Carlo Alberto MAGNI - personale UniMoRe

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Carlo Alberto MAGNI

Professore Ordinario
Dipartimento di Economia "Marco Biagi"

Pubblicazioni

2023 - The Attribution Matrix and the joint use of Finite Change Sensitivity Index and Residual Income for value-based performance measurement [Articolo su rivista]
Magni, Ca; Marchioni, A; Baschieri, D
abstract

2023 - The Split-Screen Approach for Project Appraisal (Part I: The Theory) [Articolo su rivista]
Magni, Carlo Alberto
abstract

2023 - The Split-Screen Approach for Project Appraisal (Part II: Spreadsheet Modeling) [Articolo su rivista]
Baschieri, Davide; Magni, Carlo Alberto
abstract

2022 - Impact of financing and payout policy on the economic profitability of solar photovoltaic plants [Articolo su rivista]
Magni, C. A.; Marchioni, A.; Baschieri, D.
abstract

This paper introduces an innovative comprehensive evaluation model for appraising an investment in a solar photovoltaic plant which encompasses both operational and financial management. We illustrate the intricate network of logical relations among technical (estimated) variables and financial (decision) variables and show that establishing transparent links between the former and the latter enhances the accuracy and soundness of the model. The results indicate that understanding the conceptual and formal relations of operating variables and financial decisions is necessary for correctly measuring shareholder value creation and making rational decisions, even for those projects (such as solar energy projects) where the operating, technical component is of paramount importance. We show how a firm's decision of replacing conventional energy with solar energy may be affected by managerial decisions regarding the firm's payout/retention policy and its financing policy to support the project. The model discloses insights on how to fine-tune the financing and distribution decisions in order to maximize the value creation for shareholders. We apply the model to a real-life photovoltaic project to be located in the province of Modena, in Northeast Italy, and quantify the effect of financial decisions on the project's net present value, showing that the financing and distribution policies may amplify or shrink the impact of changes in other inputs and may even revert an otherwise unprofitable project into a value-creating one. Finally, we allow operational variables as well as financial variables to change in order to measure their importance via the application of the Clean Finite Change Sensitivity Indices (Magni et al., 2020).

2022 - Mixed Integer Linear Programming for CO2 emissions minimization in a Waste Transfer Facility Location Problem [Relazione in Atti di Convegno]
Caselli, Giulia; Columbu, Giomaria; Iori, Manuel; Magni, Carlo Alberto
abstract

In this work, we solve a real-world facility location problem by means of a mixed integer linear programming model. The problem is faced by an Italian multi-utility company operating in the sector of waste management. The company works in several Italian regions to collect and treat the urban waste through a network of facilities. In this problem, a set of demand points is given with a predicted quantity of waste to be collected and a fixed number of visits required over a predetermined time horizon. The flow of different classes of recyclable waste must be optimized by deciding whether and where to open additional intermediate transfer facilities among a set of dedicated points. The aim is to minimize the CO2 emissions involved in the process, including emissions from the use of additional facilities and the transport of waste across the network. We provide a mathematical formulation for the problem, and use it to solve a real-world case study. An optimal solution is obtained with a significant reduction in CO2 emissions and a well-structured network, proving the efficacy of the model.

2022 - Mixed Integer Linear Programming for a Real-World Parallel Machine Scheduling Problem with Workforce and Precedence Constraints [Capitolo/Saggio]
Caselli, G.; Delorme, M.; Iori, M.; Magni, C. A.
abstract

In this work, we consider a real-world scheduling problem occurring in the engineering test laboratory of a multinational company producing hydraulic components for motion systems. Similar problems have been solved in the literature under the framework of resource constrained parallel machine scheduling problems. In our work, the tests on the hydraulic components are the jobs to be scheduled. Each job must be processed on a machine and requires an additional human resource to prepare the machine and supervise the job. Machine and workforce eligibility constraints are also included. Release and due dates are given for jobs. The aim is to minimize the total weighted tardiness. Each job has a processing time expressed in working days that depends on the machine and requires a fixed number of hours per day for its assigned worker. Moreover, precedence and contiguity relations between jobs must be respected. We propose a Mixed Integer Linear Programming formulation to model the problem and demonstrate its effectiveness on both real-world and randomly generated instances.

2022 - Performance attribution, time-weighted rate of return, and clean finite change sensitivity index [Articolo su rivista]
Magni, C. A.; Marchioni, A.
abstract

We propose an innovative methodology for decomposing the value added generated by a money manager within a given assessment interval into the contributions of the manager’s investment decisions made in the various periods, in order to identify the most (and the least) impactful period decisions. To this end, we benchmark an actively-managed investment against a reference portfolio replicating the client’s contributions and distributions and earning the benchmark returns. We apply the Clean Finite Change Sensitivity Index method (Borgonovo in Eur J Oper Res 200:127–138, 2010a, Risk Anal 30(3):385–399, 2010b; Magni et al. in J Oper Res Soc 71(12):1940–1958, 2020) to the investment’s value added in order to obtain a complete decomposition of it into the contributions of the investment decisions made in the various periods; we rank the period decisions according to their contributions and show that, if the contribution-and-distribution policy changes, the effect of the investment choices made in the various periods on the value added changes as well, which testifies of the interaction between the manager’s decisions and the client’s decisions, the former affecting the financial efficiency and latter affecting the investment scale. In particular, neutralizing the contributions and distributions (and, therefore, the investment scale), we show that the Time-Weighted Rate of Return (TWRR) can be arithmetically obtained from the value added and can be decomposed into the same period contributions of the value added, thereby providing a reconciliation between the value added notion and the TWRR.

2022 - Value Creation and Investment Projects: An Application of Fuzzy Sensitivity Analysis to Project Financing Transactions [Articolo su rivista]
Guerra, Ml; Magni, Ca; Stefanini, L
abstract

This paper presents a methodology which blends sensitivity analysis (SA) and fuzzy arithmetic for managing uncertainty in project financing transactions. Specifically, we adopt the perspective of the equityholders and use the average return on equity (ROE) to measure shareholder value creation and, in particular, the financial efficiency of the equity investment. We cope with uncertainty via global and local SA and fuzzy arithmetic; we use the fuzzy version of the well-known (global) Gamma indicator and we introduce the fuzzy versions of two (local) importance measures, the differential importance measures (DIM). We then apply them to the pro forma financial statements drawn up by the analyst for measuring and ranking the impact of the key accounting parameters on the resulting values and we show how the uncertain accounting and financial magnitudes of the project company affect the financial efficiency. Among the advantages of this analysis, aimed to enhance the managerial insights generated by the financial model and to lead to appropriate managerial actions, we focus on the attractiveness of fuzzy calculus and possibility theory to represent and compute all relevant financial data that appear in project financing and budgeting, where available information is characterized by incompleteness or nonstatistical uncertainty. In this context, fuzzy computing and appropriate SA techniques, based on application of the extension principle, allow complete investigation of the project characteristics.

2021 - A Decision Support System to Evaluate Suppliers in the Context of Global Service Providers [Relazione in Atti di Convegno]
Bruck, Bruno Petrato; Vezzali, Dario; Iori, Manuel; Magni, Carlo Alberto; Pretolani, Daniele
abstract

In this paper, we present a decision support system (DSS) developed for a global service provider (GSP), which solves a real-world supplier selection problem. The GSP operates in the Italian market of facility management, supplying customers with a variety of services. These services are subcontracted to external qualified suppliers spread all over Italy and chosen on the basis of several criteria, such as service quality, availability and proximity. Selecting the best supplier is a complex task due to the large number of suppliers and the great variety of facility management services offered by the GSP. In the proposed DSS, the choice of the best supplier for a certain service is made according to a thorough multi-criteria analysis. The weights for the criteria were generated by implementing both a simplified analytic hierarchy process and a revised Simos' procedure, later validated by the decision makers at the GSP. The DSS provides quick access to historical performance data, visual tools to aid decisions, and a suggested ranked list of suppliers for each given contract. The effectiveness of the proposed system was assessed by means of extensive simulations on a seven-year period of real-data and several rounds of validation with the company.

2021 - An integrated task and personnel scheduling problem to optimize distributed services in hospitals [Relazione in Atti di Convegno]
Porto Campana, Nicolas; Zucchi, Giorgio; Iori, Manuel; Magni, Carlo Alberto; Subramanian, Anand
abstract

This paper addresses a real-life task and personnel scheduling problem arising in a large Italian company that needs to provide cleaning services inside a hospital. In this case study, the challenge is to determine a schedule of the employees to clean the whole hospital aiming to minimize the total labor cost, taking into account the fact that the building is a complex structure with multiple levels and each room has different peculiarity. To solve the problem, we propose a three-step approach using mathematical models and metaheuristic algorithms. The solution obtained indicates that the schedule attained by our method is better than the one generated by the company. In addition, to test and validate our approach more thoroughly, a set of artificial instances have been created. The results indicate that our method can help organizations to quickly generate and test a large variety of solutions. Our findings can be of general interest for other personnel scheduling problems involving distributed services.

2021 - Average internal rate of return for risky projects [Articolo su rivista]
Hazen, G.; Magni, C. A.
abstract

The average internal rate of return (AIRR) fixes many deficiencies associated with the traditional internal rate of return (IRR), including apparent inconsistency with net present value (NPV). The AIRR approach breaks down project NPV into scale (the capital invested) and economic efficiency (the AIRR), and maintains NPV consistency for accept/reject decisions. Here we examine extensions of the AIRR to risky capital asset projects, a domain where the IRR appears intractable. We show that one can uniquely break down a risky NPV into a risk-sensitive project scale and a risk-sensitive extended AIRR, representing risky project efficiency, so that consistency with NPV for accept/reject decisions is maintained in the certainty-equivalent sense, in direct analogy to the deterministic case. This novel breakdown gives managerial insight by helping determine a risky project’s locus of uncertainty, be it the project scale, or economic efficiency, or both. In this way, risky features of competing projects can be explored in more detail, leading to insights substantiating the NPV ranking. We also show that under risk neutrality, the expected AIRR is equal to the AIRR of the expected cash flow, a property that notoriously fails for the stochastic IRR.

2021 - Economic profitability and (non)additivity of residual income [Articolo su rivista]
Magni, C. A.
abstract

We show that the standard notion of residual income (RI) does not fulfill additive coherence. This gives rise to ambiguities and inconsistencies. The pitfall resides in the capital charge, which blends a non-market value with a market rate. We solve the problem by using a capital charge based on economic return, obtained as the product of a market value and a market rate. The resultant economic RI enjoys additivity. The economic RI is naturally associated to the average Return on Investment (ratio of total income to total invested capital). Subtracting the respective cost of capital (ratio of total economic return to total invested capital) the marginal economic efficiency of the capital is correctly captured. Economic RI guarantees consistency among the various sets of incomes, book values, economic values, accounting rates, and costs of capital, under an investment perspective as well as a financing one, both at a period level and at an aggregate level, either assuming time-invariant or time-varying costs of capital. Therefore, the economic RI offers a coherent tool for the assessment of a project’s or firm’s economic efficiency.

2021 - Internal rates of return and shareholder value creation [Articolo su rivista]
Magni, C. A.
abstract

We propose a genuinely internal approach to project valuation and decision based on the average Return On Investment (ROI), obtained as the ratio of total operating profit (NOPAT) to total invested capital or, equivalently, as the ratio of net cash flow to total invested capital. The approach presented enables decomposing the economic value created into the project scale (total capital invested) and the economic efficiency, obtained as the difference between average ROI and a suitably adjusted weighted average cost of capital (WACC). We show that a project’s average ROI is equal to the weighted mean of the average Return On Equity (ROE) (total net income divided by total equity capital) and the average Return On Debt (ROD) (total interest expenses to total debt outstanding) and show that the average ROE itself correctly measures shareholder value creation when compared with a suitably adjusted cost of equity. We show that the internal average-based approach presented is also valid under the more general assumption of time-varying cost of capital.

2021 - Smart-Meter Installation Scheduling in the Context of Water Distribution [Abstract in Atti di Convegno]
Baschieri, Davide; Iori, Manuel; Magni, Carlo Alberto; Marchioni, Andrea; Vezzali, Dario
abstract

In this work, we propose a Mixed Integer Linear Programming (MILP) formulation to model a Smart-Meter Installation Scheduling Problem (SMISP) in the context of water distribution. The model has been used to solve a real case study from a multi-utility company operating in the Italian market. Specifically, in compliance with the European and the Italian regulations on metering, a distribution company is obligated to periodically control meters and substitute them in case they have reached their lifespan. In the examined case study, the multi-utility company has opted for a massive substitution plan in order to install innovative “walk-by smart-meters” in place of traditional mechanical meters. The MILP formulation aims at integrating both the operational and the financial perspective of the SMISP. In particular, the objective function has been carefully defined in order to maximize the Net Present Value (NPV) of the massive substitution plan, including the operational savings produced by using the walk-by smart-meters, the additional incomes originating from the gradual charge of substitution costs on customers’ invoices as considered by the Italian Authority, the depreciation of walk-by smart-meters, the investment costs, and the impact of income taxes on the objective function. The final goal of the proposed formulation is to define a scheduling for the massive substitution plan that satisfies a number of operational constraints and produces the maximum NPV.

2021 - Teaching financial statements with the split-screen Matrix. [Relazione in Atti di Convegno]
Magni, Carlo Alberto
abstract

2020 - Average rates of return, working capital, and NPV-consistency in project appraisal: A sensitivity analysis approach [Articolo su rivista]
Magni, C. A.; Marchioni, A.
abstract

In project appraisal under uncertainty, the economic reliability of a measure of financial efficiency such as a rate of return depends on its strong NPV-consistency, meaning that the performance metric (i) supplies the same recommendation in accept–reject decisions as the NPV, (ii) ranks competing projects in the same way as the NPV, (iii) has the same sensitivity to perturbations in the input data as the NPV. In real-life projects, financial efficiency is greatly affected by the management of the working capital. Using a sensitivity analysis approach and taking into explicit account the role of working capital, we show that the average return on investment (ROI) is not strongly NPV-consistent in accept–reject decisions if the working capital is uncertain and changes under changes in revenues and costs. Also, it is not strongly NPV-consistent in project ranking. We also show that the internal rate of return (IRR) is not strongly NPV-consistent and economic analysis may even turn out to be impossible, owing to possible nonexistence and multiplicity caused by perturbations in the input data, as well as to possible shifts in the financial meaning of IRR under changes in the project's value drivers. We introduce the straight-line rate of return (SLRR), based on the notion of average rate of change, which overcomes all the problems encountered by average ROI and IRR: It always exists, is unique, strongly NPV-consistent for both accept–reject decisions and project ranking, and has an unambiguous financial nature.

2020 - Comprehensive financial modeling of solar PV systems [Relazione in Atti di Convegno]
Baschieri, D; Magni, Ca; Marchioni, Andrea
abstract

2020 - Investment Decisions and the Logic of Valuation. Linking Finance, Accounting, and Engineering [Monografia/Trattato scientifico]
Magni, Ca
abstract

2020 - Investment and financing perspectives for a solar photovoltaic project [Relazione in Atti di Convegno]
Marchioni, Andrea; Magni, Ca; Baschieri, D
abstract

2020 - Practice patterns regarding drains management in breast surgery: Results of a survey of Senonetwork Italia breast centers [Articolo su rivista]
Pallara, T.; Fortunato, L.; Folli, S.; Roncella, M.; Scuderi, N.; Friedman, D.; Arnez, Z.; Ribuffo, D.; Manna, E.; Persichetti, P.; Abonante, S.; Altomare, V.; Amanti, C.; Ambrosiani, L.; Andreoli, C.; Bafile, A.; Ballardini, B.; Barbero, M.; Battaglia, C.; Bianchi, A.; Biganzoli, L.; Bortul, M.; Bravetti, P.; Burlizzi, S.; Cabula, C.; Caponi, C.; Caruso, F.; Cedolini, C.; Cianchetti, E.; Corsi, F.; Curcio, A.; Custodero, O.; D'Errico, F.; Francesconi, D.; Frassoldati, A.; Frittelli, P.; Generali, D.; Giovanazzi, R.; Grassi, M. M.; Huscher, A.; Lazzaretti, M. G.; Leone, F.; Lolli, G.; Magni, C.; Mainente, P.; Manca, L.; Mancini, S.; Massarut, S.; Massocco, A.; Menghini, L.; Mirri, M.; Mondini, G.; Monti, M.; Murgo, R.; Pacquola, M. G.; Paduos, A.; Pagani, G.; Pagliari, C.; Palli, D.; Papaccio, G.; Passamonti, M.; Pellegrini, A.; Pellini, F.; Pietribiasi, F.; Ressa, C. M.; Ricci, F.; Rovera, F.; Rubino, C.; Ruggeri, E. M.; Rulli, A.; Sanguinetti, A.; Saturno, M.; Scalco, G.; Sgarella, A.; Stefanini, P.; Svegliati, F.; Taffurelli, M.; Tazzioli, G.; Tinterri, C.; Trunfio, M.; Zagarese, P.
abstract

2020 - Rating firms and sensitivity analysis [Articolo su rivista]
Magni, C. A.; Malagoli, S.; Marchioni, A.; Mastroleo, G.
abstract

This paper introduces a model for rating a firm’s default risk based on fuzzy logic and expert system and an associated model of sensitivity analysis (SA) for managerial purposes. The rating model automatically replicates the evaluation process of default risk performed by human experts. It makes use of a modular approach based on rules blocks and conditional implications. The SA model investigates the change in the firm’s default risk under changes in the model inputs and employs recent results in the engineering literature of Sensitivity Analysis. In particular, it (i) allows the decomposition of the historical variation of default risk, (ii) identifies the most relevant parameters for the risk variation, and (iii) suggests managerial actions to be undertaken for improving the firm’s rating.

2019 - Accounting Measures and Economic Measures: An Integrated Theory of Capital Budgeting [Articolo su rivista]
Magni, Ca
abstract

Accounting measures are traditionally considered non-significant from an economic point of view. In particular, accounting rates of return are often regarded economically meaningless or, at the very best, poor surrogates for the IRR, which is held to be “the” economic yield. Likewise, residual income, another well-known accounting measure, does not enjoy, in general, periodic consistency with the project NPV, so residual income maximization is not equivalent to NPV maximization. This paper shows that the opposition accounting/economic is artificial and, taking a capital budgeting perspective, illustrates the strong (formal and conceptual) connections existing between economic measures and accounting measures. In particular, the average accounting rate of return is a correct economic yield of a project; the traditional IRR is (whenever it exists) only a particular case of it. The average accounting rate generates a decision rule which is logically equivalent to the NPV rule for both accept/reject decisions and project ranking. The paper also shows that maximization of the simple arithmetic mean of residual incomes is equivalent to NPV maximization, owing to its periodic consistency in the sense of Egginton (1995). Such an index may then be used for incentive compensation as well. Moreover, asset pricing may be interpreted in accounting terms as the process whereby the market determines the income impact on the assets’ value. As a result, the paper harmonizes the notions of accounting rate of return, internal rate of return, residual income, net present value: they are just different ways of cognizing the same notion. This conciliation stems in a rather natural way from three sources: (i) a fundamental accounting identity, which links income and cash flow in a comprehensive way, (ii) the definition of Chisini mean, (iii) a notion of residual income which takes account of a comprehensive cost of capital.

2019 - Performance measurement and decomposition of value added. [Relazione in Atti di Convegno]
Magni, Ca; Marchioni, Andrea
abstract

2019 - THE ACCOUNTING-AND-FINANCE OF A SOLAR PHOTOVOLTAIC PLANT: ECONOMIC EFFICIENCY OF A REPLACEMENT PROJECT [Relazione in Atti di Convegno]
Magni, Ca; Marchioni, Andrea
abstract

In this work we illustrate a simple logical framework serving the purpose of assessing the economic profitability and measuring value creation in a solar photovoltaic (PhV) project and, in general, in a replacement project where the cash-flow stream is nonnegative, with some strictly positive cash flows. We use the projected accounting data to compute the average ROI, building upon Magni (2011, 2019) (see also Magni and Marchioni 2018), which enables retrieving information on the role of the project's economic efficiency and the role of the project scale on increasing shareholders' wealth. The average ROI is a genuinely internal measure and does not suffer from the pitfalls of the internal rate of return (IRR), which may be particularly critical in replacement projects such as the purchase of a PhV plant aimed at replacing conventional retail supplies of electricity.

2018 - Chisini means and rational decision making: equivalence of investment criteria [Articolo su rivista]
Magni, C. A.; Veronese, P.; Graziani, R.
abstract

A plethora of tools are used for investment decisions and performance measurement, including net present value, internal rate of return, profitability index, modified internal rate of return, average accounting rate of return. All these and other known metrics are generally considered non-equivalent and some of them are regarded as unreliable or even naïve. Building upon Magni (Eng Econ 55(2):150–180, 2010a, Eng Econ 58(2):73–111, 2013)’s average internal rate of return, we show that the notion of Chisini mean enables these tools to be used as rational decision criteria. Specifically, we focus on 11 metrics and show that, if properly used, they all provide equivalent accept–reject decisions and equivalent project rankings. Therefore, the intuitive notion of mean is the founding basis of investment decision criteria.

2018 - Investment decisions and sensitivity analysis: NPV-consistency of rates of return [Articolo su rivista]
Marchioni, A.; Magni, C. A.
abstract

Investment decisions may be evaluated via several different metrics/criteria, which are functions of a vec- tor 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 NPV-consistent if it is coherent with NPV in signaling value creation. This paper makes use of Sensitivity Analysis (SA) for measuring coherence between rates of return and NPV. In particular, it introduces a new, stronger definition of NPV-consistency that takes into account the 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 (1904) correlation coefficient and Iman and Conover (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.

2018 - On “Introducing excess return on time-Scaled contributions”: A clarification [Articolo su rivista]
Magni, C. A.
abstract

2018 - PROJECT APPRAISAL AND THE INTRINSIC RATE OF RETURN [Relazione in Atti di Convegno]
Magni, Ca; Marchioni, Andrea
abstract

Building upon Magni (2011)'s approach, we propose a new rate of return measuring a project's economic profitability. It is called the intrinsic rate of return (IROR). It is defined as the ratio of project return to project's intrinsic value. The IROR approach decomposes the NPV into project scale and economic efficiency. In particular, NPV is found as the product of the project's total invested capital and the excess rate of return, obtained as the difference between the IROR and the minimum attractive rate of return (MARR). This approach provides correct project ranking and is capable of managing time-varying costs of capital. In case of levered projects, shareholder value creation is captured by the equity IROR, which we call Intrinsic Return On Equity (IROE) (net income divided by total equity capital invested). If the project is unlevered, the IROE and the IROR lead to the same decision; if the project is levered, and the nominal value of debt is not equal to the market value of debt, the IROE should be preferred to project IROR.

2018 - Some problems of the IRR in measuring PEI performance and how to solve it with the pure investment AIRR [Articolo su rivista]
Cuthbert, J; Magni, Ca
abstract

2016 - An Average-Based Accounting Approach to Capital Asset Investments: The Case of Project Finance [Articolo su rivista]
Magni, C. A.
abstract

Literature and textbooks on capital budgeting endorse net present value (NPV) and generally treat accounting rates of return as not being reliable tools. This paper shows that accounting numbers can be reconciled with NPV and fruitfully employed in real-life applications. Focusing on project finance transactions, an average return on investment (AROI) is drawn from the pro forma financial statements, obtained as the ratio of aggregate income to aggregate book value. We show that such a metric correctly captures a project's economic profitability, as long as it is compared with a comprehensive weighted average cost of capital (WACC) that includes a correction factor that takes account of the capital foregone by the investors. In contrast to the internal rate of return, AROI is unique, and we provide an explicit functional relation that links it to the NPV. The approach holds for levered and unlevered projects, constant and non-constant leverage ratios, and constant and non-constant WACCs.

2016 - Capital depreciation and the underdetermination of rate of return: A unifying perspective [Articolo su rivista]
Magni, C. A.
abstract

This paper shows that the notion of rate of return is best understood through the lens of the average-internal-rate-of-return (AIRR) model, first introduced in Magni (2010a). It is an NPV-consistent approach based on a coherent definition of rate of return and on the notion of Chisini mean, it is capable of solving the conundrums originated by the rate-of-return notion and represents a unifying theoretical paradigm under which every existing measure of wealth creation can be subsumed. We show that a rate of return is underdetermined by the project's cash-flow stream; in particular, a unique return function (not a unique rate of return) exists for every project which maps depreciation classes into rates of return. The various shapes a rate of return can take on (internal rate of return, average accounting rate of return, modified internal rate of return, etc.) derive from the (implicit or explicit) selection of different depreciation patterns. To single out the appropriate rate of return for a project, auxiliary assumptions are needed regarding the project's capital depreciation. This involves value judgment. On one side, this finding opens terrain for a capital valuation theory yet to be developed; on the other side, it triggers the creation of a toolkit of domain-specific and purpose-specific metrics that can be used, jointly or in isolation, for analyzing the economic profitability of a given project. We also show that the AIRR perspective has a high explanatory power that enables connecting seemingly unrelated notions and linking various disciplines such as economics, finance, and accounting. Some guidelines for practitioners are also provided.

2016 - Measuring the inadequacy of IRR in PFI schemes using profitability index and AIRR [Articolo su rivista]
. Cuthbert, J.; Magni, C. A.
abstract

The internal rate of return (IRR) is widely used in Private Finance Initiative (PFI) schemes in the UK for measuring performance. However, it is well-known that the IRR may be a misleading indicator of economic profitability. Treasury Guidance (2004) recognises that the IRR should not be used and net present value (NPV) should be calculated instead, unless the cash flow pattern is even. The distortion generated by the IRR can be quantified by the notion of scheduling effect, introduced in Cuthbert and Cuthbert (2012). We combine this notion with the notion of average IRR (AIRR), introduced in Magni, 2010, Magni, 2013 and show that a positive scheduling effect arises if the AIRR, relative to a flat payment stream, exceeds the project's IRR. The scheduling component can be measured in two separate ways, in terms of specific AIRRs, one of which enables the scheduling component to be decomposed into relative capital and relative rate components. We also highlight the role of average capital, whose quotation in the market, in association with IRRs or AIRRs, would deepen the economic analysis of the project

2016 - Sensitivity analysis and investment decisions: NPV-consistency of rates of return [Working paper]
Marchioni, Andrea; Magni, C. A.
abstract

2015 - Aggregate return on investment for investments under uncertainty [Articolo su rivista]
Magni, C. A.
abstract

This paper deals with capital budgeting decisions under uncertainty. We present an Aggregate Return On Investment (AROI), obtained as the ratio of total (undiscounted) cash flow to total invested capital and show that it is a genuine rate of return which, compared with the risk-adjusted cost of capital, correctly signals wealth creation. For choosing between two mutually exclusive projects, we derive an incremental AROI and an incremental risk-adjusted cost of capital, by means of which two unequal-risk projects can be correctly compared. Iterating the incremental procedure, we show that the AROI approach correctly ranks any bundle of different-risk competing projects. Relations with other criteria such as Modified Internal Rate of Return, Average Internal Rate of Return, Cash Multiple, and Profitability Index are provided. Theoretically, the AROI approach constitutes a link between arbitrage choice theory and corporate investment theory, and shows that explicit discounting is not necessary for measuring economic profitability. Practically, the AROI is a user-friendly, easy-to-compute rate of return derived from the same set of data required by the net present value (NPV). Also, it does not incur the difficulties met by the internal rate of return: in particular, it is unique and it is based on economically significant capital values (i.e., market-driven values). As such, the AROI significantly expresses the efficiency of the project's invested capital.

2015 - Introducing Aggregate Return on Investment as a Solution to the Contradiction Between Some PME Metrics and IRR [Working paper]
Altshuler, D.; Magni, C. A.
abstract

The Index Comparison Method (ICM) is a well-known approach for measuring a Private Equity Investment’s (PEI) performance. It is based on the construction of a benchmark portfolio that, each period, earns the index return. This generates a time series of interim net asset values that leads to a terminal NAV, from which an Internal Rate of Return is computed. However, the IRR is itself necessarily associated with its own time series of built-in NAVs, to which the IRR is applied. And, unfortunately, this series of values will be different from the aforementioned benchmark portfolio’s NAVs. As a result, the ICM approach rests on two contradictory sets of values, thereby rendering it illegitimate. Furthermore, the ICM approach does not preserve additivity of the rates of return, and, in principle, might even generate multiple IRRs. This paper presents the Aggregate Return on Investment (AROI), a metric which (i) uses one consistent time series of NAVs (the benchmark portfolio’s true values) (ii) preserves additivity, and (iii) does not incur the problem of multiple solutions.

2015 - Introducing Aggregate Return on Investment as a solution to the contradiction between some PME metrics and IRR [Articolo su rivista]
Magni, Carlo Alberto; Altshuler, Dean
abstract

2015 - Investment, financing and the role of ROA and WACC in value creation [Articolo su rivista]
Magni, C. A.
abstract

Evaluating an industrial opportunity often means to engage in financial modeling which results in estimationof a large amount of economic and accounting data, which are then gathered in an economically rationalframework: the pro forma financial statements. While the standard net present value (NPV) condenses allthe available pieces of information into a single metric, we make full use of the crucial information suppliedin the pro forma financial statements and give a more detailed account of how economic value is created. Inparticular, we construct a general model, allowing for varying interest rates, which decomposes the projectinto investment side and financing side and quantifies the value created by either side; an equity/debtdecomposition is also accomplished, which enables to appreciate the role of debt in adding or subtractingvalue to equityholders. Further, the major role of accounting rates of return as value drivers is highlighted,and new relative measures of worth are introduced: the project ROA and the project WACC, which aggregateinformation deriving from the period rates of return. To achieve these results, we make use of the Average-Internal-Rate-of-Return (AIRR) approach, recently introduced, which rests on capital-weighted arithmeticmeans and sets a direct relation between holding period rates and NPV

2015 - Pseudo-naïve approaches to investment performance measurement [Working paper]
Magni, C. A.
abstract

This paper makes use of Magni’s (2013. Insurance Mathematics and Economics, 53, 747−756) Average Interest Rate (AIR) in order to find a performance index which does not depend on the valuation rate (i.e., benchmark return). To this end, we distort the AIR by dropping the discount factors in the formula. The resulting modified AIR (MAIR) is the ratio of overall (undiscounted) return to overall (undiscounted) capital. While seemingly a na¨ıve metric, we show that it is a genuinely internal metric, capable of capturing an investment’s economic profitability, as long as it is compared with an appropriate cutoff rate which adequately takes account of the opportunity cost of capital. The not-so na¨ıve MAIR is then extended to several different capital bases; the result is that other well-known (allegedly na¨ıve) metrics, such as cash multiple, undiscounted profitability, Modified Dietz and Simple Dietz return are given economic significance: each such metric is a (pseudo-na¨ıve) performance index that correctly expresses the investment’s amount of return per unit of a specific capital: overall capital, initial investment, total cash outflow, average cash outflow).

2015 - ROI and profitability index: A note on managerial performance [Working paper]
Magni, C. A.
abstract

This note deals with the case of a principal (e.g., a firm’s board of directors) which delegates execution of an economic activity to a business unit (or a subsidiary firm) managed by a manager. It is assumed that the manager has no control over the cash flows injected into the unit or withdrawn from it: such decisions are made by the principal. The principal aims at measuring the manager’s performance in a given interval of time. Neither the Net Present Value (NP V ) nor its companion Net Terminal Value (NT V ) are appropriate measures for this purpose, because they depend on the cash flows injected and withdrawn by the principal. We introduce the manager’s profitability index (MP I), which is invariant under changes in the cash flows, so neutralizing the effect on value creation of the principal’s decisions. We also break down the project’s NT V into two components, which measure the manager’s contribution and the principal’s contribution to value creation.

2015 - The Term Structure of Capital Values: An accounting-based framework for measuring economic profitability [Working paper]
Magni, C. A.; Peasnell, K. V.
abstract

This paper shows how the outputs of the accounting measurement process can be translated into terms that can be used in economic decisions. We introduce the notion of Term Structure of Capital Values (TSCV), uniquely associated to a Term Structure of Interest Rates (TSIR). We show that the state of temporary disequilibrium created by an asset (project, firm, etc.) introduced in a market can be described in terms of a pair of internal TSCV and TSIR. Any internal TSCV determines a specific depreciation schedule for the asset and the corresponding TSIR determines an economic rate of return obtained as a capital-weighted average of the Return On Assets collected in the TSIR. The difference between this economic rate of return and the overall equilibrium rate (itself an average of the equilibrium forward rates) correctly captures value creation. The determination of a specific (internal) TSCV depends on the way the market sweeps away arbitrage opportunities and restores the equilibrium. As each possible accounting system can be viewed as being associated with a corresponding disequilibrium-to-equilibrium process and therefore with a corresponding economic rate of return, the paper shows that the determination of an economic rate of return is both a matter of accounting (in terms of a specific internal TSCV) and a matter of finance (in terms of a specific disequilibrium-to-equilibrium process). This evidently calls for a theory of capital valuation capable of associating the proper TSCV with each asset so that the intrinsic underdetermination of the economic rate of return can be solved

2014 - A quasi-IRR for a project without IRR [Articolo su rivista]
Pressacco, Flavio; Magni, Carlo Alberto; Stucchi, Patrizia
abstract

Discounted cash flows methods such as Net Present Value and Internal Rate of Return are often used interchangeably or even together for assessing value creation in industrial and engineering projects. Notwithstanding its diffi culties of applicability and reliability, the internal rate of return (IRR) is commonly used in real-life applications. Among other problems, a project may have no real-valued IRR, a circumstance that may occur in projects which require shutting costs or imply an initial positive cash flow such as a down payment made by a client. This paper supplies a genuine IRR for a project which has no IRR. This seemingly paradoxical result is achieved by making use of a new approach to rate of return (Magni, 2010), whereby any project is associated with a unique return function. which maps aggregate capitals into rates of return. Each rate of return is a weighted average of one-period (internal) rates of return, so it is called Average Internal Rate of Return (AIRR). We introduce a twin project which has a unique IRR and the same NPV as the original project's, and which is obtained through an appropriate minimization of the distance between the original project's cash ow stream and the twin project's. Given that the latter's IRR lies on the original project's return function, it represents an AIRR of the original project. And while it is not the IRR of the project, the measure presented is `almost' the IRR of the project, so it is actually the \quasi-IRR" of the project.

2014 - An average-based accounting approach to capital asset investments: The case of project finance (Working Paper) [Working paper]
Magni, C. A.
abstract

Literature and textbooks on capital budgeting endorse Net Present Value (NPV) and generally treat accounting rates of return as not being reliable tools. This paper shows that accounting numbers can be reconciled with NPV and fruitfully employed in real-life applications. Focusing on project finance transactions, an Average Return On Investment (AROI) is drawn from the pro forma financial statements, obtained as the ratio of aggregate income to aggregate book value. It is shown that such a metric correctly captures a project’s economic profitability, as long as it is compared with a comprehensive Weighted Average Cost of Capital that includes a correction factor which takes account of the capital foregone by the investors. Contrary to the Internal Rate of Return, AROI is unique and we provide an explicit functional relation which links it to the NPV. The approach holds for levered and unlevered projects, constant and non-constant leverage ratios, constant and non-constant WACCs.

2014 - Arithmetic returns for investment performance measurement [Articolo su rivista]
Magni, Carlo Alberto
abstract

This paper introduces new money-weighted metrics for investment performance analysis, based on arithmetic means of holding period rates weighted by the investment’s market values. This approach generates rates of return which measure a fund’s or portfolio’s performance and a fund manager’s performance. It also enables to show that the Internal Rate of Return (IRR) is a weighted mean of holding period rates associated with interim values which differ from market values, so that value additivity is violated. The manager’s Arithmetic Internal Rate of Return (AIRR) is shown to be the true period equivalent of the cumulative Time Weighted Rate of Return (TWRR), whereas the period TWRR (a geometric return) provides a different ranking. The method is easily generalized for coping with varying benchmark rates. We also cope with the practical problem of estimating interim values whenever they are not available.

2014 - Axiomatization of residual income and generation of financial securities [Articolo su rivista]
Ghiselli Ricci, R.; Magni, Carlo Alberto
abstract

This paper presents an axiomatization of residual income, also known as excess profit, and illustrates how it may univocally engenders fixed-income or variable-income assets. In the firstpart it is shown that, depending on the relations between excess profit and the investor's excess wealth, a well-specied theory of residual income is generated: one is the standard theory, whichhistorically traces back to Hamilton (1777) and Marshall (1890) and is a deep-rooted notion in economic theory, nance, and accounting. Another one is the systemic value added or lost-capital paradigm: introduced in Magni (2000, 2003), the theory is enfolded in Keynes's (1936) notion of user cost and is naturally generated by an arbitrage-theory perspective. In the secondpart, the paper reverts the usual analysis: instead of computing residual incomes prots from a pattern of cash flows, residual incomes are fixed rst to derive vectors of cash flows. It isshown that variable- or fixed-income assets may be constructed on the basis of either theory starting from pre-determined growth rates for excess prot. In particular, zero-coupon bondsand coupon bonds traded in a capital market are shown to be deducted as equilibrium vectors of residual-income-based assets.

2014 - Interval and fuzzy Average Internal Rate of Return for investment appraisal [Articolo su rivista]
Guerra, M. L.; Magni, Carlo Alberto; Stefanini, L.
abstract

In investment appraisal, uncertainty can be managed through intervals or fuzzy numbers because the arithmetical properties and the extension principle are well established and can be successfully applied in a rigorous way. We apply interval and fuzzy numbers to the Average Internal Rate of Return (AIRR), recently introduced for overcoming the problems of the traditional Internal Rate of Return (IRR). In the setting of interval and fuzzy arithmetic, we establish relations between the interim capitals invested, the profits and the cash flows, which are the ingredients of the AIRR and shed lights on the different ways uncertainty propagates depending on which variable is known and which one is derived. The relations between fuzzy AIRR and fuzzy Net Present Value are also investigated.

2014 - Mathematical analysis of average rates of return and investment decisions: The missing link [Articolo su rivista]
Magni, Carlo Alberto
abstract

This paper expands Teichroew, Robichek and Montalbano's (TRM) (1965a,b) rate-of-return model into a complete and general model of economic profitability for investment decision-making. Specifically, TRM's assumptions are relaxed and a project rate of return is derived, expressing the project's overall economic protability; direct relations among rates, costs of capital and net present value are supplied. The various value drivers are identified and isolated, and the NPV is decomposed into financing NPV and investment NPV. The approach allows for any pattern of financing rates, investment rates, and costs of capital. Relations with old literature and new literature on rates of return are shown: the link between them is obtained by making use of the mean operator (i.e., affine combinations of rates) and via the one-to-one correspondence between rates and invested capitals.

2014 - Pseudo-Naive Approaches to Investment Performance Measurement [Working paper]
Magni, Carlo Alberto
abstract

This paper makes use of Magni's (2013) Average Interest Rate (AIR) in order to fi nd a performance index which does not depend on the valuation rate (i.e., benchmark return). To this end, we distort the AIR by dropping the discount factors in the formula. The resulting modi ed AIR (MAIR) is the ratio of overall (undiscounted) return to overall (undiscounted) capital. While seemingly a na ive metric, we show that it is a genuinely internal metric, capable of capturing an investment's economic profi tability, as long as it is compared with an appropriate cuto rate which adequately takes account of the opportunity cost of capital. The not-so na ve MAIR is then extended to several di erent capital bases; the result is that other well-known (allegedly na ve) metrics, such as cash multiple, undiscounted pro tability, Modi ed Dietz and Simple Dietz return are given economic signi cance: each such metric is a (pseudo-na ve) performance index that correctly expresses the investment's amount of return per unit of a speci c capital: overall capital, initial investment, total cash out ow, average cash out ow). Keywords. Finance, investment,

2013 - Generalized Makeham's formula and economic profitability [Articolo su rivista]
Magni, Carlo Alberto
abstract

This paper generalizes Makeham’s formula, allowing for varying interest rates and for a non-flat structure of valuation rates. An average interest rate (AIR) is introduced, as well as an average valuation rate (AVR), which exist and are unique for any asset. They can be computed either as principal-weighted arithmetic means or as interest-weighted harmonic means of period rates. Economic profitability of an asset or a portfolio of assets is captured by the spread between AIR and AVR, which has the same sign as the Net Present Value. This makes (i) AIR a more reliable tool for valuation and decision than the venerable Internal Rate of Return, and (ii) AVR a natural generalization of the cost-of-capital notion. Keywords. Makeham’s formula, net present value, average interest rate.

2013 - The Internal Rate of Return approach and the AIRR paradigm: a refutation and a corroboration [Articolo su rivista]
Magni, Carlo Alberto
abstract

This paper shows that the Internal-Rate-of-Return (IRR) approach is unreliable, and that the recently introduced Average-Internal-Rate-of-Return (AIRR) model constitutes the basis for an alternative theoretical paradigm of rate of return. To this end, we divide the paper into two parts: a pars destruens and a pars construens. In the “destructive” part, we present a compendium of eighteen flaws associated with the IRR approach. In the “constructive” part, we construct the alternative approach from four (independent) economic intuitions and put the paradigm to the test by showing that it does not suffer from any of the flaws previously investigated. We also show how the IRR, as a rate of return, is absorbed into the new approach.

2012 - Average Rate of return with Uncertainty [Relazione in Atti di Convegno]
Guerra, M. L.; Magni, Carlo Alberto; Stefanini, L.
abstract

In investment appraisal, uncertainty can be managed throughintervals or fuzzy numbers. The arithmetical properties and the extension principle are well established and can be successfully applied in a rigorous way. In this paper we introduce uncertainty in the investment appraisal managed through a new criterion called Average Internal Rate of Return (AIRR), introduced in Magni (2010). The consistency of the arithmetic of variables represented with intervals of fuzzy numbers makes possible the application of the extension principle and a rigorous analysis of the investment decisions handled under uncertaintyconditions.

2012 - Economic Profitability and the Accounting Rate of Return [Working paper]
Magni, Carlo Alberto; Peasnell, K. V.
abstract

The internal rate of return (IRR) is a widely used benchmark for assessing the reliability of the accounting rate of return (ROA) as a measure of economic profitability. We turn this reasoning process on its head by demonstrating that a suitable (weighted average) aggregation of ROAs better captures what is generally meant by economic profitability than does the IRR. We show that the average ROA, when compared with the cost of capital, will always correctly signal economic profitability in the sense that it will correspond exactly with what would be obtained from a net present value calculation. We also show that the average ROA can be used to make meaningful inter-firm comparisons of profitability, when due allowance is made for differences in investment scale. Using this framework, we show how average ROA can be used to assess economic profitability for a truncated time series where the opening and closing capital stocks provided by the accounting system do not adequately represent the firm’s initial and ending endowments of resources. Finally, we suggest how this approach can be put to practical use in assessing profitability of firms on an on-going basis.

2012 - In Search of the 'Lost Capital'. A Theory for Valuation, Investment Decisions, Performance Measurement [Articolo su rivista]
Magni, Carlo Alberto
abstract

This paper presents a theoretical framework for valuation, investment decisions, and performance measurement based on a nonstandard theory of residual income. It is derived from the notion of “unrecovered” capital, which is here named “lost” capital because it represents the capital foregone by the investors. Its theoretical strength and meaningfulness is shown by deriving it from four main perspectives: financial, microeconomic, axiomatic, accounting. Implications for asset valuation, capital budgeting and performance measurement are investigated. In particular: an aggregation property is shown, which makes the simple average residual income play a major role in valuation; a dual relation between the standard theory and the lost-capital theory is proved, clarifying the way periodic performance is computed in the two paradigms and the rationale for measuring performance with either paradigm; the average accounting rate of return is shown to be more reliable than the internal rate of return as a capital budgeting criterion, and maximization of the average residual income is shown to be equivalent to maximization of Net Present Value (NPV). Two metrics are also presented: one enjoys the nice property of robust goal congruence irrespective of the sign of the cash flows; the other one enjoys periodic consistency in the sense of Egginton (1995). The results obtained suggest that this theory might prove useful for real-life applications in firm valuation, capital budgeting decisions, ex post performance measurement, incentive compensation.

2012 - Interval and fuzzy Average Internal Rate of Return for investment appraisal [Working paper]
Guerra M., L; Magni, Carlo Alberto; Stefanini, L.
abstract

In investment appraisal, uncertainty can be managed through intervals or fuzzy numbers because the arithmetical properties and the extension principle are well established and can be successfully applied in a rigorous way. We apply interval and fuzzy numbers to the Average Internal Rate of Return (AIRR), recently introduced for overcoming the problems of the traditional Internal Rate of Return (IRR). In the setting of interval and fuzzy arithmetic, we establish relations between the interim capitals invested, the profits and the cash flows, which are the ingredients of the AIRR, and shed lights on the different ways uncertainty propagates depending on which variable is known and which one is derived. The relations between fuzzy AIRR and fuzzy Net Present Value are also investigated.

2012 - The AIRR approach for investment performance measurement [Working paper]
Magni, Carlo Alberto
abstract

This paper introduces a new method of investment performance analysis, based on the recent approach of Average Internal Rate of Return (AIRR). We show that the approach generates rates of return which measure both a fund’s (portfolio’s) performance and a manager’s performance. The metrics proposed are arithmetic means of holding period rates weighted by the fund’s actual (i.e., market) values. The Internal Rate of Return (IRR) is shown to be a particular case of AIRR, that is, a weighted arithmetic mean of holding period rates associated with interim values which have not to do with market values. Relations with the Time Weighted Rate of Return (TWRR) are investigated.

2012 - Why IRR is not the rate of return for your investment: introducing AIRR to the real estate community [Articolo su rivista]
Altshuler, Dean; Magni, Carlo Alberto
abstract

The internal rate of return (IRR) is used extensively in the real estate sector, notwithstanding certain nagging deficiencies taught in most business school texts. One of those deficiencies is that the IRR may have multiple solutions which cannot be reconciled. Unbeknownst to most practitioners, this “nagging deficiency” has been refuted in the last decade, seemingly great news for IRR advocates. However, the elimination of this deficiency exposes a more fundamental criticism, one which is addressed in this article; and it is that the IRR calculation itself assumes interim investment values that are mechanically generated by the IRR equation itself and will almost surely differ from the true interim values of the project under consideration. To the extent that these values differ, the IRR result will not be an accurate rate of return. Furthermore, from an ex-post, i.e., performance reporting standpoint, such values implied by the IRR will almost certainly contradict any estimated project values being used for time-weighted rate of return (TWR) purposes. A new metric called AIRR (Average IRR) overcomes these criticisms and produces a correct money-weighted rate of return (MWR) for a project. Furthermore, AIRR has none of the other problems that the IRR has: e.g. it always exists and is unique, and it appropriately accounts for the amounts actually invested, from time to time, over the course of the investment.

2011 - A Quasi-IRR for a Project Without IRR [Working paper]
Pressacco, F.; Magni, Carlo Alberto; Stucchi, P.
abstract

The internal rate of return (IRR) and the corresponding criterion has well-known difficulties of applicability and reliability. Among other problems, a project may have no real-valued IRR. The latter problem has been recently solved by Magni (2010a), who shows that, for any project, a unique return function exists which maps aggregate capitals into rates of return, each of which is a weighted average of one-period (internal) rates of return, so it is called Average Internal Rate of Return (AIRR). Given the extent to which the IRR notion is rooted in real-life applications as well as in academia, this paper aims to supply a genuine IRR even for a project which has no IRR. This seemingly paradoxical result is achieved by introducing a twin project which has a unique IRR and the same NPV as the original project's, and which is obtained through an appropriate minimization of the distance between the original project's cash flow stream and the twin project's. Given that the latter's IRR lies on the original project's return function, it genuinely expresses a rate of return (an AIRR) of the original project. And while it is not the IRR of the project, the measure presented is 'almost' the IRR of the project, so it is actually the "quasi-IRR" of the project.

2011 - Addendum to "Average Internal Rate of Return and investment decisions: a new perspective." [Articolo su rivista]
Magni, Carlo Alberto
abstract

This note specifies some results found in [Magni 2010. The Engineering Economists, 55(2), 150-180] where the Average Internal Rate of Return (AIRR) is presented, which overcome all the IRR difficulties. In particular, the AIRR approach enables to prove that (1) a project is not uniquely associated with a return rate, but with a return function, which maps aggregate capitals to rates of return; (2) any definition of rate of return appearing in the (past and) future literature is a particular case of AIRR; (3) for choices between mutually exclusive projects, the AIRR acceptability criterion may be applied to the incremental project; (4) the AIRR approach easily provides the evaluator with the rate of return on initial investment, whereas the IRR is only the rate of return on an aggregate capital which is internally implied by the IRR itself.

2011 - Aggregate return on investment and investment decisions: A cash-flow perspective [Articolo su rivista]
Magni, Carlo Alberto
abstract

The recent notion of Average Internal Rate of Return (AIRR) [Magni 2010, The Engineering Economist, 55(2), 150-180] completely solves the long-standing problem of the internal rate of return (IRR). While the AIRR is a return measure, this paper presents a cash-flow measure, namely the ratio of net cash flow (i.e., cash inflows minus cash outflows) to capital invested, which we call Aggregate Return On Investment (AROI). It is a purely internal measure because, unlike the AIRR, it does not depend on the market rate, and is a return measure, for it is a mean of one-period return rates, weighed by the outstanding capitals. The AROI is reliable in both accept/reject decisions and project ranking, in association with an appropriate hurdle rate, economically significant: the comprehensive cost of capital (CCOC), which takes into account not only the interest foregone on the capital actually employed, but also the interest foregone on the capital that is given up by the investor. This perspective enables one to decompose the project NPV into an excess-rate share and an excess-capital share. The traditional IRR is just a particular case of both AIRR and AROI, but the latter approach has the advantage that the IRR’s nature (rate of return versus rate of cost) does not depend on the market rate and is unambiguously determined by the capital invested.

2011 - Average internal rate of return with internval arithmetic [Relazione in Atti di Convegno]
Guerra, M. L.; Magni, Carlo Alberto; Stefanini, L.
abstract

2011 - L’approccio del reddito residuale e il metodo S&A per la valutazione delle aziende [Articolo su rivista]
Magni, Carlo Alberto
abstract

2011 - Return On Equity, Internal Rate of Return and shareholder value creation [Relazione in Atti di Convegno]
Magni, Carlo Alberto
abstract

In both finance and accounting the Return On Equity (ROE) is considered a biased indicator of economic profitability. Opposing this view, this paper shows that an appropriate mean of ROEs signals shareholder value creation. This implies that the notion of Market Value Added may be replaced by an average ROE net of the cost of capital. Such a measure does not suffer from problems of existence or uniqueness and may be employed for comparing profitability. The Internal Rate of Return (IRR) itself may be interpreted as the average ROE corresponding to an idealized depreciation system: such an interpretation enables one to solve the multiple-IRR problem and use the IRR for comparing firms’ profitability.

2011 - Using Average Internal Rates of Return for investment performance measurement and attribution [Relazione in Atti di Convegno]
Magni, Carlo Alberto
abstract

The paper describes the application of the Average Internal Rate of Return approach to investment performance and attribution

2010 - Average Internal Rate of Return and investment decisions: A new perspective [Working paper]
Magni, C. A.
abstract

The internal rate of return (IRR) is often used by managers and practitioners for investment decisions. Unfortunately, it has serious flaws: (i) multiple real-valued IRRs may arise, (ii) complex-valued IRRs may arise, (iii) the IRR is, in general, incompatible with the net present value (NPV) in accept/reject decisions (iv) the IRR ranking is, in general, different from the NPV ranking, (v) the IRR criterion is not applicable with variable costs of capital. The efforts of economists and management scientists in providing a reliable project rate of return have generated over the decades an immense bulk of contributions aiming to solve these shortcomings. This paper offers a complete solution to this long-standing issue by changing the usual perspective: the IRR equation is dismissed and the evaluator is allowed to describe the project as an investment or a borrowing at his discretion. This permits to show that any arithmetic mean of the one-period return rates implicit in a project reliably informs about a project’s profitability and correctly ranks competing projects. With such a measure, which we name ”Average Internal Rate of Return”, complex-valued numbers disappear and all the above mentioned problems are wiped out. The economic meaning is compelling: it is the project return rate implicitly determined by the market. The traditional IRR notion may be found back as a particular case.

2010 - Average Internal Rate of Return and investment decisions: a new perspective [Articolo su rivista]
Magni, Carlo Alberto
abstract

The internal rate of return (IRR) is often used by managers and practitioners for investment decisions. Unfortunately, it has serious flaws: (i) multiple real-valued IRRs may arise, (ii) complex-valued IRRs may arise, (iii) the IRR is, in general, incompatible with the net present value (NPV) in accept/reject decisions (iv) the IRR ranking is, in general, different from the NPV ranking, (v) the IRR criterion is not applicable with variable costs of capital. The efforts of economists and management scientists in providing a reliable project rate of return have generated over the decades an immense bulk of contributions aiming to solve these shortcomings. This paper offers a solution to this long-standing issue by changing the usual perspective: the IRR equation is dismissed and the evaluator is allowed to describe the project as an investment or a borrowing at his discretion. This permits to show that any arithmetic mean of the one-period return rates implicit in a project reliably informs about a project’s profitability and correctly ranks competing projects. With such a measure, which we name ”Average Internal Rate of Return”, complex-valued numbers disappear and all the above mentioned problems are wiped out. The economic meaning is compelling: it is the project return rate implicitly determined by the market. The traditional IRR notion may be found back as a particular case.

2010 - CAPM and Capital Budgeting: Present/Future, Equilibrium/Disequilibrium, Decision/Valuation [Articolo su rivista]
Magni, Carlo Alberto
abstract

This paper expands on the results obtained in Magni (2009) regarding investment decisions with the Capital Asset Pricing Model (CAPM). It is shown that four different decision criteria are deductively drawn from this model: the disequilibrium Net Present Value (NPV), the equilibrium NPV, the disequilibrium Net Future Value (NFV), and the equilibrium NFV. It is shown that all of them may be used for accept-reject decisions, but only the equilibrium NPV and the disequilibrium NFV may be used for valuation, given that they have the additivity property. However, it is possible to deductively dismiss the two nonadditive indexes if the ‘accept/reject’ problem is reframed as a choice among mutually exclusive alternatives. As for the remaining (additive) measures, the equilibrium NPV and the disequilibrium NFV are unreliable for both valuation and decision, because despite their additivity, they do not signal arbitrage opportunities whenever there is some state of nature for which they are decreasing functions with respect to the end-of-period cash flow. In this case, the equilibrium value of a project is not the price it would have if it was traded in the security market. This result is the capital-budgeting counterpart of Dybvig and Ingersoll’s (1982) result.

2010 - On the long-standing issue of the Internal Rate of Return: A complete solution [Relazione in Atti di Convegno]
Magni, Carlo Alberto
abstract

The IRR problem. As widely known, the IRR has serious flaws: (i)multiple real-valued IRRs may arise, (ii) the meaning of each IRR may be ambiguous (rate of return or rate of cost?), (iii) complex-valued IRRs may arise, (iv) the IRR is, in general, incompatible with the net present value (NPV) in accept/reject decisions and the IRR ranking is, in general, different from the NPV ranking, (v) the IRR decision criterion is not applicable with variable costs of capital. Since the origins of the notions (Boulding 1935, 1936; Keynes 1936), the IRR drawbacks have stimulated an immense bulk of contributions over the decades investigating this issue and searching for some solution (see references in Gronchi 1987 and in Magni 2010a). We here present two autonomous solutions which solve the IRR problems completely by dismissing the traditional the IRR equation and considering a simple mean of period rates. As a pleasant byproduct, we find that accounting rates of return are meaningful economic rates of return, whereas the IRR is just a particular case of the mean of period rates.

2010 - ROE, Market Value Added e creazione di valore [Articolo su rivista]
Magni, Carlo Alberto
abstract

Questo articolo si propone di dimostrare che il ROE (Return On Equity) ha un significato economico pregnante, al contrario di quanto sostenuto nella letteratura finanziaria e manageriale, nonché nella prassi aziendale. In particolare, un'opportuna media dei ROE è in grado di segnalare la creazione (o distruzione) di valore da parte del management dell’azienda. In altri termini, il concetto di Market Value Added, può essere espresso, in termini di tasso, da un ROE medio al netto del costo del capitale.

2010 - Relevance or irrelevance of retention for dividend policy irrelevance. [Articolo su rivista]
Magni, Carlo Alberto
abstract

In an interesting recent paper, DeAngelo and DeAngelo (2006) highlight that Miller and Modigliani’s (1961) proof of dividend irrelevance is based on the assumption that the amount of dividends distributed to shareholders is equal or greater than the free cash flow generated by the fixed investment policy. They claim that, if retention is allowed, dividend policy is not irrelevant. This paper shows that the dividend irrelevance proposition holds even in case of retention. The key assumption has not to do with retention but with the NPV of the extra funds (either retained or raised): if NPV is zero, dividend irrelevance applies. Yet, the dichotomy retention/no-retention is useful, because if agency problems are present, managers tend to retain funds and invest them in negative-NPV projects, and therefore the zero-NPV assumption must be removed, so that dividend irrelevance does not apply any more.

2010 - Residual income and value creation: an investigation into the lost-capital paradigm [Articolo su rivista]
Magni, Carlo Alberto
abstract

This paper presents a new way of measuring residual income, originally introduced by Magni (2000a,b,c, 2001a,b, 2003). Contrary to the standard residual income, the capital charge is equal to the capital lost by investors. The lost-capital may be viewed as (a) the foregone capital, (b) the capital implicitly infused into the business, (c) the outstanding capital of a shadow project, (d) the claimholders’ credit. Relations of the lost-capital with book values and market values are studied, as well as relations of the lost-capital residual income with the classical standard paradigm; many appealing properties are derived, among which an aggregation property. Different concepts and results, provided by different authors in such different fields as economic theory, management accounting and corporate finance, are considered: O’Hanlon and Peasnell’s (2002) unrecovered capital and Excess Value Created; Ohlson’s (2005) Abnormal Earnings Growth; O’Byrne’s (1997) EVA improvement; Miller and Modigliani’s (1961) investment opportunities approach to valuation; Young and O’Byrne’s (2001) Adjusted EVA; Keynes’s (1936) user cost; Drukarczyk and Schueler’s (2000) Net Economic Income; Fernández’s (2002) Created Shareholder Value; Anthony’s (1975) profit. They are all conveniently reinterpreted within the theoretical domain of the lost-capital paradigm and conjoined in a unified view. The results found make this new theoretical approach a good candidate for firm valuation, capital budgeting decision-making, managerial incentives and control.

2009 - Accounting and Economic Measures: An integrated theory of capital budgeting [Working paper]
Magni, Carlo Alberto
abstract

Accounting measures are traditionally considered not significant from an economic point of view. In particular, accounting rates of return are often regarded economically meaningless or, at the very best, poor surrogates for the IRR, which is held to be “the” economic yield. Likewise, residual income does not enjoy, in general, periodic consistency with the project NPV, so residual income maximization is not equivalent to NPV maximization. This paper shows that the opposition accounting/economic is artificial and, taking a capital budgeting perspective, illustrates the strong (formal and conceptual) connections existing between economic measures and accounting measures. In particular, the average accounting rate of return is the correct economic yield of a project; the traditional IRR is (whenever it exists) only a particular case of it. The average accounting rate generates a decision rule which is logically equivalent to the NPV rule for both accept/reject decisions and project ranking. The paper also shows that maximization of the simple arithmetic mean of residual incomes is equivalent to NPV maximization, owing to its periodic consistency in the sense of Egginton (1995). Such an index may then be used for incentive compensation as well. Moreover, asset pricing may be interpreted in accounting terms as the process whereby the market determines the income impact on the assets’ value. As a result, the paper harmonizes the notions of accounting rate of return, internal rate of return, residual income, net present value: they are just different ways of cognizing the same notion. This conciliation stems in a rather natural way from three sources: (i) a fundamental accounting identity, which links income and cash flow in a comprehensive way, (ii) the definition of Chisini mean, (iii) a notion of residual income which takes account of the “real” (comprehensive) cost of capital.

2009 - Accounting and economic measures: An integrated theory of capital budgeting [Working paper]
Magni, C. A.
abstract

2009 - Correct or incorrect application of the CAPM? Correct or incorrect decisions with the CAPM? [Articolo su rivista]
Magni, Carlo Alberto
abstract

This paper focuses on inconsistencies arising from the use of NPV and CAPM for capital budgeting. It shows that: (i) CAPM capital budgeting decision-making based on disequilibrium NPV is deductively inferred by the capital asset pricing model, (ii) the use of the disequilibrium NPV is widespread in finance both as a decision rule and as a valuation tool, (iii) the disequilibrium NPV does not guarantee additivity nor consistency with arbitrage pricing, so that it is unreliable for valuation, (iv) Magni’s [Magni, C.A., 2002. Investment decisions in the theory of finance: Some antinomies and inconsistencies. European Journal of Operational Research 137, 206–217; Magni, C.A., 2007a. Project valuation and investment decisions: CAPM versus arbitrage. Applied Financial Economics Letters 3 (2), 137–140] criticism of the NPV criterion refers to the disequilibrium NPV, and De Reyck’s [De Reyck, B., 2005. On investment decisions in the theory of finance: Some antinomies and inconsistencies. European of Operational Research 161, 499–504] project valuation method, on the basis of which Magni’s criticism to NPV is objected, leaves decision makers open to arbitrage losses and incorrect decisions. 2007 Elsevier

2009 - Firm value and the mis-use of the CAPM for valuation and decision making [Working paper]
Magni, Carlo Alberto
abstract

The use of CAPM‐based disequilibrium betas and Net Present Value (NPV) for investment decisionsand valuations is widespread in finance. Actually, its use is logically deducted from the CAPMassumptions. This paper deals with decisions about purchase of a firm and the related issue of firmvaluation. In particular, it contrasts disequilibrium betas and NPVs with Modigliani and Miller’sProposition I, and shows that disequilibrium betas and NPVs should not be used because theylead to irrational valuations and unreliable decisions; in particular, they lead decision makers toinfringe Modigliani and Miller’s Proposition I. To prove the thesis, a counterexample is shownwhere two firms with same expected free cash flows are valued, one of which is levered, the otherone is unlevered. A formal generalization is also provided. The results indicate that the use ofdisequilibrium NPV should be avoided, because valuations are incorrect and decisions are unsafe,leaving decision makers open to framing effects and arbitrage losses.

2009 - Investment decisions, net present value and bounded rationality [Articolo su rivista]
Magni, Carlo Alberto
abstract

The Net Present Value maximizing model has a respectable ancestry and is considered by most scholars to be a theoretically sound decision model. In real-life applications, decision makersuse the NPV rule, but apply a subjectively determined hurdle rate, as opposed to the ‘correct’ opportunity cost of capital. According to a heuristics-and-biases-program approach, thisimplies that the hurdle-rate rule is a biased heuristic. This work shows that the hurdle-rate rule may be interpreted as a fruitful strategy of bounded rationality, where several domain-specificand project-specific elements are integrated and condensed into an aspiration level. The paper also addresses the issue of a productive cooperation between bounded and unbounded rationality.

2009 - Opportunity cost, excess profit, and counterfactual conditionals [Articolo su rivista]
Magni, Carlo Alberto
abstract

Counterfactual conditionals are cognitive tools that we incessantly use during our lives for judgments, evaluations, decisions. Counterfactuals are used for defining concepts as well; an instance of this is attested by the notions of opportunity cost and excess profit (residual income), two all-pervasive notions of economics: They are defined by undoing a given scenario and constructing a suitable counterfactual milieu. Focussing on the standard paradigm and Magni’s (2000, 2005, 2009a,b) proposal this paper shows that the formal translation of the counterfactual state is not univocal and that Magni’s model retains formal properties of symmetry, additive coherence, homeomorphism,which correspond to properties of frame-independence, time invariance, completeness. Two introductory studies are also presented to illustrate how people cope with these counterfactuals and ascertain whether either model is seen as more “natural”. A brief discussion of the results obtained is also provided.

2009 - Potential dividends versus actual cash flows in firm valuation [Articolo su rivista]
Magni, Carlo Alberto; Vélez Pareja, Ignacio
abstract

Practitioners and some academics use potential dividends rather than actual payments to shareholders for valuing a firm’s equity. We underline the differences between the two methods and present some arguments supporting the thesis that firm valuation with potential dividends overstate the actual value of the firm’s equity. In particular, consistently with DeAngelo and DeAngelo (2006, 2007), we underline that cash flows create value for shareholders only if they are withdrawn from the firm, and that the use of potential dividends may lead to contradictions.

2009 - Potential dividends and actual cash flows in equity valuation: a critical analysis [Articolo su rivista]
Velez Pareja, Ignacio; Magni, Carlo Alberto
abstract

Practitioners and most academics in valuation include changes in liquid assets (potential dividends) in the cash flows. This widespread and wrong practice is inconsistent with basic finance theory. We present economic, theoretical, and empirical arguments to support the thesis. Economic arguments underline that only flows of cash should be considered for valuation; theoretical arguments show how potential dividends lead to contradiction and to arbitrage losses. Empirical arguments, from recent studies, suggest that investors discount potential dividends with high discount rates, which means that changes in liquid assets are not value drivers. Hence, when valuing cash flows, we should consider only actual payments.

2009 - Rating and ranking firms with fuzzy expert systems: the case of Camuzzi [Articolo su rivista]
Malagoli, Stefano; Magni, Carlo Alberto; Buttignon, Fabio; Mastroleo, Giovanni
abstract

In this paper we present a real-life application of a fuzzy expert system aimed at rating and ranking firms. Unlike standard DCF models, it integrates financial, strategic and business determinants and processes both quantitative and qualitative variables. Twenty-one value drivers are defined, concerning the target firm (strategic assets in place and expected financial performance), the acquisition (synergies, quality of management) and the sector (intensity of competition, entry barriers). Their combination via if-then rules leads to the definition of an output represented by a real number in the interval [0,1]. Such a number expresses the valuegenerating power of the target firm inclusive of synergies with the bidder (Strategic Enterprise Value). The system may be used for rating and ranking firms operating in the same sector. A regression analysis using hostile takeovers multiples may be employed to translate the score into price. The real-life case refers to Camuzzi (a natural gas distributor), acquired by Enel, the Italian ex monopolist of electric energy.

2009 - Splitting up value: A critical review of residual income theories [Articolo su rivista]
Magni, Carlo Alberto
abstract

This paper deals with the notion of residual income, which may be defined as the surplus profit that residues after a capital charge (opportunity cost) has been covered. While the origins of the notion trace back to the 19th century, in-depth theoretical investigations and widespread real-life applications are relatively recent and concern an interdisciplinary field connecting management accounting, corporatefinance and financial mathematics (Peasnell, 1981, 1982; Peccati, 1987, 1989, 1991; Stewart, 1991; Ohlson, 1995; Arnold and Davies, 2000; Young and O’Byrne, 2001; Martin, Petty and Rich, 2003). This paper presents both a historical outline of its birth and development and an overview of the main recent contributions regarding capital budgeting decisions, production and sales decisions, implementation of optimal portfolios, forecasts of asset prices and calculation of intrinsic values. A most recent theory, the systemic-value-added approach (also named lost-capital paradigm), provides a different definition ofresidual income, consistent with arbitrage theory. Enfolded in Keynes’s (1936) notion of user cost and forerun by Pressacco and Stucchi (1997), the theory has been formally introduced in Magni (2000a,b,c;2001a,b; 2003), where its properties are thoroughly investigated as well as its relations with the standard theory; two different lost-capital metrics have been considered, for value-based management purposes, by Drukarczyk and Schueler (2000) and Young and O’Byrne (2001). This work illustrates the main properties of the two theories and their relations, and provides a minimal guide to construction of performance metrics in the two approaches.

2009 - Un ombrello logico per la valutazione di azienda: la relazione fondamentale [Articolo su rivista]
Magni, Carlo Alberto
abstract

Questo articolo analizza le relazioni tra concetti diversi quali reddito, profitto, interesse, tasso, consumo, dividendo, rata, flusso di cassa, capitale. Ci si propone di riunire tali nozioni in un unico “ombrello” interpretativo a cinque punte, ciascuna delle quali rappresenta una prospettiva disciplinare: convenzionalmente, le denomineremo (1) contabilità, (2) teoria economica, (3) teoria della finanza, (4) teoria del credito, (5) matematica finanziaria. Queste nozioni e queste aree tematiche costituiscono un apparente guazzabuglio; si tratta in realtà di un puzzle che si tenterà di ricostruire. Sulla base del quadro ottenuto si mostrerà che la valutazione di azienda si fonda su una singola relazione tra tre nozioni economiche fondamentali: capitale, profitto, flusso di cassa.

2008 - CAPM-based capital budgeting and nonadditivity [Articolo su rivista]
Magni, Carlo Alberto
abstract

Purpose – In investment decision making, the net present value (NPV) rule is often used alongside the well-known capital asset pricing model (CAPM). In particular, the use of disequilibrium NPV is endorsed in corporate finance for both valuation and decision. The purpose of this paper is to test the reliability of this approach to capital budgeting valuations and decisions.Design/methodology/approach – The use of disequilibrium values for computing a project’s NPV is considered, and the consistency with the CAPM is checked. The resulting valuation and decision arecontrasted with the no-arbitrage principle, which is universally considered a benchmark for rationality.Findings – The paper finds that the disequilibrium NPV is logically deducted from the CAPM for decision-making purposes. However, this NPV provides nonadditive values, which makes it inconsistent with the no-arbitrage principle.Practical implications – The use of the CAPM þ NPV procedure for valuing projects is invalid if disequilibrium values are used. Its use for decision making is logically valid but practically unsafe,because decision makers may frame equivalent courses of action in different ways, resulting in different decisions, which implies that they may incur arbitrage losses.Originality/value – The literature does not distinguish between equilibrium and disequilibrium NPV nor between valuation and decision. This paper explicitly makes this distinction and the resulting consequences are highlighted.Paper type - Research paper

2008 - Economic profit, NPV, and CAPM: Biases and violations of Modigliani and Miller’s Proposition I [Articolo su rivista]
Magni, Carlo Alberto
abstract

For one–period projects under certainty, the notion of Net Present Value (NPV) formally translates the notion of economic profit, where the discount rate is the cost of capital. Under uncertainty, the cost of capital is the expected rate of return of an equivalent-risk alternative that the investor might undertake and is often found by making recourse to the Capital Asset Pricing Model. This paper shows that the notions of disequilibrium NPV and economic profit for risky one-period projects are not equivalent: NPV-minded agents are open to framing effects and to arbitrage losses, which imply violations of Modigliani and Miller’s Proposition I. The notion of disequilibrium (present) value, deductively derived from the CAPM by several authors and widely used in applied corporate finance, should therefore be dismissed.

2007 - A Sum & Discount method of appraising firms: An illustrative example [Working paper]
Magni, Carlo Alberto
abstract

2007 - A Sum&Discount Method for Appraising Firms: An Illustrative Example [Working paper]
Magni, C. A.
abstract

2007 - CAPM and capital budgeting: present versus future, equilibrium versus disequilibrium, decision versus valuation [Working paper]
Magni, C. A.
abstract

2007 - Measuring Performance and Valuing Firms: In Search of the Lost Capital [Working paper]
Magni, C. A.
abstract

2007 - Project selection and equivalent CAPM-based investment criteria [Articolo su rivista]
Magni, Carlo Alberto
abstract

This article shows that the Capital Asset Pricing Model-based capital budgeting criteria proposed by Tuttle and Litzenberger (1968), Mossin (1969), Hamada (1969), Stapleton (1971), Rubinstein (1973), Bierman and Hass (1973) and Bogue and Roll (1974) are equivalent. They all state that a project is profitable if its internal rate of return is greater than the risk-adjusted cost of capital, where the latter is given by the sum of the risk-free rate and a risk-premium which is a function of the systematic risk of the project, itself a function of the project cost.

2007 - Project valuation and investment decisions: CAPM versus arbitrage [Articolo su rivista]
Magni, Carlo Alberto
abstract

This study shows that (a) project valuation via CAPM contradictsvaluation via arbitrage pricing, (b) CAPM-minded decision makers may fail to profit from arbitrage opportunities, (c) standard CAPM-based valuation violates value additivity. As a consequence, the use of CAPM for project valuation and decision making should be reconsidered.

2007 - Rating and Ranking Firms with Fuzzy Expert Systems: The Case of Camuzzi [Working paper]
Malagoli, S.; Magni, C. A.; Buttignon, F.; Mastroleo, G.
abstract

2007 - Relevance or Irrelevance of Retention for Dividend Policy Irrelevance [Working paper]
Magni, C. A.
abstract

2007 - Residual Income and Value Creation: An Investigation Into Lost-Capital Paradigm [Working paper]
Magni, C. A.
abstract

2007 - The use of fuzzy logic and expert systems for rating and pricing firms: a new perspective on valuation [Articolo su rivista]
Malagoli, Stefano; Magni, Carlo Alberto; Mastroleo, Giovanni
abstract

This paper presents an expert system aimed at evaluating firms and business units. It makes use of fuzzy logic and integrates financial, strategic, managerial aspects, processing both quantitative and qualitative information. Twenty-nine value drivers are explicitly taken into account and combined together via “if-then” rules to produce an output. The output is a real number in the interval [0,1], representing the value-creation power of the firm. The system may be used for rating, ranking and pricing firms as well as for assessing the impact of managers’ decisions on value creation and as a tool of corporate governance.

2006 - Ambiguità nell’applicazione del CAPM per la valutazione degli investimenti [Articolo su rivista]
Magni, Carlo Alberto
abstract

2006 - An alternative approach to firms' evaluation: Expert systems and fuzzy logic [Articolo su rivista]
Magni, Carlo Alberto; Malagoli, Stefano; Mastroleo, Giovanni
abstract

Discounted cash flow techniques are the generally accepted methods for valuing firms. Such methods do not provide explicit acknowledgment of the value determinants and overlook their interrelations. This paper proposes a different method of firm valuation based on fuzzy logic and expert systems. It does represent a conceptual transposition of discounted cash flow techniques but, unlike the latter, it takes explicit account of quantitative and qualitative variables and their mutual integration. Financial, strategic and business aspects are considered by focusing on 29 value drivers that axe combined together via if-then rules. The output of the system is a real number in the interval [0, 1], which represents the value-creation power of the firm. To corroborate the model a sensitivity analysis is conducted. The system may be used for rating and ranking firms as well as for assessing the impact of managers' decisions on value creation and as a tool of corporate governance.

2006 - Economic value added and systemic value added: symmetry, additive coherence and difference in performance [Articolo su rivista]
Ghiselli Ricci, Roberto; Magni, Carlo Alberto
abstract

Two measures of residual income (excess profit) are currently available in the literature: Economic Value Added (EVA) (Stewart, 1991) and Systemic Value Added (SVA) (Magni, 2003a, b, 2004; 2005). This study shows that, unlike EVA, SVA is symmetric and additively coherent. Also, EVA and SVA are not simply different in value but also convey different information about good or bad performances.

2006 - Project Selection and Equivalent CAPM-based Investment Criteria [Working paper]
Magni, C. A.
abstract

2006 - Zelig and the art of measuring excess profit [Articolo su rivista]
Magni, Carlo Alberto
abstract

This paper tells the story of a student of economics and finance who meets a couple of alleged psychopaths, suffering from the ‘syndrome of Zelig’, so that they think of themselves to be experts of economic and financial issues. While speaking, they come across the concept of excess profit. The student tells them that the formal way to translate excess profit is to apply Stewart’s (1991) EVA model and shows that this model is equivalent to Peccati’s (1987, 1991, 1992) decomposition model of a project’s Net Present (Final) Value. The ‘Zeligs’ listen to him carefully, then try to apply themselves the EVA model: Unfortunately, both She-Zelig and He-Zelig seem to feel uneasy with basic mathematics, so they make some mistakes. Consequently, each of them miscalculates the excess profit. Strangely enough, they make different mistakes but both get to the (correct) Net Final Value of the project and, in addition, their excess profits do coincide. Further, the (biased) models presented by the Zeligs, though different from the EVA model, seem to bear strong relations to the latter. The student is rather surprised. I give my version of this event, arguing that the Zeligs are offering us a rational way of measuring excess profit, alternative to the standard one (EVA) but equally valuable. As I see it, they are only adopting a different cognitive interpretation of the concept of excess profit, which is based on a counterfactual conditional that differs from Stewart’s and Peccati’s.

2005 - Firm Value and the Mis-Use of the CAPM for Valuation and Decision Making, pp. 17 [Working paper]
Magni, C. A.
abstract

2005 - Investment decisions, equivalent risk and bounded rationality [Working paper]
Magni, c. a.
abstract

2005 - La definizione di investimento e il criterio del TIR ovvero: la realtà inventata [Articolo su rivista]
Magni, Carlo Alberto
abstract

La definizione di investimento comunemente accettata in matematica finanziaria (legata ai flussi di cassa) e il problema del Tasso Interno di Rendimento (TIR) vengono rivisitati sotto nuova luce, che consente di imputare alla definizione stessa di investimento un uso improprio del criterio del TIR, attribuendo a quest’ultimo il ruolo chiaro e univoco che gli è proprio. Esso è un parametro di valutazione affidabile la cui presunta ambiguità deriva da una definizione errata di investimento. Si propone una nuova definizione di investimento, inteso come modificazione strutturale del sistema patrimoniale-finanziario del decisore, la quale rende il TIR unico, chiaro e affidabile.

2005 - Norms of Rationality and Investment Decisions: CAPM, Arbitrage and Description Invariance [Working paper]
Magni, Carlo Alberto
abstract

The classical Capital Asset Pricing Model (CAPM) represents a well-rooted paradigm of rationality for investment decision-making. Following some anticipations in Magni [European Journal of Operational Research 137 (2002) 206] and Magni [AppliedFinancial Economics Letters, forthcoming] this paper shows that decision makers abiding by the CAPM prescriptions violate two other standards of rationality: The arbitrage principle and the principle of description invariance. In particular, referring to investments, (i) the notion of value in the CAPM is incompatible with the one employed in arbitrage theory; (ii) CAPM-minded decision makers may fail to exploit arbitrage opportunities; (iii)the principle of additivity is not fulfilled in CAPM-based asset valuation; (iii) the notion of value derived from CAPM is senseless; (iv) CAPM-minded agents fall prey to framingeffects. In other terms, this paper fosters the idea that economic agents complying with this consolidated paradigm of valuation and decision-making exhibit biases in investmentvaluation and choice.

2005 - On decomposing net final values: EVA, SVA and shadow project [Articolo su rivista]
Magni, Carlo Alberto
abstract

A decomposition model of Net Final Values (NFV), named Systemic Value Added (SVA), is proposed for decision-making purposes, based on a systemic approach introduced in Magni [Magni, C. A. (2003), Bulletin of Economic Research 55(2), 149-176; Magni, C. A. (2004) Economic Modelling 21, 595-617]. The model translates the notion of excess profit giving formal expression to a counterfactual alternative available to the decision maker. Relations with other decomposition models are studied, among which Stewart's [Stewart, G.B. (1991), The Quest for Value: The EVA (TM) Management Guide, Harper Collins, Publishers Inc]. The index here introduced differs from Stewart's Economic Value Added (EVA) in that it rests on a different interpretation of the notion of excess profit and is formally connected with the EVA model by means of a shadow project. The SVA is formally and conceptually threefold, in that it is economic, financial, accounting-flavoured. Some results are offered, providing sufficient and necessary conditions for decomposing NFV. Relations between a project's SVA and its shadow project's EVA are shown, all results of Pressacco and Stucchi [Pressacco, F. and Stucchi, P. (1997), Rivista di Matematica per le Scienze Economiche e Sociali 20, 165-185] are proved by making use of the systemic approach and the shadow counterparts of those results are also shown.

2005 - Project valuation and investment decisions: CAPM versus arbitrage [Working paper]
Magni, c. a.
abstract

2005 - Reasoning the ‘net-present-value’ way: Biases and how to use psychology for falsifying decision models [Working paper]
Magni, Carlo Alberto
abstract

This paper analyzes the net-present-value (NPV) model, a keystone in economics:Behaviors and lines of reasoning of NPV-minded decision makers are observed and analyzed. As aresult, one finds out that the NPV methodology is biased and its decision makers fall prey tovarious forms of fallacies and inconsistencies. The normative side of the NPV paradigm istherefore challenged by psychological arguments. While the power of psychology in falsifyingmodels and theories from a descriptive point of view is already recognized, it may be the case thatpsychology will prove useful in normative falsification of optimization models as well.

2005 - Reasoning the ‘net-present-value’ way: Some biases and how to use psychology for falsifying decision models [Working paper]
Magni, C. A.
abstract

2005 - THEORETICAL FLAWS IN THE USE OF THE CAPM FOR INVESTMENT DECISIONS [Working paper]
Magni, C. A.
abstract

2005 - The Use of NPV and CAMP for Capitai Budgeting Is Nota Good Idea. A Reply to De Reyck (2005) [Working paper]
Magni, C. A.
abstract

2005 - The use of NPV and CAPM for capital budgeting is not a good idea: A reply to De Reyck (2005) [Working paper]
Magni, Carlo Alberto
abstract

In Magni [Eur. J. Operat. Res. 137 (2002) 206] I present some inconsistencies implicit in the net-present-value criterion, as currently used in finance. This paper shows that the standard use of CAPM for capital budgeting, based on disequilibrium values, is at odds with arbitrage theory, and that the corresponding CAPM-based NPV rule is meaningless even in the simplest case, because net present value is any number one wants it to be. Cognitively, this amounts to saying that the NPV procedure leaves decision makers subject to a framing bias; financially, this amounts to saying that additivity does not hold. De Reyck's [Eur. J. Operat. Res. 161 (2005) 499] objection to my thesis is invalid because he mistakes a project's expected rate of return for a project's cost of capital. De Reyck's Proposition, on the basis of which my thesis is criticized, leaves decision makers open to arbitrage losses, because it is an (admittedly interesting) reframing of the security market line and (as surprising as it might be) the use of the SML for project valuation is incompatible with the no-arbitrage principle. To use NPV and CAPM for capital budgeting is not a good idea.

2005 - Theoretical flaws in the use of CAPM for investment decisions [Working paper]
Magni, Carlo Alberto
abstract

This paper uses counterexamples and simple formalization to show that the standardCAPM-based Net Present Value may not be used for investment valuations. The reason is that thestandard CAPM-based capital budgeting criterion implies a notion of value which does not complywith the principle of additivity. Framing effects arise in decisions so that different descriptions ofthe same problem lead to different choices. As a result, the CAPM-based NPV as a tool for valuingprojects and making investment decisions is theoretically unsound, even if the CAPM assumptionsare met.

2004 - Antinomie e illusioni cognitive nel criterio del valore attuale netto [Articolo su rivista]
Magni, Carlo Alberto
abstract

This paper shows that the net-present-value criterion is unreliable for logical reasons: It rests on a tenet stating that a rational decision maker should never compare alternatives which are not homogeneous in terms of risk. This tenet is self-contradictory and inapplicable. It is possible to escape this contradictory tool by integrating cognitive psychology, expert systems and fuzzy logic.

2004 - Modelling excess profit [Articolo su rivista]
Magni, Carlo Alberto
abstract

This paper deals with the problem of modelling in a formal way the concept of excess profit. A common idea is that excess profit is an unequivocal concept, being the difference between profit and costs, where all types of costs are taken into account, included the opportunity cost, i.e. the profit the entrepreneur would obtain if she invested in another business. This paper aims at showing that this difference is not univocal and that different approaches may be followed to give voice to such a notion. It turns out that two different interpretations are possible. The one existing in the literature is well described by Preinreich (1938); Edwards and Bell (1961) and, more recently, in the financial literature, by Stewart (1991). The interpretation here provided gives rise to a different way of modelling the notion of excess profit. While the existing models are tied to the financial literature, the model here presented is more akin to a microeconomic perspective. The paper focuses on the formal relations among the various models and necessary and sufficient conditions are provided for the integration of all models in the systemic framework here adopted.

2004 - Strategic options and expert systems: a fruitful marriage [Articolo su rivista]
Magni, Carlo Alberto; Mastroleo, Giovanni; Vignola, Marina; Facchinetti, Gisella
abstract

Business economics does not provide anymethodology for appraising strategic investments, relyingon informal approaches. Conversely, financial economicsoffers us plenty of sophisticated mathematical modelsunsuitable for applications and based on unrealisticassumptions. This paper presents an example of howstrategic investments may be handled with a formal buteasy-to-understand tool. While this paper shows a specificapplication, a real-life case, we think the model hereproposed may be generalized, so contributing to developinga new approach to business decisions. In particular,we think of a fuzzy expert system approach as a convenienttool overwhelming many of the shortcomingsinherent in the ‘‘crisp’’ approaches of the financial literature(DCF methods, options pricing, dynamic programming),while avoiding at the same time the refusal of anymethodology (typical of business economics). The ideahere presented develops some results by Magni et al.(2001) and Facchinetti et al. (2001). An evaluation functionis drawn up via ‘‘if-then’’ rules; the latter are made towork automatically by means of an expert system, whichadequately replicates the evaluation of human experts.A sensitivity analysis is presented to test the soundness ofthe model.

2003 - Decomposition of Net Final Values: Systemic Value Added and Residual Income [Articolo su rivista]
Magni, Carlo Alberto
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 Net Present Value (NPV)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 ofshadow project the SVA model can be re-shaped so that the decomposition of the SVA can be accomplished by applyingPeccati'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 tetra-dimensional decomposition iseasily obtained.

2003 - Economic value added and systemic value added as competing measures for modelling excess profit [Relazione in Atti di Convegno]
Magni, Carlo Alberto
abstract

Investigation of the differences between "Economic value added" and "Systemic value added" for measuring excess profit

2002 - Antinomie e illusioni cognitive nel criterio del valore attuale netto [Working paper]
Magni, C. A.
abstract

2002 - Investment decisions in the theory of finance: some antinomies and inconsistencies [Articolo su rivista]
Magni, Carlo Alberto
abstract

The net-present-value rule is a pillar of modern finance theory. As known, it is a capital budgeting rule. Finance theory prescribes the investor to compare the opportunity in hand with an asset of equivalent risk, i.e. to discount cash flows with a risk-adjusted rate of return. This paper aims at showing that inconsistencies and antinomies arise when applying the above-mentioned rule. Further, it turns out that it is actually impossible to compare alternatives equivalent in risk and any decision maker cannot prevent herself to violate the above tenet.

2001 - A Fuzzy Expert System for Solving Real Option Decision Processes [Articolo su rivista]
Magni, Carlo Alberto; Mastroleo, Giovanni; Facchinetti, Gisella
abstract

This paper presents a new approach to real options. The current options-based models have provided new insights into capital-budgeting decisions. Unfortunately they are not widely used by corporate managers and practitioners as they are formally complex, rather difficult to understand and rest on strong implicit assumptions that considerably limit their scope of application. We propose a possible alternative by using a fuzzy expert system, on the basis of Mastroleo, Facchinetti and Magni (2001). We draw up a decision tree with multiple uncertain variables affecting the value of an investment opportunity, consisting of a defer option, a growth option, an abandonment option. Some simulations are conducted to test the economic soundness of the model as well as its consistency with the current models in the literature. A rather refined study can be accomplished by showing how inputs and outputs of the model interrelate one another.

2001 - A Proposal for modeling real options through fuzzy expert systems [Relazione in Atti di Convegno]
Mastroleo, Giovanni; Facchinetti, Gisella; Magni, Carlo Alberto
abstract

This paper presents a proposal for evaluating real options. The existing models are not widely used by corporate managers as they are formally complex, rather difficult to understand and rest on strong implicit assumptions. We propose a possible alternative by using a fuzzy expert system.

2001 - An application of fuzzy expert systems to strategic investments: The case of Florim S.p.a [Relazione in Atti di Convegno]
Facchinetti, G.; Magni, Carlo Alberto; Mastroleo, G.; Vignola, Marina
abstract

This paper makes use of a fuzzy expert system to evaluate a strategic investment. In particular, the model proposed aims at replicating the actual decision process accomplished by Florim S.p.a., an Italian ceramic tile firm which recently had the opportunity of buying a firm in the USA. The model is perfectly consistent with the evaluation process conducted by Florim’s experts and on the basis of the same data available to the expert’s panel, our expert system provides the same investment value and therefore the same solution to the decision process.

2001 - Scomposizione di sovraprofitti: Economic Value Added e Valore Aggiunto Sistemico [Articolo su rivista]
Magni, Carlo Alberto
abstract

The Economic Value Added formally translates the theoretical notion of excess profit (also known as residual income). Its use is so firmly entrenched in applied corporate finance and management accounting that its name is often used as a noun for denoting the concept of excess profit itself. This paper investigates the conceptual properties of such a notion and, in particular, it studies the relations between the excess profit generated in a period and the excess profit generated in the following period, showing that the classical approach forgets the past story of the project and the evolution of the capital invested. On the basis of this analysis, a new approach to residual income is offered, called Systemic Value Added (SVA). The latter takes account of the dynamic system governing the evolution of the capital invested, and is coherently additive in that the uncompounded sum of the SVAs leads to the Net Final Value. Interesting relations between the classical approach and the new approach are provided, and a final conventionalist position is endorsed: the excess profit is not an unambiguous concept and the choice between either approaches depends on the pieces of information one is willing to retrieve.

2001 - Valore aggiunto sistemico: un'alternativa all'EVA quale indice di sovraprofitto periodale [Articolo su rivista]
Magni, Carlo Alberto
abstract

In questo lavoro viene presentato un modello di sovraprofitto periodale che introduce la nozione di Valore Aggiunto Sistemico (VAS). Esso si contrappone all’EVA di Stewart (1991) pur essendo ad esso coerente in termini globali: il Valore Finale Netto (VFN) di un investimento può essere ottenuto mediante somma degli EVA capitalizzati o mediante somma dei VAS non capitalizzati. Pertanto i modelli VAS ed EVA scompongono il VFN in modo diverso. Due esempi numerici rendono agevole l’applicazione del modello proposto. I due indici sono il risultato di un approccio cognitivo differente. La possibilità di dare vita a traduzioni formali differenti del concetto di sovraprofitto induce a ritenere che tale nozione sia eminentemente convenzionale.

2001 - Valuing strategic investments with a fuzzy expert system: an italian case [Relazione in Atti di Convegno]
Facchinetti, G.; Magni, Carlo Alberto; Mastroleo, G.; Vignola, Marina
abstract

This paper makes use of a fuzzy expert system forvaluing a strategic investment. In particular, the modelproposed aims at replicating the actual decisionprocess accomplished by Florim S.p.a., an Italianceramic tile firm which recently had the opportunity ofbuying a firm in the USA. The model is perfectlyconsistent with the evaluation process conducted byFlorim's experts and on the basis of the same dataavailable to the expert's panel our expert systemprovides the same investment value and therefore thesame solution to the decision process.

2001 - Zelig an d the Art of Measuring Residual Income [Working paper]
Magni, C. A.
abstract

2000 - Decomposition of a Certain Cash Flow Stream: Differential Systemic Value and Net Final Value [Relazione in Atti di Convegno]
Magni, Carlo Alberto
abstract

2000 - Decomposition of a Certain Cash Flow Stream: Systemic Value Added and Net Final Value [Working paper]
Magni, c. a.
abstract

2000 - On Decomposing Net Final Values: Systemic Value Added and Shadow Project [Working paper]
Magni, c. a.
abstract

2000 - Real Options: a fuzzy approach for strategic investments [Working paper]
Facchinetti, G.; Magni, C. A.; Mastroleo, G.
abstract

2000 - Systemic Value Added, Residual Income and Decompasition of a Cash Flow Stream [Working paper]
Magni, c. a.
abstract

2000 - Tir, Roe, Van: convergenze formali e concettuali in un approccio sistemico [Articolo su rivista]
Magni, Carlo Alberto
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, accountingflavoured: 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.

2000 - Valore Aggiunto Sistemico: un’alternativa all’EVA quale indice di sovraprofitto periodale [Working paper]
Magni, c. a.
abstract

1999 - Un criterio strutturalista per la valutazione di investimenti [Articolo su rivista]
Magni, Carlo Alberto
abstract

1998 - A systemic rule for investment decisions: generalizations of the traditional DCF criteria and new conceptions [Working paper]
Magni, Carlo Alberto
abstract

1998 - Aspetti quantitativi e qualitativi nella valutazione di un'opzione di investimento [Articolo su rivista]
Magni, Carlo Alberto
abstract

1998 - NOTE SPARSE SUL DILEMMA DEL PRIGIONIERO (E NON SOLO) [Working paper]
Magni, Carlo Alberto
abstract

1998 - Pictures, language and research: the case of fin ance an d financial mathematics [Working paper]
Magni, c. a.
abstract

1998 - Pictures, language and research: the case of finance and financial mathamatics [Working paper]
Magni, Carlo Alberto
abstract

1997 - Critica alla Definizione Classica di Investimento: un Approccio Sistematico [Working paper]
Magni, c. a.
abstract

1997 - IRR, ROE and NPV: a Systemic Approach [Working paper]
Magni, Carlo Alberto
abstract

1997 - La Definizione di Investimento e il Criterio del Tir ovvero: la Realtà Inventata [Working paper]
Magni, c. a.
abstract

1997 - La trappola del ROE e la tridimensionalità del VAN in un approccio sistemico, [Relazione in Atti di Convegno]
Magni, Carlo Alberto
abstract

1997 - La trappola del Roe e la tridimensionalità del V an in un approccio sistemico [Working paper]
Magni, c. a.
abstract

1997 - Paradossi, inverosimiglianze e contraddizioni del VAN: operazioni certe [Working paper]
Magni, c. a.
abstract

1997 - Paradossi, inverosimiglianze e contraddizioni del Van: operazioni Aleatorie [Working paper]
Magni, Carlo Alberto
abstract

1997 - Tir, Roe e Van: Distorsioni Linguistiche e Cognitive nella Valutazione degli Investimenti [Working paper]
Magni, c. a.
abstract

1997 - Un Criterio Strutturalista per la Valutazione di Investimenti [Working paper]
Magni, Carlo Alberto
abstract

1996 - Opzioni reali d’investimento e interazione competitiva: programmazione dinamica stocastica in optimal stopping [Working paper]
Magni, Carlo Alberto
abstract

1996 - Repeatable and una tantum real options a dynamic programming approach [Working paper]
Magni, Carlo Alberto
abstract

1996 - Repeatable and una tantum real options a dynamic programming approach pp. 23 [Working paper]
Magni, C. A.
abstract

1996 - Un esempio di opzione di investimento industriale con interazione competitiva e avversione al rischi [Working paper]
Magni, C. A.
abstract

1996 - Un esempio di opzione di investimento industriale con interazione competitiva e avversione al rischio [Relazione in Atti di Convegno]
Magni, Carlo Alberto
abstract

1996 - Un semplice modello di opzione di differimento e di vendita in ambito discreto [Working paper]
Magni, c. a.
abstract

1996 - Un semplice modello di opzione di investimento e di vendita in àmbito discreto [Working paper]
Magni, Carlo Alberto
abstract

1996 - Vaghezza e logica fuzzy nella valutazione di un’opzione reale [Working paper]
Magni, C. A.
abstract

1995 - Un uso delle catene di Markov nella rappresentazione dell'intensità concorrenziale in un settore economico [Relazione in Atti di Convegno]
Magni, Carlo Alberto
abstract

1994 - Un criterio Matrioska nella selezione del portafoglio [Relazione in Atti di Convegno]
Magni, Carlo Alberto
abstract

1993 - Valore attuale netto e politiche ottimali d'indebitamento [Relazione in Atti di Convegno]
Magni, Carlo Alberto
abstract

1991 - Reputazione e credibilità di una minaccia in un gioco di contrattazione [Articolo su rivista]
Magni, Carlo Alberto; Ricci, Gianni
abstract

Il presente articolo tratta di giochi di contrattazione in cui due giocatori cooperano al fine di trarne un mutuo beneficio. Se un giocatore devia dalla strategia cooperativa l’altro pone in essere una minaccia precedentemente annunciata a titolo di ritorsione. Questo modello cerca di formalizzare il concetto di efficacia e credibilità di una minaccia. Si prende poi in considerazione la reputazione di un giocatore, la quale influenza la decisione di attuare una minaccia incrementando la funzione di utilità di colui che la pone in essere. La velocità di ritorsione e la rapidità di percezione dell’inganno mostrano come un giocatore può deviare dalla strategia cooperativa vanificando gli sforzi di ritorsione del giocatore ingannato.

Università degli studi di Modena e Reggio Emilia

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