Decomposition of Net Final Values: Systemic Value Added and Residual Income

Transcription

1 See discussions, stats, and author profiles for this publication at: Decomposition of Net Final Values: Systemic Value Added and Residual Income ARTICLE in BULLETIN OF ECONOMIC RESEARCH FEBRUARY 2003 Impact Factor: 0.19 DOI: / Source: RePEc CITATIONS 13 READS 43 1 AUTHOR: Carlo Alberto Magni Università degli Studi di M 83 PUBLICATIONS 328 CITATIONS SEE PROFILE Available from: Carlo Alberto Magni Retrieved on: 08 April 2016

2 # Blackwell Publishing Ltd and the Board of Trustees of the Bulletin of Economic Research Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA Bulletin of Economic Research 55:2, 2003, DECOMPOSITION OF NET FINAL VALUES: SYSTEMIC VALUE ADDED AND RESIDUAL INCOME Carlo Alberto Magni Dipartimento di Economia Politica Universita` di Modena e Reggio Emilia ABSTRACT This paper proposes a model aiming at decomposing the Net Final Value of a project under certainty. It makes use of a systemic outlook: the investor s net worth is regarded as a dynamic system whose structure changes over time. On this basis, a profitability index is presented, here named Systemic Value Added (SVA), which lends itself to a periodic decomposition: the periodic shares formally translate the economic concept of residual income (or excess profit). While as an overall index the Systemic Value Added coincides with the Net Final Value (NFV) of an investment, the systemic partition of a SVA is shown to differ from the NFV decomposition model proposed by Peccati (1987, 1991, 1992), which in turn bears a strong resemblance to Stewart s (1991) EVA model. The SVA model and the NFV-based model bear interesting relations: by introducing the concept of a shadow project the SVA model can be re-shaped so that the decomposition of the SVA can be accomplished by applying Peccati s argument to the shadow project, or, which is the same, by computing the shadow project s Economic Value Added. The paper then generalizes the approach allowing for a portfolio of projects, multiple debts and multiple synchronic opportunity costs of capital, for which a tetradimensional decomposition is easily obtained. 149

3 150 BULLETIN OF ECONOMIC RESEARCH I. INTRODUCTION This paper deals with investment evaluation under certainty and with the concept of residual income. The evaluator faces the opportunity of undertaking a project and aims at evaluating its periodic performance as well as its overall profitability, i.e. the so-called residual income (or excess profit). A widely accepted evaluation index for overall performance is the well-known Net Present Value (NPV), or Net Final Value (NFV) if compounded, which evaluates the (overall) differential profit of the opportunity of investing in the project as compared to an alternative course of action. Peccati (1987, 1991, 1992) proposes a way of decomposing the NPV (NFV) of a project, or, in other terms, he proposes a way of formally translating the concept of residual income. This model bears strong resemblance to Stewart s (1991) EVA model, which measures residual income mainly for evaluating firms. An alternative index, named Systemic Value Added, is here proposed by means of a systemic approach: the investor s net worth is seen as a dynamic system whose structure consists of multiple accounts (one of which is the project at hand), which are periodically activated for withdrawals and reinvestments of cash flows. The SVA is amenable to a decomposition into periodic shares representing (periodic) excess profits. Therefore, the model here proposed offers an alternative way of formally translating the classical concept of residual income. Mathematically, the SVA model and Peccati s model are reconciled by introducing a so-called shadow project. The paper is structured as follows: the second section describes the concept of Net Present (Final) Value. Section III presents Peccati s decomposition model showing his NFV-based perspective and its equivalence to Stewart s model. Section IV introduces the SVA, grounded on a different interpretation of the notion of excess profit. Section V is devoted to shedding light on the relations between the approaches. Section VI shows that systemic evaluators using the NFV-based approach contradict themselves. Section VII shows that the contradiction can be formally removed, provided that a so-called shadow project is introduced. Section VIII clarifies the reason why the NFV-based approach may be used even in a systemic perspective. Section IX takes some steps toward a generalization of the model, and a portfolio of investments, loan contracts and opportunity accounts is considered. The final section is devoted to the illustration of a simple example and some remarks conclude the paper. II. THE NPV APPROACH Assume that the evaluator currently invests funds at a rate of interest i and faces the opportunity of a non-deferrable investment, say P : 1 for the 1 We shall not be concerned with real options.

4 SYSTEMIC VALUE ADDED AND RESIDUAL INCOME 151 sake of simplicity we can assume that the project consists of an initial outlay a 0 at time 0, and that equidistant cash flows a s æ 0 will be available at time s, respectively, ; ::: n. The evaluator s initial wealth is E 0, with 0 < a 0 E 0 (the term net worth will also be used as a synonym of wealth). To enrich our analysis we can also assume that the evaluator finances the investment with a loan contract, whose cash flows are f 0 at time 0 and f s 0 at time s æ 1, with 0 f 0 a 0. According to the NPV approach, the investor will accept project P if the investment undertaking will leave her better off than investing funds at the rate i. In the latter case, denoting with E n her net worth at time n, she will hold: E n ¼ E 0 (1 þ i) n ; (1) conversely, if she decides to forego the latter opportunity in order to obtain the sequence: ( a 0 þ f 0 ; a 1 f 1 ; :::; a n f n ); her net worth, denoted with E n, will be E n ¼ (E 0 a 0 þ f 0 )(1 þ i) n þ Xn (a s f s )(1 þ i) n s (2) where we have assumed that each net cash flow will be reinvested (or withdrawn if negative) at the constant rate of interest i, which is the socalled opportunity cost of capital. 2 If (2) is greater than (1), i.e. (E 0 a 0 þ f 0 )(1 þ i) n þ Xn (a s f s )(1 þ i) n s > E 0 (1 þ i) n ; (3) the project should be accepted, otherwise it should be rejected. The comparison in (3) between two final values can be disguised as a comparison between present values by dividing both sides of (3) by (1 þ i) n : ( a 0 þ f 0 ) þ Xn (a s f s )(1 þ i) s > 0: (4a) The left-hand side of (4a) is the well-known Net Present Value (NPV) of the investment at hand; multiplying it by (1 þ i) n we get the Net Final Value (NFV). It is worthwhile noting that the NFV of an investment is nothing but the difference between the two alternative terminal net 2 To be rigorous, if the investor s net worth is negative in some periods, the rate i is not an opportunity cost, it is a genuine rate of cost. However, this is irrelevant to our ends. The assumption a 0 E 0 above is made just for the sake of a better verbal explanation of the decision process at hand. Formally, nothing would change if it did not hold, but in the latter case we could not speak of opportunity to invest at the rate i, as i would be, at least for the first period, a genuine rate of cost.

5 152 BULLETIN OF ECONOMIC RESEARCH worths (see (3)): NFV ¼ E n E n (4b) that is the difference between (2) and (1). 3 This allows us to see the NFV as an index measuring the overall residual income: the investor faces a project P which would leave her with the sum E n. She can alternatively invest the same capital at the opportunity cost of capital i. The difference between the two opportunities gives rise to the global excess profit referred to the entire length of the project. III. THE NFV-BASED DECOMPOSITION The NPV (NFV) is an overall measure showing the global value of a project (global excess profit) and is referred to the whole investment s length. But how is it generated as time passes? How much of it should be ascribed to one period or another? That is, how can we subdivide this index in order to obtain periodic residual incomes g 1, g 2 ; :::; g n, such that g s refers to the s-th period and such that NPV ¼ g 1 þ g 2 þþg n?a periodic decomposition is proposed by Peccati (1987, 1991, 1992). We can summarize this decomposition model by making use of the relations among the cash flows, the project balance and the debt balance. The project balance at time s, at the rate of interest x, is: w 0 ¼ a 0 w s ¼ w s 1 (1 þ x) a s ; 2; :::; n (5) We will also call it outstanding balance or outstanding capital, 4 interpreting it as an account yielding interest at the rate of interest x, where a 0 is invested and the subsequent a s are withdrawn. Likewise, the debt balance at time s at the rate of interest is: D 0 ¼ f 0 D s ¼ D s 1 (1 þ ) f s ; 2; :::; n: (6) We will also call it residual debt or outstanding debt. Ifx is P s internal rate of return and is the debt s contractual rate 5 it is easy to see that w n ¼ 0 and D n ¼ 0. Using (5) and (6) and manipulating, the Net Final Value boils down to: NFV ¼ Xn (w s 1 (x i) þ D s 1 (i ))(1 þ i) n s : (7) 3 Note that NFV ¼ (E n E 0 ) (E n E 0 ). 4 w s coincides in absolute value with the project balance introduced in Teichroew et al., (1965b, p. 155), but has the opposite sign. 5 Under our assumptions x exists and both x and are unique, as the project and the loan contract have strictly monotonic Discounted Cash Flows.

6 SYSTEMIC VALUE ADDED AND RESIDUAL INCOME 153 We can generalize by allowing the project balance and the outstanding debt to have periodic rates of interest x s and s such that (5) and (6) holds with x s and s replacing x and, respectively. If we fix the outstanding capitals and residual debts first (as Peccati suggests as plausible in some cases), then the rates x s and s are univocally determined representing periodic internal rates of return. So doing we rule out any problem of existence and uniqueness of the internal rate of return. Therefore we can relax one of our assumptions by allowing a s 2 R for any s æ 1 (while maintaining a 0 < 0). Equation (7) now holds with x s and s replacing x and, respectively. Letting G s : (w s 1 (x s i) þ D s 1 (i s ))(1 þ i) n s and g s : G s =(1 þ i) n, we have NFV ¼ P n G s and NPV ¼ P n g s. But why should we regard g s (G s ) as economically significant for period s? The answer given by Peccati is the following: at the outset of period s, our investor (fictitiously) invests the amount w s 1,partly financing it with the residual debt D s 1 received from her creditor. At the end of the period she takes back the outstanding capital w s net of the outstanding debt D s along with the net cash flow a s f s.sucha situation can be regarded as a (fictitious) uniperiodic sub-project whose NPV is: w s 1 þ D s 1 (1 þ i) s 1 þ w s D s þ a s f s (1 þ i) s : Using (5) and (6) (with x s and s ) the latter can be manipulated so as to obtain g s, as we wished. The decomposition by Peccati tells us that G s (or g s ) is the value added or subtracted in the s-th period by the project with respect to the alternative course of action (investing funds at the opportunity cost of capital i). When positive, it indicates that the investment is favourable in the s-th period (positive residual income), when negative it shows a decrease in net worth with respect to the alternative opportunity (negative residual income). Peccati s decomposition is formally equivalent to Stewart s Economic Value Added (EVA) model. The Economic Value Added is a periodic residual income, i.e. it is the excess profit generated in a determined period by a firm or by a project. To see the equivalence, let us compute the EVA of P. We just have to calculate the total cost of capital, given by the product of the Weighted Cost of Capital (WACC) and the total capital invested (TC). Then the total cost of capital is subtracted from the Net Operating Profit After Taxes (NOPAT). Notationally, we have, for period s, EVA s ¼ NOPAT WACC + TC (8a)

7 154 BULLETIN OF ECONOMIC RESEARCH where subscripts are omitted for the sake of convenience. It is easy to show that (8a) is just the numerator of g s. In fact, (8a) can be rewritten as whence (ROD + Debt þ i + Equity) EVA s ¼ ROA + TC + TC Debt þ Equity EVA s ¼ ROA + TC ROD + Debt i + (TC Debt) ¼ TC + (ROA i) þ Debt + (i ROD) (8b) where ROA is the Return on Assets, ROD is the Return on Debt, and i is the opportunity cost of capital. All values refer to period s. Since TC = w s 1, ROA = x s, Debt = D s 1, ROD = s, the relation between EVA and g s is established: g s (1 þ i) s ¼ EVA s. We have then NPV ¼ P n g s ¼ P n EVA s(1 þ i) s. IV. A SYSTEMIC PERSPECTIVE Let us abandon the NPV approach and shape the evaluation process differently. The investor is comparing two lines of action, both referred to time 0: (i) undertaking project P; and (ii) investing funds at the opportunity cost of capital i. To evaluate P s periodic performance, she could compute the value of her wealth at each period, for each alternative, until time n. The net worth can be seen as a dynamic system whose path and structure depend on the course of action selected by the evaluator. Graphically, a useful way to describe both the state of the system and its structure consists of making use of double-entry sheets and considering sources and uses of funds. Letting C be the asset yielding interest at the rate i, alternative (ii) leads to: Uses Sources (9a) C s E s for s ¼ 0; 1; 2; :::; n where C s represents the value of account C, where she invests her funds. Since uses and sources of funds must coincide, we have E s ¼ C s. As for case (i), our investor will record two accounts in the debit side and two accounts in the credit side, expressing the fact that she holds an asset C (whose rate of return is i), an asset P whose periodic rate of return is x s, a loan contract D whose periodic rate (of cost) is s and her

8 SYSTEMIC VALUE ADDED AND RESIDUAL INCOME 155 own net worth E. At time s we have Uses Sources C s D s (9b) w s E s where C s, w s, D s, E s are the values of accounts C, w, D, E respectively, and where s ¼ 0; 1; :::; n. For (9a) we state the relations: C 0 ¼ E 0 C s ¼ C s 1 (1 þ i) s æ 1; (10a) whereas for (9b) we have C 0 ¼ E 0 a 0 þ f 0 w 0 ¼ a 0 D 0 ¼ f 0 C s ¼ C s 1 (1 þ i) þ a s f s (10b) w s ¼ w s 1 (1 þ x s ) a s D s ¼ D s 1 (1 þ s ) f s for s æ 1: While (10a) is obvious, it is worth clarifying the meaning of (10b). In case (i) the investor records in the double-entry sheet the following facts: the cash flow f s is withdrawn from account C; the cash flow a s is invested in account C; the capital invested in the project (outstanding capital) increases by the operating profit x s w s 1 and decreases by the receipt a s ; the debt for the loan contract increases by the periodic interest s D s 1 and decreases by the payment f s made to repay the debt. Given (9) at each date s, and using (10), the two alternative dynamic systems may be expressed by the two following equations: E s ¼ E s 1 þ x s w s 1 þ ic s 1 s D s 1 (11) E s ¼ E s 1 þ ic s 1 for (i) and (ii) respectively, where we have, in general, E s 6¼ E s for all s æ 1. From (11) we draw, respectively, E n ¼ E 0 þ Xn (x s w s 1 þ ic s 1 s D s 1 ) E n ¼ E 0 þ Xn ic s 1 : (12)

9 156 BULLETIN OF ECONOMIC RESEARCH whence E n E n ¼ Xn (x s w s 1 s D s 1 i(c s 1 C s 1 )): (13) Equation (13) represents the overall differential profit of alternative (i) over (ii), that is it shows the profitability of P. If (13) is positive, the overall project s evaluation is favourable from the point of view of a wealth maximizer. We can see that this profitability index (based on the notion of wealth as a financial system evolving in time) is already naturally decomposed into n shares to be ascribed to each period. To see how, just think that we aim at answering the following question: what s the difference between what the investor earns in period s if she undertakes P (at time 0) and what she would earn should she invest her funds (at time 0) at the rate i? To answer the question we must compute the difference between net earnings sub (i) and net earnings sub (ii). The respective returns are: (E s E s 1 ) ¼ return from alternative (i) ¼ x s w s 1 þ ic s 1 s D s 1 (E s E s 1 ) ¼ return from alternative (ii) ¼ ic s 1 ; so that the differential gain (or loss) of (i) over (ii) for period s is: (E s E s 1 ) (E s E s 1 ) ¼ x s w s 1 s D s 1 i(c s 1 C s 1 ) (14) which is actually a periodic residual income. We call such an excess profit periodic Systemic Value Added (henceforth SVA s ). Using (13) and denoting with SVA (overall Systemic Value Added) the total differential gain E n E n, we have the following decomposition of SVA: SVA ¼ SVA 1 þ SVA 2 þþsva n : (15) SVA s is therefore that part of the (overall) Systemic Value Added which is generated in the s-th period. SVA s itself can be decomposed in three components: the change in wealth accomplished by the operating profit x s w s 1 (project factor), net of interest payments s D s 1 (debt factor), and the amount i(c s 1 C s 1 ) (opportunity factor), which represents an opportunity cost or return (depending on the sign of (C s 1 C s 1 )), namely the interest the investor gives up (or which accrues to the investor) in the s-th period if she undertakes the project. V. RELATIONS BETWEEN PECCATI S (STEWART S) DECOMPOSITION MODELS AND THE SVA MODEL The decomposition of the SVA in n sub-indexes is accomplished by means of a systemic outlook. Unlike Peccati, we do not rest on the

10 concept of periodic NPV (NFV) and shape the evaluation process in a different way: We regard the decision-maker s net worth as a dynamic system, whose structure consists of uses and sources of funds gradually changing in value as time passes. The net worth is periodically invested and the structure of the system takes account of reinvestments and withdrawals of cash flows. This changes periodically the decision-maker s wealth. The two different courses of action (i) and (ii) are described by the two different paths of the system (10a) and (10b). The Systemic Value Added is consistent with the Net Final Value (and P Net Present Value). The proof is straightforward, since SVA ¼ n (E s E s ) (E s 1 E s 1 ) ¼ E n E n and the latter difference is nothing but the NFV (see (4b)). This result is then consistent with the NFV (NPV) rule in that it states that the total relative gain SVA coincides with the Net Final Value, and a wealth maximizer will accept the project if and only if SVA ¼ NFV ¼ NPV(1 þ i) n > 0. Now we show why the two decomposition models differ in their periodic shares. It is easy to prove that: 0 1 X SVA s ¼ M s þ i@ SVA k A where M s : w s 1 (x s i) þ D s 1 (i s ) ¼ EVA s, (see Appendix). This reformulation enables us to intepret SVA s as the sum of a direct factor M s (the Economic Value Added) and the periodic interest on the (s 1) indirect factors SVA h : the latter represent the gain generated in period s by those shares referring to the previous periods, which yield returns at the rate i. These returns are produced in the s-th period: That is, SVA 1, SVA 2 ; :::; SVA s 1 can be considered assets that add up value to the global relative gain. Therefore, each share depends on all the preceding ones, which keep on bearing interest at the rate i. Such an imputation collides with the NFV-based imputation. To see why, let us assume, for the sake of simplicity, n ¼ 3, and let us decompose the SVA and the NFV. Remembering that G s ¼ M s (1 þ 1) n s we have the following decomposition table: G 1 ¼ M 1 þ (im 1 ) þ (im 1 þ i 2 M 1 ) G 2 ¼ M 2 þ (im 2 ) G 3 ¼ M 3 SYSTEMIC VALUE ADDED AND RESIDUAL INCOME 157 k 1 SVA 1 ¼ M 1 SVA 2 ¼ M 2 þ (im 1 ) SVA 3 ¼ M 3 þ (im 1 þ i 2 M 1 ) þ (im 2 ) (16) where the first column refers to the NFV-based decomposition, the second one refers to the systemic partition. As we can see, the former

11 158 BULLETIN OF ECONOMIC RESEARCH accomplishes a two-step evaluation. The idea is the following: M 1, M 2, M 3 are the three shares for period 1, 2, 3, respectively. As this is money referred to the dates 1, 2, 3, respectively, the basic principles of financial calculus force the evaluator to compound (or discount) flows to take time into consideration. After capitalization (and only after) the evaluator may sum the three shares. Conversely, in the light of our systemic perspective the investor constructs the three shares of the SVA in a gradual way. The first share is M 1 (=EVA 1 ), which exactly represents the difference between what the investor receives in the first period and what she would receive should she decide to forego the project opportunity and invest her funds at the rate i. In the second period the difference between what she receives and what she would have received takes into account that, in addition to M 2 (=EVA 2 ), the first share yields a (differential) interest equal to im 1. Iterating the argument, the third share considers the return on the two first shares M 1 and M 2, along with the interest gained on im 1 itself, which is produced just in the third period. Financially speaking, we could also interpret each periodic Systemic Value Added as a capital invested at time s, yielding linear interest at the rate i until n, for a total interest of (i(n s)sva s ) each. In fact, we can easily check that: NFV ¼ SVA ¼ Xn ¼ Xn SVA s M s þ Xn 0 i@ Xs 1 h ¼ 1 1 SVA h A ¼ Xn M s þ Xn 1 i(n s)sva s : You can see that a different line of argument is obeyed by the NFVbased decomposition. G 1 embodies the term im 1 which the SVA model ascribes to the second share, since it is generated in the second period. In addition, it comprehends the term im 1 þ i 2 M 1 which in turn the SVA partition considers related to the third period. At the same time G 2 includes im 2 but lacks the term im 1 (previously embodied in G 1 ). Finally, the third share G 3 does not include the return on previous periods shares. Seen with our systemic eyes, this model seems not to consider the return yielded by the preceding shares and the use of the capitalization factor (1 þ i) n s means anticipating money that will be earned in future periods. Our systemic perspective provides a tool which does not use capitalization factor, so that the n shares are homogeneous, and our investor can safely sum them.

12 SYSTEMIC VALUE ADDED AND RESIDUAL INCOME 159 Now we investigate thoroughly the assumptions implicit in the two decomposition models. For the sake of simplicity, we will assume D s ¼ 0 for every s, but the line of argument is not invalidated by this restriction, as we will see in Section IX. We have two decomposition models and both aim at answering the following question: What is the differential gain of (i) over (ii) that we are to ascribe to the s-th period? or, in other terms, What is the residual income of project P? The answer to the question above in our systemic outlook is just the difference between alternative net profits, i.e. SVA s ¼ (x s w s 1 þ ic s 1 ) ic s 1. Conversely, following the NFVbased decomposition the investor has the opportunity to invest the sum w s 1 either at the rate x (alternative (i)) or at the rate i (alternative (ii)). If (i) is selected, the net profit will be x s w s 1, if (ii) is chosen the net profit will be iw s 1 ; the differential gain is, in time s value, M s ¼ w s 1 (x s i). VI. INCURRING CONTRADICTIONS We now show that a systemic-minded evaluator measuring residual income by means of the NFV-based argument runs into logical contradictions. In Section VII we will instead show that such contradictions can be healed and the NFV-based models can be useful even for such an evaluator. In Peccati s model n (fictitious) sub-projects are introduced starting at time (s 1) and terminating at time s whose cash flows are w s 1 and w s þ a s respectively, as described in Section IV. Let NFV(s) denote this sub-project s Net Final Value, calculated at time s (or, which is the same, the Net Present Value compounded up to time s): NFV(s) ¼ w s 1 (1 þ i) þ w s þ a s ¼ w s 1 (x s i); (17) (obviously, NFV(s) = EVA s, as we expect). A systemic-minded evaluator interprets the notion of excess profit so that it coincides with a differential net profit. If she then uses (17) as an incremental income, then it must be that NFV(s) ¼ (E s E s 1 ) (E s E s 1 ): (18) But we know that a Net Final Value is the difference between alternative final net worths. So, looking back at (4b), we have NFV(s) ¼ E s E s : (19) The latter two entail E s 1 ¼ E s 1 : (20)

13 160 BULLETIN OF ECONOMIC RESEARCH Equation (20) tells us that if project P is undertaken the net worth at time (s 1) (left-hand side) coincides with the net worth produced if the project is not undertaken (right-hand side). As this is true for every s, we have, by (19), NFV(s) = 0. Assume for the moment the realistic case w s 1 6¼ 0 for all s. We distinguish two exhaustive cases: (a) x s 6¼ i for at least one s; (b) x s ¼ i for all s. If (a) holds, we have two kinds of contradictions: the mathematical and the factual contradiction. As for the mathematical contradiction, let s* be an index such that x s * 6¼ i. As we have seen, we must have NFV(s*)=0, which in turn implies x s * ¼ i, due to (17), but this contradicts the assumption. In addition, the NFV-based argument leads to a factual contradiction. In fact, the latter accomplishes the decomposition by calculating the NFV(s) for period s, which presupposes that the following assumption is made: At time 0 the investor invests her net worth E 0 in asset C at the opportunity cost of capital until time (s 1). At time (s 1) the sum w s 1 is withdrawn by account C and invested in a uniperiodic project with rate of return x s. At time s, the investor holds the final amount w s alongside the value of account C, given by C s ¼ (C s 1 w s 1 )(1 þ i) þ a s. s As s is assumed to hold for every period ; 2; :::; n, then it boils down to a set of n assumptions, 1, 2 ; :::; n, which are incompatible, since they are mutually exclusive. As for (b), it causes the decision process to be an idle issue, as alternative (i) coincides, from a mathematical-financial point of view, with alternative (ii): There is no difference, financially speaking, in investing at the opportunity cost of capital the whole net worth or only a part of it, if the remainder is invested in a project whose rate of return is the opportunity cost of capital itself (this situation can be viewed as different only under a factual perspective, for (i) and (ii), though financially equivalent, represent distinct courses of action). Therefore in case (b) the two arguments lead to the same obvious (and uninteresting) result; further, the factual contradiction persists, as s holds regardless of (a) and (b). It could seem that the contradiction is healed with the assumption of w s 1 ¼ 0 for some s. On the contrary, the contradiction remains regardless of whatsoever assumption is made on the outstanding capitals: for a systemic evaluator, to focus on the NFV(s) means to erase all the differential past prior to time (s 1), as if the evaluator had not undertaken project P at time 0, investing instead her wealth at the opportunity cost of capital. That is, the two different courses of action are made to coincide until time (s 1) (see (20)). To accept s for every s

14 SYSTEMIC VALUE ADDED AND RESIDUAL INCOME 161 means to accept the relation C s ¼ (C s 1 w s 1 )(1 þ i) þ a s for every s. But, will it or not, the systemic evaluator cannot escape the systemic s recurrence equation C s ¼ C s 1 (1 þ i) þ a s for every s. The latter two are mathematically incompatible since they can be rewritten, respectively, as 2 3 C s ¼ 4 s Y 1 s Y 1 E 0 (1 þ i) s 1 a 0 (1 þ x k ) þ a 1 (1 þ x k ) þþa s 1 5(1 þ i) þ a s k ¼ 1 C s ¼ [E 0 (1 þ i) s 1 a 0 (1 þ i) s 1 þ a 1 (1 þ i) s 2 þþa s 1 ](1 þ i) þ a s ; which differ as long as there exists at least one k such that x k 6¼ i. 6 k ¼ 2 VII. RESHAPING THE NFV-BASED ARGUMENT WITH A SYSTEMIC PERSPECTIVE: THE SHADOW PROJECT We have shown that a systemic-minded evaluator runs into logical fallacies if she adopts the NFV-based argument to measure residual income. Therefore, the systemic partition is not only different from the NFV-based one, but also incompatible. This incompatibility is the result of a set of assumptions s which cannot be accepted under a systemic perspective and implies that the evaluator is not able to get from the NFV-based model the piece of information she asks for. Therefore, the two models seem to be radically alternative. Yet, we show in this section that, in a sense, the systemic perspective and the NFV-based argument can be conciliated so that an unbiased systemic partition of the NFV is reached on the basis of a NFV(s) analysis. Let Xw s be the value of w s obtained by replacing each x s with i: Xw s : a 0 (1 þ i) s a 1 (1 þ i) s 1 a s ; 2; :::; n Note that the latter implies Xw s ¼ C s C s and Xw s ¼ Xw s 1 (1 þ i) a s. The systemic C s can then be rewritten as C 0 ¼ C 0 Xw 0 C s ¼ C s 1 (1 þ i) þ a s ¼ (C s 1 Xw s 1 )(1 þ i) þ a s ; 2; :::; n 6 The two C s can be equal for some s, but not for every s.

15 162 BULLETIN OF ECONOMIC RESEARCH where Xw 0 : a 0. Now let Ya 0 : a 0 Ya s : x s w s 1 ixw s 1 þ a s ; 2; :::; n: (21) Suppose that the investor undertakes a project KP, which we name here shadow project of P, consisting of the cash flow stream ( Ya 0 ; Ya 1 ; :::; Ya n ). It is easy to demonstrate that if we apply Peccati s decomposition (i.e. the NFV-based argument) to the shadow project KP (with no use of capitalization factors) we obtain the SVA decomposition of P. From (21) we obtain Xw 0 ¼ Ya 0 Xw s ¼ Xw s 1 (1 þ Zx s ) Ya s ; 2; :::; n (22) where Zx s : x s (w s 1 =Xw s 1 ). We can then interpret Xw s as the project balance of KP at the rate Zx s, and the Ya s as withdrawals from (if positive) or investments in (if negative) an account yielding interest at the periodic rate Zx s, ; 2; ::: n. Applying Peccati s reasoning to KP, the investor invests Xw s 1 at the beginning of the s-th period and receives the sum Xw s þ Ya s at the end of the period: time s 1 s j11111 j111 cash flows Xw s 1 Xw s þ Ya s At time s the periodic net final value of the project is: NFV(s) ¼ Xw s 1 (1 þ i) þ Xw s þ Ya s ¼ Xw s 1 (1 þ i) þ (Xw s 1 (1 þ Zx s ) Ya s ) þ Ya s ¼ Xw s 1 (Zx s i): (23) Note that (23) is just the Economic Value Added of the shadow project (henceforth EVA s ), as we expected, due to equivalence of Peccati s model and Stewart s. We state that such an EVA s coincides with the periodic SVA of project P. In fact, EVA s ¼ NFV(s) ¼ Xw s 1 (Zx s i) ¼ x s w s 1 ixw s 1 ¼ x s w s 1 i(c s 1 C s 1 ) ¼ SVA s : (24)

16 Hence, SVA ¼ P n Xw s 1(Zx s i) ¼ P n NFV(s) ¼ P n EVA s ¼ NFV and the two models are, to a certain extent, reconciled with no need of compounding. 7 It is worthwhile noting that so that a 0 (1 þ i) n þ Xn SYSTEMIC VALUE ADDED AND RESIDUAL INCOME 163 Ya s ¼ SVA s þ a s for s æ 1 a s (1 þ i) n s ¼ Ya 0 þ Xn 0 Ya s a 0 þ Xn a s 1 A (25) where the dependence of Ya s on i is pointed out. Changing i, Ya s (i) adjusts itself so as to avoid the need of compounding. Equation (25) tells us that the Net Final Value of project P, calculatedattheratei, equals the difference between the Net Final Value of the cash-flow stream produced by project KP and the Net Final Value of the cash-flow stream generated by P, both calculated at a zero rate (which means, in other terms, that we do not compound). Therefore, the shadow project is that (fictitious) project which enables us to overlook capitalization processes. It is also worthwhile noting that (24) provides us with the interesting result that the Economic Value Added of KP coincides with the Systemic 7 We can decompose P by directly applying the NFV-based argument to P provided that we take account of the proper net worth s structure by means of a correction factor, which takes into consideration that the NFV-based argument uses w s 1, whereas we should use Xw s 1. This discrepancy arises at the beginning of period s, so that what the NFV-based argument disregards is the differential interest generated during the period. The correction factor must then be i(w s 1 Xw s 1 ) and it allows us to avoid compounding once we have found the periodic NFV of P. So, we first compute G s. Since the latter does not fit the correct system s structure, we need take account of the correction factor, compounded until time n. We sum the two and then offset the previous compounding process by discounting back to time s: We obtain, as a result, We have then Periodic NFV of P ¼ G s þ i(w s 1 Xw s 1 )(1 þ i) n s SVA ¼ Xn (1 þ i) n s ¼ M s þ i(w s 1 Xw s 1 ) ¼ NFV(s) ¼ SVA s : G s þ i(w s 1 Xw s 1 )(1 þ i) n s (1 þ i) n s :

17 164 BULLETIN OF ECONOMIC RESEARCH Value Added of P. We are dealing then with three models (Peccati s, Stewart s, SVA), whose relations can be summarized as follows: G s EVA s ¼ g s (1 þ i) s ¼ EVA (1 þ i) n s s ¼ EVA s þ Xs 1 SVA s ¼ EVA s ¼ NFV(s) k ¼ 1 0 EVA s ¼ EVA s þ i@ s k 1 ieva k (1 þ i) Xs 1 k ¼ 1 1 EVA k A: We end this section by pointing out that the decomposition based on the shadow project is only a particular case of a more general scheme. This means that we have not one but infinite shadow projects. Let P be any of these, consisting of the sequence of cash flows ( a 0 ; a 1 ; :::; a n ). Denote with w s its outstanding balance at rate x s. Fix all w s arbitrarily; then pick a 0 ¼ w 0. As for the other flows, they must satisfy: whence w s 1 (1 þ i) þ w s þ a s ¼ SVA s ; a 0 ¼ a 0 ; a s ¼ w s 1 (1 þ i) w s þ SVA s : (26) The rate x s is thus univocally determined from the outstanding balance equation: so that we can rewrite (26) as: w s ¼ w s 1 (1 þ x s ) a s ; a 0 ¼ a 0 ; a s ¼ w s 1 (x s i) þ SVA s : We have found infinite sequences of cash flows a s ¼ a s (w s 1 ; w s ), depending on the outstanding capitals selected, and infinite shadow projects P. KP is just one of these, obtained by choosing w s ¼ Xw s, which implies w s ¼ w s 1 (1 þ i) a s and therefore a s ¼ a s þ SVA s ¼ Ya s for all s æ 1. VIII. THE SHADOW PROJECT AND THE FINANCIAL SYSTEM In reframing the evaluation process we have applied the NFV-based argument to project KP. In this way, the contradiction we have seen arises

18 SYSTEMIC VALUE ADDED AND RESIDUAL INCOME 165 for project KP, but not for project P. To clear the issue, we can take a look at the following double-entry sheets: Uses Sources C s ¼ C s 1 (1 þ i) þ a s E s ¼ C s 1 þ ic s 1 þ x s w s 1 (27a) w s Uses Sources C s ¼ (C s 1 w s 1 )(1 þ i) þ a s E s ¼ C s 1 þ ic s 1 þ w s 1 (x s i) w s Uses Sources (27b) JC s ¼ (C s 1 Xw s 1 )(1 þ i) þ Ya s KE s ¼ C s 1 þ ic s 1 þ Xw s 1 (Zx s i) Xw s (27c) Equation (27a) applies the systemic argument to project P, (27b) applies the NFV-based argument to project P (that is, the evaluator applies s ), (27c) applies the NFV-based argument to project KP (that is, the evaluator applies s by pretending that alternative (i) is referred to KP, not to P), where the new notations JC s, KE s remind us that account C and the net worth E are measured by pretending that alternative (i) refers to P s shadow project. (27c) implies, as we know by (20), JC s 1 ¼ C s 1 and KE s 1 ¼ E s 1 for every s, in the same sense as (27b) implies C s 1 ¼ C s 1 and E s 1 ¼ E s 1. We know that this means that (27b) leads to the contradiction we have studied for project P. Likewise, (27c) leads to the same contradiction for project KP. But the net profit in (27c) coincides with the net profit in (27a): ic s 1 þ Xw s 1 (Zx s i) ¼ ic s 1 þ x s w s 1 : This entails that while the net profit in (27c) is incorrect for KP (from a systemic view), it is correct for P. So doing, we shift the contradiction, moving it from P to KP. Peccati s NFV-based argument can now be safely applied (without capitalization) because its assumptions invalidate, from a systemic point of view, the decomposition of KP, while recovering at the same time the decomposition of P, which now coincides with the SVA model here introduced. Now the NFV-based model can be used by a systemic evaluator too. To say it in Stewart s terms: to decompose systemically a project P take EVA s not EVA s (and forget capitalization)!

19 166 BULLETIN OF ECONOMIC RESEARCH IX. GENERALIZATIONS In this section we take some generalizations of the aforementioned results. First, we relax our zero debt assumption assuming D s 6¼ 0. Secondly, we assume that the opportunity cost of capital changes over time and denote it with i s for period s. With such assumptions, if we refer alternative (i) to P and apply the systemic argument, the financial system s structure at time s is depicted as follows: Uses Sources C s ¼ C s 1 (1 þ i s ) þ a s f s D s w s E s ¼ E s 1 þ i s C s 1 þ x s w s 1 s D s 1 which is nothing but (9b). Conversely, if we refer alternative (i) to KP and apply the NFV-based argument, the financial system s structure at time s is described as follows: Uses Sources JC s ¼ (C s 1 Xw s 1 þ ID s 1 )(1 þ i s ) ID s þ Ya s Nf s Xws KEs ¼ C s 1 þ isc s 1 þ Xws 1(Zxs is) where þ ID s 1 (i s M s ) ID 0 : D 0 ¼ f 0 ; D s : D s 1 (1 þ s ) f s : ID s : ID s 1 (1 þ i s ) f s : M s : s D s 1 ID s 1 : and where Xw s is now such that Xw s : Xw s 1 (1 þ i s ) a s : Financially speaking, we introduce the shadow project KP whose cash flows Ya s are diminished by the debt cash flows Nf s, so that the net sequence is ( Ya 0 þ Nf 0 ; Ya 1 Nf 1 ; :::; Ya n Nf n ) with Ya 0 ¼ a 0, Nf 0 ¼ f 0, and, for s æ 1, we have Ya s ¼ x s w s 1 i s Xw s 1 þ a s and Nf s ¼ s D s 1 i s ID s 1 þ f s, whence Ya s Nf s ¼ a s f s þ SVA s. The Systemic Value Added for period s is now redefined by replacing i with i s : SVA s : x s w s 1 s D s 1 i s (C s 1 C s 1 ) ¼ x s w s 1 s D s 1 i s (Xw s 1 ID s 1 ): At every time (s 1) the investor invests the sum Xw s 1 in a uniperiodic project whose rate of return is Zx s, which is partly financed with debt

20 (ID s 1 ) and partly with her own net worth (Xw s 1 ID s 1 ), i.e. by a withdrawal from account C. 8 Following Peccati s NFV-based argument, the situation is time s 1 s j j11111 cash flows Xw s 1 ID s 1 Xw s þ Ya s ID s Nf s Using the fact that we get to SYSTEMIC VALUE ADDED AND RESIDUAL INCOME 167 NFV(s) ¼ EVA s Xw s ¼ Xw s 1 (1 þ Zx s ) Ya s ID s ¼ ID s 1 (1 þ M s ) Nf s ¼ (Xw s 1 ID s 1 )(1 þ i s ) þ Xw s þ Ya s ID s Nf s ¼ Xw s 1 (Zx s i s ) ID s 1 (M s i s ) which coincides with P s periodic SVA, as can be easily checked. Let us further generalize by allowing a portfolio of q projects, p opportunity accounts and m loan contracts, with respective periodic rates x r s, i j s, l s, j ¼ 1; :::; p, l ¼ 1; :::; m, r ¼ 1; :::; q.9 The latter generalization forces the evaluator to select one or more opportunity accounts K j to be activated for withdrawals and reinvestment of the cash flows released by the r projects and the l debts. Referring to time s, denote with 8 If the value of account C is negative, account C can be seen as external financing as well as the debt. But the former looks like a current account, whereas the latter is a loan contract. 9 The financial system s structure is now articulated as Uses Sources K 1 s D 1 s K 2 s D 2 s ::: ::: K p s ::: w 1 s ::: w 2 s ::: ::: D m s w q s E s

21 168 BULLETIN OF ECONOMIC RESEARCH the quota of project r s cash flow invested in (if positive) or withdrawn from (if negative) account K j. Likewise, denote with f s lj the quota of debt l s cash flow withdrawn from account K j. Let us give the following notation: 10 a rj s Xw rj 0 : w rj 0 : a rj 0 ID lj 0 : D lj 0 : f lj 0 w rj s : w rj s 1 (1 þ x r rj s ) a s D lj s : D lj s 1 (1 þ s l lj ) f s The value of K j is 0 K j s K j 0 þ Xq r ¼ 1 Ys 1 k ¼ 1 a rj s (1 þ i k ) Xq Xm l ¼ 1 f lj s : r ¼ 1 Xw rj s rj : Xw s 1 (1 þ i s j rj ) a s ID lj lj s : ID s 1 (1 þ i s j lj ) f s Xw rj s 1 þ Xm ID lj s 1 l ¼ 1 1 A(1 þ i s j ) Let us focus on a generic account K j. For it, we have q shadow projects with initial outlays Xw rj 0. It is easy to see that the NFV(s) of the q projects portfolio is X q Xw rj s 1 (Zx r s i j Xm s ) ID lj s 1 (M s l i s j ); (28) r ¼ 1 l ¼ 1 where Zx r s : (x sw rj rj s 1 )=Xw s 1 and M s l : ( sd lj lj s 1 )=ID s 1. Equation (28) is therefore that part of the portfolio s SVA s generated by account j. We can rearrange (28) so as to decompose the share according to the source of funds used. Let s rj rj : Xw s 1 =(P q rj Xw rj lj r ¼ 1 s 1 ); then s ID s 1 is that part borrowed from creditor l financing the initial outlay Xw rj s 1. Rearranging terms and manipulating we obtain: X q m s rj lj ID s 1 (Zx r s M s l ) A þ (Xw rj Xm s 1 s rj lj ID s 1 )(Zx r s i s j ) A5: r ¼ 1 l ¼ 1 ID lj Let A lrj s : s rj s 1 (Zx r s M s l m þ 1; rj ), l ¼ 1; ::: m and denote with A s the last addend in square brackets. Summing for j and s we obtain the portfolio s Net Final Value: l ¼ 1 NFV ¼ Xn X p j ¼ 1 X q r ¼ 1 Xm þ 1 l ¼ 1 A lrj s : (29) 10 Remember that we are assuming, with no loss of generality, 0 f 0 a 0.

22 is the quota of the portfolio s NFV to be ascribed to source l, to project r, to account j, to period s. The evaluation we have arrived at provides us with four types of decomposition: (I) periodic decomposition (the share of portfolio s NFV generated in period s), obtained by summing A lrj s A lrj s for all variables except s; (II) opportunity account decomposition (the share of portfolio s NFV generated by the use of account K j ), obtained by summing A lrj s for all variables except j; (III) project decomposition (the share of portfolio s NFV generated by project r), obtained by summing A lrj s for all variables except r; (IV) financing decomposition (the share of portfolio s NFV generated by the use of source l), obtained by summing A lrj s for all variables except l. Taking the sum for more than one variable we obtain various types of information. For instance, if we wish to know project r s NFV we just have to sum A lrj s for l, j and s. If we instead wish to compute what is the total contribution of account K j to the portfolio s NFV, we must sum it for l, r, and s; if we wish to calculate the total contribution of external financing to the s-th period NFV we must sum it for all l m and then for r and j; and so on. By different use of the variables we get different relevant pieces of information. We end this section by pointing out that the portfolio s NFV in (29) coincides, as we expect, with the portfolio s SVA obtained by directly using the systemic argument, as in Section IV. NFV ¼ SVA ¼ Xn SYSTEMIC VALUE ADDED AND RESIDUAL INCOME 169 SVA s ¼ Xn whose term in brackets mirrors (14). X q r ¼ 1 Xp x r s w r s 1 j ¼ 1 i j s (K j 0 Xm l ¼ 1 Ys 1 k ¼ 1 l D l s 1 1 (1 þ i j k ) K j s 1 ) A X. A NUMERICAL EXAMPLE Assume that project P consists of the sequence ( 1000; 30; 780.5; 10; ), financed by a two-year loan contract consisting of the sequence (600; 20; 770.5). Assume that the initial wealth, before the investment is undertaken, is 500 (all invested in account C) and that i s = 13 per cent for all s. As is easily verified, the internal rate of return for P is x = 20 per cent and the debt rate is = 15 per cent. Considering the net cash flows, the Net Final Value of this levered project is: NFV ¼ 400(1:13) 4 þ 10(1:13) 3 þ 10(1:13) 2 þ 10(1:13) þ 885:84 ¼ 272:148:

23 170 BULLETIN OF ECONOMIC RESEARCH For our decompositions, we apply (10) and get to C 0 ¼ ¼ 100 C 0 ¼ 500 C 1 ¼ 100(1:13) þ 10 ¼ 123 C 1 ¼ 500(1:13) ¼ 565 C 2 ¼ 123(1:13) þ 10 ¼ 148:99 C 2 ¼ 565(1:13) ¼ 638:45 C 3 ¼ 148:99(1:13) þ 10 ¼ 178:36 C 3 ¼ 638:45(1:13) ¼ 721:448 C 4 ¼ 178:358(1:13) þ 885:84 ¼ 1087:385 C 4 ¼ 721:448(1:13) ¼ 815:236 C 0 C 0 ¼ 400 C 1 C 1 ¼ 442 C 2 C 2 ¼ 489:46 C 3 C 3 ¼ 543:089 C 4 C 4 ¼ 272:148: The sequences of outstanding debts and outstanding capitals for P are: D 0 ¼ 600 w 0 ¼ 1000 D 1 ¼ 600(1:15) 20 ¼ 670 w 1 ¼ 1000(1:2) 30 ¼ 1170 D 2 ¼ 670(1:15) 770:5 ¼ 0 w 2 ¼ 1170(1:2) 780:5 ¼ 623:5 D 3 ¼ 0 w 3 ¼ 623:5(1:2) 10 ¼ 738:2 D 4 ¼ 0 w 4 ¼ 738:2(1:2) 885:84 ¼ 0; so that, applying (8b), Stewart s model gives rise to the following decomposition: EVA 1 ¼ 1000(0:2 0:13) þ 600(0:13 0:15) ¼ 58 EVA 2 ¼ 1170(0:2 0:13) þ 670(0:13 0:15) ¼ 68:5 EVA 3 ¼ 623:5(0:2 0:13) þ 0(0:13 0:15) ¼ 43:645 EVA 4 ¼ 738:2(0:2 0:13) þ 0(0:13 0:15) ¼ 51:674: Each row represents the net final value NFV(s) of the (fictitious) oneperiod project introduced by Peccati. Consider for example the second row: it refers to a project consisting of a net outlay of w 1 D 1 ¼ ¼ 500 at time 1 followed by a net receipt equal to (w 2 þ a 2 ) (D 2 þ f 2 ) ¼ (623:5 þ 780:5) 770:5 ¼ 633:5 at time 2. Its net final value is 500(1.13) þ = 68.5, as expected. As for the SVA model we

24 have, using (14), SYSTEMIC VALUE ADDED AND RESIDUAL INCOME 171 SVA 1 ¼ (0:2)1000 (0:15)600 (0:13)400 ¼ 58 SVA 2 ¼ (0:2)1170 (0:15)670 (0:13)442 ¼ 76:04 SVA 3 ¼ (0:2)623:5 (0:15)0 (0:13)489:46 ¼ 61:07 SVA 4 ¼ (0:2)738:2 (0:15)0 (0:13)543:089 ¼ 77:038: To check that SVA s is actually a differential net profit drawn from alternative dynamic systems, just consider E 0 ¼ C 0 þ w 0 D 0 ¼ 500 E 0 ¼ C 0 ¼ 500 E 1 ¼ C 1 þ w 1 D 1 ¼ 623 E 1 ¼ C 1 ¼ 565 E 2 ¼ C 2 þ w 2 D 2 ¼ 772:49 E 2 ¼ C 2 ¼ 638:45 E 3 ¼ C 3 þ w 3 D 3 ¼ 916:68 E 3 ¼ C 3 ¼ 721:448 E 4 ¼ C 4 þ w 4 D 4 ¼ 1087:385 E 4 ¼ C 4 ¼ 815:236: The sequences of net earnings for the two alternatives are, respectively, E 1 E 0 ¼ 123 E 1 E 0 ¼ 65 E 2 E 1 ¼ 149:49 E 2 E 1 ¼ 73:45 E 3 E 2 ¼ 144:19 E 3 E 2 ¼ 82:998 E 4 E 3 ¼ 170:705 E 4 E 3 ¼ 93:788: In the first column we have then the periodic net profits that the investor will benefit if she undertakes investment, the second column refers to leaving money invested in account C. The difference between these alternative profits gives us the excess profit for each period. As can be checked, we actually find back the periodic shares just obtained. 11 The periodic shares of the two models are different but, aggregating them, we have EVA 1 (1:13) 3 þ EVA 2 (1:13) 2 þ EVA 3 (1:13) þ EVA 4 ¼ 272:148 ¼ NFV and SVA 1 þ SVA 2 þ SVA 3 þ SVA 4 ¼ 272:148 ¼ NFV: It is easy to verify that a systemic evaluator obtains the same periodic SVA s by applying Stewart s (Peccati s) model to the shadow project KP 11 Note that we have rounded the numbers in this example for convenience, so we obviously do not find a perfect coincidence.

25 172 BULLETIN OF ECONOMIC RESEARCH along the guidelines we have illustrated in Section IX (we omit the simple calculations). XI. CONCLUSIONS This paper proposes an index for evaluating periodic performance of projects or, in other terms, offers a new way of formally translating the economic concept of residual income. Both Stewart s and Peccati s models aim at measuring residual income, they are just two sides of the same medal. We have shown that the two models are formally equivalent and that the SVA model coincides with them only in overall terms (actually, SVA is nothing but NFV seen from a different perspective). We have therefore provided an economic measure of excess profit differing from the ones existing in the literature. Consequently, it seems that the formal translation of the economic concept of residual income cannot be taken for granted, since it is not univocal. This induces us to think that such a notion is not objective but subjective, depending on the cognitive framework the evaluator rests on. The question the evaluator is asking is the same for both the systemic approach and the NFV-based approach (i.e. what is the excess profit in period s?), but the answer is different: one relies upon (fictitious) periodic NPV s (NFV s), the other makes use of differential incomes drawn from alternative dynamic systems, structured in multiple assets. The argument by Peccati and Stewart is the following: The investor invests w s 1 at each period at a rate x, but she could invest the same sum at a rate i. The difference between these two courses of action is the residual income. The argument from a systemic perspective is the following: Assume our investor undertakes the project at time 0. The return on her own net worth will be xw s 1 þ ic s 1, but she could select (ii) at time 0, so she would have a return of ic s 1 in period s. The difference between these two returns is the residual income. So to say, Peccati s and Stewart s models are project-oriented, the SVA model is wealth-oriented : the former focuses on the project s balance w s 1, assuming that it can be invested either at the rate x or at the rate i. Conversely, a systemic-minded evaluator focuses on wealth s diachronic evolution looking at the alternative states of the systems. Shaping the same thing differently, note that the two alternative states of the system for our systemic-minded evaluator can be rewritten as: E s 1 ¼ C s 1 þ w s 1 E s 1 ¼ C s 1 þ (C s 1 C s 1 ):

26 SYSTEMIC VALUE ADDED AND RESIDUAL INCOME 173 The term C s 1 is shared by both alternatives; so, looking at the second addends, if the evaluator undertakes the project, at time 0, then in period s she will be investing w s 1 at the rate x s while renouncing to investing the sum (C s 1 C s 1 ) at the rate i. Hence, the SVA s. So, this model has to be regarded as an alternative way of measuring residual income (i.e. an alternative way of decomposing a Net Final Value), and the choice of either model depends on the perspective the evaluator rests on: yet, the SVA model might also be seen as a complement to Peccati s and Stewart s models, 12 since the evaluator could use both models, knowing that they derive from different cognitive perspectives, different interpretations of the notion of residual income, which implies that the economic information received is obviously different; in this sense, it could be interesting to analyse thoroughly the set of economic information provided by either model. Hopefully, the chance of having (at least) two models measuring excess profit should attract attention among scholars: the notion of excess profit itself seems to be inherently ambiguous and a deeper understanding of this aspect deserves a thorough investigation from a cognitive point of view. Could it be that the classical notion of excess profit is just a conventional matter? We have seen that Peccati s and Stewart s model cannot be used by a systemic-minded evaluator. That is, the NFV-based approach provides some contradictions if analyzed with systemic eyes (or, which is the same, the systemic evaluator incurs logical contradictions if she uses the NFV-based decomposition models). The reverse is also true, that is the SVA model provides some contradictions if analyzed with Stewart s and Peccati s eyes. But it can be demonstrated that such contradictions may be healed and that not only the periodic EVA of the shadow project is the SVA of the original (as we have demonstrated here), but also the periodic EVA of P can be seen as the periodic SVA of some other project P 0 whose shadow project is P itself (we have not provided here the proof for reasons of space). This suggests that we can develop a dual theory of excess profit, which also deserves attention, in our opinion. Our ideas on such aspects are not definitive and we think it is worthwhile investigating them more thoroughly (a first attempt can be found in Magni (2001a)), but what we should like to stress is that, to a more thorough investigation, the NFV-based models may be useful even in a systemic perspective, provided that we apply it to the shadow project. The latter should then be regarded as a bridge connecting the two models. In sum, the paper has answered two questions: (a) What is the residual income for project P? (b) How can we periodically decompose the NFV 12 We thank an anonymous referee for such a remark.