Decision making in times of Knightian uncertainty: An info-gap perspective

Ben-Haim, Yakov; Demertzis, Maria Working Paper Decision making in times of Knightian uncertainty: An infogap perspective Economics Discussion Papers, No. 2015-42 Provided in Cooperation with: Kiel Institute for the World Economy – Leibniz Center for Research on Global Economic Challenges Suggested Citation: Ben-Haim, Yakov; Demertzis, Maria (2015) : Decision making in times of Knightian uncertainty: An info-gap perspective, Economics Discussion Papers, No. 2015-42, Kiel Institute for the World Economy (IfW), Kiel This Version is available at: https://hdl.handle.net/10419/110924 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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Licensed under the Creative Commons License - Attribution 3.0 Discussion Paper No. 2015-42 | June 12, 2015 | http://www.economics-ejournal.org/economics/discussionpapers/2015-42 Decision Making in Times of Knightian Uncertainty: An Info-Gap Perspective Yakov Ben-Haim and Maria Demertzis Abstract The distinction of risk vs uncertainty as made by Knight has important implications for policy selection. Assuming the former when the latter is relevant can lead to wrong decisions. With the aid of a stylized model that describes a bank’s decision on how to allocate loans, the authors discuss decision making under Knightian uncertainty. They use the info-gap robust satisficing approach to derive a trade-off between confidence and performance (analogous to confidence intervals in the Bayesian approach but without assignment of probabilities). They show that this trade off can be interpreted as a cost of robustness and that the robustness analysis can lead to a reversal of policy preference from the putative optimum. They then compare this approach to the min-max method which is another main non-probabilistic approach available in the literature. (Published in Special Issue Radical Uncertainty and Its Implications for Economics) JEL C02 C18 D81 G10 Keywords Uncertainty vs risk; confidence; robustness; satisficing; info-gap Authors Yakov Ben-Haim, Yitzhak Moda’i Chair in Technology and Economics, Technion—Israel Institute of Technology, Haifa, 32000 Israel, [email protected] Maria Demertzis, De Nederlandsche Bank, PO Box 98, 1000 AB Amsterdam, The Netherlands, and European Commission All views are the authors’ own and do not represent those of any of the institutions they are affiliated with. Citation Yakov Ben-Haim and Maria Demertzis (2015). Decision Making in Times of Knightian Uncertainty: An Info-Gap Perspective. Economics Discussion Papers, No 2015-42, Kiel Institute for the World Economy. http:// www.economics-ejournal.org/economics/discussionpapers/2015-42 1 Introduction The economic circumstances since the start of the crisis in 2007 to the present are characterized by high levels of uncertainty. What do we mean by high uncertainty and what does it imply for policy design or decision making? High uncertainty can mean one of two things: either high stochastic volatility around known (or well estimated) average future outcomes, or at least partial ignorance about relevant mechanisms and potential outcomes. The first implies that uncertainty can be probabilistically measured (what Frank Knight called ‘risk’), whereas the second implies that it cannot (what Knight called ‘true uncertainty’ and is now known as Knightian uncertainty). We often conflate these two concepts when discussing ‘uncertainty’ in general. However, it is crucial to distinguish between them for three reasons. First, the relevant methods for decision making depend on which of the two notions of ‘high’ uncertainty we address. Designing policies under the assumption of probabilistically measurable risk can lead to serious policy mistakes if the underlying uncertainty is non-probabilistic, Knightian. Second, one’s measures of confidence differ under risk or Knightian uncertainty. Finally, the use of contextual understanding is different when dealing with risk or Knightian uncertainty. In a probabilistic setting contextual understanding can be used, for example, to select an appropriate probability distribution. In a Knightian setting contextual understanding can be used to intuit a trend or to sense a pending change that is not yet manifested in data. This paper will make the following points: •When uncertainty is probabilistically measurable risk, it is possible to design policies that are optimal on average or in some quantile sense. Policy design under risk is based on first principles as expressed by economic theory. The theory underlies policy choices that are designed to optimize specified substantive outcomes (e.g. minimize a high quantile of the inflation, maximize average growth, etc.). •Under Knightian uncertainty it is not possible to optimize stochastic outcomes because at least some probabilities are unknown. Furthermore, it is unreliable to attempt to optimize substantive outcomes because the underlying models are poorly known. Instead, under Knightian uncertainty one aims to prevent bad results from occurring or at least prepare for them. Building buffers in the financial system, applying unorthodox monetary policies in the monetary system are policies of this type; they aim to provide intervention tools to deal with or prevent bad outcomes from arising, irrespective of how likely they might be. •A non-probabilistic concept of robustness is used to evaluate the confidence in achieving an outcome under Knightian uncertainty. We will discuss info-gap robustness and compare it with the min-max robustness concept. Decision making under risk relies on known probability distributions of outcomes. Policy design becomes then a question of identifying the most likely occurrence (or perhaps a quantile of the occurrence) given the underlying models, and applying measures that optimize the outcome. Risks around those most likely occurrences are described probabilistically, and confidence in one’s actions in turn is best captured with statistical intervals. An obvious example is the forecasts that central banks present and the confidence intervals around them. 2 The resulting fan charts (first used by the Bank of England) are stochastic simulations in underlying variables under assumed probability distributions. Confidence then is defined as the probability of ranges of events. However, probabilities are measures of frequencies of events that have happened in the past, and therefore, in real time we are not necessarily confident that they represent accurate descriptions of the future. In 2007 most forecasts of, for example growth in most countries, were presenting confidence bands that were quite different from ex post outcomes. The corresponding confidence was no more than simply a false sense of certainty. Naturally, 2007 was the start of two of the most difficult years for forecasting in the past 20 to 30 years. Point estimates were revised both frequently and by substantive amounts and confidence bands were a mechanical tool void of economic significance. If one were to look at other times, one would not necessarily see equal revisions in forecasts and the corresponding confidence intervals would have been useful. How could we have done it differently? The lesson that the 2007 exercise has taught us is that even though models do serve us satisfactorily most of the time, there will be times that they fail us, and they may even fail us spectacularly. It is on these occasions that probabilities do not provide reliable assessment of, or confidence about, the outcomes. Relying on them provides a false sense of security that can lead to wrong policy decisions. The problem is that the moments at which standard models fail us are moments of crisis and are not known in advance. And importantly, it is difficult to distinguish between times that models serve us well and times that they don’t.4What does this mean for policy making or more generally for decision making? How can we evaluate confidence in these decisions? In this paper we provide an info-gap approach to decision making under Knightian uncertainty. With the aid of a simplified bank loan allocation example we will describe how the decision problem is handled in the presence of Knightian uncertainty. The info-gap approach will allow the bank to rank different portfolios in a way that it can pick those that provide satisfactory outcomes for the greatest range of adverse future contingencies. Robustness provides a measure of confidence. The paper is organized as follows. Section 2 briefly reviews some literature in the economics of Knightian uncertainty. It discusses how policies change as we account for Knightian uncertainty. Section 3 uses a simple example of bank loan decisions to illustrate methodological implications of info-gap theory for decisions under Knightian uncertainty. Section 4 compares info-gap and min-max decision methodologies. Section 5 concludes. 2 Risk versus uncertainty: Implications for policy making 2.1 Risk versus uncertainty Frank Knight (1921) distinguished between ‘risk’ (for which probability distributions are known) and ‘true uncertainty’ (for which probability distributions are not known). Knightian uncertainty reflects ignorance of underlying processes, functional relationships, strategies or intentions of relevant actors, future events, inventions, discoveries, surprises and so on. Info-gap models of uncertainty provide a non-probabilistic quantification of Knightian uncertainty (Ben-Haim, 2006, 2010). An info-gap is the disparity between what you do know and what you need to 4See Ahir, H. and P. Loungani, (2014) for a recent article on how difficult it is to predict turning points. 3 know in order to make a reliable or responsible decision. An info-gap is not ignorance per se, but rather those aspects of one’s Knightian uncertainty that bear on a pending decision and the quality of its outcome. Under risk we are confident—at least probabilistically—of the underlying model or combination of models that describe the economy. By contrast, under Knightian uncertainty, the social planner lacks important knowledge of how the system works. The planner starts with a number of models that may be relevant, but cannot identify the likelihood with which they describe the economy. When designing policy under risk, the knowledge of underlying probability distributions permits the identification of policies that are optimal on average or satisfy other quantile-optimality requirements. This is not possible under Knightian uncertainty because one lacks knowledge of the underlying distributions. But if one cannot design policy based on the principle of outcome-optimality, what other principles can one follow and what would these policies look like? Two approaches have been widely used as alternatives to outcome-optimization based on a reliably known (possibly probabilistic) model: 1) robust control (also called min-max) and 2) info-gap. Neither requires knowledge of probabilities. The overarching principle behind these two approaches is to find policies that are robust to a range of different contingencies. The literature on robust control relies on identifying and then ameliorating worst outcomes (Hansen et al 2006, Sargent and Hansen 2008 and Williams 2007). The planner considers a family of possible models, without assigning probabilities to their occurrence. Then that model is identified which, if true, would result in a worse outcome than any other model in the family. Policy is designed to minimize this maximally bad outcome (hence ‘min-max’ is another name for this approach). The appeal of this technique is that it provides insurance against the worst anticipated outcome. However, this technique has also been criticized for two main reasons. First, it is unnecessarily costly to assume that the worst will happen all the time (irrespective of how it is defined). Second, the worst may be expected to happen rarely and therefore it is an event that planners know the least about. It is odd to focus the policy analysis on an event that is the least known (Sims 2001). Confidence in robust control is not measured explicitly. It manifests itself in the following form: the planner will have maximally ameliorated the worst that is thought to be possible. The optimization is not of the substantive outcome (growth, employment, etc.) but rather of ameliorating adversity. In this sense min-max is robust to uncertainty. The second approach is called info-gap (Ben-Haim 2006, 2010) and relies on the principle of robust satisficing.5The principle of satisficing is one in which the planner is not aiming at best outcomes. Instead of maximizing utility or minimizing worst outcomes, the planner aims to achieve an outcome that is good enough. For example, the planner tries to assure that loss is not greater than an acceptable level, or growth is no less than a required level. When choosing between alternative policies, the robust-satisficing planner will choose the policy that will satisfy the critical requirement over the greatest spectrum of models.6 Min-max and info-gap methods are both designed to deal with Knightian uncertainty, but they do so in different ways. The min-max approach requires the planner to identify a range of events and processes that could occur, acknowledging that likelihoods cannot be ascribed to 5The technical meaning of “satisficing” as “to satisfy a critical requirement” was introduced by Herbert Simon (1955, 1957, 1997). 6Satisficing is a strategy that seems to maximise the probability of survival of foraging animals in adverse conditions (i.e. uncertainty) Carmel and Ben-Haim 2005. There are circumstances for which this can also be proven for economic examples (see Ben-Haim and Demertzis, 2008). 4 these contingencies. The min-max approach is to choose the policy for which the contingency with the worst possible outcome is as benign as possible: ameliorate the worst case. The info-gap robust-satisficing approach requires the planner to identify the worst consequence that can be tolerated, and to choose the policy whose outcome is no worse that this, over the widest possible range of contingencies. Both min-max and info-gap require a prior judgment by the planner: identify a worst model or contingency (min-max) or specify a worst tolerable outcome (info-gap). However, these prior judgments are different, and the corresponding policy selections may, or may not, agree.7 2.2 How do policies change as we account for uncertainty? A vast literature has analyzed how policies designed to handle risk differ from those designed to handle Knightian uncertainty. In the case of designing policy under risk the most famous result is that of Brainard in his seminal paper (Brainard 1967) in which he showed that accounting for Bayesian uncertainty, in a specific class of problems, implies that policy will be more cautious.8In terms of policy changes it therefore means smaller but possibly more persistent steps, and is known as the ‘Brainard attenuation’ effect. At the limit, as risk becomes very large, the social planner abandons the use of the instrument and is faced with policy inaction.9As the social planner is more and more uncertain of the results of policy, it is used less and less. This result has been very popular with policy makers as it appeals to their sense of caution when they lack sufficient information or knowledge.10 By contrast, policies derived under the principle of min-max (or robust control), and directed against non-probabilistic uncertainty, tend to be comparatively more aggressive. The policy steps taken are typically larger in size by comparison to either risk-based policies or outcome-optimal policies in the absence of uncertainty. The intuition is that under Knightian uncertainty, and when addressing a worst case, there is little knowledge about the transmission mechanisms, and it is therefore important to strongly exercise available tools in order to learn about and manage the economy. It is not surprising that this runs against some policy makers’ natural inclination to be cautious and avoid introducing volatility. It is here that info-gap robust satisficing provides a useful operational alternative. At the heart of the method for dealing with uncertainty lies a fundamental choice: that between robustness against uncertainty and aspiration for high-value outcomes. As we become more ambitious in our aspirations, we need to compromise in the degree of confidence that we can have about achieving these aspirations. Conversely, if we require high confidence in achieving specified goals, then we need to reduce our ambitions. Info-gap is a method developed with the specific aim of capturing this trade-off. Confidence is quantified with robustness to uncertainty. The trade-off quantifies the degree of robustness with which one can pursue specified outcome requirements. Policies therefore are not automatically more or less aggressive. It depends very much on the decision maker’s preferences. Furthermore, the decision maker can 7Further discussion of this comparison appears in Ben-Haim, Dacso, Carrasco and Rajan, 2009. See also section 4 here. 8This is for uncertainty in the coefficients that enter the model multiplicatively, not the residuals which enter the model additively. 9To be fair, this attenuation effect does not hold always but also depends on the cross-correlations of error terms in the assumed model. It is possible therefore, that the policy is more aggressive than that under no uncertainty and Brainard did acknowledge that. 10As Blinder (1988, p.12) wrote, there tends to be “a little stodginess at the central bank.” 5 rank alternative policies: between policies of similar ambitions, those that provide the greater robustness (greater confidence) are preferred. In section 4 we will compare and contrast the policy implications of min-max with robust-satisficing. 3 An informative trade-off: Robustness vs performance In this section we use a highly simplified example to illustrate how a decision maker deals with the inability to measure uncertainty, to come to informed decisions. We provide a framework, based on info-gap theory, that allows us to derive a trade-off between confidence in outcomes and performance requirements. Decision makers who are ambitious in terms of requiring high-performance outcomes will have to settle for their choices being appropriate only across a small range of events or contingencies (i.e. having low robustness). On the other hand, if the decision maker wants the comfort of knowing that policies chosen will function across a wide range of contingencies (high robustness), then relatively low performance outcomes will have to be accepted. Consider a bank that aims to give out loans to potential borrowers. Part of the problem that it faces is that the premium it requires depends on the risk type of the recipient agents, where risk here refers to their likelihood to default. However, assessing this probability is subject to Knightian uncertainty and therefore the bank is not in the position to price risk based on well defined underlying distributions. Furthermore, correlations exist between the solvencies of different borrowers which are significant even when they are small. Inter-borrower correlations are typically assumed to be zero though this is quite uncertain, potentially leading to overoptimistic estimates of bank invulnerability (Ben-Haim, 2010, section 4.1). In evaluating or designing the bank’s loan portfolio, the following two questions (among others) are pertinent. First, some of the uncertainty in assessing default probabilities can be reduced. How much reduction in uncertainty is needed to substantially increase the bank’s confidence? How should uncertainty-reduction effort be allocated among different borrower profiles as characterized by their estimated default probabilities? Second, what loan-repayment programs should be used for clients with different default-probability profiles? We describe how info-gap can help banks to allocate loans and, in section 4, compare it with the robust control (min-max) approach. 3.1 Formulation Consider a bank that plans a number of loans, all with the same duration to maturity. The potential borrowers are of different risk types but all borrowers of the same risk type are identical. Let: N:number of years to loan maturity K:number or risk-types fkn :repayment in year nof risk type k f:matrix of fkn values wk:number (or fraction) of loans of risk-type k w:vector of wkvalues Nd:number of years at which default could occur tj:year at which default could occur, for j= 1, . . . , Nd pkj :probability that a client of risk-type kwill default at year tj p:matrix of default probabilities pkj i:discount rate on loans 6 In case of default at tj, no payment is made in that year and in all subsequent years, for j= 1 . . . Nd. We define tNd=N+ 1, so “default” at year tNdmeans that the loan is entirely repaid and default has not occurred. We also assume that pk1. . . pkNdis a normalized probability distribution, so that the probability that borrowers of risk-type kdo not default is: pkNd= 1 − Nd−1 ∑ j=1 pkj.(1) The present worth (P W ) of the entire loan portfolio, assuming no defaults, is: P W = N ∑ n=1 (1 + i)−n K ∑ k=1 wkfkn,(2) The no-default present worth of a single loan of risk-type kis: g P W k= N ∑ n=1 (1 + i)−nfkn.(3) Eqs. (2) and (3) can be combined to express the total no-default present worth as: P W = K ∑ k=1 wkg P W k.(4) We first formulate the probabilistic expected value of the present worth. We then define the info-gap uncertainty of the probabilistic part of the model. The expected P W of a single loan of risk-type kis: E(P Wk) = Nd−1 ∑ j=1 pkj tj−1 ∑ n=1 (1 + i)−nfkn + 1− Nd−1 ∑ j=1 pkj  N ∑ n=1 (1 + i)−nfkn (5) = N ∑ n=1 (1 + i)−nfkn − Nd−1 ∑ j=1 pkj N ∑ n=tj (1 + i)−nfkn | {z } g P W kj (6) =g P W k− Nd−1 ∑ j=1 pkj g P W kj (7) where g P W kis defined in eq.(3) and g P W kj is defined in eq.(6). From eq.(7) we obtain the following expression for the expected P W of the entire portfolio: E(P W ) = K ∑ k=1 wk g P W k− Nd−1 ∑ j=1 pkj g P W kj .(8) We note that the expected present worth, E(P W ), depends on the distribution of risk types, expressed by the vector w, and on the repayment plans for the various risk types, expressed by the matrix f, and on the matrix, p, of default probabilities. 3.2 Info-gap uncertainty and robustness The info-gap model for uncertainty in the default probabilities employs estimated default probabilities, ˜pkj. Each estimated probability is accompanied by an assessment of its accuracy, 7 skj, expressing a judgment such as “The probability could be about ˜pkj = 0.02 plus or minus skj = 0.07 or more.”11 This judgment of the error could come from an observed historical variation but, under Knightian uncertainty, the past only weakly constrains the future and the error estimate does not entail probabilistic information (such as defining a confidence interval with known probability). Or the error estimate could be a subjective assessment based on contextual understanding. The error estimate skj does not represent a maximal possible error, which is unknown. skj describes relative confidence in the various probability estimates and does not imply anything about likelihoods. There are many types of info-gap models for representing Knightian uncertainty (Ben-Haim 2006, 2010). The following info-gap model is based on the idea of unknown fractional error of the estimates, and is applied to the uncertain probabilities of default. This info-gap model is an unbounded family of nested sets, U(h), of probability distributions p. Definition 1 Info-gap model of uncertainty. For any value of h, the set U(h)contains all mathematically legitimate probability distributions whose terms deviate fractionally from their estimates by no more than h: U(h) =   p:pkj ≥0, Nd ∑ j=1 pkj = 1,|pkj −˜pkj| ≤ skj h, ∀k, j  , h ≥0.(9) The value of the fractional error, h, is unknown, and the range of uncertainty in pincreases as hincreases, thus endowing hwith its name: horizon of uncertainty. The performance requirement, at the bank’s discretion, is that the expected value of the present worth be no less than a critical value P Wc: E(P W )≥P Wc.(10) Definition 2 Info-gap robustness. Robustness is the greatest horizon of uncertainty up to which the expected present worth of the portfolio is guaranteed to be no less than the critical value P Wc, i.e.: ˆ h(P Wc, w, f) = max {h:(min p∈U(h)E(P W ))≥P Wc}.(11) Robustness is the greatest value of hup to which eq.(10) will be fulfilled for all realizations of pin U(h). If probability estimates ˜pkj were accurate (i.e. no Knightian uncertainty), then the bank would be able to give out loans in ways that would maximize the expected present worth of the portfolio. As these estimates become unreliable due to Knightian uncertainty, the bank becomes less confident that the loans would achieve the ex ante expected present worth. Intuitively, the robustness in eq.(11) answers the following question: how wrong can the estimated probability ˜pkj be, in units of skj, and still achieve outcomes that are no worse than P Wc?12 It will be evident shortly that, if the bank wants higher confidence in the sense that its choices are robust to a larger range of probability outcomes, then it will have to settle for lower 11Subject of course to the probabilistic requirements of non-negativity and normalization. 12Note that the error estimates skj are somewhat analogous to deviations around the mean in the Bayesian case, but without employing probabilities. 8 References [1] Ahir, H.. and P. Loungani, 2014. There will be growth in the spring: How well do economists predict turning points?, Vox article April 14. [2] Ben-Haim, Yakov 1999. Set-models of information-gap uncertainty: Axioms and an inference scheme, Journal of the Franklin Institute, 336: 1093–1117. [3] Ben-Haim, Yakov 2006. Info-Gap Decision Theory: Decisions Under Severe Uncertainty, 2nd Edition Academic Press, London. [4] Ben-Haim, Yakov 2010. Info-Gap Economics: An Operational Introduction, Palgrave. [5] Ben-Haim, Yakov, Clifford C. Dacso, Jonathon Carrasco and Nithin Rajan, 2009, Heterogeneous Uncertainties in Cholesterol Management, Intl J Approximate Reasoning, 50: 1046–1065. [6] Ben-Haim Yakov and M. Demertzis 2008. Confidence in Monetary Policy, De Nederlandsche Bank Working Paper, No. 192, December. [7] Blinder, Alan S. 1998. Central Banking in Theory and Practice, Lionel Robbins Lecture, MIT Press, Cambridge. [8] Brainard, W. 1967. Uncertainty and the Effectiveness of Policy, American Economic Review, 57, pp 411–425. [9] Carmel Y. and Yakov Ben-Haim 2005. Info-gap robust-satisficing model of foraging behavior: Do foragers optimize or satisfice?, American Naturalist, 166: 633-641. [10] Hansen, L. P., T. J. Sargent, G. A. Turmuhambetova, and N. Williams 2006. Robust control and model misspecification, Journal of Economic Theory 128, 45–90. [11] Hansen, L.P. and T.J. Sargent 2008. Robustness, Princeton University Press, Princeton. [12] Knight, F. H. 1921 Risk, Uncertainty, and Profit. Boston, MA: Hart, Schaffner & Marx; Houghton Mifflin Company [13] Simon, H.A. 1955. A Behavioral Model of Rational choice, Quarterly Journal of Economics, Vol.69, 174-183. [14] Simon H. A. 1957. Models of Man, New York, John Wiley and Son. [15] Simon H.A. 1997. An Empirically Based Microeconomics, Cambridge University Press, Cambridge. [16] Sims, C.A 2001. Pitfalls of a minimax approach to model uncertainty, American Economic Review, Vol 91, 2, 51054. [17] Williams, N. 2007. Robust Control: An Entry for the New Palgrave, 2nd Edition. 15 A A Special Case: One Default Time We consider a special case for simplicity, Nd= 2, meaning that if default occurs then it happens at time t1. We derive an explicit analytical expression for the inverse of the robustness function, ˆ h, thought of as a function of the critical present worth, P Wc, at fixed loan portfolio (w, f). The analytical expression for the general case is accessible but more complicated and is unneeded to achieve the goals of this example. Definition 3 Define a truncation function: x+=xif x≤1and x+= 1 otherwise. Definition 4 Let m(h)denote the inner minimum in the definition of the robustness function, eq.(11). A plot of m(h)vs his identical to a plot of P Wcvs ˆ h(P Wc). Thus m(h)is the inverse function of ˆ h(P Wc). Given that Nd= 2, the expectation of the present worth, eq.(8), becomes: E(P W ) = K ∑ k=1 wk(g P W k−pk1g P W k1).(12) From eq.(12) and the info-gap model of eq.(9) we see that the inner minimum in eq.(11) is obtained, at horizon of uncertainty h, when the probability of default of each risk type, pk1, is as large as possible. Thus: m(h) = K ∑ k=1 wk(g P W k−[˜pk1+sk1h]+g P W k1),(13) and m(h)decreases piecewise-linearly as hincreases. Hence, since m(h)is the inverse of the robustness function, ˆ h(P Wc), we see that ˆ h(P Wc) decreases piecewise-linearly as P Wc increases. To explore the significance of this we first define several quantities. Let e E(P W )denote the expectation of the present worth with the estimated probabilities, from eq.(12) with ˜pk1rather than pk1(recall that Nd= 2): e E(P W ) = K ∑ k=1 wk(g P W k−˜pk1g P W k1).(14) Let E0denote the expectation of the present worth when each probability of default equals unity (eq.(8) with pk1= 1 and Nd= 2): E0= K ∑ k=1 wk(g P W k−g P W k1).(15) Note that: E0≤e E(P W ).(16) Finally, Definition 5 Define hmax as the value of horizon of uncertainty, h, beyond which all the probabilities terms [˜pk1+sk1h]+in eq.(13) equal unity: hmax = max 1≤k≤K 1−˜pk1 sk1 .(17) 16 Now we find, from eqs.(13)–(15), that: m(h) =          e E(P W )if h= 0 piece-wise linearly decreasing if 0≤h≤hmax E0if hmax < h. (18) From this relation we see that the robustness function has the following form: ˆ h(P Wc) =          ∞, P Wc<E0 piece-wise linearly decreasing, E0≤P Wc≤e E(P W ) 0, P Wc>e E(P W ). (19) This special case is explored with a numerical example in section 3.3. 17 Please note: You are most sincerely encouraged to participate in the open assessment of this discussion paper. You can do so by either recommending the paper or by posting your comments. Please go to: http://www.economics-ejournal.org/economics/discussionpapers/2015-42 The Editor © Author(s) 2015. Licensed under the Creative Commons Attribution 3.0.