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Penn STAT 956 - SAMUELSONS FALLACY OF LARGE NUMBERS AND OPTIONAL STOPPING

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Cover PageArticle Contentsp. 1p. 2p. 3p. 4p. 5p. 6p. 7Issue Table of ContentsJournal of Risk and Insurance, Vol. 69, No. 1, Mar., 2002Front Matter [pp. i - vi]Samuelson's Fallacy of Large Numbers and Optional Stopping [pp. 1 - 7]Moral Hazard, Basis Risk, and Gap Insurance [pp. 9 - 24]Pricing Default-Risky CAT Bonds with Moral Hazard and Basis Risk [pp. 25 - 44]Insurance Contracts and Securitization [pp. 45 - 62]Life Insurance Liabilities at Market Value: An Analysis of Insolvency Risk, Bonus Policy, and Regulatory Intervention Rules in a Barrier Option Framework [pp. 63 - 91]Recent Court Decisions [pp. 93 - 110]Book Reviewsuntitled [pp. 111 - 112]untitled [pp. 112 - 113]untitled [pp. 113 - 114]untitled [pp. 114 - 115]untitled [pp. 115 - 118]Back Matter?The Journal of Risk and Insurance, 2002, Vol. 69, No. 1, 1-7 SAMUELSON'S FALLACY OF LARGE NUMBERS AND OPTIONAL STOPPING Erol A. Pekoz ABSTRACT Accepting a sequence of independent positive mean bets that are individually unacceptable is what Samuelson called a fallacy of large numbers. Recently, utility functions were characterized where this occurs rationally, and exam- ples were given of utility functions where any finite number of good bets should never be accepted.1 Here the author shows how things change if you are allowed the option to quit early: Subject to some mild conditions, you should essentially always accept a sufficiently long finite sequence of good bets. Interestingly, the strategy of quitting when you get ahead does not per- form well, but quitting when you get behind does. This sheds some light on more possible behavioral reasons for Samuelson's fallacy, as well as strategies for handling a series of sequentially observed good investments. INTRODUCTION AND OVERVIEW Samuelson (1963) told a story in which he offered a colleague a better than 50-50 chance of winning $200 or losing $100. The colleague rejected the bet, but said he would be willing to accept a string of 100 such bets. Samuelson argued that the colleague was irrationally applying the law of averages to a sum, and this perhaps has led to a more widely held perception that accepting a sequence of good bets when a single one would be rejected is a "fallacy of large numbers." Since then a number of authors have studied this phenomenon. Samuelson (1989) gave examples of utility functions where a single bet is unacceptable but a sufficiently long finite sequence of good bets will be accepted. Also given were utility functions where a long sequence of good bets is never acceptable: Consider the utility function U(x) = -2-x and bets giving a 50 percent chance of losing $1 or winning $(1 + E), for a sufficiently small e > 0. It can be shown that expected utility decreases with each additional bet made, even though the bets are favorable and the utility function is increasing. Pratt and Zeckhauser (1987) studied the related property they labeled "proper risk aversion," where investors unwilling to make a single bet will also be unwilling to make more than one independent bet of the same type. Erol Pekoz is associate professor in the School of Management at Boston University. Thanks are due for helpful comments from Zvi Bodie, Stephen Ross, Paul Samuelson, and an anonymous referee. i Ross, 1999.2 THE JOURNAL OF RISK AND INSURANCE Nielsen (1985) found necessary and sufficient conditions for a gambler with a concave utility function to eventually accept a sequence of bounded good bets, and Lippman and Mamer (1988) extended this to unbounded, identically distributed bets. Recently Ross (1999) extended this to independent but nonidentically distributed bets. The essential idea given in Lippman and Mamer (1988) and Ross (1999) is that if the utility function decreases faster than exponentially in the negative direction, the small risk of a loss can be magnified enough to overwhelm the benefits of a gain even for arbitrarily long sequences of good bets. Gollier (1996) gave some related results on how the availability of future optional bets can increase the attractiveness of a current bet, but the eventual attractiveness of a sufficiently large number of optional good bets has not been directly studied. See Bodie (1995) for a discussion of related phenomena surrounding long-term stock market investing. Here the author shows what happens when the gambler has the option to quit early: A sufficiently long sequence of good bets should always be accepted, meaning that given a sequence of positive mean bets, for sufficiently large n you should always agree to sequentially make the first n of them with the option to quit early. This holds provided the gambler's utility function is not bounded from above, the expected utility of a single bet is finite, and a condition on the bet means and variances holds. This result does not hold in the setting of Ross (1999) without a stopping option. There the total number of bets to be made is viewed as fixed in advance, while in the setting herein it is viewed as variable up to some maximum number that is fixed in advance. Note that the gambler is not allowed to play as long as it takes to get ahead, but is only allowed a maximum of n bets, which must be fixed in advance. It is interesting to note that the strategy of quitting when you reach some large wealth level does not perform well. This is because even with arbitrarily long sequences of good bets, there can always be some small chance that the game ends with a very large loss, and a utility function can always be found that magnifies this loss more than enough to make the game unacceptable. One uses the strategy of quitting whenever the gambler's wealth goes below the starting wealth, and for sufficiently long sequences of good bets, the benefit of large gains always eventually overwhelms the risk of losses. As a final note, much controversy exists over Samuelson's fallacy and the behavioral issues surrounding it. Benartzi and Thaler (1999), for example, used it as an example illustrating the limitations of expected utility theory in explaining behavior. This article shows that if faced with the opportunity to play a long sequence of favorable bets with the option to quit early, accepting the game is rational from an expected utility point of view. The organization of the article is as follows. The next section contains the main result. The "Multiplicative Gambles" section has an analogous result for multiplicative payoffs. The "Summary"


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