IRR, ROE and NPV: Formal and Conceptual Convergences in a Systemic Approach (Tir, Roe e Van: Convergenze Formali e Concettuali in un Approccio Sistemico)

Abstract

In capital budgeting, the internal rate of return (IRR) criterion and the net present value (NPV) criterion are considered incompatible in several cases. A longstanding debate developed in past years about the reliability of either method is still an issue of investigation (see, for example, Promislow and Spring, 1996). This paper shows that, employing a systemic perspective, the two models are actually always consistent. Methodologically, the idea is, so to say, accounting-flavoured: it consists of focusing on stocks as well as on flows. In particular the investor's wealth is represented as a financial dynamic system (graphically described by double-entry sheets) and attention is drawn to initial and terminal positions of the system. The equivalence of the IRR and the NPV methods extends to the use of the ROE. An illustrative example is presented where the two alternatives "accept" and "reject" differently reverberate on the system and its terminal position. The comparison between the two alternative terminal positions may equivalently be expressed in terms of the system's IRR or the system's NPV. The systemic approach naturally originates a new definition of residual income, the Systemic Value Added, which is radically different from the standard models (e.g. EVA). The Systemic Value Added (SVA) paradigm is drawn from two different evolutions of the investor's financial system: one relates to the net income in case the project is accepted at time 0, the other one relates to the counterfactual net income that would be obtained from the system if, at time 0, funds were invested in the alternative course of action. It is shown that the sum of the SVAs leads to the Net Final Value with no need of compounding, contrary to the standard residual income. [An English translation of the section introducing the SVA is provided at the end of the original paper]

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Electronic copy available at: http://ssrn.com/abstract=1104597
Tir, Roe e Van: convergenze formali e concettuali in un approccio sistemico
[Irr, Roe and Npv: Formal and Conceptual Convergences in a Systemic Approach]
Carlo Alberto Magni
University “L. Bocconi”, Milan, Italy
University of Modena and Reggio Emilia, Italy
magni@unimo.it
Finanza marketing e produzione, 18(4), 31–59, December 2000
Abstract. In capital budgeting, the internal rate of return (IRR) criterion and the net present value
(NPV) criterion are considered incompatible in several cases. A longstanding debate developed in
past years about the reliability of either method is still an issue of investigation (see, for example,
Promislow and Spring, 1996). This paper shows that, employing a systemic perspective, the two
models are actually always consistent. Methodologically, the idea is, so to say, accounting-
flavoured: it consists of focusing on stocks as well as on flows. In particular the investor’s wealth is
represented as a financial dynamic system (graphically described by double-entry sheets) and
attention is drawn to initial and terminal positions of the system. The equivalence of the IRR and
the NPV methods extends to the use of the ROE. An illustrative example is presented where the two
alternatives “accept” and “reject” differently reverberate on the system and its terminal position.
The comparison between the two alternative terminal positions may equivalently be expressed in
terms of the system’s IRR or the system’s NPV. The systemic approach naturally originates a new
definition of residual income, the Systemic Value Added, which is radically different from the
standard models (e.g. EVA). The Systemic Value Added (SVA) paradigm is drawn from two
different evolutions of the investor’s financial system: one relates to the net income in case the
project is accepted at time 0, the other one relates to the counterfactual net income that would be
obtained from the system if, at time 0, funds were invested in the alternative course of action. It is
shown that the sum of the SVAs leads to the Net Final Value with no need of compounding,
contrary to the standard residual income.
[An English translation of the section introducing the SVA is provided at the end of the original
paper]
Suggested citation:
Magni, C.A. (2000). Tir, Roe e Van: convergenze formali e concettuali in un approccio sistemico
[Irr, Roe and Npv: Formal and Conceptual Convergences in a Systemic Approach]. Finanza
marketing e produzione, 18(4), 31–59, December.

Electronic copy available at: http://ssrn.com/abstract=1104597

8. Economic Value Added (EVA) and Systemic Value Added (SVA)
The systemic approach enables one to show that Stewart’s (1991) EVA is formally compatible with
the internal rate of return (IRR), the return on equity (ROE) and the net-present-value (NPV)
approaches. It suffices to show the equivalence of the NPV model with the EVA model. But this
equivalence is already well-known (see, among others, Stewart, 1991; Esposito, 1998; Magni,
2000a). Therefore, a change of perspective gives the opportunity of condensing four value-creation
models into one. Not only: our approach enables us to introduce a new residual income model,
alternative to the EVA model. Let us consider project A with outstanding capital A
s
and internal
rate of return equal to
A
δ
, assuming it is partially financed with debt. Let D
s
be the residual debt
outstanding at time s and let
D
δ
be the interest rate on debt. According to the EVA model (in a
proprietary approach), at the beginning of each period the capital A
s
may alternatively be invested
in the project, so that the net income is
11 −−
−
sDsA
DA
δ
δ
or may be invested at the opportunity cost of capital
i , which represents the rate of return of a
feasible alternative course of action. In the latter case net income is equal to
1−s
iA
. The differences
between the two incomes is the residual income:
)(EVA
1111 −−−−
−
−
−
=
sssDsAs
DAiDA
δ
δ
. (6)
Changing perspective and assuming a systemic point of view, let us consider the evolution of the
investor’s financial system in case of project acceptance. Assuming cash flows are (withdrawn from
and) invested in a financial asset C, whose borrowing and lending rate of interest is
i , the financial
system at time s is structured in three items: asset C, liability D (debt), project A, in addition to the
investor’s net worth, whose value I
s
fulfils the accounting equation I
s
=C
s
+A
s
–D
s
holds. Thus, if the
Following is the English translation of section 8

project is undertaken, the amount A
0
=a
0
is withdrawn from item C and the cash flows a
s
are
reinvested in (or withdrawn from) item C at each date. We have, at date s,
Assets
Liabilities
Financial asset (C
s
)
Project A (A
s
)
Debt (D
s
)
Net worth (I
s
)
where
C
s
=C
s–1
(1+i)+a
s
– f
s
D
s
=D
s–1
(1+
D
δ
)– f
s
A
s
=A
s–1
(1+
A
δ
)–a
s
with f
s
being is the instalment due for debt repayment. The residual income derived from this
situation is
I
s
–I
s–1
=
A
δ
A
s–1
+i C
s–1
–
D
δ
D
s–1
. (7)
If the investor decides not to withdraw A
0
from item C and thus not to invest in A, then the financial
system is composed of a single item:
Assets
Liabilities
Financial asset (C
s
)
Net worth (I
s
)

and the net income will be
I
s
–I
s–1
= i I
s–1
= i I
0
(1+ i)
s–1
(8)
The difference between eq. (7) and eq. (8) may be interpreted as the residual income; we will call it
Systemic Value Added (SVA):
))1((SVA
0111
s
ssDsAs
iICiDA +−+−=
−−−
δδ
Since, in general,
1101
)1(
−−−
−≠+−
ss
s
s
ADiIC , we have SVA
s
≠ EVA
s
. It is worth noting that the
sum of all SVAs coincides with the compounded Net Present Value, that is the Net Final Value. As
a result, the SVA is consistent with the NPV, the IRR, the ROE and, at an aggregate level, with
EVA. The level is inconsistent with the SVA not in terms of aggregate level but in terms of residual
income in each period. It is easy to show that following relations:
ks
s
k
k
s
k
k
i
−
==
+=
∑∑
)1(EVASVA
11
for all 1≥s
NFV)1(NPV)1(EVASVA
11
=+=+=
−
==
∑∑
nsn
n
s
s
n
s
s
ii
where NFV=Net Final Value.
We cross-refer the reader to Magni (2000a, 2000b; 2000c) for formal proofs and a thorough
investigation of this model; it is worthwhile noting here that the model is the result of the (systemic)
idea of focusing attention on the investor’s endowment as a financial system. In the example we
have dealt with a simplified system, but generalizations to a more complex system is
straightforward.