(Ir)rational Exuberance: Optimism, Ambiguity and Risk Anat Bracha and Don Brown Boston FRB and Yale University October 2013 (Revised) Anat Bracha and Don Brown (Boston FRB and Yale University) (Ir)rational Exuberance October 2013 (Revised) 1 / 28
Abstract We propose a rational model of (ir)rational exuberance in asset markets. That is, the behavior of bulls and bears is rational in the standard economic sense of agents maximizing utility subject to a budget constraint, de…ned by market prices and the agent’s income. As observed by Keynes (1930): “The market price will be …xed at the point at which the sales of the bears and the purchases of the bulls are balanced.” This equilibration of optimistic and pessimistic beliefs of investors is a consequence of investors maximizing Keynesian utilities subject to budget constraints de…ned by market prices and the investor’s income. Anat Bracha and Don Brown (Boston FRB and Yale University) (Ir)rational Exuberance October 2013 (Revised) 2 / 28
Keynesian Utilities Keynesian utilities represent the investor’s preferences for optimism. Bulls are optimistic and believe that market prices will go up, but bears are pessimistic and believe that market prices will go down. Hence bulls buy long and bears sell short. Keynesian utilities are de…ned as the composition of the investor’s preferences for risk and her preferences for ambiguity, where we assume preferences for risk and preferences for ambiguity are independent. If U ( x ) denotes preferences for risk, then U maps state-contingent claims x to state-utility vectors y = U ( x ) . If J ( y ) denotes preferences for ambiguity, then J maps state-utility vectors U ( x ) to subjective values J � U ( x ) x ! J � U ( x ) is the composition of U and J , denoted J � U ( x ) . Anat Bracha and Don Brown (Boston FRB and Yale University) (Ir)rational Exuberance October 2013 (Revised) 3 / 28
Types of Keynesian Utilities In the following 2 � 2 contingency table on the types of Keynesian utilities, the rows are ambiguity-averse and ambiguity-seeking preferences and the columns are risk-averse and risk-seeking preferences. The cells are the investor’s preferences for optimism and pessimism. The diagonal cells of the table are the symmetric Keynesian utilities and the o¤-diagonal cells of the table are the asymmetric Keynesian utilities. Bears are pessimistic and have concave Keynesian utilities. Bulls are optimistic and have convex Keynesian utilities Table 1 Keynesian Preferences Risk-Averse Risk-Seeking Ambiguity-Averse Bears Asymmetric Ambiguity-Seeking Asymmetric Bulls Anat Bracha and Don Brown (Boston FRB and Yale University) (Ir)rational Exuberance October 2013 (Revised) 4 / 28
Legendre–Fenchel Conjugates of Keynesian Utilities . For pessimistic utility functions, we invoke the Legendre–Fenchel biconjugate for concave functions, where [ ∑ π � x + J � ( π )] J � U ( x ) � min π 2 R N ++ and J � ( π ) is a smooth concave function on R N ++ , the Legendre–Fenchel conjugate of J � U ( x ) , where J � ( π ) � min [ ∑ π � x + J � U ( x )] x 2 R N + For optimistic utility functions, we invoke the Legendre–Fenchel biconjugate for convex functions, where [ ∑ π � x + J � ( π )] J � U ( x ) � max π 2 R N ++ and J � ( π ) is a smooth convex function on R N ++ , the Legendre–Fenchel conjugate of J � U ( x ) , where J � ( π ) � max [ ∑ π � x + J � U ( x )] . xR N + Anat Bracha and Don Brown (Boston FRB and Yale University) (Ir)rational Exuberance October 2013 (Revised) 5 / 28
Monotone Maps and Convex (Concave) Utilities If F ( y ) is a vector-valued map from R N into R N , then F is strictly, monotone increasing (decreasing) if for all x and y 2 R N : [ x � y ] � [ F ( x ) � F ( y )] > 0 ( < 0 ) J � U ( x ) is strictly convex (concave) in x i¤ r x J � U ( x ) is a strictly, monotone increasing (decreasing) map of x . Anat Bracha and Don Brown (Boston FRB and Yale University) (Ir)rational Exuberance October 2013 (Revised) 6 / 28
Betting Odds for Bears and Bulls It follows from the envelope theorem, that for bears [ ∑ π � x + J � ( π )] = b r x J � U ( x ) = π , where arg max π 2 R N ++ [ ∑ π � x + J � ( π )] = ∑ b π � x + J � ( b J � U ( x ) = π )] max π 2 R N ++ and for bulls [ ∑ π � x + J � ( π )] = b r x J � U ( x ) = arg min π , where π 2 R N ++ [ ∑ π � x + J � ( π )] = ∑ b π � x + J � ( b J � U ( x ) = min π )] . π 2 R N ++ The expectations of investors today regarding the payo¤s of the state-contingent claim x tomorrow is the normalized marginal subjective value of x : b r x J � U ( x ) π 2 ∆ 0 , = kr x J � U ( x ) k 1 k b π k 1 Anat Bracha and Don Brown (Boston FRB and Yale University) (Ir)rational Exuberance October 2013 (Revised) 7 / 28
A Two Parameter Family of Keynesian Utilities Let u ( x s ) � x β s If β � 1, then u ( x s ) is concave in x s . If α � 1 , then j ( u ( x s )) � ( u ( x s )) α is concave in u ( x s ) . Hence j � u ( x s ) � ( x s ) βα is concave in x s , i.e., j � u ( x s ) is pessimistic If β � 1, then u ( x s ) is convex in x s .If α � 1 , then j ( u ( x s )) � ( u ( x s )) α is convex in u ( x s ) . Hence j � u ( x s ) � ( x s ) βα is convex in x s i.e., j � u ( x s ) is optimistic. Consider the following additively separable utility functions on the space of state-contingent claims x � ( x 1 , x 2 , ..., x N ) , where U ( x ) � ( u ( x 1 ) , u ( x 2 ) , ..., u ( x N )) : s = N j � u ( x s ) where j � u ( x s ) � ( x s ) βα ∑ J � U ( x ) � s = 1 Anat Bracha and Don Brown (Boston FRB and Yale University) (Ir)rational Exuberance October 2013 (Revised) 8 / 28
A Quadratic Family of Keynesian Utilities We propose quadratic speci…cations of preferences for risk and preferences for ambiguity, de…ned by scalar proxies for risk and ambiguity: β and α . Concave quadratic utility functions were introduced by Shannon and Zame (2002) in their analysis of indeterminacy in in…nite dimension general equilibrium models. f ( x ) is a concave quadratic function if for all y and z : f ( y ) < f ( z ) + r f ( z ) � ( y � z ) � 1 2 K k y � z k 2 , where K > 0 . Anat Bracha and Don Brown (Boston FRB and Yale University) (Ir)rational Exuberance October 2013 (Revised) 9 / 28
Concave Quadratic Utilities for Risk and Ambiguity J � U ( x ) is the composition of a smooth, concave quadratic map U ( x ) , where U ( x ) is a negative de…nite diagonal N � N matrix for each x 2 R N ++ and a smooth, concave quadratic function J ( y ) , where J : R N ! R . If u : R + ! R + , then U ( x ) � ( u ( x 1 ) , u ( x 2 ) , ..., u ( x N )) is the state-utility vector for the state-contingent claim x = ( x 1 , x 2 , ..., x N ) . Anat Bracha and Don Brown (Boston FRB and Yale University) (Ir)rational Exuberance October 2013 (Revised) 10 / 28
Gradients of Composite Functions as Hadamard Products If z = [ z 1 , z 2 , ..., z N ] and w = [ w 1 , w 2 , ..., w N ] , then z � w � [ z 1 w 1 , z 2 w 2 , ..., z N w N ] is the Hadamard or pointwise product of z and w . If we de…ne the gradient of state-utility vector U ( x ) as the vector r x U ( x ) � [ ∂ u ( x 1 ) , ∂ u ( x 2 ) , ..., ∂ u ( x N )] then by the chain rule r x J � U ( x ) = [ r x U ( x )] � [ r U ( x ) J ( U ( x ))] . If G ( x ) = z ( x ) � w ( x ) , where z ( x ) and w ( x ) 2 R N ++ , then Bentler and Lee (1978) state and Magnus and Neudecker (1985) prove that r x G ( x ) = r x z ( x ) diag ( w ( x )) + r x w ( x ) diag ( z ( x )) . Anat Bracha and Don Brown (Boston FRB and Yale University) (Ir)rational Exuberance October 2013 (Revised) 11 / 28
The Hessian for Keynesian Utilities r 2 x J � U ( x ) = r x ([ r x U ( x )] � [ r U ( x ) J ( U ( x )])) U ( x ) J ( U ( x ))]( diag [ r x U ( x )]) 2 + [ r 2 = [ r 2 x U ( x )] diag [ r U ( x ) J ( U ( x ))] . If U ( x ) is a concave quadratic map and J ( y ) is a convex quadratic function, then r 2 x U ( x ) = � diag ( β ) < 0 r 2 y J ( y ) = diag ( α ) > 0 . If A and B are diagonal N � N matrices then A � B is negative semide…nite i¤ E � F . Hence r 2 x J � U ( x ) is negative semide…nite i¤: diag ( α ) diag [ r x U ( x )] 2 � diag ( β ) diag [ r U ( x ) J ( U ( x ))] � 0 . Anat Bracha and Don Brown (Boston FRB and Yale University) (Ir)rational Exuberance October 2013 (Revised) 12 / 28
Keynesian Utilities for Bulls Theorem If J � U ( x ) , is the composition of U ( x ) and J ( y ) ,where (a) ( y 1 , y 2 , ..., y N ) � y = U ( x ) � ( u ( x 1 ) , u ( x 2 ) , ..., u ( x N )) is a monotone, smooth, convex, diagonal quadratic map from R N ++ onto R N ++ ,with the proxy for risk, β > 0 , (b) J ( y ) is a monotone, smooth, convex quadratic function from R N ++ into R , with the proxy for ambiguity, α > 0 , (c) x J � U ( x ) = diag ( α )( diag [ r x U ( x )]) 2 + diag ( β ) diag [ r U ( x ) J ( U ( x ))] r 2 then J � U ( x ) is convex on R N ++ . Anat Bracha and Don Brown (Boston FRB and Yale University) (Ir)rational Exuberance October 2013 (Revised) 13 / 28
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