risk_premia

risk_premia - 1 Brigham Young University Department of...

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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 one-period 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 first-order 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 first-order 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.

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risk_premia - 1 Brigham Young University Department of...

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