Feedback Models

BEHAVIORAL ECONOMICS

Fischer Black on “Noise”

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Black is a strong believer in noise and noise traders in particular

• They lose money according to him (though they may make money for a short while)

• Prices are “efficient” if they are within a factor of 2 of “correct” value

• Actual prices should have higher volatility than values because of noise

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What is a short sale?

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Example: Sell 100 shares of GOOG at 580• What happens?

• You enter an order to sell 100 shares at 580• Order is “executed” – you sold 100 shares to someone else

somewhere• Mechanically, how do you provide the 100 shares to the buyer?

• You borrow the 100 shares from an institutional holder• You provide collateral equal to the value of the stock ($58,000) and

perhaps a little more collateral in case the stock price goes up• You mark to market

• If stock goes to 585, you send $500 more in cash to lender• If stock goes to 575, lender sends you $500 in cash

• Where do you get the $58,000? • The buyer gives you $58,000 and you pass that through to the

stock lender• On some future date, you buy 100 shares at say 550, paying $ 55,000

which you receive back from the stock lender when you return the 100 shares to the lender

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Short Sale Mechanics: 100 shares of GOOG at 580

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Short seller Stock buyer100 shares

$ 58,000

XYZ University Endowment Short seller100 shares

$ 58,000

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Overlapping Generations Structure

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• All agents live two periods

• Born in period 1 and buy a portfolio (s, u)

• Live (and die) in period 2 and consume

• At time t

• The (t-1) generation is in period 2 of their life

• The (t) generation is in period 1 of their life

• So, they “overlap”

t1 t2 t3 t4

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How many are arbitrageurs? How many are noise traders?

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• Total number of traders is the same as the number of real numbers between zero and one - an infinite number

• “Measure” means the size of any interval• Examples

• The measure of the interval between 0 and ½ is ½

• The measure of a single point (a single number) is zero

• The measure of the interval between zero and one is 1

• Think of it as a fraction of the entire interval• Measure of noise traders is µ and measure of

arbitrage traders is 1 - µ. That is, the fraction of noise traders is µ and everybody else is an arbitrage traders

0 1

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What is a noise trader?

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• Pt+1 is the price of the risky asset at time t+1

• Ρt+1 is the “mean misperception” of pt+!

Ρt+!

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What is an arbitrage trader?

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• Arbitrage traders “correctly” perceive the true distribution of pt+1

• There is “systematic” error in estimation of future price, pt+1

• But, arbitrageurs face risk unrelated to the “true” distribution of pt+1

• If there were no “noise traders,” then there would be no variance in the price of the risky asset…..but, there are noise traders, hence the risky asset is a risky asset

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Arbitrageur versus Traders

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• Arbitrageurs expectations are “correct;” noise traders expectations are “biased”

Correct mean of pt+1

Difference is ρt+1

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The Main Issue

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• What happens in equilibrium

• Undetermined

• Some forces make pt > 1, some forces push pt < 1, result is indeterminant

• Who makes more profit, arbitrageurs or noise traders?

• Depends

• But, it is perfectly possible for arbitrageurs to make more!

• Survival?

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The Price of a Risky Asset

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2

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1

2

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rrrrp tt

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When Do Noise Traders Profit More Than Arbitrageurs?

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• Noise traders can earn more than arbitrageurs when ρ* is positive

• Meaning when noise traders are systematically too optimistic

• Why?• Because they have relatively more of the

risky asset than the arbitrageurs• But, if ρ* is too large, noise traders will

not earn more than arbitrageurs• The more risk averse everyone is (higher

λ in the utility function), the wider the range of values of ρ for which noise traders do better than arbitrageurs

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What Does Shleifer Accomplish?

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• Given two assets that are “fundamentally” identical, he shows a logic where the market fails to price them identically

• Assumes “systematic” noise trader activity

• Shows conditions that lead to noise traders actually profiting from their noise trading

• Shows why arbitrageurs could have trouble (even when there is no fundamental risk)

FEEDBACK MODELS

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Feedback Models

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• Central idea is that a price higher than efficient price might have “real” effects. What are they?

• Hirshleifer et al. is one example of several attempts to show feedback effects

HIRSHLEIFER ET AL.

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Hirshleifer et al. on Feedback

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• Three dates

• Eight kinds of investors

• Early and late

• Informed and uninformed

• Rational and irrational

• Mainly rational and irrational (noise) traders

• One firm with equity shares

• Trades at period 1 and period 2

• Payoff in period 3

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Hirshleifer Payoff Function

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• In period 3, payoff is:

F = θ + ε + δ

ε really plays no role, so lets simplify to:

F = θ + δ

δ depends upon workers investment in the firm

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Benefit to stakeholder

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• The stakeholder “thinks”:

μθ = E(θ|P1P2)

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How do noise traders affect the real economy?

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“ …we assume that the prices generated at dates 1 and 2 have an effect on cash flows generated at

date 3”

Profit to the “stakeholder” of investing X amount:

Maximizing this by choice of X:

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Recalling

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Now assume: δ(X) = C2X this is the reality

Date 3 cash flow to the firm:

F = θ + ε + k E(θ|P1P2)

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Irrational Traders?

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• They believe, irrationally, that the security pays off:

η + ε• Early informed irrational traders observe

the value of η at date 1• Late informed irrational traders observe

the value of η at date 2

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So what happens in Hirshleifer Etal

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• Early irrational, but informed, traders benefit because later irrational traders buy and the early traders “know it”

• Sometimes, early informed traders benefit as well

• Late irrational traders get smashed

SUBRAHMANYAM & TITMAN (2001)

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Subrahmanyam & Titman (2001)

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• Feedback into “Cascades”

• What is a cascade?

• Bandwagon effect

• Like a social network

• Could be an “operating system”

• The idea is that you benefit if others adopt

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Mechanics of S-T article

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• Payoff Function: F + δ (with no growth)

• Growth is G

• Total payoff: F + δ + G

• ρ1,ρ2 describe the fraction of stakeholder payoff ascribed to no growth and growth payoff

• Benefit = ρ1 (F + δ) + ρ2 G (this is the benefit to the stakeholder; same for each stakeholder)

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Cost of Association

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• S-T calls this a “reservation wage”• Compare B to wi for each I

• G = G* - (N-1)r where (N-1) are those who do not associate with the firm; as those increase G falls

• Benefit = ρ1 (F + δ) + ρ2 [G* -(N-1)r]