Random Risk Tolerance: a Model of Asset Pricing and Trade Volume

Transcription

1 Random Risk Tolerance: a Model of Asset Pricing and Trade Volume Fernando Alvarez U Chicago Andrew Atkeson UCLA in Honor of Bob Lucas 1 / 26

2 When I met Bob Preamble Trading Volumes in Asset Markets 2 / 26

3 When I met Bob Preamble Trading Volumes in Asset Markets Preference Shocks and Risk Sharing 2 / 26

4 When I met Bob Preamble Trading Volumes in Asset Markets Preference Shocks and Risk Sharing 25 years later put them in the same paper 2 / 26

5 Empirical Literature Introduction Large empirical literature which studies trade volume and asset prices: Expected returns and trade volume. Expected returns and trade volume autocorrelation of returns. Trade volume as a pricing factor Campbell, Grossman & Wang 93, Amihud 02, Llorente et al 02, Pastor Stambaugh 03, Lo & Wang 02, 06,... 3 / 26

6 Empirical Literature Introduction Large empirical literature which studies trade volume and asset prices: Expected returns and trade volume. Expected returns and trade volume autocorrelation of returns. Trade volume as a pricing factor Campbell, Grossman & Wang 93, Amihud 02, Llorente et al 02, Pastor Stambaugh 03, Lo & Wang 02, 06,... We develop a general equilibrium model of the pricing of the risk that one will want to trade. 3 / 26

7 Basic Idea Introduction Dispersion in idiosyncratic shocks to desired portfolios generates trade volumes Aggregate shocks to desired portfolios drive changes in asset prices Interaction of idiosyncratic and aggregate risks are priced similar to idiosyncratic endowment shocks Mankiw 86 and Constantinides and Duffie 96 4 / 26

8 Model Features Introduction General equilibrium model: prices and quantities. Three periods. t = 0 Everyone ex-ante identical t = 1 Identity" risk: idiosyncratic and aggregate shocks to risk tolerance t = 2 Outcome" risk: endowments realized and consumed Dispersion of risk-tolerance: trade volumes at t = 1 Aggregate shocks: asset prices at t = 1 How are these two risks priced at t = 0? impact of frictions on prices and welfare a transactions tax 5 / 26

9 Main Results Introduction Develop a tractable GE asset pricing framework: solve trading volumes and asset prices preferences imply a seller of risky securities has suffered a negative idiosyncratic shock precise mathematical analogy to idiosyncratic endowment shocks Mankiw 86 idiosyncratic risk measured from interaction of trading volumes and aggregate risk premia First order welfare loss from a tax on asset transactions 6 / 26

10 Plan for Talk Introduction 1 3 period model set up. 2 preferences. 3 equilibrium w/complete and incomplete market. 4 key properties of equicautious HARA preferences 5 asset pricing and trade volumes 6 Tobin tax on asset trade. 7 / 26

11 Set up Three period model: time line and shocks ex-ante identical investors at t = 0. time t = 0 time t = 1 time t = 2 aggregate shocks: z π( ) y ρ( z)

12 Set up Three period model: time line and shocks ex-ante identical investors at t = 0. time t = 0 time t = 1 time t = 2 aggregate shocks: z π( ) y ρ( z) idiosyncratic shocks: U τ ( ) w/risk tolerance τ µ(, z)

13 Set up Three period model: time line and shocks ex-ante identical investors at t = 0. time t = 0 time t = 1 time t = 2 aggregate shocks: z π( ) y ρ( z) idiosyncratic shocks: U τ ( ) w/risk tolerance τ µ(, z) C 0 shocks to risk-tolerance U τ c(τ, y, z) price P 0 (d) rebalance, price P 1 (z; d) payoff d(y, z) U τ 8 / 26

14 Set up Investors: ex-ante identical, ex-post difference in τ t = 2 t = 1 shocks to output ρ(y 1 z 1 ) y 1 τ c(τ, y 1, z 1 )µ(τ, z 1 ) = y 1 t = 0 C 0 π(z 1 ) π(z 2 ) z 1 τ µ(, z 1 ) shocks to risk tolerance z 2 τ µ(, z 2 ) y 2... y 3... y 1... y 2... ρ(y 3 z 2 ) y 3 τ c(τ, y 3, z 2 )µ(τ, z 2 ) = y 3 9 / 26

15 Investor s preferences Set up From time t = 1 to t = 2, equicautious HARA utility w/risk tolerance τ. Distribution of risk tolerance τ µ(, z) at each z in time t = 1. At t = 0 investor use expected utility on time t = 1 Certainty Equivalent (C.E.) Using C.E. isolates other attitudes of investor s preferences V (C 0 ) + β z ( ) V Uτ 1 U τ (c(τ, y, z)) ρ (y z) µ(τ, z) π(z) τ y }{{} C 1 (τ,z) : Certainty Equivalence for τ at z 10 / 26

16 Investor s preferences Set up From time t = 1 to t = 2, equicautious HARA utility w/risk tolerance τ. At t = 0 investors use expected utility on time t = 1 Certainty Equivalent (C.E.) V (C 0 ) + β z V ( C 1 (τ, z) ) µ(τ, z) π(z) τ C 1 (τ, z) U 1 τ ( ) U τ (c(τ, y, z)) ρ (y z) y Arrow-Pratt Theorem lower risk tolerance τ = lower C 1 (τ, z) from same allocation c(, y, z) at t = 2 11 / 26

17 Set up U τ equicautious HARA preferences U τ (c) = ( ) ( ) 1 γ γ c 1 γ γ + τ γ 1 U τ (c) = log(c + τ) for {c : τ + c > 0} for γ = 1 U τ (c) = τ exp ( c/τ) as γ, U τ (c) = τ shifts risk tolerance ( ) γ ( ) γ 1 c γ + τ > 0, U τ c (c) = γ + τ < 0 R τ (c) U τ (c) U τ (c) = c γ + τ 12 / 26

18 Optimum and Equilibrium Optimum and Equilibrium with Complete Markets From t = 1 onwards, given τ, maximize EU τ (c(τ, y, z)) = y U τ (c(τ, y; z))ρ(y z) p(y; z)c(τ, y; z)ρ(y z) y y p(y; z) [y + B(τ; z)] ρ(y z) implied C 1 (τ; z) 13 / 26

19 Optimum and Equilibrium Optimum and Equilibrium with Complete Markets From t = 1 onwards, given τ, maximize EU τ (c(τ, y, z)) = y U τ (c(τ, y; z))ρ(y z) p(y; z)c(τ, y; z)ρ(y z) y y p(y; z) [y + B(τ; z)] ρ(y z) implied C 1 (τ; z) Time t = 0 choose initial consumption and τ - contingent bonds s.t. C 0 + τ,z Q(τ; z)b(τ; z)µ(τ; z)π(z) = C 0 bond market clearing B(τ; z)µ(τ; z) = 0 τ 13 / 26

20 Optimum and Equilibrium Equilibrium with Incomplete Markets From t = 1 onwards, given τ, maximize EU τ (c(τ, y, z)) = y U τ (c(τ, y; z))ρ(y z) p(y; z)c(τ, y; z)ρ(y z) y y p(y; z) [y + B(z)] ρ(y z) implied C 1 (τ; z) 14 / 26

21 Optimum and Equilibrium Equilibrium with Incomplete Markets From t = 1 onwards, given τ, maximize EU τ (c(τ, y, z)) = y U τ (c(τ, y; z))ρ(y z) p(y; z)c(τ, y; z)ρ(y z) y y p(y; z) [y + B(z)] ρ(y z) implied C 1 (τ; z) Time t = 0 choose initial consumption and bonds s.t. C 0 + z Q(z)B(z)π(z) = C 0 bond market clearing B(z) = 0 14 / 26

22 Asset Prices Optimum and Equilibrium Time t = 1 risk free bond price is numeraire p(y; z)ρ(y z)dy 1 y Time t = 1 share price D 1 (z) y p(y; z)yρ(y z) Time t = 1 asset d(y; z) P 1 (z; d) y p(y; z)d(y; z)ρ(y z) Time t = 0 asset d(y; z) P 0 (d) = z Q(z)P 1 (z; d)π(z) 15 / 26

23 Optimum and Equilibrium Two Stage Budgeting At t = 1, cost of C.E. consumption C 1 : given prices p(y; z) and τ, H τ (C 1 ; z) = min p(y; z)c(y; z)ρ(y z) c(y;z) subject to c(y; z) delivers C.E. consumption C 1 for investor τ y Budget constraints in C.E. consumption H τ (C 1 (τ; z); z) = D 1 (z) + B(z) C 0 + z Q(z)B(z)π(z) = C 0 Date t = 0 asset prices with risk to τ Q(z) = β [ / ] V (C 1 (τ; z)) H V τ (C 1 (τ; z); z) µ(τ; z) (C 0 ) C 1 τ 16 / 26

24 Key Properties of HARA Preferences Conditionally Efficient Allocations of C.E. Consumption Psuedo-resource constraint for C.E. consumption Average Risk Tolerance: τ(z) = z τµ(τ; z) C.E. consumption for average investor from consuming endowment y ( ) C 1 (z) U 1 τ(z) U τ(z) (y)ρ(y z) y All conditionally optimal allocations at t = 1 satisfy C 1 (τ; z)µ(τ; z) = C 1 (z) τ 17 / 26

25 Key Properties of HARA Preferences Conditionally Efficient Allocations of C.E. Consumption Psuedo-resource constraint for C.E. consumption Average Risk Tolerance: τ(z) = z τµ(τ; z) C.E. consumption for average investor from consuming endowment y ( ) C 1 (z) U 1 τ(z) U τ(z) (y)ρ(y z) y All conditionally optimal allocations at t = 1 satisfy C 1 (τ; z)µ(τ; z) = C 1 (z) τ Implies optimal allocation as of t = 0 has C 1 (τ; z) = C 1 (z) for all τ Identity risk not priced with complete asset markets 17 / 26

26 Key Properties of HARA Preferences Gorman Aggregation At t = 1 endowment risk priced by investor with average risk tolerance τ(z) p(y; z) = p(y; z) independent of bondholdings and dispersion in τ same with share price D 1 (z) = D 1 (z) = y p(y; z)yρ(y z) C.E. cost functions H τ (C 1 ; z) pinned down and common marginal cost / J(z) 1 H τ (C 1 (τ; z); z) C 1 Equilibrium allocation (B(z) = 0) C1 e (τ; z) = C 1 (z) + τ τ(z) [ C1 (z) D 1 (z) ] + τ(z) D 1 (z) γ 18 / 26

27 Key Properties of HARA Preferences Two Fund Separation and Trade Volumes Equilibrium allocation of C.E. consumption at t = 1 C1 e (τ; z) = C 1 (z) + τ τ(z) [ C1 (z) D 1 (z) ] + τ(z) D 1 (z) γ aggregate risk premium [ C1 (z) D 1 (z) ] C 1 (z) cost of aggregate C.E. consumption in bonds D 1 (z) cost of aggregate C.E. consumption in shares equilibrium share trade volume φ e (τ; z) 1 = τ τ(z) + τ(z) D 1 (z) γ C.E. consumption risk seen in trade volumes and aggregate risk premia 19 / 26

28 Asset Prices and Trade Volumes Asset Pricing at t = 0 Date t = 0 bond prices Q e (z) = β V ( C 1 (z)) V ( C J(z)L(z) 0 ) L(z) reflects dispersion in C.E. consumption L(z) τ V (C1 e (τ; z)) V ( C µ(τ; z) 1 (z)) 20 / 26

29 Asset Prices and Trade Volumes Asset Pricing at t = 0 Date t = 0 bond prices Q e (z) = β V ( C 1 (z)) V ( C J(z)L(z) 0 ) L(z) reflects dispersion in C.E. consumption L(z) τ V (C1 e (τ; z)) V ( C µ(τ; z) 1 (z)) and thus aggregate risk premia and trade volumes L(z) 1 + V ( C 1 (z)) ( V ( C C1 (z) D 1 (z) ) 2 1 (z)) τ ( φ e (τ; z) 1 )2 µ(τ; z) and precautionary motives V ( ) > 0 aggregate trade volume TV e (z) = 1 φ e (τ; z) 1 µ(τ; z) 2 τ 20 / 26

30 Asset Prices and Trade Volumes Trade Volume as a Pricing Factor ex-ante expected excess returns in the complete markets economy E (d) 1 = Cov (Q (z), R 1 (z; d)) and in the incomplete markets economy E e 1 (d) 1 = (E (d) 1) β V ( C 0 ) Cov (J(z) (z), R 1(z; d)) 21 / 26

31 Asset Prices and Trade Volumes Trade Volume as a Pricing Factor ex-ante expected excess returns in the complete markets economy E (d) 1 = Cov (Q (z), R 1 (z; d)) and in the incomplete markets economy E e 1 (d) 1 = (E (d) 1) (z) reflects dispersion in C.E. consumption β V ( C 0 ) Cov (J(z) (z), R 1(z; d)) (z) τ [ V (C e 1 (τ; z)) V ( C 1 (z)) ] µ(τ; z) and thus aggregate risk premia and trade volumes (z) β 2 V ( C 1 (z)) ( V ( C C1 (z) D 1 (z) ) 2 0 ) τ ( φ e (τ; z) 1 )2 µ(τ; z) and precautionary motives V ( ) > 0 21 / 26

32 Transaction Tax Impact of a transactions tax We have examined an environment with no trading frictions What is the impact of trading frictions on asset prices and welfare? Example: Tobin taxes on trading shares Transaction tax ω on rebalancing trade of shares vs bonds at time t = 1. Proceeds rebated equally to all investors at time t = 1. Tax ω: wedge between the buying and selling price of shares to dividend y. W (ω) time t = 0 ex-ante welfare with tax ω: W (ω) = V (C 0 ) + V (C 1 (τ, z; ω)) µ(τ, z) π(z) z τ 22 / 26

33 Transaction Tax First order effect of transaction tax W (ω) time t = 0 ex-ante welfare with tax ω: dw dω = β z π(z) τ µ(τ; z)v (C 1 (τ; z)) d dω C 1(τ; z) initial equilibrium marginal utility of C.E. consumption V (C 1 (τ; z)) incidence of tax on sellers (τ < τ(z)) and buyers (τ > τ(z)) of shares d dω C 1(τ; z) 23 / 26

34 Transaction Tax Complete Asset Markets Complete Mkts: "standard" Ramsey-Harberger results W (ω) ω ω=0 = 0 initial equilibrium marginal utility of C.E. consumption all equal V (C 1 (τ; z)) = V ( C 1 (z)) incidence of tax averages to zero τ d dω C 1(τ; z)µ(τ; z) = 0 24 / 26

35 Transaction Tax Incomplete Asset Markets incomplete Mkts: W (ω) ω ω=0 < 0 initial equilibrium marginal utility of C.E. consumption higher for low τ V (C 1 (τ Low ; z)) > V (C 1 (τ High ; z)) incidence of tax averages falls on low τ d dω C 1(τ Low ; z) < 0 low risk tolerant investors have relatively inelastic desire to sell shares 25 / 26

36 Conclusion Conclusion General Equilibrium model of the risk that one will want to trade Wanting to sell risky assets is a negative shock analogous to a negative endowment shock risk manifest in data on trade volumes and aggregate risk premia seen in pricing if distribution of trade volumes is correlated with aggregate shocks Tobin taxes exacerbate this risk 26 / 26