CLASS 4: ASSEt pricing. The Intertemporal Model. Theory and Experiment
Lessons from the 1- period model • If markets are complete then the resulting equilibrium is Pareto- optimal (no alternative allocation could make everyone better off). • Pareto optimality implies existed of a representative agent (with a utility function that is a weighted average of the individual investors’ utility functions) and state prices are proportional to that agent’s marginal utility at that state’s aggregate wealth. • Consequently, the higher the risk aversion of the representative agent, the higher the risk premium (the difference between the return on the market portfolio and the return on a risk-free security) in the market. • In the quadratic utility (CAPM) version of the model—risk aversion is represented by the quadratic term coefficient. 2
Infinite horizon: The Lucas model • Equilibrium notions: Radner Equilibrium, rational expectations, dynamic completeness • Preserves the main cross-sectional prediction of the 1-period model that the higher the risk aversion the higher the risk premium. • Delivers time-series predictions about prices (returns) as functions of the fundamentals. More precisely—prices move with economic fundamentals and the co-movement is stronger the higher the risk aversion. • Representative agent? Pareto Optimality? 3
The lucas model: AN example Setting (Lucas original - stationary in levels!!!) Stationary, infinite horizon setting; infinite-lived agents with time-separable utility Two assets: - A tree that pays $ 0 of $ 1 with equal chance (i.i.d.) - A (consol) bond that pays $ 0.50 each period Equal number of two types of agents: Type I receives income of $ 15 in even periods (2,4,...) and Type II receives income of $ 15 in odd periods (total income is constant over time!) Type I starts with 10 trees and 0 bonds; Type II with 0 trees and 10 bonds Assets pay their dividend before a (trading) period starts; investors receive their income at that time as well 4 Dividends/income is perishable (“cash”)
The lucas model: AN example Simple, but still complex enough to get transparent and rich predictions Cross-sectional: (Assuming risk aversion), tree is less expensive than bond (tree has higher “consumption beta”) Intertemporal: price levels move with fundamentals – the level of dividends of the “tree” Cross-sectional and intertemporal predictions reinforce each other Consumption (“cash”) is perfectly (rank-)correlated across states in equilibrium – Pareto optimality reached through dynamic completeness ; investors fully insure income fluctuations by trading continuously – desired experimentally: trade is in subjects’ interest 5 Sophisticated price risk hedging...
No Trading PERIOD 1 2 3 4 5 6 State H L L H L H Initial Holdings Tree 10 10 10 10 10 10 Bond 0 0 0 0 0 0 Dividends Tree $1*10=10 $0*10=0 $0*10=0 $1*10=10 $0*10=0 $1*10=10 $0.5*0=0 $0.5*0=0 $0.5*0=0 $0.5*0=0 $0.5*0=0 $0.5*0=0 Bond Income 0 15 0 15 0 15 Initial Cash $10 $15 $0 $25 $0 $25 (=10+0+0) (=0+0+15) (=0+0+0) (=10+0+15) (=0+0+0) (=10+0+15) Trade Tree 0 0 0 0 0 0 Bond 0 0 0 0 0 0 Cash Change $0 $0 $0 $0 $0 $0 Final Holdings Tree 10 10 10 10 10 10 Bond 0 0 0 0 0 0 CASH $ 10.00 $ 15.00 $ 0.00 $ 25.00 $ 0.00 $ 25.00 6 !
trading PERIOD 1 2 3 4 5 6 State H L L H L H Initial Holdings Tree 10 5 6 4 5 3 Bond 0 5 6 4 6 4 Dividends Tree $1*10=10 $0*5=0 $0*6=0 $1*4=4 $0*5=0 $1*3=3 Bond $0.5*0=0 $0.5*5=2.5 $0.5*6=3 $0.5*4=2 $0.5*6=3 $0.5*4=2 Income $0 $15 $0 $15 $0 $15 Initial Cash $10 $17.5 $3 $21 $3 $20 (=10+0+0) (=0+2.5+15) (=0+3+0) (=4+2+15) (=0+3+0) (=3+2+15) Trade Tree -5 +1 -2 +1 -2 +1 Bond +5 +1 -2 +2 -2 +1 Cash Change $0 -$5 +$10 -$7.5 +$10 -$5 Final Holdings Tree 5 6 4 5 3 4 Bond 5 6 4 6 4 5 CASH $ 10.00 $ 12.50 $ 13.00 $ 13.50 $ 13.00 $ 15.00 7 !
The lucas model: AN example Mathematically Investor maximizes lifetime consumption: ∞ X β t − 1 u ( c ( t )) max t =1 subject to a budget constraint on consumption c ( t ), income/dividends and investment (in bonds and trees) First-order conditions from optimal plan: ∂ u ( c ( t +1)) ∂ c β E [ ( p ( t + 1) + d ( t + 1)) | I ( t )] = p ( t ) , ∂ u ( c ( t )) ∂ c where I ( t ) is information up to t (basically, only the dividend on the tree in period t ), p and d denote prices and dividends, respectively 8 Key: price anticipations are “perfect” (Radner 1972)
The lucas model: AN example Numerical Example: Log Utility, β = 5 / 6 Prices: State Tree Bond (‘Equity Premium’ ( $ )) High 2.50 3.12 (0.62) Low 1.67 2.09 (0.42) (Di ff erence) (0.83) (1.03) (0.20) In terms of returns, bigger equity premium in ‘Low’ state: State Tree Bond (‘Equity Premium’ (%)) High 3.4% -0.5% (3.9%) Low 55% 49% (6%) 9
The lucas model: AN example Numerical Example: Log Utility, β = 5 / 6 Holdings and Trading (Type I, who do not receive income in odd periods): Period Tree Bond (Total) Odd 7.57 0.62 (8.19) Even 2.03 7.78 (9.81) (Trade in Odd) (+5.54) (-7.16) Remarks: Consumption smoothing – to the point that each Type consumes a fixed fraction of total available dividend (“cash”) (Sophisticated trading: buy the tree to hedge price risk) Lots of trading, desired experimentally (see also 10 Crockett-Du ff y, 2010)
The lucas model: in the laboratory Inducing Consumption Smoothing In The Laboratory At the end of each trading period, we roll a (twelve-sided) die If outcome is either 7 or 8, then we terminate the replication and subjects take home the cash they are holding at that moment; trees and bonds are taken away If the outcome is anything else, cash is forfeited and we move to the subsequent period, carrying over trees and bonds , which pay dividends that become cash (in addition to any income) Unlike Crockett-Du ff y (2010), who use nonlinear payo ff s 11 in order to induce smoothing...
The lucas model: in the laboratory Implementing A Stationary, Infinite-Horizon Setting In The Laboratory Technique: - Infinite Horizon: the twelve-sided die has positive probability of continuing the replication - Stationarity: if a replication has not terminated when the end of the experimental session (fixed duration) has been reached, then subjects get to keep the cash for that period The latter works because of the time-separability of preferences in the Lucas model and our assumption of 12 i.i.d. dividends!
The lucas model: in the laboratory Implementing A Stationary, Infinite-Horizon Setting In The Laboratory Technique: - Infinite Horizon: the twelve-sided die has positive probability of continuing the replication - Stationarity: if a replication has not terminated when the end of the experimental session (fixed duration) has been reached, then subjects get to keep the cash for that period The latter works because of the time-separability of preferences in the Lucas model and our assumption of 13 i.i.d. dividends!
Design summary Possible Termination of Session Possible Termination of Session *If termination--keep CASH *If termination--keep CASH *If continuation--lose CASH, *If continuation--lose CASH, carry over “Trees” and “Bonds” carry over “Trees” and “Bonds” Dividends Dividends from initial allocation from carried over allocation of “Trees” and “Bonds” of “Trees” and “Bonds” Income Income Etc. Period 1 Period 2 Period 3 Trade Trade to a fi nal allocation to a fi nal allocation of “Trees,” “Bonds,” and of “Trees,” “Bonds,” and CASH CASH 14
Trading platform 15
Sessions Session Place Replication Periods Subject Number (Total, Min, Max) Count 1 Caltech ∗ 4 (14, 1, 7) 16 2 Caltech 2 (13, 4, 9) 12 3 UCLA ∗ 3 (12, 3, 6) 30 4 UCLA ∗ 2 (14, 6, 8) 24 5 Caltech ∗ 2 (12, 2, 10) 20 6 Utah ∗ 2 (15, 6, 9) 24 (Overall) 15 (80, 1, 10) 16
Prices Tree Bond ‘Equity Price Price Premium’ ( $ ) Mean 2.75 3.25 0.50 St. Dev. 0.41 0.49 0.40 High (State) 2.91 3.34 0.43 Low (State) 2.66 3.20 0.54 Di ff erence 0.24 0.14 -0.11 17
Prices • Correlation between the Equity Premium in each replication and the price differential between high/low dividend periods (what was the theoretical prediction here???) Tree Bond Correlation 0.80 0.52 (St. Err.) (0.40) (0.40) 18
Prices Figure 2: Time series of Tree (solid line) and Bond (dashed line) transaction prices; averages per period. Session numbers underneath line segments refer to Table 4. 5 4.5 4 3.5 3 Price 2.5 2 1.5 1 1 2 3 4 5 6 0.5 0 10 20 30 40 50 60 70 Period 19
Prices and fundamentals A stochastic drift model “influenced” by tree dividend best describes price evolution: Explanatory Tree Price Change Bond Price Change Variables Estim. (95% CI) Estim. (95% CI) ∆ State: (None=0; 0.19 ∗ (0.08, 0.29) 0.10 (-0.03, 0.23) High-to-Low=-1, Low-to-High=+1) R 2 0.18 0.04 Autocor. (s.e. 0.13) 0.18 -0.19 20
(Pareto) efficiency Consumption ( $ ) Consumption Ratio Type High Low High Low I 14.93 (19.75) 7.64 (4.69) 1.01 (0.52) 1.62 (3.26) II 15.07 (10.25) 12.36 (15.31) Type Odd Even I 7.69 (2.41) 13.91 (20.65) II 14.72 (20) 11.74 (5) Positive rank correlation of end-of-period cash ( vs. autarky) Sharing (of total cash) much closer to equal 21
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