measures the total contribution of αi to the output
variability, i.e., it is inclusive of the interaction effects with other parameters or groups of
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parameters. SIiT,f can be calculated as (Saltelli et al. 2008)

E(V (f |α−i ))
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SIiT,f = , (13)
V (f )

where f |α−i = f |α1 , α2 , . . . , αi−1 , αi+1 , . . . , αn (see also Sobol’ 1993, Sobol’ 2001, Saltelli
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et al. 2008).

(iii) Finite Change Sensitivity Indices (global SA)

The Finite Change Sensitivity Indices (FCSIs), introduced in Borgonovo (2010a, 2010b),
focus on the output change due to a finite input change; there exist two versions of FCSIs:
First Order FCSI and Total Order FCSI.
The First Order FCSI of a parameter measures the individual effect of the parameter’s
variation on f ; the Total Order FCSI considers the total effect of a parameter’s variation
on f , including both the individual contribution and the interactions between a parameter
and the other parameters.
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It can be shown that V (E(f |αi )) = V (f ) − E[V (f |αi )] (see Satelli et al. 2008).

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Consider a change of parameters from α0 to α1 = α11 , α21 , . . . , αn1 ∈ A. The output
variation is ∆f = f (α1 ) − f (α0 ). Let (αi1 , α(−i)
0 ) = (α10 , α20 , . . . , αi−1
0 , α1 , α0 , . . . , α0 )
i i+1 n
be obtained by varying the parameter αi to the new value αi1 , while the remaining n − 1
parameters are fixed at α0 . The individual effect of αi on ∆f is ∆i f = f (αi1 , α(−i)
0 )−f (α0 )
and the First Order FCSI of αi , denoted as Φ1,f
i , is (Borgonovo 2010a):

∆i f
Φ1,f
i = . (14)
∆f

∆f is equal to the sum of individual effects and interactions between parameters and
groups of parameters. The total effect of the parameter αi , denoted as ∆Ti f , is the sum

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of the individual effect of αi and of the interactions that involve αi . Borgonovo (2010a,

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Proposition 1) showed that ∆Ti f can be obtained as ∆Ti f = f (α1 ) − f (αi0 , α(−i)
1 ) for all
i = 1, 2, . . . , n, where (αi0 , α(−i)
1 ) is the point with all the parameters equal to the new

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value α1 , except the parameter αi , which is equal to αi0 . The Total Order FCSI of the
parameter αi , denoted as ΦT,f
i , is (Borgonovo 2010a):

f (α1 ) − f (αi0 , α(−i)

1 )
ΦT,f
i =
∆Ti f
∆f
=
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. (15)
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(iv) Helton’s index (local SA)
Helton (1993) proposed a variance decomposition of f based on Taylor approximation.
He assumed parameters are not correlated, so the variance of f can be approximated by
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n
X  2
V̂ (f ) = fα0 i (α0 ) · V (αi ). (16)
i=1
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The impact of input αi on V (f ) can be measured by

 0 2
fαi (α0 ) · V (αi )
Hif (α0 ) = . (17)
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V̂ (f )

(v) Normalized Partial Derivatives (local SA)

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Helton (1993) also proposed the adoption of normalized partial derivatives as sensitiv-
ity measures. He defined two versions of normalized partial derivatives (NPDs):
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αi0
N P D1fi (α0 ) = fα0 i (α0 ) · , (18)
f (α0 )
σ(αi )
N P D2fi (α0 ) = fα0 i (α0 ) · , (19)
σ̂(f )

where σ̂(f ) is the square root of V̂ (f ) defined in (16). N P D1fi (α0 ) measures the elasticity
of f with respect to αi in α0 assuming that the relative change in αi is fixed for i =
 
1, 2, . . . , n (Helton 1993, p. 329). N P D2fi (α0 ) is the square root of (17).

(vi) Differential Importance Measure (local SA)

The total variation f (α0 +dα)−f (α0 ) of a differentiable function f due to a local change
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dα can be approximated by the total differential df = ni=1 fα0 i (α0 ) · dαi . The Differential

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Importance Measure (DIM) of parameter αi is the ratio of the partial differential of f with
respect to αi to the total differential of f (Borgonovo and Apostolakis 2001, Borgonovo
and Peccati 2004):

dfai fα0 (α0 ) · dαi

DIMif (α0 , dα) = = Pn i 0 0
. (20)
df j=1 fαj (α ) · dαj

The DIM of a parameter represents the percentage of the function’s variation due to the
variation of that parameter (Borgonovo and Apostolakis 2001, Borgonovo and Peccati
2004).

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4 Coherence between objective functions

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Risk management problems are often characterized by the definition of more than one

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objective function (Borgonovo and Peccati 2006b, Borgonovo, Gatti and Peccati 2010).
For a given technique, the analysis can be applied using different objective functions. A
relevant aspect is the evaluation of the coherence (or compatibility) between the results
of the sensitivity analysis for different functions.
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We consider the objective functions f, g : A → R. The vector of importance measures
are respectively Rf = (R1f , R2f , . . . , Rnf ) and Rg = (R1g , R2g , . . . , Rng ); the ranking vectors
AN
are rf = (r1f , r2f , . . . , rnf ) and rg = (r1g , r2g , . . . , rng ).

Definition 3. (Coherence) Given a technique of SA and two objective functions f and g,

M

they are coherent if the ranking vectors coincide: rf = rg . If, in addition, the vectors of
the relevances coincide, Rf = Rg , they are strictly coherent.
ED

If two functions f and g are not coherent, the degree of incoherence can be alternatively
measured through Spearman’s rank correlation coefficient (Spearman 1904) or top-down
correlation coefficient (Iman and Conover 1987).
PT

Spearman’s rank correlation coefficient (henceforth, Spearman’s coefficient) between

two stochastic variables is the correlation coefficient between the ranks of the stochastic
variables (Spearman 1904). In SA, Spearman’s coefficient between two objective functions
CE

f and g, denoted as ρf,g , is the correlation coefficient of the ranking vectors rf and rg :
Pn f f
Cov(rf , rg ) i=1 (ri− rM ) · (rig − rM
g
)
AC

ρf,g = = qP
q , (21)
σ(rf ) · σ(rg ) n f f 2 P n g g
2
i=1 ri − rM · i=1 ri − rM

f g n+1
where, as seen, rM = rM = 2 . The coefficient ρf,g attributes the same weight to top
and low parameters and lies in the interval [−1, 1]. The coefficient ρf,g is equal to 1 if and
only if f and g are coherent according to Definition 3. Therefore, a value of ρf,g smaller
than 1 signals incoherence between f and g: The smaller the value of ρf,g , the higher the
degree of incoherence. The difference 1 − ρf,g can be taken as representative of the degree
of incoherence.
Iman and Conover (1987) introduced the top-down correlation coefficient, a compati-
bility measure that attributes a higher weight to top parameters than to low parameters.

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This measure is based on Savage Score (Savage 1956). The

 Savage scoreof parameter αi
Pn
f
is Si = h=rf h . The vector of Savage scores is S = S1f , S2f , . . . , Snf .4 The average
1 f
i
Pn
f Sif
Savage score is SM = i=1
n = 1.
The top-down correlation coefficient between the objective functions f and g, denoted
as ρS f ,S g , is the correlation coefficient between the Savage scores’ vectors S f and S g (Iman
and Conover 1987):
Pn f f
Cov(S f , S g ) i=1 (Si− SM ) · (Sig − SM g
)
ρS f ,S g = = qP
q , (22)
σ(S f ) · σ(S g ) 2 P g
2
n
i=1 Sif − SM
f
· n
i=1 S g
i − S M

T
f g
where SM = SM = 1. The coefficient ρS f ,S g measures the compatibility between the

IP
parameters’ ranking of f and g: The accordance between top parameters has a remarkable
influence on ρS f ,S g , while the discordance between low parameters has a weak influence

CR
on ρS f ,S g (Iman and Conover 1987).
If the aim of the analysis is factor prioritization (i.e., identification of the most relevant
parameters), the top-down coefficient should be preferred to Spearman’s coefficient.

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The maximum value of ρS f ,S g is equal to 1. In case f and g have no ties (i.e., no
relevance is equal), the minimum value is −1 for n = 2, it increases as n increases, and it
tends to −0.645 as n tends to infinity (Iman and Conover 1987).
AN
ρS f ,S g is equal to 1 if and only if f and g are coherent. Therefore, a value of ρS f ,S g
smaller than 1 signals incompatibility between f and g. The smaller the value of ρS f ,S g ,
the higher the incoherence level. The degree of incoherence of f and g can then be
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measured by 1 − ρS f ,S g .
Borgonovo et al. (2014) showed that an objective function f and a monotonic trans-
formation g of it generate the same ranking of the parameters under several techniques.
ED

This means that they are coherent according to Definition 3.

We now show that, if g is an affine transformation of f , that is, g(α) = l · f (α) + q for
PT

all α ∈ A, then f and g are strictly coherent under several techniques.

Proposition 1. A function and an affine transformation of it are strictly coherent under

CE

the following techniques:

(i) Standardized regression coefficient
(ii) Sensitivity Indices in variance-based decomposition methods
AC

(iii) Finite Change Sensitivity Indices

(iv) Helton’s index
(v) Normalized Partial Derivative (N P D2fi )
(vi) Differential Importance Measure.

Proof. By hypothesis, g(α) = l · f (α) + q. Therefore,

4
For example, if n = 5 and rf = (1, 2, 3, 4, 5), then S f = (2.283̄, 1.283̄, 0.783̄, 0.45, 0.2).

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Pn Pn
(i) g = l · (β0f + f
i=1 βi · αi + u) + q = (l · β0f + q) + i=1 (l · βif ) · αi + l · u, whence

β0g = l · β0f + q,
βig = l · βif

so that
βig · σ(αi ) l · βif · σ(αi )
SRCig = = = SRCif .
σ(g) l · σ(f )

(ii) Denoting as f |αi (and g|αi ) the function f (and g) conditional to a specific value of
αi , g|αi = (l · f + q)|αi = (l · f )|αi + q = l · f |αi + q. Therefore,

T
V (E(g|αi )) V (E(l · f |αi + q)) l2 · V (E(f |αi ))
SIi1,g = = = = SIi1,f .

IP
V (g) V (l · f + q) l2 · V (f )

Analogously, g|α−i = (l · f + q)|α−i = l · f |α−i + q. Hence,

CR
E(V (g|α−i )) E(V (l · f |α−i + q)) l2 · E(V (f |α−i ))
SIiT,g = = = = SIiT,f .
V (g) V (l · f + q) l2 · V (f )

US

(iii) Since ∆g = g(α1 ) − g(α0 ) = l · f (α1 ) + q − l · f (α0 ) − q = l · f (α1 ) − f (α0 ) = l · ∆f,
and
AN
∆i g = g(αi1 , α(−i)
0
) − g(α0 ) = l · f (αi1 , α(−i)
0
) + q − l · f (α0 ) − q


= l · f (αi1 , α(−i)
0
) − f (α0 ) = l · ∆i f,
M

then
∆i g l · ∆i f ∆i f
Φ1,g
i = = = = Φ1,f
i .
∆g l · ∆f ∆f
ED

As for the Total Indices,

∆Ti g = g(α1 ) − g(αi0 , α(−i)

1
) = l · f (α1 ) + q − l · f (αi0 , α(−i)
1
)−q
PT

= l · f (α1 ) − f (αi0 , α(−i)
1
) = l · ∆Ti f

so that
CE

∆Ti g l · ∆Ti f ∆T f
ΦT,g
i = = = i = ΦT,f
i .
∆g l · ∆f ∆f

(iv) From (16),

AC

n
X
V̂ (g) = [gα0 i (α0 )]2 · V (αi )
i=1
n
X n
X
= [l · fα0 i (α0 )]2 · V (αi ) = l2 · [fα0 i (α0 )]2 · V (αi ) = l2 · V̂ (f ).
i=1 i=1

Hence,

[gα0 i (α0 )]2 · V (αi ) l2 · [fα0 i (α0 )]2 · V (αi )

Hig (α0 ) = = = Hif (α0 ).
V̂ (g) l2 · V̂ (f )

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(v) Straightforward, since N P D2fi  is the square root of Hif (α0 ).5
(vi) From (20),

gα0 (α0 ) · dαi l · fα0 i (α0 ) · dαi fα0 i (α0 ) · dαi

DIMig (α0 , dα) = Pn i 0 0
= Pn 0 0
= Pn 0 0
j=1 gαj (α ) · dαj j=1 l · fαj (α ) · dαj j=1 fαj (α ) · dαj

= DIMif (α0 , dα).

(This result is independent of the structure of dα.)

Remark 1. While we have proved that, for several SA techniques, a function and its

T
affine transformation are coherent (even in a strict sense), it is intuitive to inductively

IP
believe that a function and its affine transformation share an absolute coherence, in that
they are coherent for every existing SA technique. We leave the proof of this more general

CR
statement for future research.

5 Coherence between return rates and NPV

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The investment risk can be defined as “the potential variability of financial outcomes”
AN
(White et al. 1997). The future outcomes of an investment are stochastic and the investor
has limited information. Referring to NPV and IRR, Joy and Bradley (1973, p. 1255)
wrote: “It has often been suggested that capital budgeting theory has over-emphasized
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the development of such techniques with little regard for the typically poor data used in
project evaluation and the effect that errors in capital budgeting inputs have on project
profitability.” The practice of valuation criteria should be corroborated by a careful
ED

investment risk analysis.

Given an investment model based on a set of value drivers, SA allows the evaluator
to identify the most relevant parameters in terms of variation of the value. The most
PT

relevant parameters are the risk factors that mainly influence the investment. After SA
has been performed, the investment risk can be reduced through information insights on
CE

the main risk factors identified by the analysis; the collection of extra information on
these parameters allows more precise estimates and a remarkable uncertainty reduction
(Borgonovo and Peccati 2006b). Furthermore, the potential investor is able to appreciate
AC

the convenience of possible hedging strategies.

As the NPV is the main decision criterion in capital budgeting theory, the analysis
of the parameters’ relevance on NPV variability is fundamental. Any relative measure of
worth should be consistent with NPV not only in terms of classical consistency but also
in terms of output variability with respect to changes in the inputs.
5
It is worth noting that f and g are coherent but not strictly coherent under N P D1fi technique:

αi0 αi0 f (α0 ) f (α0 )

N P D1gi (α0 ) = gα0 i (α0 ) · 0
= l · fα0 i (α0 ) · 0
· 0
=l· · N P D1fi (α0 )
g(α ) g(α ) f (α ) g(α0 )

so that |N P D1fi | > |N P D1fj | implies |N P D1gi | > |N P D1gj |. Therefore, the parameters’ ranking in f and g is
equal: rf = rg .

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Definition 4. (Strong NPV-consistency) Given an analysis technique T , a metric ϕ (and

its associated decision criterion) is strongly NPV-consistent (or coherent with NPV) under
T if it fulfills Definition 2 and NPV and ϕ are coherent functions. The metric ϕ is strictly
NPV-consistent if the coherence is strict.
If a metric/criterion possesses strong NPV-consistency, the investor can equivalently adopt
NPV or such criterion for measuring the value creation under uncertainty. In case a metric
is not strongly NPV-consistent, the degree of incompatibility can be measured through
Spearman’s coefficient or through top-down coefficient, as seen in section 4.
We now show that the average ROI possesses strong NPV-consistency. To this end,
we maintain the symbol α = (α1 , . . . , αn ) as the vector of the project’s value drivers and

T
α0 is the base value. We assume that the initial invested capital (or borrowed amount) is

IP
exogenously given, as well as the COC (and p). The economic profitability of P depends on
the realization of the value drivers, which affect the FCFs, as seen in section 2: Ft = Ft (α),
P

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t = 1, 2, . . . p. We now let f (α) = NPV(α) = −B0 + pt=1 Ft (α)(1 + k)−t and g(α) denotes
the average ROI.
Proposition 2. For any fixed k, p and Dep = (Dep1 , . . . , Depp ),6 average ROI and NPV

US
are strongly consistent in a strict sense under the following techniques:
(i) Standardized regression coefficient
AN
(ii) Sensitivity Indices in variance-based decomposition methods
(iii) Finite Change Sensitivity Indices
(iv) Helton’s index
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(v) Normalized Partial Derivative (N P D2fi )

(vi) Differential Importance Measure.
ED

Proof. The depreciation charge, Dept , does not depend on the value drivers; therefore, B
does not depend on α and, using (9), one can write g(α) = q + l · NPV(α) where q = k
and l = (1 + k)/B. The thesis follows from Proposition 1.7
PT

The above proposition guarantees that the value drivers’ effect on the variability of ı̄(B)
and NPV is the same, not only in terms of ranks (rnpv = rı̄(B) ) but also in terms of
CE

relevances (Rnpv = Rı̄(B) ). Therefore, ρı̄(B),npv = ρS ı̄(B) ,S npv = 1. This means that an
investor can equivalently employ average ROI or NPV to analyze an investment under
AC

uncertainty.

6 Worked examples
In the previous sections we have shown that, given a depreciation plan, the average ROI is
strongly consistent with NPV. The aim of this section is to discuss two models. The first
analyzes an example with straight-line depreciation. The second one is a real-life applica-
tion, illustrated in Hartman (2007, p. 344) and is based on declining balance depreciation.
We will accomplish a SA by focusing on two techniques: FCSI and DIM.
6
Obviously, to fix Dep is equivalent to fixing B.
7
Evidently, ı̄(B) is strongly NPV-consistent under N P D1fi as well but not in a strict sense.

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6.1 Straight-line depreciation

We discuss a simple model, consisting of a firm facing the opportunity of investing in
a 4-period project whose estimated revenues and operating costs are Rt and Ot . We
assume that the tax rate is zero, τ = 0 (it is not a risk factor). This implies, from (1),
Ft = Rt − Ot . The project’s value drivers are then αi = Ri for i = 1, 2, 3, 4 and αi = Oi−4
for i = 5, 6, 7, 8. Hence, the value driver’s vector is α = {R1 , R2 , R3 , R4 , O1 , O2 , O3 , O4 }.
NPV is computed as:

R1 − O1 R2 − O2 R3 − O3 R4 − O4
NPV(α) = −B0 + + + + .
1+k (1 + k)2 (1 + k)3 (1 + k)4

T
We assume that straight-line (SL) depreciation is employed, which implies that the in-

IP
vested capital depreciates linearly with time: Dept = γB0 where γ = 1/p. This means
P

Bt = B0 (1 − γt) and, in turn, B = B0 · pt=1 1 − t−1 p (1 + k)
−(t−1) . This implies

CR
ROIt = (Rt − Ot − γB0 )/(B0 (1 − γ(t − 1))). The average ROI can be computed as a
weighted average of the ROIs or as the ratio of overall profit to overall capital, B. Equiv-
alently, using NPV, one can compute it as the value obtained by the AIRR function at
C = B. Specifically, ı̄(B) = k + NPV(1 + k)/B.
US
Example 1. Assume B0 = 750 and k = 10%. Table 1 describes the base value
AN
α0 = (R10 , R20 , R30 , R40 , O10 , O20 , O30 , O40 )

and reports the corresponding Free Cash Flows and valuation metrics. The NPV is
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157.37 = −750+380/1.1+270/(1.1)2 +360/(1.1)3 +100/(1.1)4 . Considering that deprecia-

tion charge is 750/4 = 187.5, the vector of capitals associated with the average ROI is B =
(750, 562.5, 375, 187.5, 0) and B = 1712.15 = 750 + 562.5/1.1 + 375/(1.1)2 + 187.5/(1.1)3 .
ED

Therefore, the average ROI is equal to ı̄(B) = 10% + 157.37/1712.15 · 1.1 = 20.11%.

Table 1: Investment evaluated in α0

PT

0 1 2 3 4
Rt0 580 570 560 400
CE

Ot0 200 300 200 300

Ft −750 380 270 360 100
Valuation
NPV 157.37
AC

ı̄(B) 20.11%

Let α1 be the vector of new values of revenues and costs (see Table 2), with the
corresponding new values of Ft , NPV, and ı̄(B). In α1 , NPV is 442.92, ı̄(B) is 38.46%.
The observed variations are: ∆NPV = 285.55 = 442.92 − 157.37; ∆ı̄(B) = 18.35% =
38.46% − 20.11%. Table 3 shows the First Order FCSIs (Φ1,f f
i ), the ranks (ri ), and the
Savage scores of parameters (Sif ) for NPV and ı̄(B). The First Order FCSIs are equal:
1,ı̄(B)
Φ1,npv
i = Φi . Hence, ı̄(B) and NPV are strongly coherent in a strict sense and the
degree of coherence is maximum: ρı̄(B),npv = ρS ı̄(B) ,S npv = 1. (Note that, in this case, Total
Order FCSIs and First Order FCSIs coincide, because the value drivers do not interact
one another.)

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Table 2: Investment evaluated in α1

0 1 2 3 4
Rt1 800 810 780 630
Ot1 350 250 380 600
Ft −750 450 560 400 30
Valuation
NPV 442.92
ı̄(B) 38.46%

Table 3: Finite Change Sensitivity Indices

T
NPV Average ROI

IP
T,ı̄(B) 1,ı̄(B) ı̄(B) ı̄(B)
Parameter ΦT,npv
i = Φi1,npv rinpv Sinpv Φi = Φi ri Si
R1 70.04% 2 1.718 70.04% 2 1.718

CR
R2 69.46% 3 1.218 69.46% 3 1.218
R3 57.89% 4 0.885 57.89% 4 0.885
R4 55.01% 5 0.635 55.01% 5 0.635
O1 −47.76% 6 0.435 −47.76% 6 0.435
O2 14.47% 8 0.125 14.47% 8 0.125
O3
O4
Correlations
−47.36%
−71.76%
7
1 US
0.268
2.718
−47.36%
−71.76%
7
1
0.268
2.718
AN
ρı̄(B),npv 1
ρS ı̄(B) ,S npv 1

We now illustrate one numerical example where the DIM technique is used. It is a
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local SA technique, so it measures the value drivers’ impact on the objective function in a
neighbourhood of α0 . We assume that changes in the inputs are proportional to the base
ED

value (dαi = ξ · αi0 for some ξ 6= 0) so the resulting DIM is

fα0 (α0 ) · ξ · αi0 fα0 i (α0 ) · αi0

DIMif (α0 ) = Pn i 0 0 0 = P n 0 0 0 (23)
j=1 fαj (α ) · ξ · αj j=1 fαj (α ) · αj
PT

(Borgonovo and Apostolakis 2001, Borgonovo and Peccati 2004). In particular, the first
partial derivatives of NPV(α), evaluated in α0 , are
CE

(
(1 + k)−i , i = 1, 2, 3, 4;
NPV0αi (α0 ) = (24)
−(1 + k)−(i−4) , i = 5, 6, 7, 8.
AC

The first partial derivatives of ı̄(B), evaluated in α0 , are

1+k
ı̄(B) 0αi (α0 ) = NPV0αi (α0 ) · . (25)
B

Example 2. Consider a four-period investment P , with B0 = 900 and k = 8%. Hence,

Dept =225 which implies B = 2089.41. The base value is

α0 = (900, 1000, 1100, 1200, 600, 700, 800, 900).

The corresponding cash-flow vector is F = (−900, 300, 300, 300, 300) and NPV = 93.64,

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ı̄(B) = 12.84%. Table 4 shows the DIMs, the ranks, and the Savage scores. As expected,
the two metrics share the same rank and even the same DIMs. Therefore, they are strictly
coherent.

Table 4: Coherence under DIM technique

NPV Average ROI

ı̄(B) ı̄(B) ı̄(B)
Parameter α0 DIMinpv (α0 ) rinpv Sinpv DIMi (α0 ) ri Si
R1 900 83.87% 4 0.885 83.87% 4 0.885
R2 1000 86.28% 3 1.218 86.28% 3 1.218
R3 1100 87.88% 2 1.718 87.88% 2 1.718
R4 1200 88.77% 1 2.718 88.77% 1 2.718

T
O1 600 −55.91% 8 0.125 −55.91% 8 0.125
O2 700 −60.40% 7 0.268 −60.40% 7 0.268

IP
O3 800 −63.91% 6 0.435 −63.91% 6 0.435
O4 900 −66.58% 5 0.635 −66.58% 5 0.635

CR
Correlations
ρı̄(B),npv 1
ρS ı̄(B) ,S npv 1

6.2 Declining-balance depreciation

US
AN
We discuss a model based on (2). In particular, we borrow from Hartman (2007, p. 344)
a real-life application. In 2003, Sunoco Inc. agreed to build a coke-making plant with an
annual capacity of 550,000 tons per year in order to supply plants of International Steel
M

Group (ISG) Inc. The cost of the plant was $140 million and ISG agreed to purchase the
coke (needed for producing steel) for the next 15 years.
Table 5 collects the (stochastic and non-stochastic) relevant data affecting the project’s
ED

economic profitability. The 11 stochastic parameters are evaluated in the base case α0 .
We assume that the facility is depreciated in 15 years with a double-declining balance


PT

switching to SL depreciation (DDB-SL), that is, Dept = max 2/p · Ct−1 ; Ct−1 /(p − t + 1) .
This implies that the depreciation schedule is
CE

Dep = (18.67, 16.18, 14.02, 12.15, 10.53, 9.13, 7.91, 6.86, 6.37, 6.37, 6.37, 6.37, 6.37, 6.37, 6.37).

From (2) the after-tax operating profit is obtained as

AC

It = q · p0 (1 + gp )t − M (1 + gm )t − L(1 + gl )t − E(1 + ge )t − Ov − Dept (1 − τ ).

Table 6 describes the value drivers at α0 and α1 and the resulting value of NPV and
average ROI. The individual and total contribution of the value drivers, as well as the
ranking, are measured via the First Order FCSI and the Total Order FCSI respectively
(Tables 7 and 8). Unlike the previous example, the two FCSIs are not equal, owing to
nonzero interactions among the value drivers. As expected, the effect of each parameter
on average ROI is the same as its effect on NPV, in terms of both magnitude and direction,
which means that the average ROI and the NPV are strictly coherent.8
8
It is interesting to note that, while the change in both NPV and average ROI is not so large, the effect of

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Table 5: Sunoco project: Input data

Stochastic (value drivers)

Annual production q 0.55 million tons
Price p0 $350 per ton
Price growth rate gp 2%
Materials M $27.5 million
Materials growth rate gm 2%
Labor L $75 million
Labor growth rate gl 5%

T
Energy E $20 million
Energy growth rate ge 3%

IP
Overhead Ov $7 million
Tax rate τ 35%

CR
Non-stochastic
Investment $140 million
Salvage Value $0 million
COC 12%
Periods
Dep Method
US 15
DDB-SL
years
AN
M
ED

Table 6: Sunoco project evaluated in α0 and α1

Parameter α0 α1
PT

q 0.55 0.57
p0 $350 $340
gp 2.0% 2.5%
CE

M $27.5 $35.0
gm 2.0% 3.6%
L $75 $68
AC

gl 5.0% 4.0%
E $20 $25
ge 3.0% 2.0%
O $7 $10
τ 35.0% 38.0%
Valuation α0 α1
NPV $120.61 $128.53
ı̄(B) 34.60% 36.08%

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Table 7: Sunoco project: First Order FCSI

NPV Average ROI

1,ı̄(B) ı̄(B) ı̄(B)
Parameter Φi1,npv rinpv Sinpv Φi ri Si
q 442.23% 3 1.520 442.23% 3 1.520
p0 −347.47% 6 0.737 −347.47% 6 0.737
gp 383.60% 4 1.187 383.60% 4 1.187
M −473.82% 2 2.020 −473.82% 2 2.020

T
gm −183.24% 8 0.427 −183.24% 8 0.427
L 534.84% 1 3.020 534.84% 1 3.020

IP
gl 357.37% 5 0.937 357.37% 5 0.937
E −336.21% 7 0.570 −336.21% 7 0.570
ge 81.31% 11 0.091 81.31% 11 0.091

CR
O −167.82% 9 0.302 −167.82% 9 0.302
τ −107.70% 10 0.191 −107.70% 10 0.191
Correlations
ρı̄(B),npv
ρS ı̄(B) ,S npv
1
1
US
AN
M
ED

Table 8: Sunoco project: Total Order FCSI

PT

NPV Average ROI

T,ı̄(B) ı̄(B) ı̄(B)
Parameter ΦT,npv
i rinpv Sinpv Φi ri Si
q 422.70% 3 1.520 422.70% 3 1.520
CE

p0 −354.32% 5 0.937 −354.32% 5 0.937

gp 368.36% 4 1.187 368.36% 4 1.187
M −499.62% 1 3.020 −499.62% 1 3.020
gm −222.45% 8 0.427 −222.45% 8 0.427
AC

L 478.34% 2 2.020 478.34% 2 2.020

gl 309.06% 6 0.737 309.06% 6 0.737
E −301.30% 7 0.570 −301.30% 7 0.570
ge 96.95% 11 0.091 96.95% 11 0.091
O −160.07% 9 0.302 −160.07% 9 0.302
τ −117.75% 10 0.191 −117.75% 10 0.191
Correlations
ρı̄(B),npv 1
ρS ı̄(B) ,S npv 1

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We now use Sunoco’s example to show the behavior of the two metrics with the DIM
technique. The computation of DIMs is easy, given that the calculation of the partial
derivatives of NPV with respect to each parameter is straightforward (see Appendix B
for the list of derivatives) and the derivatives of the average ROI is obtained from (25).
The strict coherence obviously holds. It is interesting to note that, in this case, there
are ties: The first rank is shared by two key drivers, the current price, p0 , and the quantity
sold, q. An equal relative change of either parameter affects the average ROI (and the
NPV) in the same way. The operating costs related to labor are top drivers (labor cost
has rank 3 and its growth rate has rank 4). Much less impact have the growth rates in
energy and materials (rank 10 and 11, respectively).

T
Table 9: Sunoco project: DIM technique

IP
NPV Average ROI

CR
ı̄(B) ı̄(B) ı̄(B)
Parameter α0 DIMinpv (α0 ) rinpv Sinpv DIMi (α0 ) ri Si
q 0.55 93.27% 1.5 2.520 93.27% 1.5 2.520
p0 350 93.27% 1.5 2.520 93.27% 1.5 2.520
gp 2% 11.54% 6 0.737 11.54% 6 0.737
M
gm
L
gl
27.5
2%
75
5%
−13.32%
−1.65%
−43.95%
−14.26%
5
11
3
4
US 0.937
0.091
1.520
1.187
−13.32%
−1.65%
−43.95%
−14.26%
5
11
3
4
0.937
0.091
1.520
1.187
AN
E 20 −10.31% 7 0.570 −10.31% 7 0.570
ge 3% −1.95% 10 0.191 −1.95% 10 0.191
O 7 −3.00% 9 0.302 −3.00% 9 0.302
τ 35% −9.64% 8 0.427 −9.64% 8 0.427
M

Correlations
ρı̄(B),npv 1
ρS ı̄(B) ,S npv 1
ED

7 Concluding remarks
PT

Many different investment criteria are available to managers, professionals and practition-
ers. NPV is considered a theoretically reliable measure of economic profitability. Indus-
CE

trial and financial investments are often evaluated through relative measures of worth as
well. Recently, it has been introduced a new class of return rates named AIRR (Magni
AC

2010, Magni 2013). This class includes the average ROI, which plays an important role
in the appraisal of industrial investments (Magni 2015, Mørch et al. 2017). The average
ROI exists and is unique, and is coherent with NPV in the sense that it correctly signals
value creation or value destruction, just like the NPV (and, therefore, the decision made
using either metric is the same).
This work provides a new definition of NPV-consistency making use of sensitivity anal-
ysis (SA). Given an SA technique, a metric is strongly consistent or coherent with NPV if
it fulfills the classical definition of NPV-consistency and generates the same ranking of the

each parameter on the two metrics is extremely high. In this model, the NPV and the average ROI are highly
sensitive to the contributions of each driver but, overall, the parameters’ effects reciprocally compensate, in
such a way that the resulting change is “smoothed”.

21
 ACCEPTED MANUSCRIPT

value drivers as that generated by the NPV. If, in addition, the parameters’ relevances are
equal to the ones associated with NPV, then the metric and NPV are strongly consistent
in a strict form.
We assume that the COC is exogenously fixed by the decision maker, as well as the
initial investment and the lifetime of the project. After proving that an affine transfor-
mation of a function preserves the ranking, we show that the average ROI, being an affine
transformation of NPV, is strongly NPV-consistent under several (possibly, all) different
techniques of SA.
We have illustrated some simple numerical examples using FCSI (Borgonovo 2010a)
and DIM (Borgonovo and Apostolakis 2001, Borgonovo and Peccati 2004), based on differ-

T
ent depreciation plans (straight-line depreciation and accelerated depreciation). We have

IP
measured the degree of NPV-consistency via Spearman’s (1904) coefficient and Iman and
Conover’s (1987) top-down coefficient. We have found that average ROI and NPV show

CR
perfect correlation and even strict consistency. However, we stress that not all AIRRs
enjoy strong NPV-consistency, including the economic AIRR and the IRR, both showing
degrees of incoherence that may be nonnegligible (see Appendix A).

US
The findings allow us to claim that the average ROI can be reliably associated with
NPV, providing consistent pieces of information. Also, the average ROI is a good can-
didate for absolute NPV-consistency, to be intended as a strong coherence under any
AN
possible technique of SA (this should hold, given the affine relation between the average
ROI and NPV). Future researches may be devoted to finding other relative measures of
worth that enjoy strong NPV-consistency.
M

Acknowledgments. We thank four anonymous reviewers for their invaluable insights that
ED

have proved helpful in the revision of the paper.

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Appendix A. (Non)strong consistency of other AIRRs

In principle, the class of AIRRs consists of infinitely many rates of return (albeit most of
them non-economically significant), so it is no wonder that many of them are not strongly-

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consistent. In this appendix we briefly focus on four special cases of AIRR, three of which
are not strongly consistent with NPV.
Internal rate of return (IRR). Magni (2010, 2013) shows that the internal rate
of return (IRR) is a special case of AIRR. Specifically, the IRR is a weighted mean
of generally time-varying period rates, generated by any vector C fulfilling the following
P P
condition: C = pt=1 pk=t Fk (1+x)−(p−t+1) ·(1+k)−(t−1) .9 While the IRR is traditionally
NPV-consistent (Hazen 2003), it suffers from some difficulties that have been extensively
investigated in the literature. A part of it has been concerned with the necessary and
sufficient condition for existence and uniqueness (e.g., Soper 1959, De Faro 1978, Bernhard
1980. See also Magni 2010 and references therein) or with project ranking (see Ekern 1981,

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Foster and Mitra 2003 for ranking of risk-free projects. See Ben-Horin and Kroll 2017 for

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ranking of nonequivalent-risk projects). In particular, Ekern (1981) and Foster and Mitra
(2003) can be interpreted as supplying conditions of (non)existence of IRR in the interval

CR
(0, +∞), assuming that a project is ranked against the null alternative. Therefore, they
provide a tool to measure the robustness of a value-creating project under changes in
the COC and, at the same time, the conditions where IRR does not exist and cannot

US
then be employed for ranking value drivers.10 Percoco and Borgonovo (2012) show that,
if the IRR exists and is unique, the ranking of value drivers provided by IRR is not
equal to the ranking provided by the NPV, which means that the IRR is not strongly
AN
NPV-consistent. It is easy to see that its degree of NPV-inconsistency, as measured by
Spearman’s correlation coefficient or Iman and Conover’s top-down coefficient, may be
not negligible.11 Also, the financial nature of the IRR is not unambiguously determined:
M

An investment project may well turn to a financing project if value drivers change, which
makes SA meaningless.
Economic AIRR (EAIRR). Another relevant AIRR is the economic AIRR, based
ED

on market values (Magni 2013, 2014, Barry and Robison 2014, Bosch-Badia et al. 2014).
Pp t−k for all t = 1, . . . , p − 1 (while
It is generated by picking Ct = k=t+1 Fk (1 + k)
C0 = −F0 ), which represents the economic value of the project at time t. The EAIRR is
PT

NPV-consistent in a traditional sense and, unlike the IRR, this AIRR always exists and
is unique. However, just like the IRR, its financial nature may change under changes in
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the value drivers.

Strong consistency with NPV is not guaranteed because Ct (and, therefore, C) depends
on F which, in turn, depends on the value drivers. Hence, it is not an affine transformation
AC

of NPV. The degree of inconsistency may be rather high.12

9
This implies that the assumption it = x for all t is sufficient but not necessary to generate a rate of return
equal to IRR.
10
It is usually believed that the case of no-IRR is very rare. However, in some engineering projects it is
not infrequent that disposal and remedial costs occur at the terminal date, which is a necessary condition for
inexistence of IRR. Most recently, Lima e Silva et al. (2017) focus on a very common transaction where the
case of (multiple IRRs and) no IRR is the rule rather than the exception.
11
For instance, in Example 1, the ranking generated by IRR with Total Order FCSIs is (1, 2, 4, 6, 3, 8, 7, 5) and
the top-down coefficient is ρS npv ,S irr = 0.409. In Example 2, where the DIM technique is used, the parameters
ranking supplied by IRR is (1, 2, 3, 4, 8, 7, 5, 6) and the top-down coefficient is ρS npv ,S irr = 0.309.
12
In Example 1, the Total Order FCSIs for EAIRR generate the parameters’ ranking (1, 2, 4, 6, 3, 8, 5, 7) and
ρS npv ,S eairr = 0.239. In Example 2 the ranking is (1, 2, 3, 4, 8, 7, 5, 6) (equal to the ranking of IRR) and, therefore,
the top-down coefficient is equal as well: ρS npv ,S eairr = 0.309.

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Modified internal rate of return (MIRR). The MIRR approach, also known
as the external-rate-of-return approach, consists of modifying project P by discounting
and/or compounding some or all of its cash flows at an external rate so as to generate a
0 0
modified project P (with a modified cash-flow stream F ) bearing the same NPV as P
0
but such that F has only one change in sign for the cash-flow stream. This guarantees
0
that the IRR of P (i.e., the MIRR of P ) exists and is unique. The MIRR suffers from
some ambiguities of definition: (i) it is not clear what the external rate should be, (ii)
there are many ways to modify the project (resulting in different MIRRs), none of which
seems to deserve a privileged status, and (iii) it does not actually measure P ’s rate of
return (see Brealey and Myers 2000, Ross et al. 2011, Magni 2015).

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0 0
Being an IRR of P , the MIRR is an AIRR of P and is not strongly consistent.

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Further, the external rate from which it depends adds a source of uncertainty in the
valuation process (it may be equal or different from the COC). This implies that the

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MIRR may not be NPV-consistent, not even in the traditional sense (see Magni 2015,
Appendix).
Profitability Index (PI). PI is defined as P I = NPV/B0 . It is a well-known and

US
widespread metric that measures the NPV per unit of initial investment. It is strongly
consistent with NPV in a strict sense, as it is an affine transformation of NPV. It is also
easily seen that the PI is strictly linked with AIRR; namely, if cash-flow accounting is used,
AN
that is, assets are expensed immediately (whence Bt = 0 for t > 0), then B = B0 and,
from (9), ı̄(B0 ) = k + NPV(1 + k)/B0 whence P I = (ı̄(B0 ) − k)/(1 + k). Therefore, PI is
an affine transformation of the average ROI that is associated with a cash-flow-accounting
M

depreciation system.
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AC

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Appendix B. Partial derivatives

15 
X 
1 + gp t
NPV0q (α0 ) = p0 · (1 − τ ) · ,
1+k
t=1
15 
X 
1 + gp t
NPV0p0 (α0 ) = q · (1 − τ ) · ,
1+k
t=1
15
X t · (1 + gp )t−1
NPV0gp (α0 ) = p0 · q · (1 − τ ) · ,
(1 + k)t
t=1
15 
X 
0 0 1 + gm t

T
NPVM (α ) = −(1 − τ ) · ,
1+k
t=1

IP
15
X t · (1 + gm )t−1
NPV0gm (α0 ) = −M · (1 − τ ) · ,
(1 + k)t
t=1

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15 
X 
0 0 1 + gl t
NPVL (α ) = −(1 − τ ) · ,
1+k
t=1
15
X

0 0
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NPV0gl (α0 ) = −L · (1 − τ ) ·

15 
X 
1 + ge t
t · (1 + gl )t−1
(1 + k)t
t=1
,
AN
NPVE (α ) = −(1 − τ ) · ,
1+k
t=1
15
X t · (1 + ge )t−1
NPV0ge (α0 ) = −E · (1 − τ ) · ,
(1 + k)t
M

t=1
15
X 1
NPV0O (α0 ) = −(1 − τ ) · ,
(1 + k)t
t=1
ED

15
X It
NPV0τ (α0 ) = − .
(1 − τ ) · (1 + k)t
t=1
PT
CE
AC

31