Prospect theory

Before presenting our main model where VaR serves non-professional investors to allo-cate money among different utility sources, it is important to address the problem of perception. The motivation resides in the fact that these usually non-trained investors perceive financial market risks – and hence VaR – in a subjective way which may deviate substantially from the formal definitions summarized in Section 3.1.1. We formalize the perceptions of non-professional investors in terms of the prospect theory, the main issues of which are addressed in this section.

85It is also plausible to think that mangers explain to their clients some details of the procedure used for reaching the optimal asset composition. Since VaR is a key variable of this procedure, non-professional investors will be learning even more about the meaning and use of VaR.

The prospect theory (abbr. PT) of Kahneman and Tversky constitutes a corner-stone of behavioral finance. Introduced in Kahneman and Tversky (1979) and extended in Tversky and Kahneman (1992),⁸⁶ as well as in many further papers by different au-thors, PT rebuts the basic principles of theexpected utility theory(abbr. EUT), the major approach regarding decision making under risk in neoclassical Economics.⁸⁷ The need to change the traditional framework came in consequence of the fact that deviations from theoretically prescribed behaviors were repeatedly observed in real decision situations (above all in financial markets).⁸⁸

Specifically, experiments and empirical observations reveal different effects that violate the basic normative axioms imposed by EUT on human preferences and choices.⁸⁹ Tversky and Kahneman (1992) summarize them as framing effects,⁹⁰ non-linear preferences,⁹¹ source dependence,⁹² risk seeking,⁹³ and loss aversion.⁹⁴⁹⁵

86The extension is referred to as the cumulative prospect theory (abbr. CPT) and exhibits additional features, such as the usage of cumulative instead of separable decision weights, the coverage of decision problems under both risk and uncertainty and with any number of outcomes, the formulation of distinct weighting functions for gains and losses, and the introduction of diminishing sensitivity and loss aversion.

87EUT assumes that investors behave as perfectly rational maximizers of the expected utility of wealth.

They are able to process the entire information at their disposal and form unbiased judgments. Thus, they assess the expected utility as a linear combination of final states (or outcomes), weighted by the probabilities of the corresponding events with these outcomes. These probabilities are mostly updated by means of the Bayes rule and must be well-known to the investors. The utility function is taken to be unique for all possible outcomes. Moreover, investors exhibit consistent preferences and risk-averse behaviors, so that the utility function is concave in wealth.

Specifically, EUT postulates the valuation of a future situation – i.e. choice alternative or prospect – as the mathematical expectation of its monetary values. This expectation is defined as the weighted sum of the outcome utilities (reference dependence). The gains and losses are evaluated symmetrically (symmetry of valuation) and proportionally to the accordant expectations (non-proportional marginal sensitivity). The weights represent outcome probabilities that sum up to 1 and are independent of the origin of uncertainty. Accordingly, the utility function depends only on final states. The risk is captured by means of a unique and constant risk coefficient. This renders the entire utility function linear for risk-neutral subjects, concave for risk-averse, and convex for risk-seeking ones, respectively.

88Thaler (1985) devises the so-called behavioral decision research by enunciating and refuting 15 prin-ciples of the classic utility theory: the choice dependence on outcomes and generally on final states, the formulation of decision weights as outcome probabilities and their independence of the source of uncer-tainty, the independence of preferences of their representation, the preference of dominating alternatives, the equivalence of opportunity and out-of-pocket costs, the optimal search, the influence of sunk costs on decisions, the exclusive dependence of preferences on the qualities of an alternative and not on its perceived merits, the consistency and lack of bias (i.e. the full rationality) of probabilistic judgments, and the Bayesian learning.

89These axioms refer to: completeness (that also implies reflexivity), transitivity, the Archimedean property, and independence. The fulfillment of the first two is denoted as rationality.

90Different presentations (framing) of the choice problem can entail distinct preferences. This contra-dicts the assumption of description invariance in the rational theory.

91Specifically, utility is non-linear in the outcome probabilities.

92Not only the degree but also the source of uncertainty can influence preferences. One classic example is the paradox by Ellsberg (1961).

93Namely for losses and for low probable high gains relative to the expected value.

94That is, the asymmetry in the perception of gains and losses. More details are given in the text below.

95The original Kahneman and Tversky (1979) paper addresses more specifical effects, such as the

cer-PT attempts to incorporate these psychological aspects in the decision making process that is considered to rely on the individual perception of reality. In essence, PT was not intended to be a normative theory based on axioms as EUT, but rather a descriptive approach attempting to capture empirically observed behaviors.⁹⁶ Thus, PT stresses that decisions are in fact rarely based on final states (outcomes), but sooner on subjectively perceived changes in welfare generated by these outcomes. Perceptions are formulated relative to a subjective reference point,⁹⁷ so that deciders distinguish between positive and negative wealth changes (i.e. gains and losses).

According to the original Kahneman and Tversky (1979) paper, the human choice process develops in two stages. The first one refers to editing of choice alternatives (or prospects) and entails a mental representation of them. It implies different operations, such as coding,⁹⁸combination (of probabilities of prospects with identical outcomes), seg-regation (of the risk-free component from the risky one), cancellation (of components or of outcome-probability pairs that are common among prospects), simplification of prospects (e.g. by probability or outcome rounding), and detection of dominance (where the domi-nated alternatives are rejected). Naturally, the sequence of these editing operations influ-ences the final edited prospect, hence the preference order.⁹⁹ The second stage consists of the evaluation of the edited prospects and of the final choice (which is the prospect with the highest ascribed value).

The evaluation phase implies the assessment of an overall value of each choice alter-native, denoted as prospective value. Formally, it represents the weighted sum of the values subjectively assigned by each individual to the possible outcomes, where outcomes are separately treated as gains (henceforth denoted by a symbol +) or losses (denoted by –

tainty effect (i.e. the preference of certain smaller to uncertain higher gains which violates the substitution axiom and relates to the well-known Allais (1953) paradox), the reflection effect (i.e. the fact that high risky losses are preferred to certain smaller ones), the probabilistic insurance (i.e. the preference for con-tingent insurances that provides certain coverage vs. probabilistic insurances, that is due to the fact that different formulations entail distinct preferences over prospects with identical outcomes and probabili-ties), the isolation effect (according to which prospects appear to be often decomposed and people focus merely on the components that distinguish choice alternatives; as different decompositions are possible, preferences can revert and become inconsistent which violates the completeness and transitivity).

96Axiomatizations of CPT for decisions under uncertainty and risk are provided in Wakker and Tversky (1993) and Chateauneuf and Wakker (1999), respectively, and extended in Schmidt (2003) in order to capture the impact of shifting reference points.

97As underlined in Kahneman and Tversky (1979), the reference point can be the status quo (e.g. the current asset value) but also an aspiration level. Its shift is possible and affects the preference order.

98This assumes defining the reference point and perceptually separate outcomes in gains and losses with respect to it.

99This phenomenon is also known as theframingof the problem.

).¹⁰⁰ According to Tversky and Kahneman (1992), the prospective valueV of outcome i, wherei= 1. . . n, yields:

Vi =V_i⁺+V_i⁻=X

x⁺_i

π⁺_i v(x⁺_i ) +X

x⁻_i

π⁻_i v(x⁻_i ),

wherev stands for the value function andπ for the decision weights, both of which being defined below. Finally,x_i denotes the possible project outcomes i= 1, . . . n and x⁺_i (x⁻_i ) indicates the domain of gains (losses).

The subjective value of outcomes is encompassed by the so-called value function which exhibits several particular features. First, it addresses the perceptional segmen-tation into two domains with different evolutions. These domains correspond to gains and losses, so that the value function is asymmetric. The delimitation of the loss and gain domains takes place with respect to a subjective reference point.¹⁰¹ Moreover, the value function exhibitsdiminishing sensitivityin both domains, i.e. its variation decreases with the magnitude of gains and losses, respectively. In addition, as people appear to be more reluctant to incur losses compared to gains of the same size – a property denoted as loss aversion -, the value function presents a kink at the origin. Consequently, the value function has to be zero at the reference point, steeper for losses than for gains, as well as concave in the domain of gains and convex in the domain of losses (in sum, s-shaped).¹⁰² The CPT of Tversky and Kahneman (1992) formulates the value function v of an individual (investor) k. For simplicity reasons, we henceforth omit the subscript

k. This convention will apply to all variables except for the outcome x_i. The CPT-value function yields:

v(x_i) =







A(x_i−x⁰_i)^α, ifx_i ≥x⁰_i ≡(x⁺_i )

−B(x⁰_i −x_i)^β, ifx_i < x⁰_i ≡(x⁻_i ),

wherex⁰_i stands for the subjective reference point. In addition, 0< α, β ≤1 are specifical

100The prospective value is the counterpart of the expected utility function in EUT. Thus, the concept of “utility”, defined in EUT in terms of net wealth is replaced by the one of “value”, defined in PT in terms of relative wealth.

101Formally, the value function is a function of two variables: the reference point and the magnitude of changes with respect to this reference. CPT considers the reference point to be fixed, e.g. corresponding to the status quo or initial endowment. Schmidt (2003) develops an extended axiomatization for variable references, that accounts for the derivation of both value function and decision weights. Davies (2005) extends the notation in Schmidt (2003) in order to allow for the independence of the reference point and hence of the initial endowment and thus provides a basis for the unification of the frameworks for risky and risk-free choices.

102Norsworthy, Gorener, Schuler, Morgan, and Li (2004) find empirical evidence on the US-market for reference dependence, asymmetric valuation of gains and losses and diminishing sensitivity.