A Utility Function for a Risk-Seeking Individual

Dually registered-plus insurance - Yes, but in effective reality, the client seeks and obtains advice from both. I do not think it is possible for a product provider to do their job without giving some degree and form of advice. If a product provider were purely representing a single product, then perhaps. But as soon as the provider offers two or more products and offers any information to the client that could help them choose between them they are effectively, and most assuredly from the client's standpoint, giving advice. Personally I think it's foolish to try and skirt this issue—we use all sorts of games and word-play to try and dance around the issue, but the truth is, clients seek advice from financial professionals and we give it. Because of this, being held to a fiduciary standard is (or should be) a logical and necessary function of the job

etc... Frank Knight, for example, believed that “human activity is largely impulsive response to stimuli and suggestions.” Finally, it should be observed that an equilibrium point mapped onto R + is a valuation, V. The higher is the position of x in the ranking, the higher is the V of x, commanding a corresponding proportion ∝ of consumer i’s budget B. Hence, if both quantity and price could be made perfectly divisible, individual consumer demand could be a rectangular hyperbola. Indeed, in Dominique (1999, ch.3), it is shown how the invertible map M is constructed for a simple 3 goods-3 consumer economy to minimize a potential market gain, and how it easily maps ℜ onto P (the equilibrium price vector of a pure exchange model), but without the hypothesis of utility maximization. And variations in the price vector are perfectly correlated with changes in the ranking order.

Yun-Yeong Kim Department of International Trade, Dankook University, Yongin-si, South Korea Abstract In this paper, we extend Kim (2013) [9] for the optimal foreign exchange (FX) risk hedging solution to the multiple FX rates and suggest its application method. First, the generalized optimal hedging method of selling/buying of multiple foreign currencies is introduced. Second, the cost of handling for- ward contracts is included. Third, as a criterion of hedging performance evaluation, there is consideration of the Leontief utility function, which represents the risk averseness of a hedger. Fourth, specific steps are intro- duced about what is needed to proceed with hedging. There is a computation of the weighting ratios of the optimal combinations of three conventional hedging vehicles, i.e. , call/put currency options, forward contracts, and leav- ing the position open. The closed form solution of mathematical optimization may achieve a lower level of foreign exchange risk for a specified level of ex- pected return. Furthermore, there is also a suggestion provided about a pro- cedure that may be conducted in the business fields by means of Excel.

Q Z is the gradient of a convex function.  3.2.5. Partial insurance and monotone mean preserving decreases in risk. The char- acterization of monotone mean preserving increase in risk given in Theorem 3 allows us to extend to multivariate risk sharing a celebrated result of Landsberger and Meil- ijson in [21] stating that partial insurance contracts are Pareto efficient relative to second order stochastic dominance if and only if they involve a decrease in Bickel- Lehmann dispersion. Consider an individual A bearing a risk Y that she considers sharing with individual B, in the sense that A would bear X A and B would bear X B

Abstract This short paper demonstrates that the claim of Cumulative Prospect Theory (CPT) that people are risk seeking for loss prospects, which confirmed a hypothetical assumption of the earlier Prospect Theory (PT), appears to be merely a result of using a specific form of the probability weighting function to estimate the power factor of the value function. Using experimental data and the form of the probability weighting function presented by CPT gives a power factor for losses of less than 1. This would mean that people are risk seeking for loss prospects. However, once more flexible, two-parameter forms are used, the power factor takes on values between 1.04 and 1.10.

there is a direct equivalence between the utility function and the cumulative distribution of income. The idea is not particularly recent in economics as it was employed by Gregory (1980) and is also related to the “Leyden” approach mentioned in section I. Of course, there are difficulties if the equivalence is actually applied to the whole income distribution, because then (assuming the distribution unimodal) the lower income side is convex, implying risk-loving behaviour at lower incomes. 14 The concavity condition can be maintained by taking the equivalence to be between utility and the distribution “translated” to its mode as origin. Since income distributions are usually described as positively skew with long right hand tails, the GEV distribution with k ≤ 0 would seem appropriate. So too, of course, would other distributions such as lognormal, Weibull and Pareto (when lower incomes ignored). There are other themes in psychology related to choice of utility function. The random utility model of Luce and Suppes (!965) treats a person as possessing a mental distribution of utility functions, one of which is selected at random when making a decision. 15 In behavioural economics individuals are sometimes considered as being composed of ‘multiple selves’ (for example, Thaler and Shefrin, 1981) and a referee has suggested that perhaps this currently popular approach could provide a justification for GEV utility. GEV distributions arise as limiting distributions for the largest value in a sample and perhaps the extreme value drawn from the distribution of ‘multiple selves’ captures utility. Unfortunately, this author does not know enough about behavioural economics to properly assess and develop this suggestion.

Of course, distribution functions cannot represent the whole class of utility functions. Some very popular utility functions, such as the logarithmic or positive power forms, 3 are unbounded and cannot be scaled to distribution functions. There can also be limitations in terms of flexibility or convenience to employing some distributions as utility functions. It is easily shown that some imply increasing absolute risk aversion (IARA) over the whole range of income or wealth, a property also associated with the quadratic utility function. Because of the perceived implausibility of this property, textbooks tend to dismiss quadratic utility and the same would presumably apply to the employment of these distributions as utility functions. Again, many distribution functions, including the normal, cannot be expressed in a closed form, even if they can be evaluated numerically. While the familiarity of the normal may make it the exception, 4 most economic researchers would consider it inconvenient not to have explicit algebraic expressions for their utility functions. 5

The point estimates for the relative risk aversion coefficient φ are negative but they are not significantly different from zero. Thus, a reduction to Cobb-Douglas preferences cannot be rejected. 13 The estimated values for φ are smaller than the ones reported by Reis et al. (1998) [between 3.6 and 6.43], and by Issler and Piqueira (2000) [median value of 1.7 for quarterly data with seasonal dummies]. The comparison, however, is not appropriate due to the inclusion of different arguments in the utility function for each study.

A derivative is a contract that derives its value from the performance of an underly- ing entity [15]. Financial derivatives enable parties to trade specific financial risks to other entities to manage these risks. The risk embodied in a derivatives contract can be traded by trading the contract itself, such as with options. As a matter of fact, op- tion is one of the more common derivatives [10]. The value of the financial derivative derives from the reference price. Because the future reference price is not known, the value of the financial derivative at maturity can only be estimated. The assumptions made by Black and Scholes [3] when they derived their option pricing formula were volatility σ was constant and there were no transaction costs or taxes. The volatility σ here follows a stochastic differential equation of the geometric Brownian motion

Moreover, most of the individually elicited utility functions were actually not inverse-S shaped. However, this is obviously not sufficient to discredit the ‘ruinous losses’ hypothesis. Simply, the 6-point elicitation process may have prevented most subjects from reaching their ‘ruin point’. Obviously, the ‘ruin point’ is unlikely to be the same for all the subjects; instead, it can be expected to be a personal feature, depending on socio-demographic, financial, and psychological characteristics. More systematic investigation of the utility function as well as of the financial and personal background of the subjects is warranted to investigate whether the ‘ruinous losses’ phenomenon should be considered as a general human feature or not, as well as to capture the location of the inflection point for each subject if it may exist.

3 The Theory Tzeng et al. (2012) modify the almost SD rule developed by Leshno and Levy (2002) so that the almost SD rule for risk averters possesses the property of expected-utility maximization. In this paper we will show that the almost SD rule for risk seekers also possesses the property of expected-utility maximization. Here, we state both results in the following theorem:

for n = 1, 2, 3 are defined in (2.2), ϵ-almost FDSD, SDSD, and TDSD stand for almost first-, second-, and third-order DSD, respectively. Now let’s turn back to the Example 2.2. As discussed before, B cannot dominate A by SDSD. However, for this example, we can conclude that B ≽ almost(ϵ) 2D A. The SD approach is regarded as one of the most useful tools for ranking investment prospects when there is uncertainty, since ranking assets has been proven to be equivalent to expected- utility maximization for the preferences of investors/decision makers with different types of utility functions. It is interesting to examine whether almost SD possesses a property of expected-utility maximization similar to SD. Before we carry on our discussion, we first specify different types of utility functions as shown in the following definition:

Example 3.1 consider u(x) = 2x − x 2 , x ∈ [0, 1]. we can have u ′ (x) = 2 − 2x and u ′′ (x) = −2. Clearly, u ∈ U A ∗ 2 (ϵ), while it does not belong to U 1 A ∗ (ϵ) since inf {u ′ (x) } = 0. We note that Theorem 3.1 shows that both almost ASD and DSD rules possess the property of expected-utility maximization. It is well-known that both ASD and DSD possess the hierarchy property (Levy, 1992, 1998) that FASD implies SASD, which, in turn, implies TASD and also FDSD implies SDSD, which, in turn, implies TDSD. Thus, the lowest order of SD relationship is reported and any higher order SD relationship is “trivial” since it can be implied by a lower order SD. For example, if we find X FASD Y , then it is trivial that X SASD Y and X TASD Y . Guo, et al. (2013) find that the almost ASD defined in Definition 2.2 does not possess the hierarchy property such that almost FASD does not imply almost SASD, which also does not imply almost TASD. Similarly, one could easily show that the almost DSD defined in Definition 2.3 does not possess the hierarchy property such that almost FDSD does not imply almost SDSD, which also does not imply almost TDSD. To illustrate the non-hierarchy of almost DSD, we give a simple example to show that almost FDSD does not imply almost SDSD:

their choices, there are important reasons not to reward them. The first reason is that we want people to consider the alternatives as if they were investing their own money. By awarding people for their choices, we create an endowment effect. 16 Besides that, we want people to think in terms of substantial amounts of money. This is virtually impossible in an incentive-induced experiment, simply because it would get too costly. A solution that is often chosen for the second problem is to give all the participants a chance to win their choice. In such a case, one person would be allowed to play the game of her choice. However, Rabin [2000] argues that this lottery procedure is known to only be sufficient when the expected-utility hypothesis is maintained. Since in this paper we are testing the implications from this hypothesis against competing hypotheses, this procedure would not be useful for our purpose. 17 In addition, Camerer [1995] argues that persons with well-formed preferences are likely to express these truthfully, whether they are paid or not. 18

If < is a CEU ordering with a nonconstant u, it is immediate to check that V is a canon- ical representation of <, and u is a canonical utility index of <. In fact, monotonicity is well known, and Eq. (3) holds with ρ(A) = ν(A) for all A ∈ Σ. (ii) A popular generalization of CEU is the CPT of Tversky and Kahneman (1992). In CPT some consequence is established to be the DM’s reference point. The consequences which are better than the reference point are called ‘gains’, and those which are worse are called ‘losses’. The preferences over F are represented as follows: Given a nonconstant utility function u, normalized so that the reference point has utility 0, every act f is split into its ‘gain’ part f + (of the payoffs with positive utility) and its ‘loss’ part f − (of the payoffs with negative utility). V (f ) is the sum of the Choquet integral of u(f + ) w.r.t. a capacity ν + and the Choquet integral of u(f − ) w.r.t. another capacity ν − . (CEU corresponds to the special case in which ν − = ν + .) A CPT preference has a canonical representation only if it is also CEU. However, a CPT preference has a canonical representation on the sets of acts which only yield gains (or only losses). Also, it would be easy to generalize the notion of biseparable preference to allow willingness to bet to be different depending on whether gains or losses are considered.

Translation equivariance might be considered desirable for a risk measure used to calculate provisions or premiums. Indeed it seems to be reasonable when the risk contains a certain amount b that [ b + X ] = b + [ X ] in both cases. This is of course not the case for risk measures used for the calculation of solvency (regulatory or economic) capital. These quantities constitute amounts of money for safety, in addition to the provisions and premiums. Such risk measures are of the form [ X ] = [( X − [ X ]) + ] for some appropriate risk measures and . Consequently, if is translation equivariant then satisfies [ X + b ] = [ X ] , so that we have translation invariance in this case. It is of course not forbidden to take translation equivariance as an axiom for the construction of risk measures in this situation, but it cannot be interpreted in the economic concept of solvency. The prescribed rules in the framework of Solvency 2 for the calculation of solvency capital (the difference between the Value-at-Risk at level 99.5 % and the Value-at-Risk at level 75 %) recognizes this reality. Related of course is the requirement [ b ] = b which can be defended in the case of insurance pricing (to avoid free lunches and/or the no-ripoff condition) but in the definition of a risk measure [ b ] might be equal to a function u ( b ) , the utility.

to choose among risky prospects. In CEU, decision-makers use their own subjective probabilities, derived from some capacity. 6.2 Risk aversion As RDEU distinguishes attitudes to outcomes and attitudes to probabilities, risk aver- sion in RDEU must combine two different concepts. First, there is outcome risk aver- sion, associated with the idea that the marginal utility of wealth is declining. This is the standard notion of risk aversion from the EU theory defined by concavity of the utility function. Second, there are attitudes specific to probability preferences. Risk aversion in probability weighting corresponds to pessimism: the decision-maker adopts a set of decision weights that yields an expected value for a transformed risky prospect lower than the mathematical expectation. An alternative, more restrictive, character- ization of pessimism leads to a definition of risk aversion in terms of  icv . Note that the restriction to concave utility functions does not prohibit risk seeking behaviors in RDEU. It is indeed possible to model risk seeking with diminishing marginal utility of wealth, an attractive feature of RDEU.

Although affect and risk perception are increasingly mentioned in the literature, the focus has mostly been on the influence of mood or affective states on risky decision-making (e.g. Isen, Nygren, & Ashby, 1988; Mano, 1994; Wright & Bower, 1992). In this work we consider the impact of intuitive or deliberative decision-making based on the idea that the information used for a judgment varies with respect to the individually preferred habitual decision mode. While deliberative people rather use the stated information, intuitives seem to process not only the stated values but also their subjective feeling of how safe or how good a lottery is. People using affective information (i.e. people with a preference for intuition) may be more prone to the effects of mood on their decisions in risky situations. Future studies might attempt to control for mood effects to rule out this explanation.

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The brain basis of this personality trait therefore has high relevance for understanding both healthy human behaviour and several prevalent disease states. This review first discusses insights into differences in neurobiology underlying differences in SS behaviour derived from studies in both humans and animal models, particularly with respect to midbrain dopamine systems. Evidence for how these differences might relate to differential risk for addic- tive and gambling disorders is then considered, as well as the role high SS may play in more functionally adaptive behaviour involving exploration and stress resiliency. Finally, we briefly touch upon the importance of considering individual differences such as SS in per- sonalising both treatment and targeted intervention programmes for relevant psychopathologies.