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Unformatted text preview: 1 Brigham Young University Department of Economics Economics 459  International Monetary Theory Derivation of the Risk Premium in Covered vs Uncovered Investments Let the oneperiod investor maximize utility from consuming using next period’s wealth. Let utility take the following form: C C U ln ) ( = . Let the returns on all potential assets be distributed normally, i.e. ) , ( ~ 2 i i i N r σ μ . The investor starts off with initial wealth of S . Hence consumption next period is ∑ = + = I i i i r w S C 1 ) 1 ( Which we can rewrite as follows: ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = + = ∑ ∑ = = I i i i I i i i w r S r w S C 1 1 1 ) 1 ( The investor’s problem is: 1 . . )}; ( { 1 = = ∑ = I i i w w t s C U E U Max i With these this utility function and normal distributions for the returns, we can rewrite a utility maximization problem as follows: 1 . . }; {ln } {ln 1 2 = − = ∑ = I i i w w t s C V C E U Max i γ ; ∑ = + = I i i i r w S C 1 ln ln Taking the appropriate expected values of ln C gives: ∑ = = I i i i w C E 1 } {ln μ ∑∑ = = = I i I j ij j i w w C V 1 1 } {ln σ ; ( 2 i ii σ σ = ) Hence, the Lagrangian for this problem is given by: ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = ∑ ∑∑ ∑ = = = = 1 1 1 1 2 1 I i i I i I j ij j i I i i i w w w w L λ σ μ γ A generic firstorder condition (with respect to element n ) is: ∑ = = − − I i in i n w 1 λ σ γ μ Note given the construction of ln C , the second term is the covariance of asset n with ln C . Suppose the return on asset number S is riskless. Then all the covariance terms ( σ iS ) will be zero. And we get λ μ = S . 2 Substituting this back into the generic firstorder condition gives: } ln , { 1 C r Cov w n I i in i S n γ σ γ μ μ = = − ∑ = (1) The above is the “excess return” or “absolute risk premium”, i.e the additional return The above is the “excess return” or “absolute risk premium”, i....
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This note was uploaded on 02/29/2012 for the course ECON 459 taught by Professor Phillips,k during the Winter '08 term at BYU.
 Winter '08
 Phillips,K
 Utility

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