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Risk pricing in a non-expected utility framework
European Journal of Operational Research 246 (2015) 944–948
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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Short Communication
Risk pricing in a non-expected utility framework Gebhard Geiger∗ Technical University of Munich, Faculty of Economics, Institute of Financial Management and Capital Markets, Arcisstrasse 21, 80333 München, Germany
a r t i c l e
i n f o
Article history: Received 17 May 2014 Accepted 17 April 2015 Available online 30 April 2015 Keywords: Risk analysis Risk pricing Certainty equivalent Utility theory Non-expected utility
a b s t r a c t Risk prices are calculated as the certainty equivalents of risky assets, using a recently developed non-expected utility (non-EU) approach to quantitative risk assessment. The present formalism for the pricing of risk is computationally simple, realistic in the sense of behavioural economics and straightforward to apply in operational research and risk and decision analyses. © 2015 Elsevier B.V. and Association of European Operational Research Societies (EURO) within the International Federation of Operational Research Societies (IFORS). All rights reserved.
1. Introduction The intrinsic price of a risky asset of random monetary value X is the certainty equivalent cp of the probability distribution p(X). The term “intrinsic” refers to risk quantification within a given accounting system rather than to risk prices extrinsically determined by the market for the asset. In the framework of utility theory, the certainty equivalent of a risk p is implicitly defined by the requirement
U (p) = U (cp ),
p ∈ P,
(1)
where U is a real-valued utility functional defined on a space P of probability distributions and cp , cp ࢠ P, is the degenerate risk which gives the result X = cp with certainty. In Eq. (1) and throughout this research note, we do not accordingly distinguish between real numbers x and degenerate probability distributions which give X = x with certainty. We also abbreviate p(X = x) as p(x), as is usual elsewhere. Procedures to specify certainty equivalents are useful in theoretical and applied risk research. They can help to rank-order risk preferences in simple and consistent ways, to determine intrinsic risk prices in finance and insurance applications and to provide experimental tests of theories of utility (Becker, DeGroot, & Marschak, 1964; Farquhar, 1984; Denuit, Dhaene, Goovaerts, Kaas, & Laeven, 2006). But within the theoretical frameworks of EU and non-EU theory, they rarely admit explicit solutions of Eq. (1) for cp (Denuit et al., 2006). Alternatively, to solve Eq. (1) for cp by numerical approximation, the utility and probability weighting functions on which representations of U are typically based must often be determined empirically (for a
∗
Tel.: +49 089 289 25489; fax: +49 089 289 25488. E-mail address: [email protected]
compilation of the extensive literature, see van de Kuilen & Wakker, 2011) or chosen ad hoc as parametric functions (Stott, 2006) and then fitted to the experimental data. The latter step demands, in addition, considerable methodological effort (for review, see Abdellaoui, Bleichrodt, & L’Haridon, 2008). As for general methodological foundations, recent experimental results and reference to the literature on risk-pricing behaviour, see Blavatskyy and Köhler (2009, 2011). In the following, Eq. (1) will be solved for cp within a recently developed axiomatic framework of status quo dependent risky choice involving a non-EU utility functional U. U accommodates systematic violations of EU theory of various kinds observed in risky choice experiments (Geiger, 2008, 2012). It will be shown that cp possesses an explicit representation solely involving the cumulative probabilities of gain and loss associated with p and a few exogenous parameters that are not related to p. This result will then be extended to utility preferences for multivariate risks. Applications to recent experimental results on risk pricing behaviour will also be indicated. The present formalism for the intrinsic pricing of risk may thus be of theoretical and practical use in wide areas of operational research and risk management.
2. The analytic framework We consider a convex set P of simple probability distributions p defined on a compact real interval I. A person’s attitude towards a given risk p is assumed to be governed, besides by p, by his or her neutral reference point x0 , x0 ࢠ I, status quo risk s, s ࢠ P, and relative persistence ε of p in the presence of s, that is, overall probability ε of Tp > Ts , where Tp and Ts are the uncertain times to resolution of p and s. Accordingly, ε (Tp > Ts ) generally varies with p for given status
http://dx.doi.org/10.1016/j.ejor.2015.04.032 0377-2217/© 2015 Elsevier B.V. and Association of European Operational Research Societies (EURO) within the International Federation of Operational Research Societies (IFORS). All rights reserved.
G. Geiger / European Journal of Operational Research 246 (2015) 944–948
p 3 = p3 p p (0
)> 0
p2 = 0
0
p (0
Z
)=
p q = q+
.p p
p 1 = p1
Fig. 1. Indifference lines (dashed) in a probability triangle . The “fanning out” of the indifference lines is familiar from systematic violations of EU theory in risky choice experiments (Starmer, 2000). The risk p with p ࣔ p± , p(0) > 0, is indifferent to the “pure chance” q = q + .
quo s.1 To avoid trivial cases, s is non-degenerate and involves at least some chance of gain s(x) > 0 for some x > x0 , x ࢠ I, and some risk of loss s(x´) > 0 for some x´ < x0 , x´ ࢠ I. In applications of the formalism, the parameters x0 , ε and s will normally be measurable or can be estimated with some confidence by standard statistical methods (status quo risk) and multivariate survival, or hazard rate, analysis (probability ε (Tp > Ts )). Without loss of generality, x0 is normalised to x0 = 0. As an option of choice, to stay in one’s status quo (i.e., choose s given s) amounts to selecting the degenerate risk 0 given s, that is, adding nothing to s with certainty. Hence, cs = 0. As for the axiomatic foundations and for more detailed explanations of the approach, see Geiger (2002, 2008). To state the basic equations of utility preference for which Eq. (1) is to be solved, the following variables must be defined (Fig. 1): λp = (1 – Fp (0))/(1 – p(0)) and 1 – λp respectively are the overall relative probabilities of gain and loss (Fp is the cumulative distribution of p) so that p = p+ λp + p– (1 – λp ), where p+ (x) = p(x)/λp and p– (x) ≡ 0 for x > 0, p– (x) = p(x)/(1 – λp ) and p+ (x) ≡ 0 for x < 0, and p+ (0) = p– (0) = p(0) if 0 < λp < 1. If, on the other hand, λp = 1 or λp = 0, then ± p± = p, respectively. The expected gain (loss) is μ± p = x∈Sp xp (x),
μ+p λp
where Sp is the support of p and μp = + μ− p (1 − λp ). Finally, for every p ࢠ P with 0 < λp < 1, there exists a unique p0 , p0 = p+ λ0p + p− (1 − λ0p ), so that p0 and s are “isoneutral” in preference (i. e., cp0 = cs = 0), where λ0p is determined by
μ+p λ0p + μ−p (1 − λ0p ) μs = (μ − μ ) λs (1 − λs ) μ+p − μ−p λ0p 1 − λ0p + s
− s
(2)
(Geiger, 2008, Sec. 5). The basic equations of the approach involve a probabilitydependent utility function u(p, x) and the functional U: P → R so that
u(p, μ
) − = u(p, μ ) + p − p
ε(1 − λp ) + (1 − ε) 1 − λ0p ελp + (1 − ε)λ0p
,
μ <0 − p
+ U (p) = (1 − p(0))(u(p, μ− p )(1 − λp ) + u(p, μp )λp ),
U (p) = U (q) ⇔ u(p, x) = u(q, x), u(p, −x) = A(p)u(p, x),
p ∈ P, q ∈ P, x ∈ I
x ≥ 0, A > 0
(3) (4) (5)
ε(Tp > Ts ) = ε(Tq > Ts ) is tacitly assumed. U and u are unique up to positive affine transformations of the utility scale. They are normalised to u(p, 0) = 0, u(p, –1) = –1, U(0) = 0 and U(–1) = –1. It is important to note that in Eq. (4) the assessment of p in terms of cumulative gains and losses is a consequence of the general principles underlying the approach, notably the assumption of a neutral reference point and an axiom of status quo dependence of risk preferences. As for the empirical significance of the cumulative probabilities of success and failure in risky choice, see Fennema and Wakker (1997), Payne (2005) and Diecidue and van de Ven (2008). The dependence of u on x has been made explicit elsewhere. Here, it suffices to note that u(p, x) is everywhere smooth and strictly increasing in x so that cp exists for all p ࢠ P (Geiger, 2002, 2008). For a > 0, let Xa = aX and pa (Xa = ax) = p(X = x) so that pa (0) = p(0), λap = λp and λp0,a = λ0p by construction, where “λap ” and “λp0,a ” are obvious notations. Then,
U (pa ) = U (p),
u(pa , x) = u(p, x),
p ∈ P, x ∈ I
(7)
in agreement with the equivalence (5). Eqs. (7) imply that utility preferences are invariant to positive homogeneous linear transformations of the x-axis. Respectively denoting the certainty equivalents of p± and p0 by cp± and cp0 , one altogether has
μ−p = cp− < cp0 = cs = 0 < cp+ = μ+p
(8)
± Note that cp± = μ± p does not necessarily mean risk neutrality of p since, in the representation p = p+ λp + p– (1 – λp ), p± are degenerate risks which respectively give μ± p with probabilities λp = 1 and λp = 0. For a degenerate risk x, one always has cx = μx = x even if the associated utility function is convex or concave, that is, non-neutral.
3. The pricing of risk Fig. 1 shows the indifference pattern in a probability triangle , + ࣪ P, with the vertices p1 = p− 1 , p2 = 0, p3 = p3 . The indifference lines are straight and intersect in a point Z outside on the extended
isoneutral line connecting 0 and p0 . Z is uniquely determined, besides by p, by the exogenous parameters ε and s. For every p, pࢠ, with 0 < λp < 1, there exists a unique q, qࢠ, so that q = q ± and U(p) = U(q ± ) if λp > λ0p . In either case, <
μ μ = μ μ + p − p
+ q − q
(9)
If λp = λ0p , then q = 0. Considering (8), one also has
cp = cq = cq± = μ± q =
(1 − q(0))μ±p (1 − p(0))
(10)
± ± Letting a = −1/μ− p , replacing μp by aμp in Eq. (4) and considering Eqs. (7) gives
U (p
a
) = U(p) = (1 − p(0))u(p, −μ μ ) + p/
− p
−1 + λp − + λp u(p, −μ+ p /μp )
(11) (6)
where A(p) is a parameter-dependent elementary algebraic function (parameters ε , s; see Geiger, 2008, pp. 131–132). In Eq. (5), The interpretation of ε as ε (Tp > Ts ) is as in Geiger (2012), but differs from that in Geiger (2002, 2008), where ε is defined as the complementary probability ε (Tp ࣘ Ts ) obtained by substituting ε → 1 – ε . Our formal results remain unaffected by this change in denotation, but the present specification of ε is appropriate in applications as it gives the correct limiting behaviour of the utility function in the boundary cases ε = 0 and ε = 1 (Geiger 2012). 1
945
where u(p, –1) = –1 has been used. Assume q = q+ so that λq = 1, and let λp > λ0p . It follows
U (p) = U (q+ ) = (1 − q(0))u(q, μ+ q ) by Eq. (4) − − = (1 − q(0))u(qa , −μ+ q /μq ), a = −1/μq
by Eqs. (7), (11)
− = (1 − q(0))u(pa , −μ+ p /μp ) by Eqs. (5), (9) − = (1 − q(0))u(p, −μ+ p /μp )
by Eq. (7)
− + = (1 − p(0))u(p, −μ+ p /μp )cp /μp
by Eq. (10)
(12)
946
G. Geiger / European Journal of Operational Research 246 (2015) 944–948 risk p
unacceptable cp < 0
acceptable
unacceptable c <0
cp > 0
p
300
300 0,9
= 0.9
x0 = 0
250
0 p
200
250 200
rp = µp- cp "risk premium" 150
cp
µp p
100
µp
p
100
cp
rp
50
50
µp
-50
0
50
100
150
µp
0
250
-100 300
µp
0,3
µp < 0
-0,3
-50
200
and
-0,3
-0,9 -100 -100
0,6
cp
cp > 0
0,0
-0,6
µp > cp risk aversion
-50
µp
1/3
0,0
0
0
0=
1 0,9
0,3
150
µ < c risk proneness
acceptable cp > 0
= 0.5 x0 = 0
0,6
= 0.7
risk p
-1 -0,9
risk aversion
risk proneness
-0,6
µp0 -0,6
cp
-0,9 -0,3
0,0
0,3
0,6
0,9
µp
(a)
(b)
Fig. 2. Certainty equivalent cp as a function of μp . (a) Example of risk averse status quo s, μs > cs = 0, or, equivalently, μ0p > cp0 = 0. (b) Risk prone status quo s with μs < cs = 0 (μ0p < cp0 ).
Eqs. (12) and (11) successively give
cp
μ+p
−1 + λp U (p) = + λp + − = (1 − p(0))u(p, −μp /μp ) u(p, −μ+p /μ−p )
− Finally, Eq. (3) can be used to eliminate u(p, −μ+ p /μp ). After rearranging terms,
cp =
μ+p
(1 − ε) λp − λ0p
ε(1 − λp ) + (1 − ε) 1 − λ0p
,
λp ≥ λ0p ,
(13)
is obtained. Now let q = q– , with λq = 0. Analogously to the derivation of Eq. (13), one gets
cp = −μ− p U (p) = μp
(1 − ε) λ0p − λp
ελp + (1 − ε)λ0p
λ0p ≥ λp .
(14)
Eqs. (13) and (14) can be rewritten conjointly as
cp = μp f
λ0p , λp
(15)
where f gives the deviation of cp from μp ,
f
=
f
λ0p , λp
(1 − ε)(λp − λ0p ) , λp ≥ λ0p , (λp + (1 − λp )μ μ+p ) ε(1 − λp ) + (1 − ε) 1 − λ0p − p/
λ0p , λp
1 − ε λ0p − λp , = (λp μ+p /μ−p + 1 − λp ) ελp + (1 − ε)λ0p
λ0p ≥ λp .
The initial value of x0 must be added to the right-hand side of Eq. (15) if our previous normalisation x0 = 0 is finally to be cancelled. Eq. (15) is the intended result. The right-hand side of (15) does not explicitly depend on the utility function. Rather, cp can be directly and easily computed from p and the exogenous parameters ε , x0 and s determining λ0p . One has f(λ0p , 1) = f(λ0p , 0) = 1 and f(λ0p , λp 0 ) = 0. The former two cases respectively correspond to pure chance p = p+ and pure risk p = p– , in which cp = μ;p . If p persists while s resolves first with certainty (ε = 1), one has cp = cp0 = 0. Conversely, if ε = 0, cp is a linear function of λp (EU limit; see Eqs. (13) and (14)). For λp = λ0p , cp = cp0 = 0 obtains although μp 0 may not vanish. In particular, decision makers are risk averse (prone) towards p0 if μ0p > 0 (μ0p < 0), and similarly for the status quo s with μs > 0 (μs < 0). Generally, risk aversion (proneness) prevails if f(λ0p , λp ) < 1 (f(λ0p , λp ) > 1). The example of Fig. 2(a) shows cp as a function of μp for a risk averse status quo, μs > cs = 0 and, consequently, μ0p > cp0 = 0. It demonstrates that there may be risks with positive μp for which
cp is negative. Conversely, if his status quo is unfavourable (μs < 0, μ0p < 0), the decision maker may feel forced to “gamble for resurrection” and turn risk prone (cp > μp ). He thus assigns a positive cp to p even though μp is negative (Fig. 2(b), hatched area). According to Eq. (15), cp satisfies the “positive homogeneity” condition cp0 = acp , a > 0 (Tversky & Kahneman, 1992) – in agreement with Eqs. (7). Fig. 3(a) shows the typical risk-seeking (risk-averse) attitude in experimental pricing tasks for lotteries with a low (high) probability of gain (Blavatskyy & Köhler, 2009). Individuals were asked to state selling prices for binary lotteries presented to them in the experiments. The statements were subject to limited ranges of admissible prices imposed on subjects via the experimental setup. Blavatskyy and Köhler (2009) offer an explanation for the observed variability in risk attitude. It is based on a constant relative risk aversion utility function combined with a model of random risk-pricing behaviour. Fig. 3(b) suggests that the experimental results may admit an alternative, much simpler explanation in terms of Eq. (15) once the elicited median selling prices of the test lotteries are interpreted as certainty equivalents. Blavatskyy and Köhler (2009) point out the conditions under which this interpretation would be correct, and discuss the relevant literature. The method they employed in the experiment may indeed elicit the true certainty equivalents even in non-EU applications. Comparison between Fig. 3(a) and (b) suggests that the utility model behind Eq. (15) can, in principle, fit the data with some reasonable accuracy. The approximation in Fig. 3(b) has been achieved by specifying the parameters x0 , λ0p and ε in an ad hoc fashion for demonstration purposes. 4. The multi-attribute utility case Under moderately restrictive conditions of utility independence2 of multivariate random variables, the non-EU model of Eqs. (3) and (4) can be extended to multi-dimensional, though not necessarily monetary, risk attributes (Geiger, 2012, App. B). Let P be a convex set of simple probability distributions defined on a compact interval I, I ⊂ RN , N ≥ 1. If p is the joint finite discrete probability distribution of a random vector (X1 , . . . , XN ) in I, a certainty equivalent of p always exists (Geiger, 2012, Theorem 2). It is a real vector cp ࢠ I so that (X1 , . . . , XN ) = cp obtains with certainty and U(p) = U(cp ). The problem, then, is how to define the intrinsic price of p, as measured by a single
2 Of two or more real random variables, one variable is said to be utility independent of the others if the risk of change in this variable (e. g., decreased health) is assessed, in utility terms, independently of the values at which the other variables (e. g., one’s economic wealth or income) are currently held.
G. Geiger / European Journal of Operational Research 246 (2015) 944–948
947
Fig. 3. Median selling price as a function of the probability of gain (a) in experimental pricing tasks according to Blavatskyy and Köhler (2009) and (b) in the interpretation as a certainty equivalent calculated according to Eq. (15). The gain is in Swiss francs (CHF).
real number c, and which of the Xi ´ s should that be, that is, take on the value c with certainty. To solve this problem, observe that cp is no longer uniquely determined if N ࣙ 2. In fact, I can always be chosen so that, for every i ࣘ N, there exists a vector c(pi) ∈ I, c(pi) = (0, . . . , γpi , 0, . . .), so that γpi
is the ith component of c(pi) and U (c(pi)) = U (p). To see this, put hp = U(p), and let UXi be the single-attribute utility functional associated with Xi . Then there exists a simple univariate probability distribution qXi (xi ) and a real number γqi so that
hp = UXi (Geiger,
γqi = U 0, . . . , γqi , 0, . . . = UXi (qXi )
2012,
(0, . . . γpi , 0, . . .) to
γpi = γqi and c(pi) = satisfy U (cp ) = U (p). For each i, γpi can accordingly
proof
of
Theorem
2).
Put
(i)
be defined as the intrinsic price of p, as given in the “currency” Xi . It can be converted into the other “currencies” Xj , with the respective equivalent prices γ p j ,
j U (p) = U 0, . . . , γpi , . . . , 0 = U 0, . . . , γp , . . . , 0 ,
i ≤ N, j ≤ N.
In practice, the γpi ’s are straightforward to determine. On the utility independence assumption of the approach, U(p) has been given a multilinear representation in terms of the single-attribute utility functions uXi (pXi , xi ) associated with the Xi -marginals pXi of p (Geiger, 2012, App. B). This representation allows for calculating U(p) with the use of the single-attribute utility functions of Eq. (3). Consider the case a a U(p) ࣙ 0 first. In Eq. (3), replace p by pxii to obtain uxi (pxii , ai μ− pX ) = i
a
− uxi (pxii , −1) = −1, where ai = −1/μ− pX > 0 and “μpX ” is an obvious i
i
a
i notation. Further let λpXi = 1 so that pXi = ai μ+ pX = ai γp . Eq. (3) now
gives uXi (ai γpi , ai γpi ) as a function of “uXi (ai γpi , ai γpi )”
i
λ0pXi .
i
Note that in the expression
the first component continues to be reserved for the probability function variable, or degenerate probability distribution ai γpi for that matter. By Eq. (4),
hp = U (p) = UXi ai γpi = (1 − pXi (0))uXi ai γpi , ai γpi ,
i ≤ N (16)
If hp = U(p) = 0 in Eq. (16), one trivially has γpi = 0. For U(p) > 0 one has, in our slight abuse of notation for degenerate probability distributions, pxi = γpi and γpi > 0 and, hence, pXi (0) = 0. In either case Eq. (16) reduces to hp = uXi (ai γpi , ai γpi ). Solving this equation for λ0pX with the use of Eq. (3) gives λ0pX as an elementary i
i
algebraic function of hp . Substituting λ0pX for λ0p in Eq. (2), letting i
μ−p = ai μ−pXi = −1 and μ+p = ai μ+pXi = ai γpi eventually gives a simple quadratic equation for γpi . A similar elementary calculation can be done if U(p) < 0 and γpi < 0, using Eq. (6).
5. Discussion and conclusion Eq. (15) has been argued to be accessible to empirical test, with the use of standard certainty equivalent techniques, as are reviewed in Blavatskyy and Köhler (2009, 2011). Tests can be based directly on stated indifferences between risks (e. g., lotteries offered to individuals in choice experiments) and certainty equivalents. The tests dispense with cumbersome elicitations of utility and probability weighting functions. Yet interpretations of the experimental results in terms of the conceptual framework of Eq. (15) require measurements or estimates of a few exogenous parameters. These can, in principle, be obtained by standard statistical methods. Schmidt and Zank (2012) have questioned the observability of one such parameter in behavioural experiments, namely, the reference point x0 . They suggested that reference dependence can be derived from behaviour and that the reference point arises endogenously. In contrast, the present utility model presumes that the existence and location of reference points can often be observed although they may show high subjective variability. Thus, often people quite consciously and positively state their aspiration levels in medical decision making (Bleichrodt, 2007); or in business applications reference points may be exogenously imposed on decision makers as the buying prices of risky goods or services, or as limits of profitability.
References Abdellaoui, M., Bleichrodt, H., & L’Haridon, O. (2008). A tractable method to measure utility and loss aversion under prospect theory. Journal of Risk and Uncertainty, 36, 245–266. Becker, G. M., DeGroot, M. H., & Marschak, J. (1964). Measuring utility by a singleresponse sequential method. Behavioral Science, 9, 226–232. Blavatskyy, P. R., & Köhler, W. R. (2009). Range effects and lottery pricing. Experimental Economics, 12, 332–349. Blavatskyy, P. R., & Köhler, W. R. (2011). Lottery pricing under time pressure. Theory and Decision, 70, 431–445. Bleichrodt, H. (2007). Reference-dependent utility with shifting reference points and incomplete preferences. Journal of Mathematical Psychology, 51, 266–276. Denuit, M., Dhaene, J., Goovaerts, M., Kaas, R., & Laeven, R. (2006). Risk measurement with equivalent utility principles. Statistics and Decisions, 24, 1–25. Diecidue, E., & van de Ven, J. (2008). Aspiration level, probability of success and failure, and expected utility. International Economic Review, 49, 683–699. Farquhar, P. (1984). Utility assessment methods. Management Science, 30, 1283–1300. Fennema, H., & Wakker, P. (1997). Original and cumulative prospect theory: A discussion of empirical differences. Journal of Behavioral Decision Making, 10, 53–64. Geiger, G. (2002). On the statistical foundations of non-expected utility theory. European Journal of Operational Research, 136, 449–465. Geiger, G. (2008). An axiomatic account of status quo-dependent non-expected utility. Mathematical Social Sciences, 55, 116–142. Geiger, G. (2012). Multi-attribute non-expected utility. Annals of Operations Research, 196, 263–292.
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Payne, J. W. (2005). It is whether you win or lose: The importance of the overall probabilities of winning or losing in risky choice. Journal of Risk and Uncertainty, 30, 5–19. Schmidt, U., & Zank, H. (2012). A genuine foundation for prospect theory. Journal of Risk and Uncertainty, 45, 97–113. Starmer, C. (2000). Developments in non-expected utility theory: The hunt for a descriptive theory of choice under risk. Journal of Economic Literature, 38, 332–382.
Stott, H. P. (2006). Cumulative prospect theory’s functional menagerie. Journal of Risk and Uncertainty, 32, 101–130. Tversky, A., & Kahneman, D. (1992). Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5, 297–323. van de Kuilen, G., & Wakker, P. P. (2011). The midweight method to measure attitudes toward risk and ambiguity. Management Science, 57, 582–598.
Contents lists available at ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Short Communication
Risk pricing in a non-expected utility framework Gebhard Geiger∗ Technical University of Munich, Faculty of Economics, Institute of Financial Management and Capital Markets, Arcisstrasse 21, 80333 München, Germany
a r t i c l e
i n f o
Article history: Received 17 May 2014 Accepted 17 April 2015 Available online 30 April 2015 Keywords: Risk analysis Risk pricing Certainty equivalent Utility theory Non-expected utility
a b s t r a c t Risk prices are calculated as the certainty equivalents of risky assets, using a recently developed non-expected utility (non-EU) approach to quantitative risk assessment. The present formalism for the pricing of risk is computationally simple, realistic in the sense of behavioural economics and straightforward to apply in operational research and risk and decision analyses. © 2015 Elsevier B.V. and Association of European Operational Research Societies (EURO) within the International Federation of Operational Research Societies (IFORS). All rights reserved.
1. Introduction The intrinsic price of a risky asset of random monetary value X is the certainty equivalent cp of the probability distribution p(X). The term “intrinsic” refers to risk quantification within a given accounting system rather than to risk prices extrinsically determined by the market for the asset. In the framework of utility theory, the certainty equivalent of a risk p is implicitly defined by the requirement
U (p) = U (cp ),
p ∈ P,
(1)
where U is a real-valued utility functional defined on a space P of probability distributions and cp , cp ࢠ P, is the degenerate risk which gives the result X = cp with certainty. In Eq. (1) and throughout this research note, we do not accordingly distinguish between real numbers x and degenerate probability distributions which give X = x with certainty. We also abbreviate p(X = x) as p(x), as is usual elsewhere. Procedures to specify certainty equivalents are useful in theoretical and applied risk research. They can help to rank-order risk preferences in simple and consistent ways, to determine intrinsic risk prices in finance and insurance applications and to provide experimental tests of theories of utility (Becker, DeGroot, & Marschak, 1964; Farquhar, 1984; Denuit, Dhaene, Goovaerts, Kaas, & Laeven, 2006). But within the theoretical frameworks of EU and non-EU theory, they rarely admit explicit solutions of Eq. (1) for cp (Denuit et al., 2006). Alternatively, to solve Eq. (1) for cp by numerical approximation, the utility and probability weighting functions on which representations of U are typically based must often be determined empirically (for a
∗
Tel.: +49 089 289 25489; fax: +49 089 289 25488. E-mail address: [email protected]
compilation of the extensive literature, see van de Kuilen & Wakker, 2011) or chosen ad hoc as parametric functions (Stott, 2006) and then fitted to the experimental data. The latter step demands, in addition, considerable methodological effort (for review, see Abdellaoui, Bleichrodt, & L’Haridon, 2008). As for general methodological foundations, recent experimental results and reference to the literature on risk-pricing behaviour, see Blavatskyy and Köhler (2009, 2011). In the following, Eq. (1) will be solved for cp within a recently developed axiomatic framework of status quo dependent risky choice involving a non-EU utility functional U. U accommodates systematic violations of EU theory of various kinds observed in risky choice experiments (Geiger, 2008, 2012). It will be shown that cp possesses an explicit representation solely involving the cumulative probabilities of gain and loss associated with p and a few exogenous parameters that are not related to p. This result will then be extended to utility preferences for multivariate risks. Applications to recent experimental results on risk pricing behaviour will also be indicated. The present formalism for the intrinsic pricing of risk may thus be of theoretical and practical use in wide areas of operational research and risk management.
2. The analytic framework We consider a convex set P of simple probability distributions p defined on a compact real interval I. A person’s attitude towards a given risk p is assumed to be governed, besides by p, by his or her neutral reference point x0 , x0 ࢠ I, status quo risk s, s ࢠ P, and relative persistence ε of p in the presence of s, that is, overall probability ε of Tp > Ts , where Tp and Ts are the uncertain times to resolution of p and s. Accordingly, ε (Tp > Ts ) generally varies with p for given status
http://dx.doi.org/10.1016/j.ejor.2015.04.032 0377-2217/© 2015 Elsevier B.V. and Association of European Operational Research Societies (EURO) within the International Federation of Operational Research Societies (IFORS). All rights reserved.
G. Geiger / European Journal of Operational Research 246 (2015) 944–948
p 3 = p3 p p (0
)> 0
p2 = 0
0
p (0
Z
)=
p q = q+
.p p
p 1 = p1
Fig. 1. Indifference lines (dashed) in a probability triangle . The “fanning out” of the indifference lines is familiar from systematic violations of EU theory in risky choice experiments (Starmer, 2000). The risk p with p ࣔ p± , p(0) > 0, is indifferent to the “pure chance” q = q + .
quo s.1 To avoid trivial cases, s is non-degenerate and involves at least some chance of gain s(x) > 0 for some x > x0 , x ࢠ I, and some risk of loss s(x´) > 0 for some x´ < x0 , x´ ࢠ I. In applications of the formalism, the parameters x0 , ε and s will normally be measurable or can be estimated with some confidence by standard statistical methods (status quo risk) and multivariate survival, or hazard rate, analysis (probability ε (Tp > Ts )). Without loss of generality, x0 is normalised to x0 = 0. As an option of choice, to stay in one’s status quo (i.e., choose s given s) amounts to selecting the degenerate risk 0 given s, that is, adding nothing to s with certainty. Hence, cs = 0. As for the axiomatic foundations and for more detailed explanations of the approach, see Geiger (2002, 2008). To state the basic equations of utility preference for which Eq. (1) is to be solved, the following variables must be defined (Fig. 1): λp = (1 – Fp (0))/(1 – p(0)) and 1 – λp respectively are the overall relative probabilities of gain and loss (Fp is the cumulative distribution of p) so that p = p+ λp + p– (1 – λp ), where p+ (x) = p(x)/λp and p– (x) ≡ 0 for x > 0, p– (x) = p(x)/(1 – λp ) and p+ (x) ≡ 0 for x < 0, and p+ (0) = p– (0) = p(0) if 0 < λp < 1. If, on the other hand, λp = 1 or λp = 0, then ± p± = p, respectively. The expected gain (loss) is μ± p = x∈Sp xp (x),
μ+p λp
where Sp is the support of p and μp = + μ− p (1 − λp ). Finally, for every p ࢠ P with 0 < λp < 1, there exists a unique p0 , p0 = p+ λ0p + p− (1 − λ0p ), so that p0 and s are “isoneutral” in preference (i. e., cp0 = cs = 0), where λ0p is determined by
μ+p λ0p + μ−p (1 − λ0p ) μs = (μ − μ ) λs (1 − λs ) μ+p − μ−p λ0p 1 − λ0p + s
− s
(2)
(Geiger, 2008, Sec. 5). The basic equations of the approach involve a probabilitydependent utility function u(p, x) and the functional U: P → R so that
u(p, μ
) − = u(p, μ ) + p − p
ε(1 − λp ) + (1 − ε) 1 − λ0p ελp + (1 − ε)λ0p
,
μ <0 − p
+ U (p) = (1 − p(0))(u(p, μ− p )(1 − λp ) + u(p, μp )λp ),
U (p) = U (q) ⇔ u(p, x) = u(q, x), u(p, −x) = A(p)u(p, x),
p ∈ P, q ∈ P, x ∈ I
x ≥ 0, A > 0
(3) (4) (5)
ε(Tp > Ts ) = ε(Tq > Ts ) is tacitly assumed. U and u are unique up to positive affine transformations of the utility scale. They are normalised to u(p, 0) = 0, u(p, –1) = –1, U(0) = 0 and U(–1) = –1. It is important to note that in Eq. (4) the assessment of p in terms of cumulative gains and losses is a consequence of the general principles underlying the approach, notably the assumption of a neutral reference point and an axiom of status quo dependence of risk preferences. As for the empirical significance of the cumulative probabilities of success and failure in risky choice, see Fennema and Wakker (1997), Payne (2005) and Diecidue and van de Ven (2008). The dependence of u on x has been made explicit elsewhere. Here, it suffices to note that u(p, x) is everywhere smooth and strictly increasing in x so that cp exists for all p ࢠ P (Geiger, 2002, 2008). For a > 0, let Xa = aX and pa (Xa = ax) = p(X = x) so that pa (0) = p(0), λap = λp and λp0,a = λ0p by construction, where “λap ” and “λp0,a ” are obvious notations. Then,
U (pa ) = U (p),
u(pa , x) = u(p, x),
p ∈ P, x ∈ I
(7)
in agreement with the equivalence (5). Eqs. (7) imply that utility preferences are invariant to positive homogeneous linear transformations of the x-axis. Respectively denoting the certainty equivalents of p± and p0 by cp± and cp0 , one altogether has
μ−p = cp− < cp0 = cs = 0 < cp+ = μ+p
(8)
± Note that cp± = μ± p does not necessarily mean risk neutrality of p since, in the representation p = p+ λp + p– (1 – λp ), p± are degenerate risks which respectively give μ± p with probabilities λp = 1 and λp = 0. For a degenerate risk x, one always has cx = μx = x even if the associated utility function is convex or concave, that is, non-neutral.
3. The pricing of risk Fig. 1 shows the indifference pattern in a probability triangle , + ࣪ P, with the vertices p1 = p− 1 , p2 = 0, p3 = p3 . The indifference lines are straight and intersect in a point Z outside on the extended
isoneutral line connecting 0 and p0 . Z is uniquely determined, besides by p, by the exogenous parameters ε and s. For every p, pࢠ, with 0 < λp < 1, there exists a unique q, qࢠ, so that q = q ± and U(p) = U(q ± ) if λp > λ0p . In either case, <
μ μ = μ μ + p − p
+ q − q
(9)
If λp = λ0p , then q = 0. Considering (8), one also has
cp = cq = cq± = μ± q =
(1 − q(0))μ±p (1 − p(0))
(10)
± ± Letting a = −1/μ− p , replacing μp by aμp in Eq. (4) and considering Eqs. (7) gives
U (p
a
) = U(p) = (1 − p(0))u(p, −μ μ ) + p/
− p
−1 + λp − + λp u(p, −μ+ p /μp )
(11) (6)
where A(p) is a parameter-dependent elementary algebraic function (parameters ε , s; see Geiger, 2008, pp. 131–132). In Eq. (5), The interpretation of ε as ε (Tp > Ts ) is as in Geiger (2012), but differs from that in Geiger (2002, 2008), where ε is defined as the complementary probability ε (Tp ࣘ Ts ) obtained by substituting ε → 1 – ε . Our formal results remain unaffected by this change in denotation, but the present specification of ε is appropriate in applications as it gives the correct limiting behaviour of the utility function in the boundary cases ε = 0 and ε = 1 (Geiger 2012). 1
945
where u(p, –1) = –1 has been used. Assume q = q+ so that λq = 1, and let λp > λ0p . It follows
U (p) = U (q+ ) = (1 − q(0))u(q, μ+ q ) by Eq. (4) − − = (1 − q(0))u(qa , −μ+ q /μq ), a = −1/μq
by Eqs. (7), (11)
− = (1 − q(0))u(pa , −μ+ p /μp ) by Eqs. (5), (9) − = (1 − q(0))u(p, −μ+ p /μp )
by Eq. (7)
− + = (1 − p(0))u(p, −μ+ p /μp )cp /μp
by Eq. (10)
(12)
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G. Geiger / European Journal of Operational Research 246 (2015) 944–948 risk p
unacceptable cp < 0
acceptable
unacceptable c <0
cp > 0
p
300
300 0,9
= 0.9
x0 = 0
250
0 p
200
250 200
rp = µp- cp "risk premium" 150
cp
µp p
100
µp
p
100
cp
rp
50
50
µp
-50
0
50
100
150
µp
0
250
-100 300
µp
0,3
µp < 0
-0,3
-50
200
and
-0,3
-0,9 -100 -100
0,6
cp
cp > 0
0,0
-0,6
µp > cp risk aversion
-50
µp
1/3
0,0
0
0
0=
1 0,9
0,3
150
µ < c risk proneness
acceptable cp > 0
= 0.5 x0 = 0
0,6
= 0.7
risk p
-1 -0,9
risk aversion
risk proneness
-0,6
µp0 -0,6
cp
-0,9 -0,3
0,0
0,3
0,6
0,9
µp
(a)
(b)
Fig. 2. Certainty equivalent cp as a function of μp . (a) Example of risk averse status quo s, μs > cs = 0, or, equivalently, μ0p > cp0 = 0. (b) Risk prone status quo s with μs < cs = 0 (μ0p < cp0 ).
Eqs. (12) and (11) successively give
cp
μ+p
−1 + λp U (p) = + λp + − = (1 − p(0))u(p, −μp /μp ) u(p, −μ+p /μ−p )
− Finally, Eq. (3) can be used to eliminate u(p, −μ+ p /μp ). After rearranging terms,
cp =
μ+p
(1 − ε) λp − λ0p
ε(1 − λp ) + (1 − ε) 1 − λ0p
,
λp ≥ λ0p ,
(13)
is obtained. Now let q = q– , with λq = 0. Analogously to the derivation of Eq. (13), one gets
cp = −μ− p U (p) = μp
(1 − ε) λ0p − λp
ελp + (1 − ε)λ0p
λ0p ≥ λp .
(14)
Eqs. (13) and (14) can be rewritten conjointly as
cp = μp f
λ0p , λp
(15)
where f gives the deviation of cp from μp ,
f
=
f
λ0p , λp
(1 − ε)(λp − λ0p ) , λp ≥ λ0p , (λp + (1 − λp )μ μ+p ) ε(1 − λp ) + (1 − ε) 1 − λ0p − p/
λ0p , λp
1 − ε λ0p − λp , = (λp μ+p /μ−p + 1 − λp ) ελp + (1 − ε)λ0p
λ0p ≥ λp .
The initial value of x0 must be added to the right-hand side of Eq. (15) if our previous normalisation x0 = 0 is finally to be cancelled. Eq. (15) is the intended result. The right-hand side of (15) does not explicitly depend on the utility function. Rather, cp can be directly and easily computed from p and the exogenous parameters ε , x0 and s determining λ0p . One has f(λ0p , 1) = f(λ0p , 0) = 1 and f(λ0p , λp 0 ) = 0. The former two cases respectively correspond to pure chance p = p+ and pure risk p = p– , in which cp = μ;p . If p persists while s resolves first with certainty (ε = 1), one has cp = cp0 = 0. Conversely, if ε = 0, cp is a linear function of λp (EU limit; see Eqs. (13) and (14)). For λp = λ0p , cp = cp0 = 0 obtains although μp 0 may not vanish. In particular, decision makers are risk averse (prone) towards p0 if μ0p > 0 (μ0p < 0), and similarly for the status quo s with μs > 0 (μs < 0). Generally, risk aversion (proneness) prevails if f(λ0p , λp ) < 1 (f(λ0p , λp ) > 1). The example of Fig. 2(a) shows cp as a function of μp for a risk averse status quo, μs > cs = 0 and, consequently, μ0p > cp0 = 0. It demonstrates that there may be risks with positive μp for which
cp is negative. Conversely, if his status quo is unfavourable (μs < 0, μ0p < 0), the decision maker may feel forced to “gamble for resurrection” and turn risk prone (cp > μp ). He thus assigns a positive cp to p even though μp is negative (Fig. 2(b), hatched area). According to Eq. (15), cp satisfies the “positive homogeneity” condition cp0 = acp , a > 0 (Tversky & Kahneman, 1992) – in agreement with Eqs. (7). Fig. 3(a) shows the typical risk-seeking (risk-averse) attitude in experimental pricing tasks for lotteries with a low (high) probability of gain (Blavatskyy & Köhler, 2009). Individuals were asked to state selling prices for binary lotteries presented to them in the experiments. The statements were subject to limited ranges of admissible prices imposed on subjects via the experimental setup. Blavatskyy and Köhler (2009) offer an explanation for the observed variability in risk attitude. It is based on a constant relative risk aversion utility function combined with a model of random risk-pricing behaviour. Fig. 3(b) suggests that the experimental results may admit an alternative, much simpler explanation in terms of Eq. (15) once the elicited median selling prices of the test lotteries are interpreted as certainty equivalents. Blavatskyy and Köhler (2009) point out the conditions under which this interpretation would be correct, and discuss the relevant literature. The method they employed in the experiment may indeed elicit the true certainty equivalents even in non-EU applications. Comparison between Fig. 3(a) and (b) suggests that the utility model behind Eq. (15) can, in principle, fit the data with some reasonable accuracy. The approximation in Fig. 3(b) has been achieved by specifying the parameters x0 , λ0p and ε in an ad hoc fashion for demonstration purposes. 4. The multi-attribute utility case Under moderately restrictive conditions of utility independence2 of multivariate random variables, the non-EU model of Eqs. (3) and (4) can be extended to multi-dimensional, though not necessarily monetary, risk attributes (Geiger, 2012, App. B). Let P be a convex set of simple probability distributions defined on a compact interval I, I ⊂ RN , N ≥ 1. If p is the joint finite discrete probability distribution of a random vector (X1 , . . . , XN ) in I, a certainty equivalent of p always exists (Geiger, 2012, Theorem 2). It is a real vector cp ࢠ I so that (X1 , . . . , XN ) = cp obtains with certainty and U(p) = U(cp ). The problem, then, is how to define the intrinsic price of p, as measured by a single
2 Of two or more real random variables, one variable is said to be utility independent of the others if the risk of change in this variable (e. g., decreased health) is assessed, in utility terms, independently of the values at which the other variables (e. g., one’s economic wealth or income) are currently held.
G. Geiger / European Journal of Operational Research 246 (2015) 944–948
947
Fig. 3. Median selling price as a function of the probability of gain (a) in experimental pricing tasks according to Blavatskyy and Köhler (2009) and (b) in the interpretation as a certainty equivalent calculated according to Eq. (15). The gain is in Swiss francs (CHF).
real number c, and which of the Xi ´ s should that be, that is, take on the value c with certainty. To solve this problem, observe that cp is no longer uniquely determined if N ࣙ 2. In fact, I can always be chosen so that, for every i ࣘ N, there exists a vector c(pi) ∈ I, c(pi) = (0, . . . , γpi , 0, . . .), so that γpi
is the ith component of c(pi) and U (c(pi)) = U (p). To see this, put hp = U(p), and let UXi be the single-attribute utility functional associated with Xi . Then there exists a simple univariate probability distribution qXi (xi ) and a real number γqi so that
hp = UXi (Geiger,
γqi = U 0, . . . , γqi , 0, . . . = UXi (qXi )
2012,
(0, . . . γpi , 0, . . .) to
γpi = γqi and c(pi) = satisfy U (cp ) = U (p). For each i, γpi can accordingly
proof
of
Theorem
2).
Put
(i)
be defined as the intrinsic price of p, as given in the “currency” Xi . It can be converted into the other “currencies” Xj , with the respective equivalent prices γ p j ,
j U (p) = U 0, . . . , γpi , . . . , 0 = U 0, . . . , γp , . . . , 0 ,
i ≤ N, j ≤ N.
In practice, the γpi ’s are straightforward to determine. On the utility independence assumption of the approach, U(p) has been given a multilinear representation in terms of the single-attribute utility functions uXi (pXi , xi ) associated with the Xi -marginals pXi of p (Geiger, 2012, App. B). This representation allows for calculating U(p) with the use of the single-attribute utility functions of Eq. (3). Consider the case a a U(p) ࣙ 0 first. In Eq. (3), replace p by pxii to obtain uxi (pxii , ai μ− pX ) = i
a
− uxi (pxii , −1) = −1, where ai = −1/μ− pX > 0 and “μpX ” is an obvious i
i
a
i notation. Further let λpXi = 1 so that pXi = ai μ+ pX = ai γp . Eq. (3) now
gives uXi (ai γpi , ai γpi ) as a function of “uXi (ai γpi , ai γpi )”
i
λ0pXi .
i
Note that in the expression
the first component continues to be reserved for the probability function variable, or degenerate probability distribution ai γpi for that matter. By Eq. (4),
hp = U (p) = UXi ai γpi = (1 − pXi (0))uXi ai γpi , ai γpi ,
i ≤ N (16)
If hp = U(p) = 0 in Eq. (16), one trivially has γpi = 0. For U(p) > 0 one has, in our slight abuse of notation for degenerate probability distributions, pxi = γpi and γpi > 0 and, hence, pXi (0) = 0. In either case Eq. (16) reduces to hp = uXi (ai γpi , ai γpi ). Solving this equation for λ0pX with the use of Eq. (3) gives λ0pX as an elementary i
i
algebraic function of hp . Substituting λ0pX for λ0p in Eq. (2), letting i
μ−p = ai μ−pXi = −1 and μ+p = ai μ+pXi = ai γpi eventually gives a simple quadratic equation for γpi . A similar elementary calculation can be done if U(p) < 0 and γpi < 0, using Eq. (6).
5. Discussion and conclusion Eq. (15) has been argued to be accessible to empirical test, with the use of standard certainty equivalent techniques, as are reviewed in Blavatskyy and Köhler (2009, 2011). Tests can be based directly on stated indifferences between risks (e. g., lotteries offered to individuals in choice experiments) and certainty equivalents. The tests dispense with cumbersome elicitations of utility and probability weighting functions. Yet interpretations of the experimental results in terms of the conceptual framework of Eq. (15) require measurements or estimates of a few exogenous parameters. These can, in principle, be obtained by standard statistical methods. Schmidt and Zank (2012) have questioned the observability of one such parameter in behavioural experiments, namely, the reference point x0 . They suggested that reference dependence can be derived from behaviour and that the reference point arises endogenously. In contrast, the present utility model presumes that the existence and location of reference points can often be observed although they may show high subjective variability. Thus, often people quite consciously and positively state their aspiration levels in medical decision making (Bleichrodt, 2007); or in business applications reference points may be exogenously imposed on decision makers as the buying prices of risky goods or services, or as limits of profitability.
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