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612 A Digression: Derivation of Arrrow-Pratt Coef cient of Absolute Risk Aversion (CARA) This subsection considers a derivation of a coef cient of absolute risk aversion (CARA) If we know the functional form of the utility function U(X), we can compute this measure Most investors are risk averse in some sense Is there a formal measure of risk aversion related to the utility function Some readers may consider an answer to this question to be a digression and skip this subsection However, we remind the reader that CARA will be relevant in Section 635, which lays the groundwork for fourth-order stochastic dominance (4SD) Consider a random variable Z ~ N(0, 1) similar to a unit normal, such that E(Z) = 0 and variance is unity Thus Z represents a zero expectation gamble, which yields on an average no gain or loss and X represents current wealth.

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Figure 1.1 Venus of Willendorf, a Palaeolithic gurine carved out of ne-grained limestone, was found near Willendorf in the Wachau region of Lower Austria in 1908. It can be seen in the Natural History Museum, Vienna. Image reproduced courtesy of the Natural History Museum, Vienna.

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Note that Z can be negative and we are offering the choice between a certain payoff of X similar to choice (a) above and an uncertain payoff of X + Z, a fair gamble similar to choice (b) above The Risk Premium is the amount of side-payment p necessary to make the risk averse individual treat the fair gamble as equivalent to the certain outcome Thus p(X, Z) depends on the current wealth and the zero expectation random variable Z It satis es the following equality: E[U(X + Z)] + p = U(X) That is, the risk premium p is positive, if the following inequality is satis ed: U(X) - E[U(X + Z)] > 0 Since individuals are assumed to prefer larger income, U(X) is an increasing function of X, and we have U (X) > 0, denoting derivatives by primes.

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Thus the Fock matrix in the l basis is diagonalized by standard methods to yield the MO energies e and the matrix V from which the coe cient matrix c may be obtained by c uV. There are several ways in which the matrix u and its inverse may be determined. The most commonly used is the symmetric othogonalization due to Lowdin, which in volves diagonalization of the overlap matrix. We will not discuss this further. INTEGRALS Solution of the Fock equations requires integrals involving the basis functions, either in pairs or four at a time. Some of these we have already seen. The simplest are the overlap integrals, stored in the form of the overlap matrix S, whose elements are given by equation (A.48) as Sij wi 1 wj 1 dt1 The Fock integrals rst encountered in equation (A.45) are constructed from kinetic energy integrals, nuclear-electron attraction integrals, and two-electron repulsion integrals, as follows, continuing from equation (A.49): 4 5 4 5 NN n n Ne ZI e 2 q 2 2 ckb clb wj 1 dt1 1 wj 1 dt1 wi 1 Fij wi 1 2m r1I I 1 k 1 l 1 b 1 ! e2 e2 AX56 wi 1 wk 2 wl 2 wj 1 dt2 dt1 wi 1 wk 2 wj 2 wl 1 dt2 dt1 r12 r12

Note that any inequality remains valid if we multiply both sides by a positive quantity Hence we can divide both sides of the inequality by the positive quantity U (X) to yield the condition for positive risk premium to be U ( X ) E[U ( X + Z )] > U ( X ) U ( X ) (612).

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if(Time < 2300 and sellsToday = 0 and longLiqPoint <> EntryPrice and BarsSinceEntry >= 4) then SellShort( LongLiqRev ) next bar at longLiqPoint stop;

proper relation to the partition model entity, all needed services to represent mesh partitioning and support interpartition communications are easily supported. Given this the partition classi cation is de ned as the unique association of mesh topological entities of dimension di , Midi , to the topological entity of the partition model of

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