CLASS 1: ASSEt pricing. CAPM. Theory and Experiment Theory: The - - PDF document

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CLASS 1: ASSEt pricing.

• CAPM. Theory and

Experiment Theory: The economy

• Two dates (today, tomorrow), two period economy.
• J investors, I assets. is the number of units of asset i that

investor j holds. ( is the vector of i’s holdings).

• is the price of asset i today. ( is the price vector).
• is the payoff of asset i tomorrow. ( is the payoff vector).
• is the endowment of asset i. ( is i’s endowment vector).
• is agent j’s utility function. w is wealth/consumption.

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xi

j

pi Ri ei

j

u j(w) x j p R e j

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Theory: The INVEstor’s portfolio problem

• Solve:
• FOC:

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x j

Max E[u j(R'x j)]

{ }

s.t. p'x j ≤ p'e j E ∂[u j(R'x j)] ∂xi

j

" # $ % & ' ~ pi

Therefore

( → ((( xi

j = xi j(p)

Theory: Details

• FOC:
• Therefore

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∂E[u j(R'x j)] ∂xi

j

" # $ % & ' = λ jpi

MRSi,k

j =

∂E[u j(R'x j)] ∂xi

j

∂E[u j(R'x j)] ∂xk

j

= pi pk ∂E[u j(R'x j)] ∂xi

j

= ∂ ∂xi

j

π su j(R

s

∑

'(s)x j) = π sRi(s)u j '(R

s

∑

'(s)x j)

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Theory: Equilibrium

• Equilibrium consists of asset prices p and portfolio choices

such that:

• 1. Given prices choices are budget optimal
• 2. Prices are such that asset markets clear:

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x j x

j

∑

j(p) =

e

j

∑

j

Theory: Arrow-debreu Securities

• AD securities: iff i=s.
• Portfolio Choice FOC implications:
• Therefore

, or

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∂E[u j(R'x j)] ∂xi

j

= π sRi(s)u j '(R

s

∑

'(s)x j) = πiu j '(xi

j)

Ri(s) =1 πiu j '(xi

j)

π ku j '(xk

j) = pi

pk u j '(xi

j)

u j '(xk

j) =

pi πi pk π k

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Theory: Equilibrium

• Properties of equilibrium
• 1. State price probabilities are inversely related to aggregate wealth.
• 2. Representative agent when markets are complete.
• 3. Final wealth/consumption is perfectly rank-correlated among

individuals and with aggregate wealth.

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Theory: CAPM

• Take quadratic utility function, so that marginal utility, or MU
• 1. FOC:
• 2. Apply to the risk-free security:
• 3. Substitute
• 4. Define and , where is the aggregate

wealth and is the price of the market portfolio.

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MU = a + bw E[(a + bw)Ri]= λpi E[(a + bW)rf ]= λ aE(Ri pi )+ bE(w Ri pi ) = rfa +rfbE(w) r

i = Ri

pi r

M = 

w pM

 w

pM

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Theory: CAPM

• Then
• Use
• To derive
• r
• Apply to market

, or

• Therefore

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E(r

i)+ b

a E(  wr

i) = rf +rf

b a E(  w) E(  wr

i) = E( 

w)E(r

i)−Cov( 

w,r

i)

E(r

i)(1+ b

a E(  w))− b a Cov(  w,r

i) = rf (1+ b

a E(  w)) (E(r

M )−rf )(1+ b

a E(w)) = b a Cov(  w pM ,r

M )pM

E(r

i)−rf =

Cov(  w pM ,r

i)

(1+ b a E(  w)) b a pM

E(r

i)−rf = Cov(r M,r i)

Var(r

M )

E(r

M )−rf

( )

E(r

M )−rf

Var(r

M ) =

1 (1+ b a E(  w)) b a pM

CAPM

• Theory: In equilibrium, the

expected return on risky securities is solely determined by their covariance with aggregate risk (“beta”)

• Equivalent: the “market

portfolio” will give you maximum expected reward for its risk (risk=return variance), or the Sharpe ratio

• f the market portfolio is

maximal

• Sharpe ratio = Expected

return on portfolio minus riskfree rate / return standard deviation

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Experiments

• I. The Experiments
• Three markets (one riskfree, shortsales allowed), several pe-

riods.

• Three states; determine liquidation value at end of each pe-

riod; known probabilities. Security State X Y Z A 170 370 150 B 160 190 250 Notes 100 100 100 (Note: complete markets...)

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Experiments

Example Of This Experiment: 011126

Draw Subject Signup Endowments Cash Loan Exchange Type Type Reward A B Notes Rate (#) (franc) (franc) (franc) $/franc D 18 125 5 4 400 2200 0.04 18 125 2 8 400 2310 0.04

See web pages...

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Logout Exit Market

Markets -- Actual

Marketplace closes in 01:56:25 Cash on hand: $500.0 A (5) Item: A B (4) Item: B Notes (0) Item: Notes

• Navigation

Order Form Messaging View Transaction Table Marketplace A B Notes

Order Type: Market: A Price: $195.00 Units: 1 Total Value: $195.00

Messaging

Received Messages

mc: Markets will be called in 10 minutes mc: Markets will be called in 9 minutes mc: Markets will be called in 8 minutes mc: Markets will be called in 7 minutes

Message History

Price: 195.00

• CAPM PREDICTIONS
• The expected returns of each of securities A and B are

positively related to the betas of the securities

• The market portfolio is mean-variance efficient.
• The final portfolio of each individual should have risky

securities in the same proportion as in the market portfolio.

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Prices

• Prices of the two

securities are virtually the same.

• Expected payoff
• f A is higher,

i.e., expected return of A is higher.

• Precisely because

Beta(A)>Beta(B)

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Prices

1000 2000 3000 4000 5000 6000 7000 8000 9000 0.5 1 1.5 2 2.5 3 3.5 time (in seconds) state price density

State pricing: X becomes most expensive; Y cheapest; Z in-between

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Efficiency of market portfolio

• Compute Sharpe

ratio of market portfolio with each transaction.

• Take the

difference between Sharpe

• f the market

and the highest Sharpe ratio.

• CAPM predicts

this difference should be 0.

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INDIVIDUAL Behavior

• For each of the 8

periods plot each subject’s final holdings of asset A.

• Red dot indicates

the proportion of A in the market portfolio.

• Individuals are all
• ver the place.

Allocations do not improve with time.

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1 2 3 4 5 6 7 8 0.2 0.4 0.6 0.8 1 period proportion

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INDIVIDUAL Behavior

1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930313233343536 8 7 6 5 4 3 2 1 1 −1 subject period

AD holdings: no improvement in poor rank correlations among final wealths

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