Utility and Portfolio Choice

They are the solutions to recall uw euw 2 euw

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Unformatted text preview: pay an insurance risk premium • These two premiums are closely related but not identical. They are the solutions to (recall: u(W ) > Eu(W + ε)):2 E[u(W + πc + ε)] = u(W ) E[u(W + ε)] = u(W − πi ) (risk premium) (insurance premium; certainty equivalent) RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 10/23/2012. © P. KOLM. 7 When the utility function is sufficiently smooth and the risk is small the risk premium can be determined approximately. By expanding each side of E[u(W + ε)] = u(W − πi ) in Taylor series, for some α and β we have E [u(W ) + εu ′(W ) + 1 ε 2u ′′(W ) + 1 ε 3u ′′′(W + αε)] = u(W ) − πiu ′(W ) + 1 πi2u ′′(W − βπi ) 2 6 2 Canceling terms on each side, we get3 1 2 var(ε)u ′′(W ) ≈ −πiu ′(W ) so 1 ⎢⎡ u ′′(W ) ⎤⎥ πi ≈ − var(ε) ⎢ u ′(W ) ⎥ 2⎣ ⎦ RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 10/23/2012. © P. KOLM. 8 1 ⎢⎡ u ′′(W ) ⎤⎥ πi ≈ − var(ε) ⎢ u ′(W ) ⎥ 2⎣ ⎦ The term in brackets is an appropriate measure of infinitesimal or local risk aversion • rA (W ) = − u ′′(W ) is cal...
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