The Behavioral Economics of Insurance


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1 DEPARTMENT OF ECONOMICS The Behavioral Economics of Insurance Ali alnowaihi, University of Leicester, UK Sanjit Dhami, University of Leicester, UK Working Paer No. 0/2 Udated Aril 200
2 The Behavioral Economics of Insurance and the Comosite Prelec Probability Weighting Function. Ali alnowaihi Sanjit Dhami y 28 Aril 200 Abstract We focus on four stylized facts of behavior under risk. Decision makers: () Overweight low robabilities and underweight high robabilities. (2) Ignore events of extremely low robability and treat extremely high robability events as certain. (3) Buy inadequate insurance for very low robability events. (4) Keeing the exected loss xed, there is a robability below which the takeu of insurance dros dramatically. Exected utility (EU) fails on 4. Existing models of rank deendent utility (RDU) and cumulative rosect theory (CP) satisfy but fail on 2, 3, 4. We roose a new class of axiomaticallyfounded robability weighting functions, the comosite Prelec weighting functions (CPF) that simultaneously account for and 2. When CPF are combined with RDU and CP we get resectively, comosite rank deendent utility (CRDU) and comosite cumulative rosect theory (CCP). Both CRDU and CCP are able to successfully exlain 4. CCP is, however, more satisfactory than CRDU because it incororates the emirically robust henomena of reference deendence and loss aversion. Keywords: Decision making under risk, Insurance, Comosite Prelec robability weighting functions, Comosite rank deendent utility theory, Comosite cumulative rosect theory, ower invariance, local ower invariance. JEL Classi cation: C60(General: Mathematical methods and rogramming), D8(Criteria for decision making under risk and uncertainty). Deartment of Economics, University of Leicester, University Road, Leicester. LE 7RH, UK. Phone: Fax: y Deartment of Economics, University of Leicester, University Road, Leicester. LE 7RH, UK. Phone: Fax:
3 ... eole may refuse to worry about losses whose robability is below some threshold. Probabilities below the threshold are treated as zero. Kunreuther et al. (978,. 82). Obviously in some sense it is right that he or she be less aware of low robability events, other things being equal; but it does aear from the data that the sensitivity goes down too raidly as the robability decreases. Kenneth Arrow in Kunreuther et al. (978,. viii). An imortant form of simli cation involves the discarding of extremely unlikely outcomes. Kahneman and Tversky (979,. 275). Individuals seem to buy insurance only when the robability of risk is above a threshold... Camerer and Kunreuther et al. (989,. 570).. Introduction The insurance industry is of tremendous economic imortance. The total global gross insurance remiums for 2008 were 4:27 trillion dollars, which accounted for 6:8% of global GDP (Plunkett, 200). The study of insurance is crucial in almost all branches of economics. Yet, desite imressive rogress, we show that existing theoretical models are unable to exlain the stylized facts on the takeu of insurance for low robability events. The main motivation for the aer is to rovide a theory that exlains the stylized facts on insurance for events of all robabilities, articularly those occurring with low robabilities. We highlight the following four robust emirical stylized facts, SS4, about the behavior of decision makers under risk. S. Decision makers overweight low robabilities but underweight high robabilities. S2. Decision makers ignore very low robability events and treat very high robability events as certain. S3. Decision makers buy inadequate insurance against low robability events. S4. Whether remiums are actuarially fair, unfair or subsidized, there is a robability below which the takeu of insurance dros dramatically, as the robability of the loss decreases and the loss increases, keeing the exected loss constant. We discuss these stylized facts at greater length, below. For the moment, note that S and S2 are generic stylized facts about human behavior under risk. However, S3, S4 aly seci cally in an insurance context. 2 See, for instance, Kahneman and Tversky (979), Kahneman and Tversky (2000), Starmer (2000). 2 See Kunreuther (978). S3 and S4 are not the only known behavioral attitudes towards insurance. Individuals might also be in uenced by framing e ects and exhibit the conjunction fallacy when they view damages broken into their subarts; see Camerer and Kunreuther (989). These lie outside the scoe of this aer.
4 The most oular decision theory in economics, exected utility theory (EU), redicts too much insurance and it violates all of SS4. For examle, a well known result under EU is that a riskaverse decision maker will buy full insurance if the remium is actuarially fair. By contrast, Kunreuther et al. (978,.69) found that only 20% of exerimental subjects buy insurance, at the actuarially fair remium, if the robability of the loss is 0:00. We shall consider the insurance imlications of EU in more detail in Section 2.3. Rank deendent utility theory (RDU) and cumulative rosect theory (CP) were both develoed to give a better exlanation of behavior under risk. RDU and CP, unlike EU, incororate robability weighting functions 3 that incororate stylized fact S. An examle is the axiomaticallyfounded Prelec (998) robability weighting function that is consistent with S (but, crucially, not S2). We show that this feature leads to the excessive takeu of insurance. In this sense, RDU and CP actually do worse than EU. The main aim of our aer is to roose a theoretical solution that enables RDU and CP to exlain the stylized facts SS4. However, it is not ossible to rescue EU. To coe with the stylized fact S2 (and other henomena) Kahneman and Tversky (979) roosed an editing hase where decision makers decide which events to ignore and which to treat as certain. This is followed by an evaluation hase in which the decision maker chooses from among the sychologicallycleaned lotteries from the rst stage. Insired by Kahneman and Tversky s (979) idea, 4 we roose a new class of robability weighting functions which are simultaneously able to account for S and S2. We call these comosite Prelec robability weighting functions (CPF) because they are comosed of several segments of Prelec (998) robability weighting functions, each aroriate for the relevant robability range. The CPF s are arsimonious, exible, comatible with the emirical evidence and are axiomaticallyfounded (see Aendix 2, Proosition 4, for an axiomatic derivation of CPF). We use the term comosite rank deendent utility theory (CRDU) to refer to (otherwise standard) RDU when combined with a CPF. Similarly, we use the term comosite rosect theory (CCP) to refer to (otherwise standard) CP when combined with a CPF. We show how CCP and CRDU can exlain all the stylized facts SS4. We know of no other decision theory under risk that can relicate this erformance. Furthermore, we argue that CCP is likely to be suerior to CRDU because it also incororates reference deendence and loss aversion which are not only sychologically salient but are also robust emirical ndings. 5 We conjecture that CCP can be fruitfully alied to all situations of risk, esecially those 3 By a robability weighting function we mean a strictly increasing function w() : [0; ] onto! [0; ]. w() is the subjective robability assigned by an individual to an objective robability,. 4 Incidentally Kahneman and Tversky (979) is the second most cited aer in Econometrica and in all of economics. We are grateful to Peter Wakker for ointing this out. 5 See, for instance, Kahneman and Tversky (2000), Starmer (2000) and Dhami and alnowaihi (2007, 200b). 2
5 where the robability of an event is very low. 6 The structure of the aer is as follows. In Section 2 we discuss, heuristically, human behavior for low robability events, and the relative failure of EU, RDU and CP to exlain observed insurance behavior. Section 3 formally describes the basic insurance model. Section 4 introduces some essentials of nonlinear robability weighting functions that are required to aly RDU and CP. Section 5 describes the Prelec robability weighting function (Prelec, 998), which lays a fundamental role in this aer. In Sections 6 and 7 we formally derive our results on insurance behavior under RDU and CP, resectively. Section 8 introduces our roosed new robability weighting functions, which we call comosite Prelec robability weighting functions (CPF). Section 8 then combines CPF with, resectively, RDU and CP to form comosite rank deendent theory (CRDU) and comosite rosect theory (CCP). Section 9 shows that CRDU and CCP successfully exlain SS4. Section 0 concludes. Proofs are relegated to Aendix. Aendix 2 contains an axiomatic derivation of the CPF (Proosition 4) and some useful information on the Prelec function (Proositions 2 and 3). 2. The insurance roblem 2.. Insurance for low robability events (Stylized facts S3, S4) The seminal study of Kunreuther et al. (978) rovides striking evidence of individuals buying inadequate, nonmandatory insurance against low robability events, e.g., earthquake, ood and hurricane damage in areas rone to these hazards (stylized fact S3). This was a major study, with 35 exert contributors, involving samles of thousands, survey data, econometric analysis and exerimental evidence. All three methodologies gave rise to the same conclusion, see Kunreuther et al. (978). 7 EU redicts that a decision maker facing an actuarially fair remium will buy full insurance for all robabilities, however small. Kunreuther et al. (978, chater 7) reort the following exerimental results. They resented subjects with varying otential losses with various robabilities, keeing the exected value of the loss constant. Subjects faced actuarially fair, unfair or subsidized remiums. In each case, they found that there is a oint below which the takeu of insurance dros dramatically, as the robability of the loss decreases and as the magnitude of the loss increases, keeing the exected loss constant (stylized fact S4 ). These results were robust to changes in subject oulation, exerimental format and order of resentation, resenting the risks searately or simultaneously, bundling the risks, comounding over time and introducing no claims bonuses. 6 See alnowaihi and Dhami (200) for further details. 7 In the foreword, Arrow (Kunreuther et al.,978,. vii) writes: The following study is ath breaking in oening u a new eld of inquiry, the large scale eld study of risktaking behavior. 3
6 Remarkably, the lack of interest in buying insurance arose desite active government attemts to (i) rovide subsidy to overcome transaction costs, (ii) reduce remiums below their actuarially fair rates, (iii) rovide reinsurance for rms and (iv) rovide relevant information. Hence, one can safely rule out these factors as contributing to the low takeu of insurance. Arrow s own reading of the evidence in Kunreuther et al. (978) is that the roblem is on the demand side rather than on the suly side. Arrow writes (Kunreuther et al., 978,.viii) Clearly, a good art of the obstacle [to buying insurance] was the lack of interest on the art of urchasers. A sketical reader might question this evidence on the grounds that otential buyers of insurance could have had limited awareness of the losses or that they might be subjected to moral hazard (in the exectation of federal aid). Both exlanations are rejected by the data in Kunreuther et al. (978). Furthermore, there is no evidence of rocrastination (arising from say, hyerbolic discounting) in the Kunreuther et al. (978) data Other examles of individual resonse to low robability events In diverse contexts, eole ignore low robability events that could, in rincile imose huge losses to them (stylized fact S2). Since many of these losses are selfimosed, because of individual actions, eole are choosing not to selfinsure. It is beyond the scoe of this aer to go through the relevant evidence, but we make some suggestive remarks here. 8 Peole were reluctant to use seat belts rior to their mandatory use desite ublicly available evidence that seat belts save lives. Prior to 985, only 020% of motorists wore seat belts voluntarily, hence, denying themselves selfinsurance; see Williams and Lund (986). Even as evidence accumulated about the dangers of breast cancer (low robability event) women took u the o er of breast cancer examination, only saringly. 9 BarIlan and Sacerdote (2004) estimate that there are aroximately 260,000 accidents er year in the USA caused by redlight running with imlied costs of car reair alone of the order of $520 million er year. It stretches lausibility to assume that these are simly mistakes. In running red lights, there is a small robability of an accident, however, the consequences are self in icted and otentially have in nite costs. A user of mobile hones, while driving, faces otentially in nite costs (e.g. loss of one s and/or the family s life) with low robability, in the event of an accident. Survey evidence in UK indicates that u to 40% of individuals drive and talk on mobile hones; see, Royal Society for the Prevention of Accidents (2005). Pöystia et al. (2004) reort that two thirds of Finnish drivers and 85% of American drivers use their hone while driving. Mobile hone usage, while driving, increases the risk of an accident by two to 8 See Dhami and alnowaihi (200a) for further examles. 9 In the US, this changed after the greatly ublicised events of the mastectomies of Betty Ford and Hay Rockefeller; see Kunreuther et al. (978,. xiii and. 34). 4
7 six fold. Handsfree equiment, although now obligatory in many countries, seems not to o er essential safety advantage over handheld units. A natural exlanation is that the individuals simly ignore or substantially underweight the low robability of an accident The failure of exected utility theory (EU) to exlain the stylized facts It is a standard theorem of EU that eole will insure fully if, and only if, they face actuarially fair remiums. Since insurance rms have to at least cover their costs, market remiums have to be above the actuarially fair ones. Thus, EU rovides a comletely rational exlanation of the widely observed henomenon of underinsurance. This has the olicy imlication that if fullinsurance is deemed necessary (because of strong externalities for examle), then it has to be encouraged through subsidy or stiulated by law. However, EU is unable to exlain several stylized facts from insurance, as we now outline. 0 First, Kunreuther et al. (978, ch.7) found that only 20% of exerimental subjects insure, at the actuarially fair remium, if the robability of the loss is 0:00. Second, EU cannot exlain why many eole simultaneously gamble and insure. The size of the gambling/insurance industries makes it di cult to dismiss such behavior as quirky. Third, EU redicts that a risk averse decision maker always buys some insurance, even when remiums are unfair. However, many eole simly do not buy any insurance, even when it is available, esecially for low robability events. Fourth, when faced with an actuarially unfair remium, EU redicts that a decision maker, who is indi erent between fullinsurance and not insuring, would strictly refer robabilistic insurance to either. This is contradicted by the exerimental evidence (Kahneman and Tversky, 979: ). What accounts for the low takeu of insurance for low robability events? There is some evidence of a bimodal ercetion of risks that could o er a otential exlanation. 2 Some individuals focus more on the robability and others on the size of the loss. The former do not ay attention to losses that fall below a certain robability threshold, while for the latter, the size of the loss is relatively more salient. Hence, the former are likely to ignore insurance while the latter might buy it for low robability but highly salient events. We focus here on the former set of individuals. 0 There are also well known roblems with EU in noninsurance contexts; see Kahneman and Tversky (2000), Starmer (2000) and Dhami and alnowaihi (2007, 200b). For examle, according to EU, an individual who is indi erent between full insurance and not insuring at all should strictly refer being covered on (say) even days (but not odd days) to either. 2 See Camerer and Kunreuther (989) and for the evidence, see McClelland et al (993) and Schade et al (200). 5
8 2.4. The failures of alternative theories to exlain the stylized facts The di culties arising from the use of EU have motivated a number of alternatives. The most imortant of these are rank deendent utility theory (RDU), see Quiggin (982, 993), and cumulative rosect theory (CP), see Tversky and Kahneman (992). 3 These two theories o er several imrovements over EU, e.g., CP can exlain the anomalous result arising from robabilistic insurance that EU cannot. 4 Unlike EU, both RDU and CP use robability weighting functions, w(), to overweight low robabilities and underweight high robabilities (stylized fact S). One such w() function, that is consistent with much of the evidence on nonextreme robability events, and has axiomatic foundations, is the Prelec (998) function lotted in Figure 2. below. 5. w() Figure 2.: The Prelec (998) function, w () = e ( ln ) 2. Remark (In nite overweighting of in nitesimal robabilities): Several weighting functions, in addition to Prelec s (998), have been roosed; we discuss some below. However, they all have the feature that the decision maker, () in nitely overweights in nitesimal robabilities in the sense that the ratio w()= goes to in nity as goes to zero, and (2) in nitely underweight nearone robabilities in the sense that the ratio [ w()]=[ ] goes to in nity as goes to. We call the set of these functions as standard robability weighting functions. They underin all models of RDU and CP and violate stylized fact S2. 3 Mark Machina (2008) has recently argued that... the Rank Deendent form has emerged as the most widely adoted model in both theoretical and alied analyses. However, CP, by adding the notions of reference deendence, loss aversion and searate attitudes to risk in the domain of gains and losses, can exlain everything that RDU can do and, in addition, exlain henomena of major economic imortance that RDU cannot. 4 For other seci c imrovements o ered by RDU and CP over EU, see Starmer (2000). 5 See De nition 5, below. 6
9 We make three contributions, CC3, in this aer. The rst contribution, C, is listed immediately below, followed by contributions C2 and C3 in the next subsection. C. Failure of RDU and CP: Recall that standard robability weighting functions in nitely overweight in nitesimal robabilities. Hence, we show that decision makers who use RDU or CP, insure fully against a su ciently low robability loss even under actuarially unfair remiums and xed costs of insurance, rovided a mild articiation constraint is satis ed. Thus, CP and RDU cannot account for stylized facts S3 and S4. This is a big blow to RDU and CP, because both were roosed recisely to overcome the emirical refutations of EU in the domain of choice under risk Towards an exlanation of the stylized facts from insurance We now describe how RDU and CP might be modi ed to exlain the stylized facts SS4. We begin with the insight of Kahneman and Tversky (979) in rosect theory (PT), which describes two sequential hases.. In the editing hase, decision makers choose which imrobable events to treat as imossible and which robable events to treat as certain. In the insurance context, decision makers might ignore events below a robability threshold (stylized fact S3). Kahneman and Tversky (979, ) wrote: Because eole are limited in their ability to comrehend and evaluate extreme robabilities, highly unlikely events are either ignored or overweighted, and the di erence between high robability and certainty is either neglected or exaggerated. 2. In the decision/evaluation hase, which follows the editing hase, decision makers aly rosect theory to the sychologicallyedited lotteries. Having decided to ignore low robability events in the editing hase, decision makers demand zero insurance for such events in the decision hase, in conformity with stylized fact S3. We can now describe our remaining two main contributions, C2 and C3. C2. Comosite Prelec robability weighting functions (CPF): Recall that the Prelec weighting function (see Figure 2.) catures stylized fact S but fails on stylized fact S2. We modify the end oints of the Prelec function that enables us to address, simultaneously, S and S2. Intuitively, we combine the editing and the evaluation hases of PT. The resulting function, the comosite Prelec robability weighting function (CPF) is sketched in Figure 2.2. It is consistent with the emirical evidence, is arsimonious but exible and has an axiomatic foundation (see Aendix 2, Proosition 4, for an axiomatic derivation of CPF). In Figure 2.2, decision makers heavily underweight 7
10 Figure 2.2: A comosite Prelec robability weighting function (CPF) very low robabilities in the range [0; ] (by contrast the Prelec weighting function in Figure 2., in nitely overweights nearzero robabilities). Akin to Kahneman and Tversky s (979) editing hase, decision makers who use the CPF in Figure 2.2 would tyically ignore very low robability events by assigning low subjective weights to them (stylized fact S2). Hence, in conformity with the evidence, they do not buy insurance for very low robability events (unless mandatory). 6 Over the robability range [ 3 ; ], decision makers overweight the robability, as suggested by the evidence; see, for instance, Kahneman and Tversky s (979, ). In the middle segment, 2 [ ; 3 ], the CPF is identical to the Prelec function, and so addresses stylized fact S. C3. Exlanation of SS4 using comosite rosect theory (CCP: CP+CPF) and comosite rank deendent utility (CRDU: RDU+CPF): Tversky and Kahneman (992) introduced cumulative rosect theory (CP). This relaced Kahneman and Tversky s (979) rosect theory (PT) in two resects. (i) The sychologicallyrich editing hase that determined, among other things, which low robability events to ignore (stylized fact S2) was eliminated. 7 (ii) Cumulative transformations of robability relaced the oint transformations under PT. 8 Hence, a decision maker using CP 6 In other contexts, they are unlikely to be dissuaded from lowrobability highunishment crimes, reluctant to wear seat belts (unless mandatory), reluctant to articiate in voluntary breast screening rograms (unless mandatory) and so on. 7 The reason for this was that the cumulative transformations of robabilities in CP requires a continuous,, and onto robability weighting function, w(), on [0; ] such that w(0) = 0 and w() =. The editing hase in PT, however, creates discontinuities in the weighting function, which are not ermissible under CP. 8 The introduction of cumulative transformation of robabilities was an insight borrowed from RDU. 8
11 never chooses stochastically dominated otions (unlike PT). When CP is augmented with CPF (instead of a standard robability weighting function; see remark ) we refer to it as comosite cumulative rosect theory (CCP). Analogously, when combined with CPF, RDU is referred to as comosite rank deendent utility, CRDU. CCP and CRDU are able to address all four stylized facts, SS4. In articular, a decision maker who uses CCP and CRDU will not buy insurance against an exected loss of su ciently low robability; in agreement with the evidence. To quote from Kunreuther et al. (978, 248) This brings us to the key nding of our study. The rincial reason for a failure of the market is that most individuals do not use insurance as a means of transferring risk from themselves to others. This behavior is caused by eole s refusal to worry about losses whose robability is below some threshold Are these low robabilities economically relevant? In section 8., we t two comosite Prelec robability weighting functions (CPF) to two searate data sets from Kunreuther et al. (978). The range of low robabilities that are underweighted by the CPF (the range (0; ] in Figure 2.2) are, resectively, (0; 0:95] and (0; 0:006]. The uer ends of these intervals involve robabilities that would comfortably seem to accommodate many insurance contexts. The decision maker might, however, still want to buy insurance even if he/she underweights these robabilities. This brings us to the issue of the exact amount of underweighting and the robability of a loss for which insurance is sought. These are some of the issues addressed in our aer. 3. The Model Suose that a decision maker can su er the loss, L > 0, with robability 2 (0; ). She can buy coverage, C 2 [0; L], at the cost rc + f, where r 2 (0; ) is the remium rate, and f 0 is a xed cost of buying insurance. 9 We allow deartures from the actuarially fair condition. We do so in a simle way by setting the insurance remium rate, r, to be r = ( + ). (3.) Thus, = 0 corresonds to the actuarially fair condition, > 0 to the actuarially unfair condition and < 0 to the actuarially overfair condition. Hence, the decision maker s See, for examle, Quiggin (993) or Starmer (2000) for arguments for referring the cumulative transformation of robability to the oint transformation. 9 The xed costs include various transactions costs of buying insurance that are not re ected in the insurance remium. For instance, the costs of information acquisition about alternative insurance olicies, the oortunity costs of one s time sent in buying insurance and so on. 9
12 wealth is: W rc f with robability W rc f L + C W rc f with robability Let U I (C) be the utility of the decision maker if she decides to buy an amount of coverage, C > 0. Let C be the otimal level of coverage. Denote by U NI the utility of the decision maker from not buying any insurance. Then, a decision maker who buys coverage C, satis es the articiation constraint, U NI U I (C ). 3.. Some observations on limiting rocesses In this aer, we shall make heavy use of limiting arguments. The decision maker faces a loss, L, with robability,. Hence, the exected value of the loss is L = L. We shall take the limit! 0, keeing L xed. It follows that L!. One can object by saying that losses can never be in nite. 20 At this level of generality, one could argue against the use of asymtotic theory in statistics on the grounds that samles can never be in nite. One could also argue against using in nite horizon models, against in nitely divisible quantities and so on. 2 A further examle may hel. The early literature that studied convergence of Cournot cometition to erfect cometition assumed that market size remained xed while the number of roducers tended to in nity. 22 A consequence was that the size of each roducer had to aroach zero. Further consequences were that the average cost curve had to be ositive sloing and that any transaction cost, however small, would block trade. These consequences were regarded as unsatisfactory. Therefore, the subsequent literature considered roducers of xed size with Ushaed average cost curves, ossibly with xed costs. As the number of rms tended to in nity, the total size of the market had to also tend to in nity. Although no market can, in fact, be in nite, the later literature was regarded as more satisfactory Actually, loss of life, which can be associated with many kinds of low robability disasters is arguably an in nite loss. Loss of honour or severe injustice (in many societies), or su ering a serious lifelong handica could also aroximate in nite losses. 2 Continuoustime trading and an exogenous rocess for the underlying asset are fundamental assumtions of the BlackScholesMerton model, one of the fundamental models in nance. However, any transaction costs, no matter how small, would block continuous time trading. An exogenous rocess for the underlying asset (una ected by trade in the nancial derivative) imlies an in nite quantity traded in the underlying asset. Similarly, in hysics, any article following a Brownian motion must cover an in nite distance in any time eriod, no matter how brief, which in itself is not a hysically accetable result. But the assumtion of Brownian motion is of immense utility, as it facilitates derivation of refutable emirically imortant consequences. 22 See, for examle, Frank (965) and Ru n (97). 23 See, for examle, Novshek (980). 0
13 More seci cally, one can object by saying that we are considering the wrong limiting argument. In articular, that we should take the limit! 0, keeing L xed. It then follows that L! 0. However, in fact, the rst tye of limiting argument (! 0, keeing L xed) is the relevant one in our case. The limiting rocess we consider is the theoretical analogue of the exerimental treatments of Kunreuther et al. (978, chater 7). They resented subjects with varying otential losses with various robabilities, keeing the exected value of the loss constant. Subjects faced actuarially fair, unfair or subsidized remiums. In each case they found that there is a oint below which the takeu of insurance dros dramatically, as the robability of the loss decreases and as the magnitude of the loss increases, keeing the exected loss constant. 4. Nonlinear transformation of robabilities The main alternatives to exected utility (EU) under risk, i.e., rank deendent utility (RDU) and cumulative rosect theory (CP), introduce nonlinear transformation of the cumulative robability distribution. In this section, we introduce the concet of a robability weighting function and some other concets which are crucial for the aer. De nition (Probability weighting function): By a robability weighting function we mean a strictly increasing function w() : [0; ] onto! [0; ]. Proosition : A robability weighting function, w(), has the following roerties: (a) w (0) = 0, w () =. (b) w has a unique inverse, w, and w is also a strictly increasing function from [0; ] onto [0; ]. (c) w and w are continuous. De nition 2 : The function, w(), (a) in nitelyoverweights in nitesimal robabilities, if w() =, and (b) in nitelyunderweights nearone robabilities, if lim =.! w() lim!0 De nition 3 : The function, w(), (a) zerounderweights in nitesimal robabilities, if w() = 0, and (b) zerooverweights nearone robabilities, if lim = 0.! w() lim!0 Data from exerimental and eld evidence tyically suggest that decision makers exhibit an inverse Sshaed robability weighting function over outcomes. See Figure 5. for an examle. Tversky and Kahneman (992) roose the following robability weighting function, where the lower bound on comes from Rieger and Wang (2006). De nition 4 : The Tversky and Kahneman robability weighting function is given by w () =, 0:5 <, 0. (4.) [ + ( ) ]
14 Proosition 2 : The Tversky and Kahneman (992) robability weighting function (4.) in nitely overweights in nitesimal robabilities and in nitely underweights nearone rob w() abilities, i.e., lim!0 = and lim! w() =, resectively. Remark 2 (Standard robability weighting functions): A large number of other robability weighting functions have been roosed, e.g., those by Gonzalez and Wu (999) and Lattimore, Baker and Witte (992). Like the Tversky and Kahneman (992) function, they all in nitely overweight in nitesimal robabilities and in nitely underweight nearone robabilities. We shall call these as the standard robability weighting functions. 5. Prelec s robability weighting function The Prelec (998) robability weighting function has the following merits: arsimony, consistency with much of the available emirical evidence (at least away from the endoints of the interval [0; ]) and an axiomatic foundation. De nition 5 (Prelec, 998): By the Prelec function we mean the robability weighting function w() : [0; ]! [0; ] given by w (0) = 0, w () = ; (5.) w () = e ( ln ), 0 <, > 0, > 0. (5.2) Proosition 3 : The Prelec function (De nition 5) is a robability weighting function in the sense of De nition. The arameter controls the convexity/concavity of the Prelec function. If <, then the Prelec function is strictly concave for low robabilities but strictly convex for high robabilities, i.e., it is inverse Sshaed as in w () = e ( ln ) 2 ( =, = ), and 2 sketched in Figure 5.. The converse holds if >, in which case, the Prelec function is strictly convex for low robabilities but strictly concave for high robabilities, i.e., it is Sshaed. An examle is w () = e ( ln )2 ( = 2, = ), sketched in Figure 5.2 as the light, lower, curve (the straight line in Figure 5.2 is the 45 o line). Between the region of strict convexity (w 00 > 0) and the region of strict concavity (w 00 < 0), for each of the cases in Figures 5. and 5.2, there is a oint of in exion (w 00 (e) = 0). The arameter in the Prelec function controls the location of the in exion oint relative to the 45 0 line. Thus, for =, this oint of in exion is at = e and lies on the 45 0 line, as in Figures 5. and 5.2 (light curve), above. However, if <, then the oint of in exion lies above the 45 0 line, as in w () = e 2 ( ln )2 ( = 2; = ), sketched as 2 2
15 w Figure 5.: A lot of the Prelec (998) function, w () = e ( ln ) 2. w Figure 5.2: A lot of w () = e 2 ( ln )2 and w () = e ( ln )2. the thicker, uer, curve in Figure 5.2. In this case, the xed oint, w ( ) =, is at ' 0:4 but the oint of in exion, w 00 (e) = 0, is at e ' 0:20. The full set of ossibilities for the Prelec function is established by Proositions 2 and 3 of Aendix 2. In Figure 5. (and rst row in Table 2 of Aendix 2), where <, note that the sloe of w () becomes very stee near = 0. By contrast, in gure 5.2 (and last row in Table 2 of Aendix 2), where >, the sloe of w () becomes very gentle near = 0. This is established by the following roosition, which will be imortant for us. Proosition 4 : (a) For < the Prelec function (De nition 5): (i) in nitelyoverweights w() in nitesimal robabilities, i.e., lim =, and (ii) in nitely underweights nearone robabilities, i.e., lim = (Prelec, 998, 505); see De nition 2 and Figure 5..!0 w()! (b) For > the Prelec function: (i) zerounderweights in nitesimal robabilities, i.e., w() w() lim = 0, and (ii) zerooverweights nearone robabilities, i.e., lim = 0; see De !0! 3
16 nition 3 and gure 5.2. According to Prelec (998,.505), the in nite limits in Proosition 4a cature the qualitative change as we move from certainty to robability and from imossibility to imrobability. On the other hand, they contradict stylized fact S2, i.e., the observed behavior that eole ignore events of very low robability and treat very high robability events as certain; see, e.g., Kahneman and Tversky (979). In sections 6 and 7, below, we show that this leads to eole fully insuring against all losses of su ciently low robability, even with actuarially unfair remiums and xed costs of insurance. This is contrary to the evidence, e.g., that in Kunreuther et al. (978). These seci c roblems are avoided for >. However, for >, the Prelec function is Sshaed, see Proosition 3d and Figure 5.2. This is in con ict with the emirical evidence, which indicates an inverses shae for robabilities bounded away from the end oints of the interval [0; ] (see stylized fact S). Hence, the two cases, < or >, by themselves, are unable to simultaneously address stylized facts S and S2. 6. Rank deendent utility (RDU) and insurance We now model the behavior of an individual using rank deendent utility theory (RDU). RDU may be regarded as a conservative extension of exected utility theory (EU) to the case where robabilities are transformed in a nonlinear manner using decision weights. To illustrate RDU, consider a decision maker with utility of wealth, u, and a robability weighting function, w(), where u is strictly concave, di erentiable, and strictly increasing, i.e., u 0 > 0. In addition to these standard assumtions, we shall assume that u 0 is bounded above 24 by (say) u 0 max. Suose that the decision maker faces the lottery (x ; ; x 2 ; 2 ), i.e., the wealth level, x, occurs with robability and the wealth level, x 2, occurs with robability 2, x < x 2, 0 i, + 2 =. A decision maker using RDU evaluates the utility from the lottery (x ; ; x 2 ; 2 ), as follows: U (x ; ; x 2 ; 2 ) = [ w ( )] u (x ) + w ( 2 ) u (x 2 ). (6.) For w () =, RDU reduces to standard exected utility theory. 25 Note that the higher outcome, x 2, receives weight 2 = w ( 2 ), while the lower outcome, x, receives weight 24 The boundedness of u 0 is needed for art (b) of Proosition 5. This seems feasible on emirical grounds, since eole do undertake activities with a nonzero robability of comlete ruin, e.g., using the road, undertaking dangerous sorts, etc. However, the boundedness of u 0 excludes such tractable utility functions as ln x and x, 0 < <. By contrast, the boundedness of u 0 is not a requirement in CP, as we shall see. 25 From Proosition 2, this is the case for = = (see Aendix 2). 4
17 = w ( + 2 ) w ( 2 ) = w () w ( 2 ) = w ( 2 ) = w ( ). and 2 are the decision weights associated with the relevant outcomes 26 We now aly this framework to the insurance model outlined in section 3. If the decision maker buys insurance, then the utility under RDU is given by: U I (C) = w ( ) u (W rc f) + [ w ( )] u (W rc f L + C). (6.2) Since U I (C) is a continuous function on the nonemty comact interval [0; L], an otimal level of coverage, C, exists. For full insurance, C = L, (6.2) gives: U I (L) = u (W rl f). (6.3) On the other hand, the decision maker s utility from not buying insurance is: U NI = w ( ) u (W ) + [ w ( )] u (W L). (6.4) The decision maker must satisfy the following articiation constraint to buy a level of insurance coverage C : U NI U I (C ). (6.5) Proosition 5 : Suose that the decision maker follows RDU. (a) A su cient condition for the articiation constraint in (6.5) to hold is that xed costs of insurance, f, be bounded above by LF (), where L = L and F () = w ( ) ( + ). (6.6) (b) If the robability weighting function in nitelyunderweights nearone robabilities (De nition 2b) then, for a given exected loss, the decision maker will insure fully for all su  ciently small robabilities. This holds even in the resence of actuarially unfair insurance and xed costs of insurance. (c) If a robability weighting function zerooverweights nearone robabilities (De nition 3b) then, for a given exected loss, a decision maker will not insure, for all su ciently small robabilities. The intuition behind Proosition 5 is as follows. First, consider art (b). Suose that the robability weighting function in nitelyunderweights nearone robabilities (De nition 2b). This is the case for all the standard robability weighting functions and, in 26 The framework is easily extended to any nite number of outcomes; see Quiggin (982, 993) for the details. We now clarify a common source of confusion. The decision weights, i (not w ( i )), sum u to one under RDU. This is no longer the case under CP when there are both gains and losses (de ned below). However, if all outcomes under CP are in the domain of gains or all are in the domain of losses, then the decision weights will sum to. 5
18 articular, for the TverskyKahneman robability function (De nition 4 and Proosition 2) and for the Prelec function with < (De nition 5 and Proosition 4(aii)). In this case, and as illustrated in Figure 5., the robability weighting function is very stee near (becoming in nitely stee in the limit) and considerably underweights robabilities close to. Now return to (6.). Recall that x < x 2, so that (in the resent context) x is total wealth if the loss occurs (with robability, ) and x 2 is total wealth if the loss does not occur (with robability 2 = ). As! 0, 2! and, hence, as required in Proosition 5(b), w ( 2 ) underweights 2. This reduces the relative salience of the weighted utility of wealth, w ( 2 ) u (x 2 ), if the loss does not occur but increases the relative salience of the weighted utility of wealth, [ w ( 2 )] u (x ), if the loss does occur. This makes insurance even more attractive under RDU than under EU (which was already too high, given the evidence). The reverse occurs in case (c). Here the robability weighting function zerooverweights nearone robabilities (De nition 3b). This is the case for the Prelec function with > (De nition 5 and Proosition 4(bii)). In this case, and as illustrated in Figure 5.2, the robability weighting function is very shallow near (with the sloe, in fact, aroaching zero) and considerably overweights robabilities close to. Now return to (6.). As 2!, as required in Proosition 5(c), w ( 2 ) overweights 2. This increases the relative salience of the weighted utility of wealth, w ( 2 ) u (x 2 ), if the loss does not occur. But it also reduces the relative salience of the weighted utility of wealth, [ w ( 2 )] u (x ), if the loss does occur. This makes insurance against very low robability events unattractive under RDU, in conformity with the evidence. From Proosition 4(aii) and Proosition 5(b), a decision maker using a Prelec robability weighting function (De nition 5), will fully insure against all losses of su ciently small robability. It is of interest to get a feel for how restrictive this articiation constraint is. Examle (), below, suggests that it is a weak restriction. Examle : To check the restrictiveness of the articiation constraint we use the result in Proosition 5(a), i.e., the su cient condition f LF () for the articiation constraint to hold. The rst row of the following Table gives losses, L (in dollars, say), from 0 to 0; 000; 000, with corresonding robabilities,, in row 2, ranging from 0: to 0:000; 000;. Hence, the exected loss in each case is L = and so the su cient condition is simly f w ( ) ( + ) F () : w( ) In row 3 are the corresonding values of for the Prelec function w () = e where the values = 0:65 and = are suggested by Prelec (998). Row 4 of the table ( ln )0:65, 6
19 reorts the corresonding value for F () for the case of a relatively high ro t rate for w( ) insurance rms of 00% (i.e., = ) so that F () = 2: Row 5 gives the uer bound on xed costs as a ercentage of exected losses.. L w( ) 2: :903 9 :6 25:088 56:29 25:88 28:83 F () 0: : :6 23:088 54:29 23:88 279:83 F () L 00 6:74 290:3 9 96: 2308:8 542: We see, from the table, that the uer bound on (i) xed costs, and (ii) xed cost as a ercentage of the exected loss, is hardly restrictive for low robabilities. Thus, from Proosition 5(a), we see that using RDU in combination with the Prelec weighting function is likely to lead to misleading results, in that it would redict too much insurance. Proosition 5(c) will enable us to demonstrate that the decision maker will not insure against any loss of su ciently small robability, in agreement with observation. We further discuss this in section 8, below. 7. Cumulative rosect theory (CP) and insurance The four central features of cumulative rosect theory (CP) all based on strong emirical evidence, are as follows. 27 () In CP, unlike EU and RDU, the carriers of utility are not levels of wealth, assets or goods, but di erences between these and a reference oint (reference deendence). The reference oint is often (but not necessarily) taken to be the status quo. (2) The utility function in rosect theory is concave for gains but convex for losses (declining sensitivity). 28 (3) The disutility of a loss is greater than the utility of a gain of the same magnitude (loss aversion). (4) Probabilities are transformed, so that small robabilities are overweighted but high robabilities are underweighted. We now exlain these features in greater detail. Consider a level of wealth y and a reference oint r. Then de ne the transformed variable x = y r as the wealth relative to the reference oint. The utility function in CP is de ned over x; x > 0 is a gain while x < 0 is a loss. De nition 6 (Tversky and Kahneman, 979). The utility function under CP, v(x) : ( ; )! ( ; ), is a continuous, strictly increasing, maing that satis es:. v (0) = 0 (reference deendence) 27 See, for instance, Kahneman and Tversky (2000), Starmer (2000). 28 This feature of CP can also exlain the observation that individuals can simultaneously gamble and insure. CP can also exlain the unoularity of robabilistic insurance (see section 2.3).. 7
20 2. v (x) is concave for x 0 (declining sensitivity for gains) 3. v (x) is convex for x 0 (declining sensitivity for losses) 4. v ( x) > v (x) for x > 0 (loss aversion). Examle 2 : The roerties in De nition 6 are satis ed by v (x) = e x and v (x) = x, 0 < < but not by v (x) = ln x, since ln 0 is not de ned. A oular utility function in CP that is consistent with the evidence and is axiomatically founded is the ower form: 29 v (x) = x if x 0 ( x) if x < 0, 0 < <, >. (7.) 7.. Illustration of CP for the case of a rosect with two outcomes Consider a decision maker who faces the lottery (x ; ; x 2 ; 2 ), i.e., win x with robability or x 2 with robability 2, x < x 2, 0 i, + 2 =. The case of most interest to us is that when both x and x 2 are in the domain of losses, i.e., x < x 2 < 0. In this case, the decision maker using CP evaluates the lottery (x ; ; x 2 ; 2 ), by forming the value function, V, as follows: V (x ; ; x 2 ; 2 ) = w ( ) v (x ) + w ( ) v (x 2 ). (7.2) Note that here, the lower outcome, x, receives weight = w ( ), while the higher outcome, x 2, receives weight 2 = w ( + 2 ) w ( ) = w () w ( ) = w ( ) The insurance decision of an individual who uses CP Consider a decision maker whose behavior is described by CP and who faces the insurance roblem described in Section 3. Take the statusquo wealth, W, of the decision maker as the reference oint. With robability, her wealth relative to the reference wealth is (W rc f) W = rc f < 0. (7.5) 29 For an axiomatic derivation of this value function, see alnowaihi et al. (2008). 30 If 0 < x < x 2, then the value function under CP is: V (x ; ; x 2 ; 2 ) = w + ( 2 ) v (x 2 ) + w + ( 2 ) v (x ), (7.3) which is similar to the corresonding formula for RDU in (6.) excet that here x i are deviations from reference wealth, while in (6.) they are absolute levels of wealth. However, if x is in the domain of losses while x 2 is in the domain of gains, so that x < 0 < x 2, then V (x ; ; x 2 ; 2 ) = w + ( 2 ) v (x 2 ) + w ( ) v (x ). (7.4) Note that the robability weighting function for losses, w, may be di erent from that for gains w +, although, emirically, it aears that w = w + = w; see Prelec (998). This is the assumtion that we shall also make. 8
21 With the comlementary robability, her wealth relative to the reference oint is (W rc f L + C) W = rc f (L C) rc f < 0. (7.6) Notice from (7.5), (7.6) that the decision maker is in the domain of loss in both states. De nition 7 (Major and Minor loss): We call L + f + rc the major loss and f + rc the minor loss. These losses occur with resective robabilities and. The decision maker s value function under CP when some level of insurance coverage C 2 [0; L] is urchased is given by: V I (C) = w () v ( rc f (L C)) + [ w ()] v ( rc f). (7.7) Since V I (C) is a continuous function on the nonemty comact interval [0; L], an otimal level of coverage, C, exists. For full insurance, C = L, (7.7) gives: V I (L) = v ( rl f). (7.8) On the other hand, if the decision maker does not buy any insurance coverage (i.e., C = 0), and so also does not incur the xed cost (f = 0), the value function is (recall that v (0) = 0): V NI = w () v ( L). (7.9) For the decision maker to buy insurance coverage C 2 [0; L], the following articiation constraint (the analogue here of (6.5)) must be satis ed: V NI V I (C ). (7.0) Proosition 6 : Suose that a decision maker uses CP, then the following hold. (a) There is a corner solution in the following sense. A decision maker will either choose to insure fully against any loss, i.e., C = L, or choose zero coverage, i.e., C = 0, deending on the satisfaction of the articiation constraint. 3 This holds even in the resence of actuarially unfair remiums and a xed cost of insurance. (b) For Prelec s robability weighting function (De nition 5), with <, for the value function (7.) and for a given exected loss, the articiation constraint (7.0) is satis ed for all su ciently small robabilities. A su cient condition for the satisfaction of the articiation constraint is that f < LF () where F () = e ( ln ) ( + ), 0 < <, > 0, > 0. (7.) (c) If a robability weighting function zerounderweights in nitesimal robabilities (De  nition 3a) then, for a given exected loss, a decision maker will not insure against any loss of su ciently small robability. 3 The answer, therefore, deends on the arameters of the model and so, we require simulations. See Examle 3, below. 9
22 The reason that art (a) holds is very simle. Since the value function is strictly convex for losses, a decision maker will always insure fully, if he insures at all. This is, of course, at variance with the evidence from artial coverage. The intuition behind art (b) is as follows. For <, the Prelec function in nitelyoverweights in nitesimal robabilities (De nitions 2a and 5 and Proosition 4(ai)). In this case, and as illustrated in Figure 5., the robability weighting function is very stee near 0 (becoming in nitely stee in the limit) and considerably overweights robabilities close to 0. Now return to (7.2). Recall that x < x 2 < 0 in this case. Hence, x is the di erence between wealth if the major loss occurs (with robability, ) and reference wealth. On the other hand, x 2 is the di erence between wealth if the minor loss occurs (with robability 2 = ) and reference wealth. As! 0, and as required by Proosition 6(b), w ( ) increasingly overweights. This increases the relative salience of the weighted major loss, w ( ) v (x ) but reduces the relative salience of the weighted minor loss, [ w ( )] u (x ). This makes insurance even more attractive under CP than under EU (which was already too high, given the evidence). The reverse occurs in case (c). Here, the robability weighting function zerounderweights in nitesimal robabilities (De nition 3a). This is the case for the Prelec function with > (De nition 5 and Proosition 4(bii)). In this case, and as illustrated in Figure 5.2, the robability weighting function is very shallow near 0 (the sloe, in fact, aroaches zero) and considerably underweights robabilities close to 0. Now return to (7.2). As! 0, and as required by Proosition 6(c), w ( ) underweights. This reduces the relative salience of the weighted major loss, w ( ) v (x ) but reduces the relative salience of the weighted minor loss, [ w ( )] u (x 2 ). This makes insurance against very low robability events unattractive under CP, in conformity with the evidence. By Proosition 6(a), a decision maker will insure fully against any loss, rovided the articiation constraint (7.0) is satis ed, even with xed costs of insurance and an actuarially unfair remium. By Proosition 6(b), for Prelec s robability weighting function (De nition 5), for the value function (7.) and for a given exected loss, the articiation constraint (7.0) is satis ed for all su ciently small robabilities. It is of interest to get a feel for how restrictive this articiation constraint is. Examle (3), below, suggests that it is a weak restriction. Examle 3 : The the rst row of the following Table gives losses (in dollars, say) from 0 to 0; 000; 000, with corresonding robabilities (row 2) ranging from 0: to 0:000; 000; ; so that the exected loss in each case is L =. In row 3 are the corresonding values of e ( ln ), where the values = 0:65 and = are suggested by Prelec (998) and = 0:88 is suggested by Tversky and Kahneman (992). Row 4 gives F () (see (7.)) for the high ro t rate of 00% ( = ) for the insurance rm, so that F () = e ( ln ) 2. 20
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