Stochastic Dominance Investment Decision Making under Uncertainty
Stochastic Dominance is devoted to investment decision-making under uncertainty. The book covers three basic approaches to this process: The stochastic dominance approach; the mean-variance approach; and the non-expected utility approach, focusing on pros
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Stochastic Dominance, Second Edition by Haim Levy Myles Robinson Professor of Finance The Hebrew University of Jerusalem Previously published books in the series: Viscusi, W. Kip: STUDIES IN RISK AND UNCERTAINTY Luken, R.: ENVIRONMENTAL REGULATION: TECHNOLOGY, AMBIENT AND BENEFITS-BASED APPROACHES Shubik, M.: RISK, ORGANIZATIONS AND SOCIETY Edwards, W.: UTILITY THEORIES: MEASUREMENTS AND APPLICATIONS Martin, W.: ENVIRONMENTAL ECONOMICS AND THE MINING INDUSTRY Kunreuther, H. and Easterling, D.: THE DILEMMA OF A SITING OF HIGH-LEVEL NUCLEAR WASTE REPOSITORY Christe, N.G.S. and Soguel, N.C.: CONTINGENT VALUATION, TRANSPORT SAFETY Battigalli, P., Montesano, A. and Panunzi, F.: DECISIONS, GAMES AND MARKETS Freeman, P. and Kunreuther, H.: MANAGING ENVIRONMENTAL RISK THROUGH INSURANCE Kopp, R.J., Pommerehne, W. W. and Schwartz, N.: DETERMINING THE VALUE OF NON-MARKETED GOODS Bishop, R.C. and Romano, D.: ENVIRONMENTAL RESOURCE VALUATION: APPLICATIONS OF THE CONTINGENT VALUATION METHOD IN ITALY Jeanrenaud, C ; Soguel, N.C.: VALUING THE COST OF SMOKING Eeckhoudt, L.: RISK AND MEDICAL DECISION MAKING Viscusi, W. Kip: THE RISKS OF TERRORISM
STOCHASTIC DOMINANCE Investment Decision Malka} 0. Select k a = (|LI - d) to obtain: 2
pjx-^|>(tx-d)}< ^ (^-dr and, a fortiori: a' (H-d)2 where jLi is the expected value and a is the standard deviation (a is the square root of a^) of the distribution of returns as defined above. p{(^i-x)>M}=P(x- L2 , L2 >- L3 . Then, by the transitivity axiom, Li >- L3. Similarly, if Li ~ L2 and L2 ~ L3 then, by this axiom , Li ~ L3. Axiom 5: Decomposability. A complex lottery is one in which the prizes are lotteries themselves. A simple lottery has monetary values Ai, A2 etc. as prizes. Suppose that there is a complex lottery L such that: L*=(qL,,(l-q)L2) where Li and L2 themselves are (simple) lotteries. Li and L2 are given by: U- {Pi A i , ( l - p i ) A2} L2= {P2A1,(1-P2)A2} Then, by this axiom, the complex lottery L can be decomposed into a simple lottery L having only A] and A2 as prizes. To be more specific: L * ~ L i = {p*Ai, (l-p*)A2} where
p = qpi + (1-q) P2
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STOCHASTIC DOMINANCE Axiom 6: Monotonicity If there is certainty, then the monotonicity axiom determines that if A2 > A] then A2 >^Ai. If there is an uncertainty, the monotonicity axiom can be formulated in two alternate ways. First: Let and then Second: Let and
Li = {pA,,(l-p)A2}, L2 = {p Ai, (1-p) A3}. If A3 > A 2 , hence A3 >- Ai L2 )- Li .
L,= {pA,,(l-p)A2}, L2 = {q Ai, (1-q) A2}, and A2 > A] (hence A2 >- Ai). If p(l-q)]
then L] >- L2.
Each of these six axioms can be accepted or rejected. However, if they are accepted, then we can prove that the MEUC should be used to choose among alternative investments. Any other investment criterion will simply be inappropriate and may lead to a wrong investment decision. We turn next to this proof. c) Proof that the Maximum Expected Utility Criterion (MEUC) is Optimal Decision Rule Theorem 2.1: The MEUC. The optimum criterion for ranking alternative investments is the expected utility of
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