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Real Option Exercise Decisions in Information Technology Investments: a Comment

A Correction to this article was published on 04 November 2020

This article has been updated

Abstract

The paper comments on Khan et al. (J Assoc Inf Syst 18(5):372–402, 2017), who study real option exercise decisions in the context of a single IT project and in a portfolio setting, respectively. The issues identified concern the concept of (economic) rationality and the treatment of project interdependencies. The explanations provided may prove useful in other contexts.

Introduction

Khan et al. [1] study real option exercise decisions in the context of a single IT project and in a portfolio setting, respectively. In particular, they “investigate the vulnerability of real option exercise decisions to decision biases” (p. 373). The authors provide a valuable service in bringing to the attention of the Information Systems (IS) community a few of the problems associated with IT investment decisions. However, I think some of the points raised by Khan et al. [1] are open to discussion. In detail, I question the classification of “rational” and “biased” decisions and the treatment of resource interdependencies. The facts required to clarify the problems referred to are distributed throughout a large number of publications from different streams of literature. Therefore, a mere reference to individual sources, such as textbooks on the valuation of real options [2, 3] or comprehensive articles, is not possible. The communication at hand collects and integrates the relevant arguments. It pays particular attention to the relationship between market-based and preference-based valuation and the distinction between linear and nonlinear dependencies. Against this background, suggestions for further research are derived. The explanations may be of use to the entire field of Operations Research.

Problems

My first objection concerns the classification of “rational” and “biased” decisions. Khan et al. [1, p. 374] use the following classification: “We use the term ‘managerial bias’ or ‘bias’ to describe any deviation from the decision that a rational economic agent would make in a similar situation. Given that real options theory and its application in IS uses a risk-neutral framework for valuing options (…), we consider a rational economic agent as a risk-neutral investor. Hence, a rational economic agent would exercise real options optimally, and bias would result in a suboptimal exercise decision.”

However, in option pricing theory “risk-neutral valuation” is to be understood in a different way. This is briefly outlined below. In general, the value of risky cash flows depends on the consumption possibilities associated with them (taking into account potential interactions with exogenous and market-determined income) and the preferences of the decision maker. I choose a one-period framework to formalize this.Footnote 1 Here, the decision maker’s preference value Φ(c0,c1) is defined over current consumption c0 and uncertain future consumption c1.Footnote 2 In this setting, the value of the option in question is understood as a certain current amount v whose preference value equals the preference value of the risky future cash flow x associated with holding the option when exogenous income sources and trading opportunities in market instruments are taken into account. Formally, this implies

$$ \max_{\mathbf{\psi}} \varPhi \left( v+e_{0}-\mathbf{\psi}^{T} \mathbf{m}_{0}, e_{1}+\mathbf{\psi}^{T} \mathbf{m}_{1} \right) = \max_{\mathbf{\xi}} \varPhi \left( e_{0}-\mathbf{\xi}^{T} \mathbf{m}_{0}, x+e_{1}+\mathbf{\xi}^{T} \mathbf{m}_{1} \right), $$

where ψ and ξ denote portfolios in financial market instruments, m0 and m1 represent vectors of securities prices, e0 and e1 describe exogenous (non-market) income. Variables relating to the future are to be interpreted as random. Financial instruments permit the decision maker to optimally spread consumption over time and states of nature according to his/her preferences. Let η denote a particular portfolio that replicates the returns of the option in each state, i.e., \(\mathbf {\eta }^{T} \mathbf {m}_{1} = x\). Setting ξ = ψη yields

$$ \begin{array}{@{}rcl@{}} &&\underset{\mathbf{\psi}}{\max} \varPhi \left( v+e_{0}-\mathbf{\psi}^{T} \mathbf{m}_{0}, e_{1}+\mathbf{\psi}^{T} \mathbf{m}_{1} \right) \\ &=& \underset{\mathbf{\psi}}{\max} \varPhi \left( e_{0}- \left[ \mathbf{\psi}-\mathbf{\eta} \right]^{T} \mathbf{m}_{0}, x+e_{1}+ \left[ \mathbf{\psi}-\mathbf{\eta} \right]^{T} \mathbf{m}_{1} \right). \end{array} $$

With the definition of the replicating portfolio, we obtain

$$ \underset{\mathbf{\psi}}{\max}\ \varPhi \left( v + e_{0}-\mathbf{\psi}^{T} \mathbf{m}_{0}, e_{1} + \mathbf{\psi}^{T} \mathbf{m}_{1} \right) \!= \underset{\mathbf{\psi}}{\max} \ \varPhi \left( \mathbf{\eta}^{T} \mathbf{m}_{0} + e_{0}-\mathbf{\psi}^{T} \mathbf{m}_{0}, e_{1} + \mathbf{\psi}^{T} \mathbf{m}_{1} \right)\!. $$

Obviously, \(v=\mathbf {\eta }^{T} {\mathbf {m}}_{0}\), meaning the value of the option to the decision maker equals the current market value of the replicating portfolio irrespective of the decision maker’s preferences or his/her exogenous income (so-called no-arbitrage principle [7]). Given a replicating portfolio, the valuation (and the determination of the optimal exercise strategy) are preference-free.

A numerical calculation requires the specification of the cash flows associated with the option and the financial instruments. Following Cox et al. [8], I assume a binomial development, meaning two possible states at the end of the period. The states 1 and 2 occur with (empirical) probabilities p and 1 − p, respectively.Footnote 3 The financial market contains two instruments, one risky (shares of common stock) and one riskless (bonds). Both positive and negative positions are allowed. The current price of a share equals s0 = s. At the end of the period, one share either trades at s11 = us (state 1) or s12 = ds (state 2). Bonds are priced at face value, b0 = 1, and promise a state-independent repayment, b11 = b12 = 1 + r. To prevent riskess arbitrage, u > 1 + r > d holds. With recourse to the notation developed above, we obtain m0 = (s0,b0)T and m1 = (s1,b1)T. The option to be valued is associated with returns x1 (state 1) and x2 (state 2). To determine the replicating portfolio, we set up the following systems of equations,

$$ \begin{array}{@{}rcl@{}} x_{1}&=&\eta_{s} us + \eta_{b} (1+r) \\ x_{2}&=&\eta_{s} ds + \eta_{b} (1+r), \end{array} $$

and obtain

$$ \eta_{s} = \frac{x_{1}-x_{2}}{(u-d)s}, \eta_{b}=\frac{ux_{2}-dx_{1}}{(u-d)(1+r)}. $$

The option value is now given by v = ηss + ηb. Simple transformations lead to

$$ v=\frac{1}{1+r} \left[ \frac{(1+r)-d}{u-d}x_{1} + \frac{u-(1+r)}{u-d}x_{2} \right]. $$

Since

$$ q=\frac{(1+r)-d}{u-d}, 1-q=\frac{u-(1+r)}{u-d} $$

are positive and add up to one, they are interpreted as “artificial” probabilities.

In traditional approaches to capital budgeting (such as Discounted Cash Flow (DCF)), risk is accounted for by adding a premium to the discount rate. As shown above, option pricing proceeds differently. Under the assumption that a portfolio of traded instruments can be constructed to replicate the returns of the option in question, no-arbitrage leads to an adjustment of the probability measure that is used to describe the uncertainty associated with future cash flows. Empirical probabilities (here: p,1 − p) are replaced by “artificial” probabilities (here: q,1 − q). The value of the option can then be obtained as follows. The “artificial” probabilities are used to calculate expected values. The latter are discounted using the risk-free rate. Due to the relationship to the risk-free rate, the “artificial” probabilities are usually referred to as “risk-neutral” probabilities. Personally, I prefer the term “risk-neutralized” probabilities.Footnote 4

Where the investment-specific risks cannot be replicated, this preference-free, i.e., purely market-based, valuation fails. Here, no-arbitrage arguments alone are not strong enough to produce unique valuations. Only (sometimes rather trivial) bounds on the project’s value can be obtained [12]. Instead, the decision maker has to account for individual circumstances (preferences, exogenous income [13]). Besides, the concept of “shareholder value”,Footnote 5 often referred to in the context of “value-based management” cannot provide an alternative justification for disregarding preferences. Its validity as a meaningful and (in the context of multiple shareholders) unanimously supported objective of the firm rests on the same terms as the “risk-neutral valuation” approach [16, 17]. Either way, it is the replicating condition that allows for a preference-independent ordering of the desirability of the actions available.

As far as IT investments are concerned, firm-specific and technological uncertainties play a major role. It is thus implausible to assume that the cash flows associated with real options can be (fully) replicated [18,19,20]. Therefore, literature points to the adverse consequences of undue simplifications [21] and calls for a preference-based valuation of real options [22, 23].

Let me come back to the article of Khan et al. [1]. As the authors do not refer to replicating markets instruments nor to a preceding (risk-neutralizing) adjustment of probabilities, a “risk-neutral valuation” in the sense of standard option pricing theory cannot be equalized with rationality. On the contrary, risk-averse or risk-seeking behavior consistent with individual preferences should be seen as fully rational. According to Simon [24, p. 294], “(t)he term ‘rational’ denotes behavior that is appropriate to specified goals in the context of a given situation.” With unobservable goals (i.e., preferences) and fragmentary information provided with regard to the characteristics of the investments in question (i.e., replicability, preceding risk adjustment, etc.) decision makers cannot be judged negatively by scientists.Footnote 6

Inversely, to explain behavior simply means to show how decision makers could have rationally chosen their action. This has been vividly elaborated by Allison and Zelikow [26, p. 25/26]: “It must be noted, however, that an imaginative analyst can construct an account of preference-maximizing choice for any action or set of actions performed (…). Putting the point more formally, if somewhat facetiously, we can state a ‘Rationality Theorem’: There exists no pattern of activity for which an imaginative analyst cannot write a large number of objective functions such that the pattern of activity maximizes each function.” Cochrane [25, p. 966] assists here: “(A)bsent arbitrage opportunities, there is always a ‘rational’ model, a specification of [preferences] that can rationalize any data.” An impressive example has been recently provided by Schneider and Day [27]. The authors demonstrate how to explain the classical paradoxes of decisions under risk using preferences that are linear in (empirical) probabilities and that maximize an expected utility function referring to an endogenous target return.

My second objection concerns the modeling of resource interdependencies. Khan et al. [1] use the following description: “Resource interdependency: we created resource interdependency in one of the portfolios by explicitly mentioning the flexibility in resources (funds and human) use between the projects” (p. 396). Later on, they write: “Both projects in the portfolio have resource dependency. This means each project in the portfolio share the same pool of financial and human resources. This sharing of resources among the projects within the portfolio means that the resources from one project can be utilized in another project” (p. 400).

However, this dependency cannot be identified in terms of the quantitative information provided (probabilities and state-dependent cash flows). Cash flow distributions are obviously identical for both independent and dependent portfolios. Consequently, Khan et al. [1, p. 387] did not find significant effects attributable to resource dependencies: “Interdependency among project resources in a portfolio had no significant impact on the narrow framing.”

From my perspective, an examination of project interdependencies requires the specification of both linear and nonlinear dependencies. Linear dependencies (i.e., correlations) indicate the degree of how the probabilities of success or failure of two projects move together. Within the previously developed binomial model, two projects (and the payments associated with them) can develop either in unison or in opposite direction. Linear dependencies form the basis of the classical Markowitz portfolio model. In the context of IT investments, this type of dependencies arises where different projects have common resource inputs or use common technology [28]. Nonlinear dependencies (i.e., synergies or complementarities) are the result of sub-additive cost or super-additive benefit structures [29, 30]. Sub-additive cost structures denote cost savings that occur when the use of resources common to different projects reduces the joint implementation costs of the respective projects. Super-additive benefit structures describe the additional benefits created when the joint benefits of different projects are greater than the sum of the projects’ individual benefits. In both cases of nonlinear dependencies, the cash flows linked to a concerted implementation of two projects a and b,xa+b, exceed the sum of the cash flows associated with the individual projects, xa + xb.

Conclusions

Given the fundamental problems in identifying “rational” and “biased” decisions, and in the spirit of Operations Research, I propose a more action-oriented perspective. This implies an intensified analysis of the components of the decision problem and the development of suitable valuation techniques. The following topics might be worth thinking about.

  1. (a)

    As has been explained, the allocative properties of financial markets, meaning the opportunities to acquire, reduce, or reshape risks, are of paramount importance. It is thus necessary to determine whether (and to which extent) the consequences of (IT) investments can be linked to the risks that are priced in financial markets. Since securities markets are changing—what is firm-specific or technological risk today may well be securitized in the future—this is an ongoing task. Case-based studies may be a suitable instrument for doing so.

  2. (b)

    Generally, people have reasons for what they do. Hence, research has to gather data that give clues as to what these reasons are [24]. Notwithstanding the problems associated with an empirical estimation [31], where replication fails, it is necessary to identify the preferences of those on whose behalf to invest. Thereby, it is an open question whether the results of general studies [32, 33] apply in the context of IS or other subfields of Operations Research. In this respect, the article of Khan et al. [1] can serve as a starting point for future research.

  3. (c)

    Since firm-specific or technological risks may be present in most applications, the use and development of “hybrid” models should be encouraged. These approaches provide for a tailored mix of market-based and preference-based valuation. The highly cited work of Smith and Nau [4] can serve as an example. The authors demonstrate for so-called partially complete markets and under specific preferential conditions that non-replicable and market risks can be priced separately. In my opinion, research should adopt the spirit of Smith and Nau [4] and initiate refinements or extensions.

Change history

  • 04 November 2020

    The original version of this article unfortunately contained a mistake in the reference section, specifically references 7 and 9. Reference 7 should be changed from “Nau, Robert F, McCardle, Kevin F (1992) Arbitrage, rationality, and equilibrium, Decision making under risk and uncertainty (Durham, NC, 1990), Theory Decis. Lib. Ser. B Math. Statist. Methods, 22 pp 189–199, Kluwer Acad. Publ., Dordrecht” to “Nau RF, McCardle KF (1991) Arbitrage, rationality, and equilibrium. Theory Decis 31(2-3):199–240”. Reference 9 should be changed from “Dybvig PH, Ross SA (1987) Arbitrage. In: Eatwell J, Milgate M, Newman P (eds) The New Palgrave: a dictionary of economics, vol 1, A to D Macmillan Press, London, pp 100–106” to “Dybvig PH, Ross SA (1987) Arbitrage. In: Eatwell J, Milgate M, Newman P (eds) The New Palgrave: a dictionary of economics, vol 1, A to D. Macmillan Press, London, pp 100–106”.

Notes

  1. 1.

    The setting can be generalized to multiple periods [4, 5].

  2. 2.

    Preferences are assumed to exhibit monotonicity [6]. A validity of the axioms underlying expected utility is, however, not supposed.

  3. 3.

    Empirical probabilities can be interpreted as unanimously held (i.e., homogeneous) probability assessments concerning the states at the end of the period [6]. Assuming homogeneous expectations does not limit the validity of my results: Where replication is possible, empirical probabilities are irrelevant (as will be shown below). Where replication fails, diverging subjective probabilities would only represent an additional source of individually differing valuations [7].

  4. 4.

    See also Dybvig and Ross [9] and Sundaram [10] for treatments at a general but relatively informal level. The concept of “risk-neutral valuation” is not fully understood in prior studies cited by Khan et al. [1]. I refer to the work of Miller and Shapira [11], who state that “normative option pricing models in finance assume investors are uniformly risk neutral and consistently apply the market’s risk neutral discount rate” (p. 271). Clearly, the authors misconceive the role played by the allocative properties of financial markets.

  5. 5.

    Examples within the IS literature include the papers of Benaroch and Kauffman [14] and Benaroch [15].

  6. 6.

    Cochrane [25] deals with comparable problems in the intersection of macroeconomics and finance. He concludes (p. 967): “So the question ‘are those true-measure [i.e., empirical] or risk-neutral probabilities?’ is not a technicality, it is the whole question.”

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The original online version of this article was revised: due to an error in references 7and 9.

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Schosser, J. Real Option Exercise Decisions in Information Technology Investments: a Comment. SN Oper. Res. Forum 1, 27 (2020). https://doi.org/10.1007/s43069-020-00032-1

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Keywords

  • Real options
  • Rationality
  • Investment decisions