fin5c - Portfolio Tools 2"A bird in the hand is worth two...

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Econ 333 Spring 08 1 Portfolio Tools 2 “A bird in the hand is worth two in the bush” Danthine and Donaldson, p. 83
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Econ 333 Spring 08 2 Portfolio Tools 2: A brief revue of concepts In individual’s attitude toward risk can be measured in two ways: The Arrow-Pratt measure of absolute risk aversion: Ra (w) = - U’’(w) / U’(w) The Arrow-Pratt measure of relative risk-aversion: Rr (w) = - w U’’(w) / U’(w) The sign of these measures immediately tells us the basic attitude toward risk For a risk-averse individual U’’< 0 (strictly concave), so Ra > 0 and Rr > 0 are positive For risk-neutral individual U’’= 0 (linear), so Ra = Rr =0 For risk-lovers U’’> 0 (strictly convex), so Ra <0 and Rr <0. Ra (w) is clearly a local measure of risk aversion, i.e. it need not be the same at every level of wealth The Von Newman Morgensten (VNM) utility of an individual is said to display: Constant Absolute Risk Aversion (CARA) if Ra (w) remains constant with an increase in wealth Decreasing Absolute Risk Aversion (DARA) if Ra (w) decreases with an increase in wealth Increasing Absolute Risk Aversion (IARA) if Ra (w) increases with an increase in wealth
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Econ 333 Spring 08 3 Portfolio Tools 2: the basic portfolio problem In this section we investigate investors’ demand for assets of different classes, specially their demand for risk-free vs risky assets. Consider an investor with wealth Y0, deciding what amount, a, to invest in a risky portfolio with uncertain return r . The alternative is to invest in a risk-free asset that pays a certain rate rf. The investor wealth at the end of the second period is: Letting U(.) be the utility of money and E the expectation operator, the choice problem can be expressed as With risk-averse individuals, i.e. U’’(.)<0, the necessary and sufficient FOC is: ) ~ ( ) 1 ( ) ~ 1 ( ) )( 1 ( ~ 0 0 1 f f f r r a r Y r a a Y r Y + + = + + + = [ ] ) ~ ( ) 1 ( max ) ~ ( max 0 1 rf r a r Y EU Y EU f a + + = [ ] 0 ) ~ ))( ~ ( ) 1 ( ( ' 0 = + + f f f r r r r a r Y U E
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Econ 333 Spring 08 4 Portfolio Tools 2: the basic portfolio problem Result (theorem): Assume U’(.)>0, U’’(.)<0 and let denote the solution to the preceding maximization program. Then: Proof: (in class)
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This note was uploaded on 04/01/2008 for the course ECON 3330 taught by Professor Mbiekop during the Spring '08 term at Cornell.

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fin5c - Portfolio Tools 2"A bird in the hand is worth two...

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