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Risk Aversion 21 u(x) U(i) x-£ CEfl) X x+£ Fig. 1. Risk averse utility function The risk premium and the certainty equivalent of a gamble x are defined with respect to a specific utility function u . We now intend to put the risk premium in relation with the random variable x and the utility function u. To this end, we consider two particular cases: a gamble described by a random variable with additive noise and a gamble described by a random variable with multiplicative noise. Let x = x + i ; where € is a random variable with zero mean and variance (J'2 and therefore E[x] = x.
Starting from enough wealth, it is possible to obtain consumption plan c by means of a portfolio choice for the N assets. The N assets are said to be non redundant if their returns are linearly independent (the vector which describes the return of an asset cannot be written as a linear combination of the returns of the other N -1 assets). The property is verified if N :::; Sand rank(R) = N; when N = S this property is equivalent to the condition det(R) t- O. The N assets enjoy the uniqueness of representation property if Vc E I(R) there is a unique vector w E ~N such that Rw = c.
7) This condition implies the following risk premium implicit in the optimal portfolio choice: - ]- _ cov(u'(Wof*) , f n ) E[ r n rf E[u'(Wof*)]' n = 1, ... , N . 8) The denominator in the right side is positive, therefore the double implication of the proposition is easily verified. D. The risk premium of an asset is positive if the covariance of its return with the marginal utility of the wealth associated with the optimal portfolio of the agent is negative. Otherwise, the premium will be negative.