CLASS 4: ASSEt pricing. The Intertemporal Model. Theory and - - PowerPoint PPT Presentation

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-
• ptimal (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.

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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
• f the fundamentals. More precisely—prices move with economic

fundamentals and the co-movement is stronger the higher the risk aversion.

• Representative agent? Pareto Optimality?

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The lucas model: AN example

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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

• f $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 Dividends/income is perishable (“cash”)

The lucas model: AN example

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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 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 Dividends Tree $1*10=10 $0*10=0 $0*10=0 $1*10=10 $0*10=0 $1*10=10 Bond $0.5*0=0 $0.5*0=0 $0.5*0=0 $0.5*0=0 $0.5*0=0 $0.5*0=0 Income 15 15 15 Initial Cash $10 (=10+0+0) $15 (=0+0+15) $0 (=0+0+0) $25 (=10+0+15) $0 (=0+0+0) $25 (=10+0+15) Trade Tree Bond Cash Change $0 $0 $0 $0 $0 $0 Final Holdings Tree 10 10 10 10 10 10 Bond CASH $ 10.00 $ 15.00 $ 0.00 $ 25.00 $ 0.00 $ 25.00

!

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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 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 (=10+0+0) $17.5 (=0+2.5+15) $3 (=0+3+0) $21 (=4+2+15) $3 (=0+3+0) $20 (=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

!

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The lucas model: AN example

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Mathematically

Investor maximizes lifetime consumption: max

∞

X

t=1

βt−1u(c(t)) subject to a budget constraint on consumption c(t), income/dividends and investment (in bonds and trees) First-order conditions from optimal plan: βE[

∂u(c(t+1)) ∂c ∂u(c(t)) ∂c

(p(t + 1) + d(t + 1))|I(t)] = p(t), 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 Key: price anticipations are “perfect” (Radner 1972)

The lucas model: AN example

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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) (Difference) (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%)

The lucas model: AN example

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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 Crockett-Duffy, 2010)

The lucas model: in the laboratory

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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-Duffy (2010), who use nonlinear payoffs in order to induce smoothing...

The lucas model: in the laboratory

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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 i.i.d. dividends!

The lucas model: in the laboratory

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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 i.i.d. dividends!

Design summary

Period 1 Period 2 Period 3 Dividends from initial allocation

• f “Trees” and “Bonds”

Income Trade to a final allocation

• f “Trees,” “Bonds,” and

CASH Possible Termination of Session *If termination--keep CASH *If continuation--lose CASH, carry over “Trees” and “Bonds” Dividends from carried over allocation

• f “Trees” and “Bonds”

Income Trade to a final allocation

• f “Trees,” “Bonds,” and

CASH

Possible Termination of Session *If termination--keep CASH *If continuation--lose CASH, carry over “Trees” and “Bonds”

Etc.

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Trading platform

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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)

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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 Difference 0.24 0.14

• 0.11

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Prices

• Correlation between the Equity Premium in each

replication and the price differential between high/low dividend periods (what was the theoretical prediction here???)

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Tree Bond Correlation 0.80 0.52 (St. Err.) (0.40) (0.40)

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.

10 20 30 40 50 60 70 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Period Price 1 2 3 4 5 6

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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) R2 0.18 0.04

• Autocor. (s.e. 0.13)

0.18

• 0.19

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(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

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What is happening

Subjects did not hedge price risk (much) – they did not expect prices to move with fundamentals Resulting equilibrium is VERY different from Lucas model! ... but very much like in our experiments (stochastic drift, etc.) (Significant correlation between prices and fundamentals cannot easily be detected in 10-15 rounds)

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Price expectations

Adam, Marcet and Nicolini (2012) also point out that even with only small mistakes in expectations about prices (assuming everyone knows underlying dividend processes!), equilibrium prices may look very different from the Lucas equilibrium – much more like in “the real world.” But Adam, Marcet and Nicolini (2012) do not point out that equilibrium allocations could still be pretty much the same as in the Lucas equilibrium – and close to optimal! ... because our agents trade consistent with their expectations, and their expectations are almost self-fulfilling.

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Price expectations

Figure 3: Time series of period-average Tree (red line) and Bond (blue line) transaction price changes. Changes are concatenated across all replications and all sesions, but exclude inter-replication observations. State is indicated by black solid line on top; state = 2 when High (tree dividend equals $1); state = 0 when Low (tree dividend equals $0).

10 20 30 40 50 60 −2 −1.5 −1 −0.5 0.5 1 1.5 2 period price change (blue; red) and state (black) bond tree

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Price expectations

Figure 4: Simulated equilibrium Tree (blue line) and Bond (green line) prices, first 100

• periods. State is indicated by red solid line on bottom; state = 1 when High (tree dividend

equals $1); state = 0 when Low (tree dividend equals $0). Equilibrium is based on (false) agent beliefs that the past prices are the best prediction of future prices.

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In conclusion

The cross-sectional pricing implications of the Lucas model are born out in the experimental data The intertemporal variation (predictability) in asset prices is far less than predicted (given cross-sectional difference) Subjects seem to have anticipated this and therefore reduce their demands to hedge against price risk; still, these anticipations are inconsistent in equilibrium (prices will – and do – depend on tree dividends even if this is not anticipated...) Nevertheless, the risk sharing properties of the Lucas equilibrium emerge: allocations are OK even if prices are wrong in one important dimension...

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