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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.