Chapter V. Risk Aversion and Investment Decisions, Part I June 20, 2006
The Canonical Portfolio Problem The various problems considered in this chapter (and the next) max a EU(Ỹ1) = max EU (Y 0 (1 + r f ) + a ( r r f )), (1) Consider first an agent solving the following two period consumption-savings problem: maxe{u(y 0 s) + δu(s R)}, s s.t. Y 0 s 0 (2) max U(Y 0 s) + δeu(s(1 + r f ) + a( r r f )), (3) {a,s}
The Canonical Portfolio Problem max a EU(Ỹ1) = max EU (Y 0 (1 + r f ) + a ( r r f )), (4) First order condition (FOC): E [ U (Y 0 (1 + r f ) + a ( r r f )) ( r r f ) ] = 0 (5) Theorem (Theorem 5.1:) Assume U ( ) > 0, and U ( ) < 0 and let â denote the solution to problem (1). Then â > 0 E r > r f â = 0 E r = r f â < 0 E r < r f
The Canonical Portfolio Problem 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), W (a) = E U (Y 0 (1 + r f ) + a ( r r f )) ( r r f ) 2] < 0, that is, W (a) is everywhere decreasing. It follows that â 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 â > 0 if and only if E ( r r f ) > 0. The other assertion follows similarly.
The Canonical Portfolio Problem U(Y ) = ln Y a = (1 + r f )[E r r f ] > 0. (6) Y 0 (r 1 r f )(r 2 r f )
Theorem (5.2:) Suppose, for all wealth levels Y, R 1 A (Y ) > R2 A (Y ) where Ri A (Y ) is the measure of absolute risk aversion of investor i, i = 1, 2. Then â 1 (Y ) < â 2 (Y ) Theorem ( 5.3:) Suppose, for all wealth levels Y > 0, RR 1 (Y ) > R2 R (Y ) where RR i (Y ) is the measure of relative risk aversion of investor i, i = 1, 2. Then â 1 (Y ) < â 2 (Y ).
Theorem (5.4 (Arrow, 1971)) Let â = â (Y 0 ) be the solution to problem (1) above; then: (i) R A (Y ) < 0 â (Y 0 ) > 0 (ii) R A (Y ) = 0 â (Y 0 ) = 0 (iii) R A (Y ) > 0 â (Y 0 ) < 0.
η(y, â) = dâ/â dy /Y = Y dâ â dy Theorem (5.5 (Arrow, 1971):) If, for all wealth levels Y, (i) R R (Y ) = 0 (CRRA) then η = 1 (ii) R R (Y ) < 0 (DRRA) then η > 1 (iii) R R (Y ) > 0 (IRRA) then η < 1
Theorem (5.6 (Cass and Stiglitz, 1970):) â 1 (Y 0 ) Let the vector.. denote the amount optimally â J (Y 0 ) invested in the J risky assets if the wealth level is Y 0. â 1 (Y 0 ) a 1 Then.. =.. f (Y 0) â J (Y 0 ) a J (for some arbitrary function f ( )) if and only if either (i) U (Y 0 ) = (θy 0 + κ) or (ii) U (Y 0 ) = ξe vy 0
maxe{u(y 0 s) + δu(s R)}, s s.t. Y 0 s 0 (7) U (Y 0 s) = δe{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.
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, and conversely, s A s B iff c P(c) 2, s A < s B iff c P(c) > 2
Assume CRRA max U(Y 0 s) + δeu(s(1 + r f ) + a( r r f )), (9) {a,s} s : (Y 0 s) γ ( 1) + δe ( [s(1 + r f ) + a( r r f )] γ (1 + r f ) ) = 0 a : E [ (s(1 + r f ) + a( r r f )) γ ( r r f ) ] = 0 Solution: a/s independent of s
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