LOSS AVERSION: AN EVOLUTIONARY PERSPECTIVE

Transcription

1 LOSS AVERSION: AN EVOLUTIONARY PERSPECTIVE Abstract Loss aversion is a central element of modern theories of choice. While loss aversion has been etensively documented eerimentally and emirically, and is emloyed to elain imortant economic henomena such as the equity remium, its origins are not yet well understood. We suggest that loss aversion is a consequence of the evolutionary objective of minimizing the robability of etinction of one's line of descendants. A simle relationshi is derived between the equilibrium loss aversion coefficient and the etinction robability. Emirical estimates of the etinction robability imly a loss aversion coefficient of 2.17, a value close to the estimates obtained eerimentally. Keywords: loss aversion, reference, evolution, etinction, having descendants forever. JEL classification: D81, D03.

2 1. Introduction One of the key elements of Prosect Theory is that the choice among risky rosects is based on the change of wealth, rather than on the total wealth. In addition, gains and losses are evaluated differently, and losses are overweighed relative to gains of the same magnitude (Kahneman and Tversky 1979, Tversky and Kahneman 1992). These two roerties are catured by the following streamlined value function introduced by Benartzi and Thaler (1995): for 0 V ( ) (1) for 0 where is the change of wealth relative to the current wealth, is the loss-aversion coefficient, and the objective is the maimization of E [V( )]. 1 An individual with these references is indifferent to a gamble with a 50% robability of losing $1 and a 50% robability of gaining $. Loss aversion imlies 1, i.e. losses are weighed more heavily than gains. Eerimental and emirical studies tyically estimate to be in the range (see Table I). (Please insert Table I about here) The evidence for loss aversion is overwhelming. Loss aversion is generally inconsistent with the standard eected utility framework, in which choice is based on total wealth, and it has numerous imlications for economics, sychology, sociology, marketing and even olitics. For eamle, Benartzi and Thaler (1995) show that loss aversion can rovide an elanation for the equity remium uzzle. The combination of loss aversion and myoia have a dramatic affect on savings behavior (Gneezy and Potters (1997), Benartzi and Thaler (1999), and Haigh and List (2005)). Barberis and Huang (2001) and Berkelaar, Kouwenberg, and Post (2004) derive the imlications of loss aversion for otimal ortfolio choice and for stock returns. Hardie, Johnson, and Fader (1993) and Ho and Zhang (2008) study the imlications of loss aversion for marketing. Genesove and Mayer (2001) eamine the effects of loss aversion on housing markets. Jervis (1992) analyzes the olitical imlications of loss aversion. Rabin (2000) and Rabin 1 In addition to loss aversion, Prosect Theory also imlies risk-aversion for gains and risk-seeking for losses (i.e. V() that is concave for >0 and conve for <0), and also allows for subjective robability weighting. As our focus in this aer is loss aversion, we abstract from these features to simlify the analysis and we emloy the Benartzi and Thaler (1995) iecewise linear value function. 1

3 and Thaler (2001) show that loss aversion offers a solution to a aradoical set of choices that arises in the eected utility framework. While loss aversion is a cornerstone of the current theory of choice, very little is known about the origin of this behavior. Choices between risky alternatives are robably affected by education, age and life eerience. However, more and more evidence is accumulating about the central role of genetics in determining references. For instance, emirical studies have shown that the asset allocation of identical twins are much more correlated than those of twins who are not identical. Furthermore, the asset allocations of identical twins who where raised aart are also highly correlated (see Cesarini et. al. 2009, 2010, and Barnea, Cronqvist, and Siegel 2010). In recent years biologists and economists have identified secific risk-attitude genes (see Kuhnen and Chiao 2009, and Zhong et. al. 2009). If references are (at least artially) genetically determined, one can view the observed loss aversion as the result of an evolutionary rocess selecting for the evolutionary most advantageous risk attitude. 2 This is the idea followed in the resent aer. While there is a raidly growing literature on the evolution of references, most of these studies analyze the imlications for risk aversion, time reference, and grou effects. 3 In contrast, the focus of the resent aer is on loss aversion. In the net section we discuss several ossible evolutionary objective functions, with a secial emhasis on the objective of "having descendants forever", i.e. the objective of not having one's line of descendants cut-off. While there is no single objective function that is dominant over all others, evolutionary biologists argue that the having-descendants-forever objective has likely layed a central role in human evolution. In Section 3 we develo the imlications of this evolutionary objective function to risky choice. We show that it imlies basing choices on change in wealth rather than total wealth, eactly as found in Prosect Theory. It also yields loss aversion. Furthermore, we develo a simle theoretical relationshi between the equilibrium loss aversion coefficient and the robability of etinction of one's line of descendants. In Section 4 we eamine this theoretical relationshi by comaring the emirical etinction robabilities reorted in the literature with the estimates of. Perhas surrisingly, we 2 It is interesting to note that loss aversion has been documented not only in humans, but in monkeys as well (Chen, Lakshminarayanan, and Santos 2006). Tom, Fo, Treel, and Poldrack (2007) identify secific regions in the brain that corresond to loss averse behavior. 3 A comrehensive review of this literature is beyond the scoe of this aer. An ecellent overview of the literature of the evolution of references can be found in Robson and Samuelson (2010). 2

4 find good agreement between these aarently comletely unrelated arameters. Section 5 concludes. 2. The Evolutionary Objective Function In an evolutionary contet, organisms can be viewed as vessels for carrying their genes (Dawkins 1989). Genes are successful if they manage to eretuate from one generation to the net. Thus, one ossible evolutionary objective function that can be considered is the maimization of the eected number of offsring the more offsring, the more coies of the organism s genes are transmitted to the net generation. While this is a simle and intuitively aealing objective function, it may lead to unreasonable results. To illustrate, consider the following simlified eamle. Suose that there are only two ossible reroduction rosects or gambles to choose from. Gamble A yields 0 offsring with robability 0.3 and 2 offsring with robability 0.7. Gamble B yields 0 offsring with robability 0.8 and 9 offsring with robability 0.2. Suose also that there are two tyes of reference genes: gene A that imlies the reference of gamble A, and gene B that imlies the reference of gamble B. 4 Assume that the gamble realizations are indeendent across individuals. 5 Then, after T generations the eected number of individuals with gene A will be T T , while the eected number of individuals with gene B will be T 1.8. Clearly, after some time the eected number of individuals with gene B becomes much larger than that of gene A, and the ratio of the eected numbers goes to infinity as T. This may be interreted as B dominates the oulation in the long run, and it is the motivation for the criterion of maimizing the eected number of offsring. However, it is far from obvious that this criterion is evolutionary advantageous. Note that in the above eamle oulation B has a much larger robability than oulation A of becoming comletely etinct. Let us elaborate. A direct calculation of the robability that A s line becomes etinct is quite cumbersome, because there are infinitely many ossible realizations that lead to etinction. 6 Fortunately, it is much easier 4 As is tyical in this literature, reroduction is assumed to be aseual, i.e. the offsring have the same references as their arent. 5 In addition to this idiosyncratic randomness, individuals and secies may also be eosed to macro or aggregate environmental risk. Robson (1996) and Brennan and Lo (2011) rovide illuminating discussions of aggregate risk effects. 6 For eamle, one ossible scenario for etinction is that the original arent has two offsring, each one of these offsring has two offsring, but in the third generation all four offsring die. Of course, there are 3

5 to solve this roblem recursively. Let us denote the robability that the line of descendants of an individual with gene A will become etinct by. It is ossible, with A robability 0.3, that the initial individual will have no offsring, and this will imly the end of his line of descendants. However, even if he survives to have 2 offsring, (and this occurs with robability 0.7), it is ossible that the lines of both of these offsring will eventually become etinct. As the offsring carry gene A, for each one of them the robability that his line of descendants will eventually become etinct is also by definition. As the gambles are assumed to be indeendent, the etinction of one offsring is A indeendent of the etinction of the other, and therefore the robability that both lines eventually become etinct is A A 2. Thus, is the solution to: A A, (2) which yields A This value catures all of the ossible scenarios leading to eventual etinction. The robability of A Having Descendants Forever, ( HDF ), is the robability that A's line of descendants does not become etinct, and it is given by ( HDF ) Similarly, the robability that individual B s line of A A descendants eventually becomes etinct is given by the solution to: B. (3) B Solving eq.(3) numerically yields B , which imlies ( HDF ) Thus, while the ratio of the eected oulation of A to B B the eected oulation of B converges to zero as T, tye A has a much higher robability of surviving forever. How can these two facts be reconciled? Note that as T becomes large the robability distribution of the number of B descendants becomes very skewed there is a large robability that B will become etinct, but there is a small robability that B will have a very large number of descendants. This etreme lowrobability event drives the high eected value of B descendants. Etinction lays an obvious central role in the evolutionary dynamics. One may susect, though, that once the oulation of a given tye reaches a certain size, the robability of etinction in the i.i.d. reroduction framework is negligible. However, evolutionary biologists show that the (HDF) criterion may be very imortant even after A infinitely many such scenarios leading to etinction, and the robability of eventual etinction, A in the notation below, is the sum of robabilities for all these events. 7 Equations (2) and (3) are secial cases of the general Galton-Watson (1875) equation for the robability of etinction of a family line. 4

6 the number of individuals in each reference tye has become large, for at least three reasons: i) genetic diversity 8, ii) oulation "bottlenecks" 9, and iii) the founder effect 10. Another evolutionary objective function suggested in the literature is the geometric mean growth rate. Note, however, that this criterion is alied to the growth rate of the entire oulation, rather than to individual choice, and these two frameworks can be very different, deending on the correlation between the individuals. Moreover, if there is any robability of etinction, the geometric mean is 0, and the comarison of gambles becomes meaningless. Each of the different evolutionary objective functions emloyed in the literature has its ros and cons. Evolutionary biologists argue that the (HDF) objective lays a central role (see, for eamle, Cohen 1993, and references therein). We do not resume to determine that maimizing (HDF) is the correct objective function, nor that it is the only one that should be considered. Our standoint is that the (HDF) criterion likely lays an imortant role in the evolutionary rocess, and should therefore be given careful consideration. This is the ath followed in this aer. The viewoint that both the number of descendants and the (HDF) are imortant is beautifully catured by the following biblical blessing: 8 Consider, for eamle, the tyes A and B discussed above. Suose that there are 100 individuals with risk reference gene A, and 100 individuals with gene B. Furthermore, assume that individuals carry many other genes in addition to the risk reference gene, i.e. each individual reresents a unique combination of genes (or alternatively, each individual actually reresents a sub-tye). After many generations, out of the 100 sub-tyes carrying gene A, on average 57.2 will survive (recall that A ( HDF ) ). In contrast, out of the initial 100 sub-tyes carrying gene B, on average only 15.7 will survive. Thus, the reference for higher (HDF) maintains more genetic diversity, which is an obvious evolutionary advantage. For eamle, environmental conditions may drastically change, making only a very small number of sub-tyes viable. The tye with more genetic diversity has a higher robability to survive such a change. 9 When viewed in a short-run ersective, oulations usually grow at rather steady rates. However, when viewed at a longer-run ersective, the oulation size sometimes changes abrutly and dramatically, tyically due to changes in the environmental conditions. There are known instances where oulations that were very large underwent drastic declines, reaching the verge of etinction. If etinction is eventually avoided, these eisodes are called "oulation bottlenecks". Evolutionary biologists believe that the human oulation eerienced a oulation bottleneck some 60,000-70,000 years ago, a rather short time in evolutionary terms, ossibly due to the erution of the Toba suer-volcano in Indonesia (see Ambrose 1998, Hawks et. al. 2000, and Dawkins 2004). This may elain the fact that all human males can trace their ancestry back to a single male, the so-called Y-chromosomal Adam that lived around 60,000 to 90,000 years ago (Dawkins 2004). Clearly, in a near-etinction situation maimization of (HDF) becomes a very imortant evolutionary objective function: the family lines that survived the bottleneck are likely those with genes "rogrammed" to maimize the robability of having descendants forever (Cohen 1993). 10 This effect refers to the situation where a small grou becomes searated from the main oulation, and elains, for eamle, the very limited gene ool in Iceland, or the etraordinarily high ercentage of deaf individuals in Martha s Vineyard (Mayr 1959). 5

7 Your descendants would have been like the sand, your children like its numberless grains; their name would never be cut off nor destroyed from before me. Isaiah 48:19 This blessing is comosed of two arts: the first is the romise of many descendants (like the number of grains of sand); the second is the romise of Having Descendants Forever (their name would never be cut off). The two arts have different meaning, and both arts are imortant. To the best of our knowledge, the first to introduce the concet of HDF to economics was Meginniss (1977), who analyzed (HDF) in the framework of a constant birth robability er unit time and a constant death robability er unit time. Perhas surrisingly, in the long time that has assed since Meginniss s innovative work, the (HDF) criterion has not received much attention 11. The urose of this aer is to develo the concet of (HDF) in a general setting, and to eamine the imlications of this criterion for the evolutionary foundations of loss aversion. 3. Loss Aversion and Minimization of the Etinction Probability From an evolutionary standoint, the objective function is defined in terms of the number of offsring. Standard economics defines utility in terms of consumtion. Of course, there is a close relation between consumtion and the number of offsring the more resources at an individual s disosal, the more offsring s/he can raise. While the eact relation between the level of consumtion and the number of offsring is not obvious, here we adot the standard, and the most simle, assumtion that raising each offsring requires a certain level of consumtion, C, and hence the number of offsring is roortional to consumtion 12. Consider an individual with N eisting offsring, each of which has a robability of having his line of descendants eventually becoming etinct. The individual faces a robability of N that his line of descendants will become etinct (i.e. (HDF)=1- N ; as 11 A few ecetions are Lesourne (1977) and Rubin and Paul (1979). 12 This assumtion clearly does not hold in modern human societies, where the relationshi between wealth and number of offsring is often reverse. However, it is robably a reasonable aroimation for the era during which our references have evolved, and indeed, this is the assumtion made in most studies. For eamle, Sinn (2003) writes: it is assumed that the number of children a arent has is roortional to the amount of resources he commands. 6

8 before, we assume indeendence across offsring). Now, the following risky gamble resents itself: gain m additional offsring with robability m, m-1 additional offsring with robability m1,, lose k offsring with robability k. I.e. the gamble is: ( m,m; m1,m 1; 0, 0; 1, 1; k, k ). 13 From the ersective of minimizing the robability of etinction, when should such a gamble be acceted? Without the gamble the robability of etinction is robability of N m for having N+m offsring, a robability of m1. With the gamble, there is a for having N+m-1 offsring, etc., and therefore the etinction robability is: m N m N m1 N k m1 k. Thus, the gamble should be acceted if and only if: or: m N m N m1 N k N m1 k, (4) m m1 k 1. (5) m m1 k Equation (5) imlies that the decision about the gamble is indeendent of the number of eisting offsring, N. This result has a rofound imlication. Just as in the Prosect Theory framework a gamble is evaluated based on the change of wealth, indeendently of the eisting wealth, in the evolutionary framework we find the same result: gambles are evaluated based on the change in the number of offsring, indeendently of the eisting number. This suggests an evolutionary basis for the eerimentally revealed referenceoint behavior as described by Prosect Theory. We would like to go further, and relate the loss aversion coefficient,, to the evolutionary objective function. Note, however, that there is no analytical solution to the criterion for a general gamble as described in eq.(5). Hence, we will restrict ourselves to a more basic gamble which can be treated analytically. Namely, consider a gamble by which there is a robability of gaining one additional offsring, and a robability of 1- of losing one offsring. In this case, eq.(4) becomes: ( ) N 1 N 1 N 1, and 1 eq.(5) becomes ( 1 ) 1, or: 2 ( 1 ) 0, (6) which yields: 13 As the number of offsring can not be negative, we have a restriction k<n. 7

9 (7) Thus, the gamble should be acceted if and only if the success robability eceeds where is the robability of eventual etinction for each offsring. 1, 1 Note that for 1/2 the gamble is refused for any value of 0 1, i.e. if the gamble is fair (or worse) it is refused, in line with the notion of loss aversion. Eq.(7) imlies that the smaller is, the larger the robability required to make the gamble accetable. The intuition for this is as follows. An addition of one offsring increases (HDF) less than the reduction of one offsring diminishes it. This is why a fair gamble is not acceted, and why must be larger than 1/2 for the gamble to be acceted. If is n1 n1 close to 1, the difference between and is not very large, and therefore does not have to eceed 1/2 by much. However, if is small the above difference is (relatively) large, and must be large to comensate. In other words, when most offsring are likely to have descendants forever, an additional offsring does not contribute much to (HDF), but a reduction of one offsring does have a large effect on (HDF). Thus, a large is required to make the gamble accetable. How would the above gamble be evaluated in the Prosect Theory framework? Given that an additional offsring corresonds to an additional consumtion of C, and that a loss of an offsring corresonds to C, the value function (1) imlies that the gamble should be acceted if and only if: C ( 1 ) C 0, (8) or: 1. (9) 14 The eression on the r.h.s. of eq.(6) is an uward facing arabola, so the 's solving eq.(6) are those 2 between the two roots of the equation ( 1 ) 0. These roots are: ( 1 ) ( 2 1) 1 1, Thus, any in the relevant range 0<<1 that satisfies ( 1 ) / also satisfies eq.(6). Rearranging, we obtain:

10 the role of Comaring eqs.(7) and (9) reveals that and, we obtain: 1 1 in the evolutionary framework lays in the Prosect Theory framework. Simlifying the relation between (10) Hence, 1 lays the role of the loss-aversion arameter. The value of has been eerimentally and emirically estimated to be aroimately 2.2 (see Table I). In the net section we discuss the emirical estimates of the etinction robability that aear in the literature, and evaluate the theoretical relation between and develoed above. 4. Emirical Estimates of the Etinction Probability The robability of a human family line becoming etinct was discussed as early as 1845 by Bienaymé, who analyzed the etinction of noble families, and later by de Candolle (1873), Galton and Watson (1875), Lotka (1931), Steffenson (1933), and Kolmogorov (1931) (for an ecellent review of the literature on this subject see Albertsen (1995)). The most rigorous emirical analysis of the etinction robability was conducted by Keyfitz and Tyree (1967) and Keyfitz (1968) who estimate the etinction robability by emloying statistics on the robability i of having i children. They use these robabilities to find the etinction robability as the solution to the Galton -Watson (1875) equation 15. Keyfitz and Tyree (1967) and Keyfitz (1968) estimate the etinction robabilities for family lines in two develoing countries (Meico and Israel), and three develoed countries (U.S., Hungary, and Jaan). 16 The values they find are reorted in Table II. The develoing countries are resumably closer to the environment in which references 15 The Galton-Watson equation is given by: m where i is the robability of having i offsring and is the robability that the line of descendants eventually becomes etinct. This equation is a generalization of equations (2) and (3) emloyed to solve for the etinction robabilities in the secific eamles of Section This is the country categorization at the time corresonding to the samle eriod of the Keyfitz and Tyree study. See htt:// 2 m 9

11 have evolved over the long history of human evolution. The average etinction robability for the develoing countries is This value corresonds by the theoretical rediction of eq.(10) to a loss aversion 1 1 coefficient of This value is very close to the value of 2.20, which is the tyical estimate for the loss aversion coefficient obtained in the very different eerimental framework of Prosect Theory (see Table I). It is encouraging and erhas even surrising that we obtain values that are so similar from two comletely different and indeendent aroaches. (Please insert Table II about here) 5. Conclusion While most studies take references as eogenously given, an innovative strand of the economics literature suggests viewing references as the result of the rocess of evolution. In this literature the number of offsring is tyically assumed to be roortional to the level of consumtion, and thus there is a corresondence between consumtion gambles and gambles on the number of offsring. Most of the studies in this strand take the evolutionary objective function as the maimization of the eected number of offsring, or alternatively, as the maimization of the geometric mean growth rate of the entire oulation. While these objectives are aealing and intuitive, in this aer we suggest that careful consideration should also be given to another evolutionary objective function advocated by evolutionary biologists: minimizing the robability of etinction of one s line of descendants (or, equivalently, maimizing the robability of Having Descendants Forever, (HDF)). This objective catures elements that likely lay an imortant role in the evolutionary rocess, and are absent in the more standard objective functions. We show that maimizing the robability of Having Descendants Forever imlies loss aversion. The criterion for acceting/rejecting a reroduction gamble in the evolutionary framework is indeendent of the number of eisting offsring, which can elain why wealth (or consumtion) gambles are evaluated indeendently of current 17 For the develoed countries the average etinction robability is somewhat higher at (because in the develoed countries the average number of children is lower than in the develoing countries). This 1 value corresonds to a lower loss aversion coefficient of Note, however, that the environmental conditions in develoed countries are robably reresentative only of the last few decades, before which they likely resembled those in develoing countries. 10

12 wealth in the Prosect Theory framework. Moreover, we find a simle theoretical relationshi tying the equilibrium loss aversion arameter,, with the robability of 1 etinction of one's line of descendants, :. Emirical estimates of the family-line etinction robabilities in develoing countries yield an average value of This corresonds to a loss aversion 1 arameter of This value is very close to the eerimental and emirical estimates of, which are tyically in the range While various caveats obviously aly, it is quite surrising and encouraging to find such an agreement between these two a-riori comletely unrelated arameters, from two different scientific areas. These findings suggest an evolutionary elanation for the eerimentally and emirically observed loss aversion. The evolutionary origin of loss aversion may be further elored via several different aths. First, it may be ossible to identify the eact gene resonsible for loss aversion behavior, directly demonstrating that loss aversion is (at least artially) a genetically transferable trait. It may also be ossible to find secluded oulations that have develoed in different environmental conditions with different etinction robabilities, and to eamine whether the different robabilities also corresond to different average degrees of loss aversion across the different oulations. Finally, agent-based simulation studies of the evolution of a heterogeneous oulation may shed light on the time it would take for the equilibrium loss aversion arameter to arise as a result of the evolutionary rocess. 11

13 References Abdellaoui, M., Bleichordt, H., and L Haridon, O., A Tractable Method to Measure Utility and Loss Aversion Under Prosect Theory, Journal of Risk and Uncertainty, 36, Abdellaoui, M., Bleichordt, H., and Paraschiv, C Loss Aversion Under Prosect Theory: A Parameter-Free Measurement, Management Science, 53, Albertsen, K., The Etinction of Families, International Statistical Review, 63, Ambrose, S.H., Late Pleistocene Human Poulation Bottlenecks, Volcanic Winter, and Differentiation of Modern Humans, Journal of Human Evolution, 34, Barberis, N., and Huang, M., Mental Accounting, Loss Aversion, and Individual Stock Returns, Journal of Finance, 56, Benartzi, S., and Thaler, R., Myoic Loss Aversion and the Equity Premium Puzzle, Quarterly Journal of Economics, 110, Benartzi, S., and Thaler, R., Risk Aversion or Myoia? Choices in Reeated Gambles and Retirement Investments, Management Science, 45, Berkelaar, A., Kouwenberg, R., and Post, T., Otimal Portfolio Choice Under Loss Aversion, Review of Economics and Statistics, 86, Bienaymé, I.J., De la loi de Multilication et de la Durée des Familles. Soc. Philomath. Paris, Etraits, 5, Cesarini, D., Dawes, C.T., Johannesson, M., Lichtenstein, P., and Wallace, B., Genetic Variation in Preferences for Giving and Risk Taking, Quarterly Journal of Economics, 124, Cesarini, D., Johannesson, M., Lichtenstein, P., Sandewall, O., and Wallace, B., Genetic Variation in Financial Decision Making, Journal of Finance, 65, Cohen, D Fitness in Random Environments, in J. Yoshimura and C.W. Clark (Eds.), Adatation in Stochastic Environments. New York: Sringer-Verlag. Dawkins, R., The Selfish Gene. Oford: Oford University Press. Dawkins, R., The Ancestor's Tale, A Pilgrimage to the Dawn of Life. Boston: Houghton Mifflin Comany. de Candolle, A., Histoire des Sciences et des Savants Deuis Deu Siecles. Geneve. Genesove, D., and Mayer, C., Loss Aversion and Seller Behavior: Evidence from the Housing Market, Quarterly Journal of Economics, 116, Gneezy, U., and Potters, J., An Eeriment on Risk Taking and Evaluation Periods, Quarterly Journal of Economics, 112, Haigh, M.S., and List, J.A., Do Professional Investors Ehibit Myoic Loss Aversion? An Eerimental Analysis, Journal of Finance, 60, Hardie, B.G.S., Johnson, E.J., and Fader, P.S., Modeling Loss Aversion and Reference Deendence Effects on Brand Choice, Marketing Science, 12,

14 Hawks, J., Hunley, K., Lee, S-H., and Woloff, M., Poulation Bottlenecks and Pleistocene Human Evolution, Molecular Biology and Evolution, 17, Ho, T-H., and Zhang, J., Designing Price Contracts for Boundedly Rational Customers: Does the Framing of the Fied Fee Matter? Management Science, 54, Jervis, R., Political Imlications of Loss Aversion, Political Psychology, 13, Kahneman, D., Knetsch, J.L., and Thaler, R.H., Eerimental Tests of the Endowment Effect and the Coase Theorem, Journal of Political Economy, 98, Kahneman, D., and Tversky, A., Prosect Theory: An Analysis of Decision Under Risk, Econometrica, 47, Keyfitz, N, and Tyree, A., Comuterization of the Branching Process. Behavioral Science, 12, Kolmogorov, A., Zur Lösung Einer Biologischen Aufgabe. In Jahrbuch Uber die Fortschritte der Mathematik, 64, Kuhnen, C.M., and Chiao, J., Genetic Determinants of Financial Risk Taking. PLoS ONE, 4, 1-4. Lesourne, J., A Theory of the Individual for Economic Analysis, Amsterdam. Levy, M., Loss Aversion and the Price of Risk. Quantitative Finance, Lotka, A.J., The Etinction of Families. Journal of the Washington Academy of Sciences, 21, Mayr, E., Isolation as an Evolutionary Factor. Proceedings of the American Philosohical Society, 103, Meginniss, J.R Alternatives to the Eected Utility Rule. Unublished dissertation, University of Chicago. Pennings, J.M.E., and Smidts. A The Shae of Utility Functions and Organizational Behavior, Management Science, 49, Rabin, M., Risk Aversion and Eected Utility Theory: A Calibration Theorem. Econometrica, 68, Rabin, M., and Thaler, R., Anomalies: Risk Aversion. Journal of Economic Persectives, 15, Robson, A.J., A Biological Basis for Eected and Non-Eected Utility. Journal of Economic Theory, 68, Robson, A.J., and Samuelson, L., The Evolutionary Foundations of Preferences. In The Social Economics Handbook. Benhabib, J., Bisin, A., and Jackson, M., (Eds.). Elsevier Press. Rogers, A.R., Evolution of Time Preference by Natural Selection. American Economic Review, 84, Rubin, P.H., and Paul, C.W., An Evolutionary Model of Taste for Risk. Economic Inquiry, 17,

15 Sinn, H.W., Weber s Law and the Biological Evolution of Risk Preferences. Geneva Paers on Risk and Insurance Theory, 28, Steffensen, J.F., Deu Problèmes du Calcul des Probabilités. Ann. l Inst. Henri Poincaré, 3, Tom, S.M., Fo, C.R., Treel, C., and Poldrack, R.A., The Neural Basis of Loss Aversion in Decision-Making under Risk. Science, 315, Tversky, A. and Kahneman, D., Loss Aversion in Riskless Choice: A Reference Deendent Model, Quarterly Journal of Economics, 106, Tversky, A. and Kahneman, D Advances in Prosect Theory: Cumulative Reresentation of Uncertainty, Journal of Risk and Uncertainty, 5, Watson, H.W., and Galton, F., On the Probability of the Etinction of Families, Journal of the Anthroological Institute of Great Britain, 4, Zhong, S., Israel, S., Xue, H., Ebstein, R.P., and Chew, S.H., Monoamine Oidase A Gene (MAOA) Associated with Attitude Towards Longshot Risks. PLoS ONE, 4.

16 Table I Eerimental and emirical estimates of the loss aversion arameter,. All of the studies below are eerimental, ecet Levy (2009) which is based on emirical data of risk and return arameters in international caital markets. See Abdellaoui, Bleichordt, and Paraschiv (2007) for alternative definitions and methods of estimating loss aversion. λ Kahneman, Knettsch, and Thaler (1990) 2 + Tversky and Kahneman (1991) 2 + Kahneman and Tversky (1992) 2.25 Pennings and Smidts (2003) 1.81 Abdellaoui, Bleichordt, and L Haridon (2008) 2.61 Levy (2009) 2.3 Table II The robability of a erson's line of descendants eventually becoming etinct,, for different countries, as estimated by Keyfitz and Tyree (1967) and Keyfitz (1968). The average value for the develoing countries (shaded) is =0.46. Meico Israel USA Hungary Jaan