(Ir)rational Exuberance: Optimism, Ambiguity and Risk Anat Bracha - - PowerPoint PPT Presentation

(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 J U(x) min

π2R N

++

[∑ π x + J(π)] and J(π) is a smooth concave function on RN

++, the Legendre–Fenchel

conjugate of J U(x), where J(π) min

x2R N

+

[∑ π x + J U(x)] For optimistic utility functions, we invoke the Legendre–Fenchel biconjugate for convex functions, where J U(x) max

π2R N

++

[∑ π x + J(π)] and J(π) is a smooth convex function on RN

++, the Legendre–Fenchel

conjugate of J U(x), where J(π) max

xR N

+

[∑ π x + J U(x)].

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 RN into RN,then F is strictly, monotone increasing (decreasing) if for all x and y 2 RN: [x y] [F(x) F(y)] > 0 (< 0) J U(x) is strictly convex (concave) in x i¤ rxJ 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 rxJ U(x) = arg max

π2R N

++

[∑ π x + J(π)] = b π, where J U(x) = max

π2R N

++

[∑ π x + J(π)] = ∑ b π x + J(b π)] and for bulls rxJ U(x) = arg min

π2R N

++

[∑ π x + J(π)] = b π, where J U(x) = min

π2R N

++

[∑ π x + J(π)] = ∑ b π x + J(b π)]. The expectations of investors today regarding the payo¤s of the state-contingent claim x tomorrow is the normalized marginal subjective value of x : rxJ U(x) krxJ U(x)k1 = b π kb πk1 2 ∆0,

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(xs) x β

s If β 1, then u(xs) is concave in xs. If α 1, then

j(u(xs)) (u(xs))α is concave in u(xs). Hence j u(xs) (xs)βα is concave in xs, i.e., j u(xs) is pessimistic If β 1, then u(xs) is convex in xs.If α 1, then j(u(xs)) (u(xs))α is convex in u(xs). Hence j u(xs) (xs)βα is convex in xs i.e., j u(xs) is optimistic. Consider the following additively separable utility functions on the space of state-contingent claims x (x1, x2, ..., xN),where U(x) (u(x1), u(x2), ..., u(xN)) : J U(x)

s=N

∑

s=1

j u(xs) where j u(xs) (xs)βα

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) + rf (z) (y z) 1 2K ky zk2 , 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 RN

++ and a smooth, concave quadratic function J(y), where J : RN

! R. If u : R+ ! R+, then U(x) (u(x1), u(x2), ..., u(xN)) is the state-utility vector for the state-contingent claim x = (x1, x2, ..., xN).

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 = [z1, z2, ..., zN] and w = [w1, w2, ..., wN], then z w [z1w1, z2w2, ..., zNwN] 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 rxU(x) [∂u(x1), ∂u(x2), ..., ∂u(xN)] then by the chain rule rxJ U(x) = [rxU(x)] [rU(x)J(U(x))]. If G(x) = z(x) w(x), where z(x) and w(x) 2 RN

++, then Bentler and Lee (1978) state and

Magnus and Neudecker (1985) prove that rxG(x) = rxz(x)diag(w(x)) + rxw(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

r2

xJ U(x) = rx([rxU(x)] [rU(x)J(U(x)]))

= [r2

U(x)J(U(x))](diag[rxU(x)])2 + [r2 xU(x)]diag[rU(x)J(U(x))].

If U(x) is a concave quadratic map and J(y) is a convex quadratic function, then r2

xU(x) = diag(β) < 0

r2

yJ(y) = diag(α) > 0.

If A and B are diagonal N N matrices then A B is negative semide…nite i¤ E F. Hence r2

xJ U(x) is negative semide…nite i¤:

diag(α)diag[rxU(x)]2 diag(β)diag[rU(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) (y1, y2, ..., yN) y = U(x) (u(x1), u(x2), ..., u(xN)) is a monotone, smooth, convex, diagonal quadratic map from RN

++ onto RN ++ ,with the

proxy for risk, β > 0, (b) J(y) is a monotone, smooth, convex quadratic function from RN

++ into R,with the proxy for ambiguity, α > 0, (c)

r2

xJ U(x) = diag(α)(diag[rxU(x)])2 + diag(β)diag[rU(x)J(U(x))]

then J U(x) is convex on RN

++.

Anat Bracha and Don Brown (Boston FRB and Yale University) (Ir)rational Exuberance October 2013 (Revised) 13 / 28

The Optimal Investment Problem for Bulls

If the investor’s income today is I and she is endowed with convex Keynesian utilities, WBulls(x), then her optimal investment problem is (P): maxfWBulls(x) j x1 0, x2 0, p x I 0g where the Fritz John Lagrangian for constrained maximization L(x1, x2, λ0, λ1, λ2, λ3) λ0WBulls(x) λ1[x1] λ2[x2] λ3[p x I].

Theorem

[Fritz John ]: If xis a local maximizer of (P) then there exists multipliers λ (λ

0, λ 1, λ 2, λ 3) 0 such that:

λ

0(∂x1WBulls(x), ∂x2WBulls(x)) = (λ 1 + λ 3p1, λ 2 + λ 3p2),

where λ

0 = 1, by Theorems 19.12 in Simon and Blume.

Anat Bracha and Don Brown (Boston FRB and Yale University) (Ir)rational Exuberance October 2013 (Revised) 14 / 28

Betting Odds of Bulls and Market Odds

If x = (0, x

2 ), then

λ

0(∂x1WBulls((0, x 2 )), ∂x2WBulls((0, x 2 )) = (λ 1 + λ 3p1, λ 3p2)

It follows that some bulls are more optimistic than the market that tomorrow’s state of the world is state 2. That is,

∂x2WBulls((0,x

2 ))

∂x1WBulls((0,x

2 )) =

λ

3p2

λ

1+λ 3p1 > p2

p1 .

Anat Bracha and Don Brown (Boston FRB and Yale University) (Ir)rational Exuberance October 2013 (Revised) 15 / 28

Betting Odds of Bulls and Market Odds [Continued]

If x = (x

1 , 0), then

λ

0(∂x1WBulls((x 1 , 0)), ∂x2WBulls((x 1 , 0)) = (λ 3p1, λ 2 + λ 3p2)

It follows that the other bulls are more optimistic than the market that tomorrow’s state of the world is state 1:

∂x1WBulls((x

1 ,0))

∂x2WBulls((x

1 ,0)) =

λ

3p1

λ

2+λ 3p2 > p1

p2 .

Anat Bracha and Don Brown (Boston FRB and Yale University) (Ir)rational Exuberance October 2013 (Revised) 16 / 28

Keynesian Utilities for Bears

Theorem

If J U(x), is the composition of U(x) and J(y),where (a) (y1, y2, ..., yN) y = U(x) (u(x1), u(x2), ..., u(xN)) is a monotone, smooth, concave, diagonal quadratic map from RN

++ onto RN ++, with the

proxy for risk, β < 0, (b) J(y) is a monotone, smooth,concave quadratic function from RN

++ into R, with the proxy for ambiguity, α < 0, (c)

r2

xJ U(x) = diag(α)(diag[rxU(x)])2 diag(β)diag[rU(x)J(U(x))]

then J U(x) is concave on RN

++.

Anat Bracha and Don Brown (Boston FRB and Yale University) (Ir)rational Exuberance October 2013 (Revised) 17 / 28

The Optimal Investment Problem for Bears

If the investor’s income today is I and she is endowed with concave Keynesian utilities WBears(x), then her optimal investment problem is (P): maxfWBears(x) j x1 0, x2 0, I p x 0g where the KKT Lagrangian for constrained maximization L(x1, x2, λ) WBears(x) + λ3[I p x] + λ1x1 + λ2x2.

Theorem

[Karush-Kuhn-Tucker] If Slater’s constraint quali…cation is satis…ed then xis a maximizer of (P), where x 2 RN

+, i¤ there exists a multipliers

λ (λ

3, λ 1, λ 2) 0 such that:

(∂x1WBears(x), ∂x2WBears(x)) = (λ

3p1 λ 1, λ 3p2 λ 2).

Anat Bracha and Don Brown (Boston FRB and Yale University) (Ir)rational Exuberance October 2013 (Revised) 18 / 28

Betting Odds of Bears and Market Odds

If x = (0, x

2 ), then

(∂x1WBears((0, x

2 )), ∂x2WBears((0, x 2 )) = (λ 3p1 λ 1, λ 3p2) and

It follows that some bears are more pessimistic than the market that tomorrow’s state of the world is state 1. That is,

∂x1WBears((0,x

2 ))

∂x2WBears((0,x

2 )) = λ 3p1λ 1

λ

3p2

< p1

p2 .

Anat Bracha and Don Brown (Boston FRB and Yale University) (Ir)rational Exuberance October 2013 (Revised) 19 / 28

Betting Odds of Bears and Market Odds [Continued]

If x = (x

1 , 0), then

(∂x1WBears((x

1 , 0)), ∂x2WBears((x 1 , 0)) = (λ 3p1, λ 3p2 λ 2)

It follows that the other bears are more pessimistic than the market that tomorrow’s state of the world is state 2. That is,

∂x2WBears((x

1 ,0))

∂x1WBears((x

1 ,0)) = λ 3p2λ 2

λ

3p1

< p2

p1 .

Anat Bracha and Don Brown (Boston FRB and Yale University) (Ir)rational Exuberance October 2013 (Revised) 20 / 28

Trades Between Bulls and Bears

Theorem

At the market prices (p1, p2), some bulls trade Arrow–Debreu state-contingent claims for state 2 with some bears for Arrow–Debreu state-contingent claims for state 1. That is, ∂x2WBulls((0, x

2 ))

∂x1WBulls((0, x

2 )) > p2

p1 ∂x2WBears((x

1 , 0))

∂x1WBears((x

1 , 0)).

At the market prices (p1, p2), other bulls trade Arrow–Debreu state-contingent claims for state 1 with other bears for Arrow–Debreu state-contingent claims for state 2. That is,

∂x1WBulls((x

1 ,0))

∂x2WBulls((x

1 ,0)) > p1

p2 ∂x1WBears((0,x

2 ))

∂x2WBears((0,x

2 )). Anat Bracha and Don Brown (Boston FRB and Yale University) (Ir)rational Exuberance October 2013 (Revised) 21 / 28

Existence of Equilibrium in an Edgeworth Box

Here is an example of a competitive equilibrium in an exchange economy with two states of the world. There is a continuum of bulls indexed on [0, 1] and a continuum of bears indexed on [0, 1]. The sum of the average endowments of the bulls, ΘBulls, and the average endowments of the bears, ΘBears, de…ne the average social endowment Θ ΘBulls + ΘBears. We construct the associated Edgeworth box, where the X-axis is the payo¤ of the average social endowment in state 1 and the Y -axis is the payo¤ of the average social endowment in state 2. Zero is the origin of the positive orthant for bulls, i.e., x 0 and Θ, the average social endowment, is the origin of the positive orthant for bears, i.e., y Θ.

Anat Bracha and Don Brown (Boston FRB and Yale University) (Ir)rational Exuberance October 2013 (Revised) 22 / 28

Existence of Equilibrium in an Edgeworth Box [Continued]

If p = (p1, p2) is a vector of positive state prices where p ΘBulls I , p ΘBears J, and a fraction ρ 2 (0, 1) of bulls who demand the asset with payo¤s (I/p1, 0) and a fraction (1 ρ) 2 (0, 1) of bulls who demand the asset with payo¤s (0, I/p2), then aggregate demand of the bulls at state prices p is z

• ρ I

p1 , (1 ρ) I p2

• .

In the Edgeworth box, z is a point on the interior of the budget line p x = I, where x = (x1, x2) is a state-contingent claim in the positive

• rthant for bulls. In this example, if every bear maximizes utility subject to

the budget constraint p y = J, where y = Θ x and z is a state-contingent claim in the positive orthant for the bulls , then [p; z, Θ z] is a competitive equilibrium in the exchange economy, where all bears are endowed with the same concave utility function U(y) and Θ z = arg max

py=J U(y).

Anat Bracha and Don Brown (Boston FRB and Yale University) (Ir)rational Exuberance October 2013 (Revised) 23 / 28

Optimality and Existence of Equilibrium in Asset Markets

Optimality follows from Aumann’s celebrated (1964) core equivalence theorem for exchange economies with a continuum of traders, where traders may be endowed with nonconcave utility functions, e.g., bulls, but consumption sets are assumed to be bounded below,i.e.,short sales are not

• allowed. Existence follows from Aumann’s (1966) existence theorem for

exchange economies with a continuum of traders, where traders may be endowed with nonconcave utility functions,e.g.,bulls, but consumption sets are assumed to be bounded below,i.e.,short sales are not allowed.

Anat Bracha and Don Brown (Boston FRB and Yale University) (Ir)rational Exuberance October 2013 (Revised) 24 / 28

Asymmetric Keynesian Utilities

For asymmetric Keynesian utilities there exists a state-contingent claim b x, “the reference point,” where for quadratic utilities of ambiguity and quadratic utilities of risk, J U(x) is concave or pessimistic on [b x, +∞] fx 2 RN

+ : x b

xg and J U(x) is convex or optimistic on (0, b x] fx 2 RN

+ : x b

xg.

Anat Bracha and Don Brown (Boston FRB and Yale University) (Ir)rational Exuberance October 2013 (Revised) 25 / 28

Risk-Seeking and Ambiguity-Averse Keynesian Utilities

Theorem

If J U(x), is the composition of U(x) and J(y),where (a) (y1, y2, ..., yN) y = U(x) (u(x1), u(x2), ..., u(xN)) is a monotone, smooth, concave, diagonal quadratic map from RN

++ onto RN ++ ,with the

proxy for risk, β < 0 (b) J(y) is a monotone, smooth, convex quadratic function from RN

+ into R,with the proxy for ambiguity, α > 0,(c)

r2

xJ U(x) = diag(α)(diag[rxU(x)])2 diag(β)diag[rU(x)J(U(x))]

then there exists a reference point ˆ x such that the …nancial market data D is rationalized by the composite function J U(x) with two domains of convexity: (ˆ x, +∞] and (0, ˆ x], where J U(x) is concave on (ˆ x, +∞] and J U(x) is convex on (0, ˆ x].

Anat Bracha and Don Brown (Boston FRB and Yale University) (Ir)rational Exuberance October 2013 (Revised) 26 / 28

Risk-Averse and Ambiguity-Seeking Keynesian Utilities

Theorem

If J U(x), is the composition of U(x) and J(y),where (a) (y1, y2, ..., yN) y = U(x) (u(x1), u(x2), ..., u(xN)) is a monotone, smooth, convex, diagonal quadratic map from RN

++ onto RN ++ with the

proxy for risk, β > 0, (b) J(y) is a monotone, smooth, concave quadratic function from RN

+ into R with the proxy for risk,α < 0, (c)

r2

xJ U(x) = diag(α)(diag[rxU(x)])2 + diag(β)diag[rU(x)J(U(x))]

then there exists a reference point b x such that the …nancial market data D is rationalized by the composite function J U(x) with two domains of convexity: (b x, +∞] and (0, b x] , where J U(x) is concave on (ˆ x, +∞] and J U(x) is convex on (0, ˆ x].

Anat Bracha and Don Brown (Boston FRB and Yale University) (Ir)rational Exuberance October 2013 (Revised) 27 / 28

Asymmetric Keynesian Utilities and Prospect Theory

Here is the 2 2 contingency table for an investor endowed with asymmetric Keynesian utilities. We divide RN

+ into the standard 4

quadrants with the reference point, b x, as the origin: Table 2 r2

xJ U(x) is inde…nite

J U(x) is concave

• n Quadrant II
• n Quadrant I

J U(x) is convex r2

xJ U(x) is inde…nite

• n Quadrant III
• n Quadrant IV

That is, the investor is a bull for “losses,” quadrant III, but a bear for “gains,” quadrant I. In prospect theory, preferences for risk have a similar “shape,” see …gure 10 in Kahneman (2011).

Anat Bracha and Don Brown (Boston FRB and Yale University) (Ir)rational Exuberance October 2013 (Revised) 28 / 28