Risk-Analysis Techniques for the Economic Evaluation of Investment Projects

Gobbi (1919, p. 49) wrote: «An event may be economically advantageous or disadvantageous for a person depending on whether he or she is more or less equipped with means for his/her life; it is indifferent if the event has neither one nor the other effect. The word risk is sometimes used in the sense of chance that has, good or bad, economic consequences; most often in the sense of an economically unfavorable eventuality».

Nor does this meaning contradict the distinction for business activities proposed by Mowbray and Blanchard (1961) in:

Uncertainty indicates a “generic state” that does not support complete knowledge of the development of the facts. Resuming Spencer and Siegelman (1964, p. 9): «Uncertainty has been defined as a state of knowledge in which one or more alternatives result in a set of possible specific outcomes, but where the probabilities of the outcomes are neither known nor meaningful».

The same explanation is given by Saraceno (1970, p. 122): «It can be seen that:

Panati and Golinelli (1991) use the terms deterministic certainty, probability certainty and absolute uncertainty.

The same exegesis is also given by Zerbe and Bellas (2006, p. 256) for the purposes of the cost-benefit analysis of investment projects: «Uncertainty refers to the idea that planners do not know for certain what the state of the world will be. While they realize that different states of the world may occur, the relative probabilities of these states of the world may be unknown. Risk is a condition where probabilities are assigned to these different states of the world and active consideration is given to how good or bad the outcomes are in each state of the world».

In this sense, Pennisi and Scandizzo (2003) on p. 227 give an interesting definition of uncertainty and risk: «There is usually uncertainty when it comes to an objective condition, characterized by the possibility that future events may differ according to the different states of nature that may occur. We speak instead of risk to denote a subjective condition in which, as a consequence of uncertainty, the well-being of one or more subjects depends on the particular state of nature that will occur. It is said in this case that the subject runs a risk to indicate that his well-being can be jeopardized (but may also be exalted) by the realization of one or more particular events».

Among them, we find: Keynes (1921, 2nd ed. 1929), Keynes (1936), Borel (1924‐34), Von Neumann and Morgenstern (1944), Samuelson (1952), Hicks (1959) and Saraceno (1970, op. cit.).

It should be noted, however, that studies related to the criteria useful for the so-called uncertainty economy date back to Fermat and Pascal (1654), Huygens (1657), Bernoulli (1730, 1731, 1738) and Laplace (1814). For further information on the historical evolution of risk analysis and treatment, see Massè (1965).

It is worth emphasizing the different conceptual meaning often attributed to the terms “economic appraisal” and “economic analysis” of the risk. As Dezzani F. explains (1971, p. 20): «While the studies that specifically investigate the economic reflection of future events are studies on the risk’s “economic analysis”, those aiming at researches on the occurrence of future events’ “state of knowledge” are instead studies on the risk’s “economic evaluation”».

The economic-corporate literature deals with risk management in business activities since the early decades of the last century. At first, the issue was addressed in terms of insurance management, since it was almost exclusively tended to forms of axiomatic cover; Only since the 1950s was there the introduction of the term risk management, currently understood as a process to permeate the entire decision-making process. On historical-evolutionary aspects, just mentioned here, see Bernstein (2002) and Beretta (2004, in particular pages 20–33).

Equivalent is the definition that Amelotti and Valcalda (1998, pp. 253–280) given to risk management for investment projects. In particular, the authors recognize moments, temporally successive and interrelated, so understood:

As Mishan (1974, p. 295) claims, «the problem concerning how to make decisions in those situations where past experience has little or no orientation value cannot be satisfactorily solved either in logical or empirical terms, and all of the rules that have been formulated are applicable in a limited way or without a practical value».

With this tool, a mathematical law, called objective function, which corresponds to the project’s result with its representative variables, is defined. Constraints, associated with the function and expressed in the form of equations or inequalities, define the eligibility set for the values ​​of the variables. The problem to solve is therefore to optimize a function with multiple variables subjected to the predetermined constraint system.

With regard to the purely mathematical aspects, see, among many, Hillier and Lieberman (2006). For an application to the investment programs definition, see Morano and Nesticò (2007).

A critical analysis on the topic is in Mishan (1974, op. cit.). About the game theory use to support the decisions, see Bennion (1956), Luce and Raiffa (1967).

In Anglo-Saxon countries, we often talk about bop analysis, where bop is the acronym of best, optimistic, pessimistic.

On this topic, see Chandler and Cockle (1982) and Loasby (1990). Examples of both sensitivity analysis and evaluation in various scenarios are in Brealey et al. (2006, pp. 234–240). It is worth noting that the authors found in break-even point analysis a different way of applying the sensitivity analysis: «When we submit a project to sensitivity analysis or when we consider alternative scenarios, we wonder what would happen if sales or costs were different from forecasts. Managers sometimes prefer to ask this question in different terms and ask how much sales may decrease before the project may be in loss. This exercise is known as a break even point analysis» (p. 238).

Other less-known approaches are also proposed for risk analysis. One is that of mathematical programming, already cited as a useful tool for the uncertainty treating. See Mao (1969).

Further models are derived from financial risks’ evaluation techniques; See, for example, Basile and Erzegovesi (1989), Penati (1991) and Cherubini and Della Lunga (2001). Equally, references arise from studies on the portfolio-decisions theory, due to Markowitz H. M. (March 1952), originally intended for the selection of securities lending, and subsequently employed to generalize real-estate investments. In this matter: Markowitz (1959), Champernowne (1969), Bicksler and Samuelson (1974), Peterson (1974); and Copeland and Weston (1992).

Interesting reading is the Arrow K. J.’s work (1951), which reviews the economic, philosophical, mathematical and statistical literature on the choice of theme of risky solutions.

Of great value is S. Reutlinger’s text (1970).

It is pleonastic to say that the project’s features arise, at least in part, from those of the reference production sector.

Think of a project that employs “mature” technologies or, otherwise, innovative and untried technical solutions. In the latter circumstance, it may be overly simplistic to exclude future variants in some of the dimensions that draw the original structure of the intervention.

The rational basis of certain equivalent lies in the concept of risk aversion. See Dallocchio (1995, p. 339).

On the topic, see Schlaifer (1969), Allen (1983) and Ingersoll (1987).

As it is written in Pivato (1993, p. 934), CAPM’s thesis is that «in a “perfect” financial market and in equilibrium conditions, the yield of each title is equal to the sum of the interest rate and to a “premium” for systematic risk. This increase is much higher as the bond yield itself’ aptitude to fluctuate in harmony with the market is lower. The random risk component, which can be eliminated with a suitable portfolio diversification, does not have any influence on the examined rate».

And then it is specified: «Risk is divided into two parts: the diversifiable (or non-systematic) component is eliminated by the same operators, with the formation of the optimal market portfolio; the non-diversifiable (or systematic) component is related, instead, to the greater or lesser discrepancies that may occur between the yield fluctuations of the particular bond under consideration, on the one hand, and the optimal portfolio on the other. The equation expressing the model is the following one:

where μ(r _j ) is the expected yield, relative to the price, for the j-th bond; i is the interest rate for risk-free financial assets; μ(r _m ) is the expected return of the optimum market portfolio; β _j is a coefficient that expresses the magnitude of the expected systematic risk of the title. Precisely, it is:

For a clear discussion on CAPM, see Copeland and Weston (1992, op. cit.), Brealey et al. (2006, op. cit.). See also Lumby (1991), Damodaran (1996a, b), Massari (1998) and Pastorello (2001).

According to Ventriglia (2005, pp. 140 and 141), who also takes Sobrero M. under exam (1999), «higher risk premiums are assigned to riskier investment projects, which are characterized by a high degree of uncertainty relative to the magnitude and temporal distribution of the expected cash flows. This approach, however, often leads to excessive distortions, stemming from the fact that the financial flows of an investment are as much penalized by the discount rates, as they are distant from the time of the decision. The use of arbitrarily high and consistent over time risk premium, gives birth to geometric increasing discount rates, while it is almost never true that an investment is characterized by uncertainty having the same dynamics. It is true, however, that for some investments most of the uncertainty extinguishes itself, normally, in the early years of life of the project. As a result, the risk decreases over time. Ultimately, the risk adjustment according to the subjective approach often results in arbitrary increments of discount rates, which grow geometrically, reducing with the same progression the values of the most distant financial flows over time». Among the texts dealing with the theme of the risk premium, see Cornell (1999), Ibbotson et al. (2004).

On adjusting the risk discount rate according to the CAPM, see Ventriglia (2005, pp. 140–148).

The expected average yield and the yield variance’s concepts are of primary importance for the economic evaluation since, considered jointly, they enable the correct interpretation of the invested capital’s risk.

In statistical terms, the expected average yield R formula—estimated through the NPV or IRR or other indicator more suited to interpret the analysis results—is expressed as the weighted average of the various yields R _s (s = 1, …, n) that the intervention can generate when various scenarios occur, where the weighting factor P _s is given by the probabilities associated with each of the n scenarios:

\(R = \sum\limits_{s = 1}^{n} {P_{s} \cdot R_{s} }\).

The variance σ² is the average of the squared of yield deviations R _s associated with the initiative related to the average value R:

\(\sigma^{2} = \sum\limits_{s = 1}^{n} {P_{s} \cdot (R_{s} - R)^{2} }\).

The square root of the variance is the standard deviation, also known as the root mean square deviation. Although variance and standard deviation give the same information about the returns dispersion around the average, the second has the advantage to return the measure of the risk in the same unit of measure in which the expected or observed values and their average are provided.

For a simple reading of the stochastic dominance concept, refer to Dallocchio (1995, op. cit., pp. 320–324). For further information: Goodwin and Wright (2004).

It is important to emphasize that the applicability of statistical criteria depends on the availability of objective data and the ability to handle them. On the subject, Dallocchio (1995, pp. 311–312) adds: «Assigning a probability distribution to a project’s outcomes therefore implies a certain degree of subjectivity from the decision maker. In order to reduce the uncertainty characterizing the formulating estimates process, there is often an analysis of the projects’ historical performance that present a risk level similar to the one of the project to evaluate».

Further reflections can be found in Azzini (1982) and Piccolo and Vitale (1984).