5.2 Risk Aversion and Portfolio Allocation; Risk Free vs. Risky Assets 5.3 Portfolio Composition, Risk Aversion and Wealth 5.4 Risk Aversion and Risky Portfolio Composition 5.5 Risk Aversion and Saving Behavior 5.6 Key Concepts and Results Asset Pricing Chapter V. Risk Aversion and Investment Decisions, Part I June 20, 2006 Asset Pricing
5.2 Risk Aversion and Portfolio Allocation; Risk Free vs. Risky Assets 5.3 Portfolio Composition, Risk Aversion and Wealth 5.4 Risk Aversion and Risky Portfolio Composition The Canonical Portfolio Problem 5.5 Risk Aversion and Saving Behavior 5.6 Key Concepts and Results The various problems considered in this chapter (and the next) EU ( ˜ Y 1 ) = max EU ( Y 0 ( 1 + r f ) + a (˜ max r − r f )) , (1) a Consider first an agent solving the following two period consumption-savings problem: s E { U ( Y 0 − s ) + δ U ( s ˜ max R ) } , (2) s.t. Y 0 ≥ s ≥ 0 { a , s } U ( Y 0 − s ) + δ EU ( s ( 1 + r f ) + a (˜ max r − r f )) , (3) Asset Pricing
5.2 Risk Aversion and Portfolio Allocation; Risk Free vs. Risky Assets 5.3 Portfolio Composition, Risk Aversion and Wealth 5.4 Risk Aversion and Risky Portfolio Composition The Canonical Portfolio Problem 5.5 Risk Aversion and Saving Behavior 5.6 Key Concepts and Results EU ( ˜ Y 1 ) = max EU ( Y 0 ( 1 + r f ) + a (˜ max r − r f )) , (4) a First order condition (FOC): U ′ ( Y 0 ( 1 + r f ) + a (˜ � r − r f )) (˜ � E r − r f ) = 0 (5) Theorem (Theorem 5.1:) ′ ( ) > 0 , and U ′′ ( ) < 0 and let ˆ Assume U a denote the solution to problem (1). Then ˆ 0 ⇔ E ˜ a r > r f > ˆ 0 ⇔ E ˜ a = r = r f ˆ 0 ⇔ E ˜ a r < r f < Asset Pricing
5.2 Risk Aversion and Portfolio Allocation; Risk Free vs. Risky Assets 5.3 Portfolio Composition, Risk Aversion and Wealth 5.4 Risk Aversion and Risky Portfolio Composition The Canonical Portfolio Problem 5.5 Risk Aversion and Saving Behavior 5.6 Key Concepts and Results Theorem (Proof of Theorem 5.1:) Define W ( a ) = E { U ( Y 0 ( 1 + r f ) + a (˜ r − r f )) } . The FOC (5) can then be written W ′ ( a ) = E [ U ′ ( Y 0 ( 1 + r f ) + a (˜ r − r f )) (˜ r − r f )] = 0 . By risk aversion ( U ′′ < 0 ) , � r − r f ) 2 � W ′′ ( a ) = E U ′′ ( Y 0 ( 1 + r f ) + a (˜ r − r f )) (˜ < 0 , that is, W ′ ( a ) is everywhere decreasing. It follows that ˆ a will be positive if and only if W ′ ( 0 ) = U ′ ( Y 0 ( 1 + r f )) E (˜ r − r f ) > 0 (since then a will have to be increased from the value of 0 to achieve equality in the FOC). Since U ′ is always strictly positive, this implies ˆ a > 0 if and only if E (˜ r − r f ) > 0 . � The other assertion follows similarly. Asset Pricing
5.2 Risk Aversion and Portfolio Allocation; Risk Free vs. Risky Assets 5.3 Portfolio Composition, Risk Aversion and Wealth 5.4 Risk Aversion and Risky Portfolio Composition The Canonical Portfolio Problem 5.5 Risk Aversion and Saving Behavior 5.6 Key Concepts and Results U ( Y ) = ln Y = − ( 1 + r f )[ E ˜ a r − r f ] ( r 1 − r f )( r 2 − r f ) > 0 . (6) Y 0 Asset Pricing
5.2 Risk Aversion and Portfolio Allocation; Risk Free vs. Risky Assets 5.3 Portfolio Composition, Risk Aversion and Wealth 5.4 Risk Aversion and Risky Portfolio Composition 5.5 Risk Aversion and Saving Behavior 5.6 Key Concepts and Results Theorem (5.2:) Suppose, for all wealth levels Y, R 1 A ( Y ) > R 2 A ( Y ) where R i A ( Y ) is the measure of absolute risk aversion of investor i , i = 1 , 2 . Then ˆ a 1 ( Y ) < ˆ a 2 ( Y ) Theorem ( 5.3:) Suppose, for all wealth levels Y > 0 , R 1 R ( Y ) > R 2 R ( Y ) where R i R ( Y ) is the measure of relative risk aversion of investor i , i = 1 , 2 . Then ˆ a 1 ( Y ) < ˆ a 2 ( Y ) . Asset Pricing
5.2 Risk Aversion and Portfolio Allocation; Risk Free vs. Risky Assets 5.3 Portfolio Composition, Risk Aversion and Wealth 5.4 Risk Aversion and Risky Portfolio Composition 5.5 Risk Aversion and Saving Behavior 5.6 Key Concepts and Results Theorem (5.4 (Arrow, 1971)) Let ˆ a = ˆ a ( Y 0 ) be the solution to problem (1) above; then: R ′ a ′ ( Y 0 ) > 0 0 ⇔ ˆ (i) A ( Y ) < R ′ a ′ ( Y 0 ) = 0 0 ⇔ ˆ (ii) A ( Y ) = R ′ a ′ ( Y 0 ) < 0 . 0 ⇔ ˆ (iii) A ( Y ) > Asset Pricing
5.2 Risk Aversion and Portfolio Allocation; Risk Free vs. Risky Assets 5.3 Portfolio Composition, Risk Aversion and Wealth 5.4 Risk Aversion and Risky Portfolio Composition 5.5 Risk Aversion and Saving Behavior 5.6 Key Concepts and Results a ) = d ˆ a / ˆ a dY / Y = Y d ˆ a η ( Y , ˆ ˆ a dY Theorem (5.5 (Arrow, 1971):) If, for all wealth levels Y, R ′ (i) R ( Y ) = 0 ( CRRA ) then η = 1 R ′ (ii) R ( Y ) < 0 ( DRRA ) then η > 1 R ′ (iii) R ( Y ) 0 ( IRRA ) then η < 1 > Asset Pricing
5.2 Risk Aversion and Portfolio Allocation; Risk Free vs. Risky Assets 5.3 Portfolio Composition, Risk Aversion and Wealth 5.4 Risk Aversion and Risky Portfolio Composition 5.5 Risk Aversion and Saving Behavior 5.6 Key Concepts and Results Theorem (5.6 (Cass and Stiglitz, 1970):) ˆ a 1 ( Y 0 ) . Let the vector denote the amount optimally . ˆ a J ( Y 0 ) invested in the J risky assets if the wealth level is Y 0 . ˆ a 1 ( Y 0 ) a 1 . . Then = f ( Y 0 ) . . ˆ a J ( Y 0 ) a J (for some arbitrary function f ( · ) ) if and only if either (i) U ′ ( Y 0 ) = ( θ Y 0 + κ ) ∆ or (ii) U ′ ( Y 0 ) = ξ e − vY 0 Asset Pricing
5.2 Risk Aversion and Portfolio Allocation; Risk Free vs. Risky Assets 5.3 Portfolio Composition, Risk Aversion and Wealth 5.4 Risk Aversion and Risky Portfolio Composition 5.5 Risk Aversion and Saving Behavior 5.6 Key Concepts and Results s E { U ( Y 0 − s ) + δ U ( s ˜ max R ) } , (7) s.t. Y 0 ≥ s ≥ 0 U ′ ( Y 0 − s ) = δ E { U ′ ( s ˜ R )˜ R } (8) Theorem (5.7 (Rothschild and Stiglitz,1971):) Let ˜ R A , ˜ R B be two return distributions with identical means such that ˜ R A SSD ˜ R B , and let s A and s B be, respectively, the savings out of Y 0 corresponding to the return distributions ˜ R A and ˜ R B . If R ′ R ( Y ) ≤ 0 and R R ( Y ) > 1, then s A < s B ; If R ′ R ( Y ) ≥ 0 and R R ( Y ) < 1, then s A > s B . Asset Pricing
5.2 Risk Aversion and Portfolio Allocation; Risk Free vs. Risky Assets 5.3 Portfolio Composition, Risk Aversion and Wealth 5.4 Risk Aversion and Risky Portfolio Composition 5.5 Risk Aversion and Saving Behavior 5.6 Key Concepts and Results P ( c ) = − U ′′′ ( c ) U ′′ ( c ) P ( c ) c = − cU ′′′ ( c ) U ′′ ( c ) Theorem (5.8) Let ˜ R A , ˜ R B be two return distributions such that ˜ R A SSD ˜ R B , and let s A and s B be, respectively, the savings out of Y 0 corresponding to the return distributions ˜ R A and ˜ R B . Then, s A ≥ s B iff c P ( c ) ≤ 2 , and conversely, s A < s B iff c P ( c ) > 2 Asset Pricing
5.2 Risk Aversion and Portfolio Allocation; Risk Free vs. Risky Assets 5.3 Portfolio Composition, Risk Aversion and Wealth 5.4 Risk Aversion and Risky Portfolio Composition 5.5 Risk Aversion and Saving Behavior 5.6 Key Concepts and Results { a , s } U ( Y 0 − s ) + δ EU ( s ( 1 + r f ) + a (˜ max r − r f )) , (9) Assume CRRA ( Y 0 − s ) − γ ( − 1 ) + δ E r − r f )] − γ ( 1 + r f ) � � [ s ( 1 + r f ) + a (˜ s : = 0 r − r f )) − γ (˜ � ( s ( 1 + r f ) + a (˜ � a : E r − r f ) = 0 Solution: a/s independent of s Asset Pricing
5.2 Risk Aversion and Portfolio Allocation; Risk Free vs. Risky Assets 5.3 Portfolio Composition, Risk Aversion and Wealth 5.4 Risk Aversion and Risky Portfolio Composition 5.5 Risk Aversion and Saving Behavior 5.6 Key Concepts and Results Increasing, decreasing, constant abosolute/relative risk aversion and their effects on portfolio composition ( risk free asset vs. Risky portfolio) Risk aversion and the composition of the optimal risky portfolio (Cass-Stiglitz) Prudence and Savings behavior Asset Pricing
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