Loss Aversion and Asset Prices

Transcription

1 Loss Aversion and Asset Prices Marianne Andries Toulouse School of Economics June 24,

2 Preferences In laboratory settings, systematic violations of expected utility theory Allais Paradox M. Rabin (2000) D. Kahneman and A. Tversky: Prospect Theory: An analysis of Decision under Risk, Econometrica

3 Allais Paradox Experiment 1 Experiment 2 Gamble 1A Gamble 1B Gamble 2A Gamble 2B Winnings Chance Winnings Chance Winnings Chance Winnings Chance $1 million 100% $1 million 89% Nothing 89% Nothing 90% Nothing 1% $1 million 11% $5 million 10% $5 million 10%! In lab, most prefer 1A to 1B 3

4 Allais Paradox Experiment 2 1B Gamble 2A Gamble 2B hance Winnings Chance Winnings Chance 9% Nothing 89% Nothing 90% % $1 million 11% 0% $5 million 10% In lab, most prefer 2B to 2A 4

5 Allais Paradox Experiment 1 Experiment 2 Gamble 1A Gamble 1B Gamble 2A Gamble 2B Winnings Chance Winnings Chance Winnings Chance Winnings Chance $1 million 100% $1 million 89% Nothing 89% Nothing 90% Nothing 1% $1 million 11% $5 million 10% $5 million 10%! In lab, most prefer 1A to 1B In lab, most prefer 2B to 2A Violation of the independence axiom 5

6 Rabin: Small Gambles versus Large Gambles M. Rabin: Risk Aversion and Expected-Utility Theory: A Calibration Theorem, Econometrica 2000 in the table 4 All entries are rounded down to an even dollar amount. $101 $105 $110 $125 $ ,250 $ $ ,050 2,090 $1,000 1,010 1,570 $2,000 2,320 $4,000 5,750 $6,000 11,810 $8,000 34,940 $10,000 $20,000 If averse to lose $100/gain bets for all wealth levels, will turn down lose /gain bets; s entered in table. So, for instance, if a person always turns down a lose /gain gamble, she will always turn down a lose $800/gain $2,090 gamble. Entries of are literal: Somebody who EU, concave and increasing utility function always turns down lose $100/gain $125 gambles will turn down any gamble with a 50% Similar results if the small gamble is rejected only up to given wealth chance of losing $600. This is because the fact that the bound on risk aversion holds everywhere if ( $100/ + $125) is rejected up to wealth $300, 000 reject implies that is bounded above. ( $1000/ + $160 bn) The theorem and corollary are homogenous of degree 1: If we know that turning down lose /gain gambles implies you will turn down lose /gain,thenforall, turning down lose /gain gambles implies you will turn down lose /gain.hence 6

7 Prospect Theory D. Kahneman and A. Tversky: Prospect Theory: An analysis of Decision under Risk, Econometrica 1979 Consider a (x, p; y, q) gamble Under EU, its value is Under Prospect Theory, its value is pu (W + x) + qu (W + y) π (p) v (x) + π (q) v (y) Simplest functional form to represent lab decisions 7

8 176 Journal of Economic Perspectives Prospect Theory Value Function v Figure 1 The Prospect Theory Value Function v(x) x Notes: The graph plots the value function proposed by Tversky and Kahneman (1992) as part of cumulative prospect theory, namely v(x) = x α for x 0 and v(x) = λ( x) α for x < 0, where x is a dollar gain or loss. The authors estimate α = 0.88 and λ = 2.25 from experimental data. The plot uses α = 0.5 and λ = 2.5 so as to make loss aversion and diminishing sensitivity easier to see. 1 Valuation on gains and losses, irrespective of current wealth (also, narrow framing) 2 Losses generate more disutility than comparable gains, no matter how small they are kink at the value zero 3 concave for gains, convex for losses The fourth and final component of prospect theory is probability weighting. In prospect theory, people do not weight outcomes by their objective probabilities p i but rather by transformed probabilities or decision weights π i. The decision weights are computed with the help of a weighting function w( ) whose argument is an objec- tive probability. The solid line in Figure 2 shows the weighting function proposed by Tversky and Kahneman (1992). As is visible in comparison with the dotted line a 45 degree line, which corresponds to the expected utility benchmark the weighting function overweights low probabilities and underweights high probabilities. 8

9 Thirty Years of Prospect Theory in Economics: A Review and Assessment 177 Prospect Theory Probability Weighting π Figure 2 The Probability Weighting Function w(p) P Notes: The graph plots the probability weighting function proposed by Tversky and Kahneman (1992) as part of cumulative prospect theory, namely w(p ) = P δ /(P δ + (1 P ) δ ) 1/δ, where P is an objective probability, for two values of δ. The solid line corresponds to δ = 0.65, the value estimated by the authors from experimental data. The dotted line corresponds to δ = 1, in other words, to linear probability weighting. 1 Over-weighting of tail events 2 Justifies both attraction to gambles and purchases of insurance part, from the fact that people like both lotteries and insurance they prefer a chance of $5,000 to a certain gain of $5, but also prefer a certain loss of $5 to a chance of losing $5,000 a combination of behaviors that is difficult to explain with expected utility. Under cumulative prospect theory, the unlikely state of the world in which the individual gains or loses $5,000 is overweighted in his mind, thereby explaining these choices. More broadly, the weighting function 3 Not erroneous beliefs, but decision weights 9

10 Prospect Theory in Finance 1 Loss Aversion Equity premium puzzle in C-CAPM with CCRA EU: E(Rm ) R f σ(r m ) < γσ c Locally infinite risk aversion at the kink in the value function Participation puzzle 2 Over-weighting of low probability events Over-pricing of deep out of the money options Low or negative returns for right-skewed assets (IPO firms, single-segment firms, OTC traded assets) Under-diversified portfolios 3 Risk-aversion for gains, risk-seeking for losses Disposition effect N. Barberis: Thirty Years of Prospect Theory in Economics: A Review and Assessment, JEP

11 Loss Aversion - Some intuitions Valuation of 50:50 gamble A σ, A + σ Utility! "#! A! +"! " with EU, U (payoff) v (A) 1 v 2 σ2 (A) with Loss Aversion, U (payoff) v (A) 1 ) (v 2 σ (A) v + (A) 11

12 Loss Aversion - Some intuitions 1 1st order pricing of risk relative to 2d order pricing of risk equity premium puzzle cross-sectional implications M. Andries: Consumption-based Asset Pricing with Loss Aversion, 2012 (CAPLA) 2 Role of frequency and information iid process with instantaneous growth rate µ and standard deviation σ expected growth increases with time interval T, standard deviation increases with T the pricing of risk becomes proportionally larger and larger the smaller the time interval T M. Andries and V. Haddad: Information Aversion, 2014 (IA) 12

13 Outline 1 Consumption-based Asset Pricing with Loss Aversion 2 Information Aversion 13

14 Plan 1 Consumption-based Asset Pricing with Loss Aversion 2 Information Aversion 14

15 CAPLA- Model of Preferences Agents are loss averse: consumption outcomes are valued relative to a reference point losses relative to the reference create more disutility than comparable gains Agents value the consumption stream recursively: V t = f (C t, E t (g (V t+1))) Loss aversion on the uncertain V t+1 Reference point as an endogenous expectation 15

16 CAPLA- Main Results A tractable consumption-based asset pricing model Impact of loss aversion on expected excess returns: Level effect Cross sectional effect Empirical implications: Negative Premium for skewness Security Market Line flatter than the CAPM Dynamic implications for the pricing of risk 16

17 CAPLA- One-Period Model At t = 1, the agent receives uncertain consumption C Standard CRRA model: γ > 1: risk aversion Loss Aversion model: ( ) C 1 γ U 0 = E 1 γ I0 1 homogeneous CRRA model above and below a reference point 2 continuous at the reference point 3 kink at the reference point (ratio of slopes) determined by a loss aversion coefficient α [0, 1) 17

18 CAPLA- One-Period Model Utility! Ref! C! " C 1"# 1"#! 18

19 CAPLA- One-Period Model Utility! Ref! C! " C 1"# 1"#! 18

20 CAPLA- One-Period Model with Loss Aversion ( ) C 1 γ U 0 = E 1 γ I0 C 1 γ for C Ref C 1 γ = C 1 γ (Ref) γ γ for C Ref }{{} scaling factor γ > γ determined by the ratio of slopes 1 α = 1 γ 1 γ Kahneman and Tversky (1979): α =

21 CAPLA- Multi-Period Model Standard Epstein-Zin (1989) preferences: V t = ( (1 β) C 1 ρ t + β (h (V t+1)) 1 ρ) 1 1 ρ h (V t+1) = ( ( E t V 1 γ)) 1 1 γ t+1 γ > 1: the risk aversion,β: the discount factor, 1 : the EIS ρ Add loss aversion on the CRRA model with reference point as an expectation 20

22 CAPLA- Properties of Multi-Period Model h (V t+1) = (E t (V t+1 1 γ )) 1 1 γ V 1 γ t+1 for v t+1 E t (v t+1) 1 γ V t+1 = V 1 γ t+1 exp [(γ γ) Et (vt+1)] for v t+1 E t (v t+1) }{{} scaling factor 1 if the outcome V t+1 is certain, then h (V t+1) = V t+1 2 h is increasing (first order stochastic dominance) 3 h is concave (second order stochastic dominance) 4 h is homogeneous of degree one (V t homogeneous of degree one in (C t, V t+1)) 21

23 CAPLA- Representative Agent Uniqueness of the solution to the optimization problem Time consistency h is concave (second order stochastic dominance) Assume agents differ in their wealth only = with homothetic preferences, the representative agent assumption is justified 22

24 CAPLA- Stochastic Discount Factor S + t,t+1 S t,t+1 exp [E t (v t+1)] = 1 α ( ) 1 exp [E t (v t+1)] + αe t 1 vt+1 E t(v t+1) Vt+1 1 γ Discontinuity in the stochastic discount factor when α > 0 Discontinuity increases with loss aversion coefficient α 23

25 CAPLA- Main Results Discontinuity in the SDF generates both a level and a cross-sectional effect 0.08 Annual Expected Excess Returns Risk Price Elasticities standard model model with loss aversion standard model model with loss aversion Loadings on the consumption shocks Loadings on the consumption shocks I use the parameters from Hansen, Heaton and Li (2008) for the consumption process and β = 0.999, γ = 10, α =

26 CAPLA- Equity Premium Calibration CAPLA- Equity Premium Calibration model with loss aversion standard model α = 0.10 α = 0.25 α = 0.55 risk aversion γ γ = % 1.29% 2.14% 0.72% γ = % 1.72% 2.74% 1.11% γ = % 2.16% 3.39% 1.50% Equity Premium from CRSP ( ) = 6.09% I use the quarterly parameters from Hansen, Heaton, and Li (2008) and β = (0.999) 1 4 Marianne Andries (TSE) Loss Aversion and Asset Prices June / 23 25

27 CAPLA- Value Premium Calibration CAPLA- Value Premium Calibration model with loss aversion standard model α = 0.10 α = 0.25 α = 0.55 risk aversion γ γ = % 2.68% 5.20% 0.65% γ = % 3.29% 5.98% 1.23% γ = % 4.85% 8.05% 2.70% Value Premium from Fama-French ( ) = 4.22% I use the quarterly parameters from Hansen, Heaton, and Li (2008) and β = (0.999) 1 4 Marianne Andries (TSE) Loss Aversion and Asset Prices June / 23 26

28 CAPLA- Prediction for CAPM The model with loss aversion qualitatively predicts a security market line flatter than the CAPM 16 Annual Expected Excess Returns R i R f in % Positive Intercept CAPM! I use the parameters from Hansen, Heaton and Li (2008) for the consumption process and β = 0.999, γ = 10, α =

29 CAPLA- Conclusion Tractable consumption-based asset pricing model with loss aversion and recursive utility Level effect on risk prices allows to match or improve on calibration exercises that use moments in asset returns Cross-sectional effect is a testable implication of my model Empirical evidence on the fit of the CAPM model provides strong support for my model with loss aversion 28

30 Plan 1 Consumption-based Asset Pricing with Loss Aversion 2 Information Aversion 29

31 IA- This Paper Why don t agents pay attention to information? Micro founded models of risk attitude towards information Information aversion Preference-based explanation of the cost of information Characterize risk and information decisions when information costs are endogenous: Properties of optimal attention to savings: Consumer Expenditure Survey (Dynan and Maki 2000): through a 15% rise in the market, 1/3 of stockholders report no change to their portfolio value. Alvarez, Guiso and Lippi (2012): household surveys in Italy, observe portfolios 4 times a year. Portfolio choice: home bias, underdiversification 30

32 Information Aversion Model Disappointment aversion 31

33 Information Aversion Model Disappointment aversion Ability to close your eyes 31

34 Information Aversion Model Disappointment aversion Recursive dynamic implementation of piecewise linear case of Gul (1991) Partial releases of information have a utility cost (Dillenberger 2010) Micro evidence and successful macro applications (Ang et al. 2005,2006, Routledge and Zin 2010, Bonomo et al. 2011, Lettau et al. 2013) Ability to close your eyes 31

35 Information Aversion Model Disappointment aversion Recursive dynamic implementation of piecewise linear case of Gul (1991) Partial releases of information have a utility cost (Dillenberger 2010) Micro evidence and successful macro applications (Ang et al. 2005,2006, Routledge and Zin 2010, Bonomo et al. 2011, Lettau et al. 2013) Ability to close your eyes No monetary or time cost of information No limited cognition Bayesian updating 31

36 IA- Results Natural theory of the cost side of information acquisition Which information flows are more costly? Higher frequency Higher risk Infinite aversion to continuous Brownian flow, not to jumps 32

37 IA- Results Natural theory of the cost side of information acquisition Which information flows are more costly? Higher frequency Higher risk Infinite aversion to continuous Brownian flow, not to jumps Information choice in a consumption-saving problem Infrequent observation of portfolio position Tradeoff for optimal frequency of information. At lower frequency: Misallocation of savings Less stressful flow of information More inattention in risky environments 32

38 IA- Results Natural theory of the cost side of information acquisition Which information flows are more costly? Higher frequency Higher risk Infinite aversion to continuous Brownian flow, not to jumps Information choice in a consumption-saving problem Infrequent observation of portfolio position Tradeoff for optimal frequency of information. At lower frequency: Misallocation of savings Less stressful flow of information More inattention in risky environments Other features of portfolio allocation Diversification Background risk Information delegation Asymmetry between good and bad news 32

39 IA Preferences: Disappointment Aversion Piecewise linear case of Gul (1991) Lottery over final outcome X Certainty equivalent: µ(x) = E [( 1 + θ1 X µ(x) ) X ] E [ 1 + θ1 X µ(x) ] Overweight disappointing outcome θ > 0, coefficient of disappointment aversion only source of aversion to risk comes from disappointment aversion Certainty equivalent µ(x) is unique solution to a fixed-point problem 33

40 IA- Disappointment Aversion, dynamic implementation Dynamic implementation: recursion on certainty equivalents Value at time t, V t, of lottery over continuation value V t+1: V t = µ (V t+1) V t = E [( 1 + θ1 Vt+1 V t ) Vt+1 I t ] E [ 1 + θ1 Vt+1 V t I t ] If no news is revealed: V t = V t+1 In continuous time: take the limit of discrete time sampling of information 34

41 Two-stage Lottery αif i = F α 1 α2 α 3 F Signal 1 Signal 2 Signal 3 F 1 F 2 F 3 A B C D A B C D 35

42 Two-stage Lottery αif i = F α 1 α2 α 3 μ(f) F Signal 1 Signal 2 Signal 3 F 1 F 2 F 3 A B C D A B C D 35

43 Two-stage Lottery α 1 α2 α 3 αif i = F μ(f) F μ(f 1 ) Signal 2 Signal 3 F 1 F 2 F 3 A B C D A B C D 35

44 Two-stage Lottery α 1 α2 α 3 αif i = F μ(f) F μ(f 1 ) μ(f 2 ) μ(f 3 ) F 1 F 2 F 3 A B C D A B C D 35

45 Two-stage Lottery αif i = F μ({f i, α i }) α 1 α2 α 3 μ(f) F μ(f 1 ) μ(f 2 ) μ(f 3 ) F 1 F 2 F 3 A B C D A B C D 35

46 Information Aversion Disappointment aversion information aversion Agent prefers not to observe the signal µ({f i, α i}) µ(f ) Dillenberger (2010): Negative Certainty Independence Preference for One-Shot Resolution of Uncertainty 36

47 Information Aversion Disappointment aversion information aversion Agent prefers not to observe the signal µ({f i, α i}) µ(f ) Dillenberger (2010): Negative Certainty Independence Preference for One-Shot Resolution of Uncertainty Agent fears possibility of repeated changes in certainty equivalent { µ (F i) = µ (F ) or µ({f i, α i}) = µ(f ) i, F i is degenerate 36

48 IA- Endogenous Information Costs Information aversion versus exogenous costs models Endogenous information cost is zero if all or no information is revealed Not monotonic increasing in quantity of information Information aversion versus cognitive constraints Endogenous information cost is zero for either fully informative or fully uninformative signals For any level of mutual information, we can construct signals with zero endogenous cost: reveal the final value of the lottery with some probability 37

49 IA- Role of Frequency Process X t with i.i.d. growth Observe its value at intervals of length T Receive value of the process X τ at time τ 38

50 IA- Role of Frequency Process X t with i.i.d. growth Observe its value at intervals of length T Receive value of the process X τ at time τ Information aversion Prefer never to observe the intermediate values Gneezy and Potters (1997),... How is the valuation of the lottery affected by the observation interval? the distribution of the process? Input for consumption-savings problem 38

51 IA- Role of Frequency Process X t with i.i.d. growth Observe its value at intervals of length T Receive value of the process X τ at time τ Because growth is i.i.d ( ) XT µ X 0 ( ) ( ) X2T X(k+1)T = µ =... = µ X T X kt Define instantaneous certainty equivalent rate v(t ): ( ) XT µ = exp(v(t )T ) X 0 Value at time 0 for payoff at time τ: V 0,τ (T ) = exp(v(t )τ) 38

52 IA- Role of Frequency Process X t with i.i.d. growth Observe its value at intervals of length T Receive value of the process X τ at time τ Because growth is i.i.d ( ) XT µ X 0 ( ) ( ) X2T X(k+1)T = µ =... = µ X T X kt Define instantaneous certainty equivalent rate v(t ): ( ) XT µ = exp(v(t )T ) X 0 Value at time 0 for payoff at time τ: V 0,τ (T ) = exp(v(t )τ) With drift g and martingale component Y : v X(T ) = g + v Y (T ) 38

53 IA- Frequency and Geometric Brownian Motion dx t X t = σdw t 0 Certainty Equivalent Rate, θ=1, σ=1 Certainty equivalent rate v(t) Observation interval T Equivalent rate as a function of observation interval 39

54 IA- Frequency and Geometric Brownian Motion dx t X t = σdw t Distaste for frequent partial information: equivalent rate increasing in observation interval optimally choose never to look at any information Certainty equivalent rate v(t) Certainty Equivalent Rate, θ=1, σ= Observation interval T Equivalent rate as a function of observation interval 39

55 IA- Frequency and Geometric Brownian Motion dx t X t Risk aversion: = σdw t equivalent rate decreasing in risk σ equivalent rate decreasing in risk aversion θ Certainty equivalent rate v(t) Certainty Equivalent Rate, θ=1, σ= Observation interval T Equivalent rate as a function of observation interval 39

56 IA- Frequency and Geometric Brownian Motion dx t X t = σdw t Infinite risk aversion at high frequency: Value for t = τ payoff equals lowest possible outcome in the continuous information limit expansion around 0: v(t ) 0 κ(θ)σ T first-order risk aversion: σ T τ/t }{{}}{{} observation discount # observations Certainty equivalent rate v(t) Certainty Equivalent Rate, θ=1, σ= Observation interval T Equivalent rate as a function of observation interval 39

57 IA- Frequency and Jump process dx t = λσdt σdn t X t N t: Poisson counting process, intensity λ Distaste for frequent partial information Risk aversion Certainty equivalent rate v(t) Certainty Equivalent Rate, θ=1, σ=.2, λ= Observation interval T Equivalent rate as a function of observation interval 40

58 IA- Frequency and Jump process dx t = λσdt σdn t X t N t: Poisson counting process, intensity λ Finite limit at high frequency: limiting behavior v(t ) θσλ T 0 no first order risk aversion: infrequent large risks vs. frequent small risks Certainty equivalent rate v(t) Certainty Equivalent Rate, θ=1, σ=.2, λ= Observation interval T Equivalent rate as a function of observation interval 40

59 IA - Portfolio problem How does the disappointment averse agent decide to consume, save, and observe information? Choice between risk-free and risky savings Setup of the fixed cost of information/transaction literature: Duffie and Sun (1990), Gabaix and Laibson (2001), Abel et al. (2007, 2013), Alvarez et al. (2013). Baumol-Tobin model (1952, 1956) No exogenous cost of information/transaction, but agent free to close her eyes 41

60 Setup Preferences: V 1 α t 1 α = C1 α t 1 α dt + (1 ρdt) (µ θ [V t+dt F t]) 1 α. 1 α θ: coefficient of disappointment aversion 1/α: intertemporal elasticity of substitution ρ: rate of time discount 42

61 Setup Preferences: V 1 α t 1 α = C1 α t 1 α dt + (1 ρdt) (µ θ [V t+dt F t]) 1 α. 1 α θ: coefficient of disappointment aversion 1/α: intertemporal elasticity of substitution ρ: rate of time discount Opportunity sets: Information: choose time until next observation T Investment: Instantaneous consumption C t Buy S t shares of the risky asset, price X t, instantaneous certainty equivalent rate v(t ) Remainder in risk-free asset, rate of return r Budget constraint: dw t = C tdt + S tdx t + r(w t S tx t)dt 42

62 Preferences: V 1 α t 1 α = T 0 ρτ C1 α t+τ e Setup θ: coefficient of disappointment aversion 1/α: intertemporal elasticity of substitution ρ: rate of time discount Opportunity sets: 1 α dτ + (µ θ [V t+t F t]) 1 α e ρt. 1 α Information: choose time until next observation T Investment: Instantaneous consumption C t Buy S t shares of the risky asset, price X t, instantaneous certainty equivalent rate v(t ) Remainder in risk-free asset, rate of return r Budget constraint: dw t = C tdt + S tdx t + r(w t S tx t)dt 42

63 Basic Properties Homothetic preferences Linear opportunity set for consumption i.i.d. dynamics Constant observation interval T Consumption-wealth ratio and asset allocation functions of wealth at last observation and time since last observation 43

64 Basic Properties Homothetic preferences Linear opportunity set for consumption i.i.d. dynamics Constant observation interval T Consumption-wealth ratio and asset allocation functions of wealth at last observation and time since last observation Remark: Fixed cost models lose homotheticity or use ad hoc assumptions on the scaling of the cost 43

65 IA- Consumption and Investment Decisions Given observation interval T : Consumption between observations deterministic, financed at the risk-free rate r Inter-observation savings: all risk-free if r > v(t ) all risky if r < v(t ) 44

66 IA- Consumption and Investment Decisions Given observation interval T : Consumption between observations deterministic, financed at the risk-free rate r Inter-observation savings: all risk-free if r > v(t ) all risky if r < v(t ) Fraction of wealth allocated to consumption: [( C (T ) = 1 exp ρ α + 1 α ) ] max(v(t ), r) T α Consumption path, for τ [0, T ]: C t+τ C(T )e ρ+r α τ 44

67 IA- Role of Observation Interval Geometric brownian motion: dx/x = gdt + σdw t 2.3 Value Function 0.7 Consumption allocation V C Observation interval T Observation interval T Parameters values: θ = 1, α = 0.5, σ = 1, g r = 1, ρ =

68 IA- Role of Observation Interval Geometric brownian motion: dx/x = gdt + σdw t 2.3 Value Function 0.7 Consumption allocation V C Observation interval T Observation interval T Parameters values: θ = 1, α = 0.5, σ = 1, g r = 1, ρ = 0.1. Infrequent observation and investment in risky asset iff g > r 45

69 IA- Role of Observation Interval Geometric brownian motion: dx/x = gdt + σdw t 2.3 Value Function 0.7 Consumption allocation V C Observation interval T Observation interval T Parameters values: θ = 1, α = 0.5, σ = 1, g r = 1, ρ = 0.1. Infrequent observation and investment in risky asset iff g > r More generally, need v(0) < r < v( ) 45

70 IA- Optimal Information Choice Key result: Optimal observation interval exists and is such that: ( v(t ) f v (T ) ρ ) ( = v (T ) ρ ) ( f v (T ) ρ ) ( r ρ ) (( f r ρ )) log(t ) 1 α 1 α 1 α 1 α 1 α ( ) ( ( )) where f (x) = exp 1 α α xt / 1 exp 1 α α xt Marginal cost of infrequent observation (RHS) lost consumption through financing risk-free rather than risky between observations increasing in equivalent rate differential between v(t ) and r Marginal benefit of infrequent observation (LHS) higher certainty equivalent for higher observation interval increasing in certainty equivalent elasticity v(t )/ log(t ) missing elasticity of fixed cost models 46

71 IA- Optimal Information Choice Key result: Optimal observation interval exists and is such that: ( v(t ) f v (T ) ρ ) ( = v (T ) ρ ) ( f v (T ) ρ ) ( r ρ ) (( f r ρ )) log(t ) 1 α 1 α 1 α 1 α 1 α ( ) ( ( )) where f (x) = exp 1 α α xt / 1 exp 1 α α xt Marginal cost of infrequent observation (RHS) lost consumption through financing risk-free rather than risky between observations increasing in equivalent rate differential between v(t ) and r Marginal benefit of infrequent observation (LHS) higher certainty equivalent for higher observation interval increasing in certainty equivalent elasticity v(t )/ log(t ) missing elasticity of fixed cost models Certainty equivalent elasticity Independent of the drift Typically decreasing in the observation interval Non-trivial dependence to the shape of the return distribution 46

72 IA- Role of risk Geometric brownian motion: dx/x = gdt + σdw t Optimal observation interval increasing in risk σ: Standard effect: equivalent rate of return v(t ) decreases. Can be compensated by higher average rate g Information aversion effect: less willingness to take on the information flow, higher elasticity v/ log(t ) 1.4 Observation interval 2.5 Value 0.5 Consumption Allocation T 1 V C Volatility Volatility Volatility 47

73 Predictions Observation interval decreasing in expected stock returns Observation interval increasing in volatility Even when compensated by higher expected returns Scary information flow Ostrich effect (Karlson et al. 2009), follow-up paper on VIX level and inattention (Sicherman et al. 2014) More disappointment averse agent observe their portfolios less frequently Alvarez et al. (2013): more risk averse agents check their accounts less often All else equal, in response to exogenous decrease in observation interval, increase in stock holdings Driven by corner solution in asset holdings Consistant with Beshears et al. (2012), also finding an increase in trading activity 48

74 IA- Diversification What if you can get 1 2 X(1) t X(2) t rather than X (1) t? Standard benefit: less risk 49

75 IA- Diversification What if you can get 1 2 X(1) t X(2) t rather than X (1) t? Standard benefit: less risk Limits of diversification: If asynchronous forced information arrival, increase in scope for disappointment 49

76 IA- Diversification What if you can get 1 2 X(1) t X(2) t rather than X (1) t? Standard benefit: less risk Result: Limits of diversification: If asynchronous forced information arrival, increase in scope for disappointment With independent Brownian motions, diversification is still valuable with non-instrumental information. The gains to diversification go to 0 as observation becomes continuous. 49

77 IA- Diversification What if you can get 1 2 X(1) t X(2) t rather than X (1) t? Standard benefit: less risk Result: Limits of diversification: If asynchronous forced information arrival, increase in scope for disappointment With independent Brownian motions, diversification is still valuable with non-instrumental information. The gains to diversification go to 0 as observation becomes continuous. Work in progress: Role of background risk: risky portfolio can be decreasing in risk in presence of background risk Home/local bias: anchor on forced information flows vs diversification benefits 49

78 IA- More General Information Choices So far, limited to simple information structure: open or closed eyes With the help of machines or people, can better taylor the information flow Result: Simple alarm when the risky asset reaches some thresholds provides more utility State-dependent trading rules do better than time-dependent rules, in contrast to fixed information cost (Abel et al. 2013) In practice: Useful to have your broker send you an following extreme performance, good or bad Media reporting large events 50

79 IA- Information Intermediaries Other individuals can not only curate information, but also take actions for the agent Portfolio managers, investment funds,... Optimal opaqueness: complex or illiquid securities hard to mark-to-market,... Information sets differ need to appropriately incentivize the informed decision maker 51

80 IA- Conclusion Information aversion: a novel foundation for inattention Disappointment aversion creates information aversion, fear of repeated disappointment Without use of information agent always prefers to close her eyes More averse to flows: about more risky outcome with frequent small news than infrequent large news about likely bad news than likely good news Simple way to summarize information aversion: certainty equivalent rate v(t ) More questions: Multi-asset decisions, background risk Delegated management Combined learning and frequency decisions Multiple agents 52