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

7/26/12 ¡ CLASS 1: ASSEt pricing. CAPM. Theory and Experiment Theory: The economy • Two dates (today, tomorrow), two period economy. j • J investors, I assets. is the number of units of asset i that x i x j investor j holds. ( is the vector of i ’s holdings). p i • is the price of asset i today. ( is the price vector). p R i • is the payoff of asset i tomorrow. ( is the payoff vector). R j • is the endowment of asset i . ( is i ’s endowment vector). e i e j u j ( w ) • is agent j ’s utility function. w is wealth/consumption. 2 1 ¡

7/26/12 ¡ Theory: The INVEstor’s portfolio problem • Solve: Max E [ u j ( R ' x j )] { } x j s . t . p ' x j ≤ p ' e j • FOC: " % E ∂ [ u j ( R ' x j )] j = x i j ( p ) Therefore ~ p i ((( x i # & ( → j ∂ x i $ ' 3 Theory: Details " % ∂ E [ u j ( R ' x j )] = λ j p i • FOC: # & j ∂ x i $ ' ∂ E [ u j ( R ' x j )] j ∂ x i = p i j = • Therefore MRS i , k ∂ E [ u j ( R ' x j )] p k j ∂ x k ∂ E [ u j ( R ' x j )] = ∂ π s R i ( s ) u j '( R ∑ π s u j ( R '( s ) x j ) = ∑ '( s ) x j ) j j ∂ x i ∂ x i s s 4 2 ¡

7/26/12 ¡ Theory: Equilibrium x j • 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: j ( p ) = j ∑ ∑ x e j j 5 Theory: Arrow-debreu Securities • AD securities: iff i=s . R i ( s ) = 1 • Portfolio Choice FOC implications: ∂ E [ u j ( R ' x j )] π s R i ( s ) u j '( R '( s ) x j ) = π i u j '( x i ∑ j ) = j ∂ x i s • Therefore p i π i π i u j '( x i u j '( x i j ) j ) j ) = p i , or j ) = π k u j '( x k u j '( x k p k π k p k 6 3 ¡

7/26/12 ¡ 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. 7 Theory: CAPM • Take quadratic utility function, so that marginal utility, or MU MU = a + bw E [( a + bw ) R i ] = λ p i 1. FOC: 2. Apply to the risk-free security: E [( a + bW ) r f ] = λ aE ( R i ) + bE ( w R i 3. Substitute ) = r f a + r f bE ( w ) p i p i M =  i = R i w  r 4. Define and , where is the aggregate r w p i p M p M wealth and is the price of the market portfolio. 8 4 ¡

7/26/12 ¡ Theory: CAPM i ) + b b a E (  a E (  • Then E ( r wr i ) = r f + r f w ) E (  i ) = E (  i ) − Cov (  • Use wr w ) E ( r w , r i ) i )(1 + b w )) − b i ) = r f (1 + b a E (  a Cov (  a E (  E ( r w , r w )) • To derive Cov (  w , r i ) or p M b E ( r i ) − r f = a p M (1 + b a E (  w )) • Apply to market a Cov (  M ) − r f )(1 + b a E ( w )) = b w E ( r M ) − r f 1 b ( E ( r , r M ) p M M ) = a p M , or (1 + b p M Var ( r a E (  w )) i ) − r f = Cov ( r M , r i ) • Therefore ( ) E ( r E ( r M ) − r f Var ( r M ) 9 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 of the market portfolio is maximal • Sharpe ratio = Expected return on portfolio minus riskfree rate / return standard deviation 10 5 ¡

7/26/12 ¡ 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...) 11 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 0 400 2200 0.04 18 125 2 8 0 400 2310 0.04 See web pages... 12 6 ¡

7/26/12 ¡ Logout Exit Market Marketplace closes in 01:56:25 Navigation Markets -- Actual Cash on hand: $500.0 Marketplace A B Notes A (5) B (4) Notes (0) View Transaction Table Item: A Item: B Item: Notes Order Form Order Type: ������ Price: 195.00 � ���� �� � Market: A ������������ Price: $195.00 ����������� Units: 1 �������������� ����� Total Value: $195.00 Messaging 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 13 ��������������� ��������������� ������������������������������ ����������������������������� 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. 14 7 ¡

7/26/12 ¡ Prices • Prices of the two securities are virtually the same. • Expected payoff of A is higher, i.e., expected return of A is higher. • Precisely because Beta(A)>Beta(B) 15 Prices 3.5 3 2.5 state price density 2 1.5 1 0.5 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 time (in seconds) State pricing: X becomes most expensive; Y cheapest; Z in-between 16 8 ¡

7/26/12 ¡ Efficiency of market portfolio • Compute Sharpe ratio of market portfolio with each transaction. • Take the difference between Sharpe of the market and the highest Sharpe ratio. • CAPM predicts this difference should be 0. 17 INDIVIDUAL Behavior • For each of the 8 periods plot each 1 subject’s final holdings of asset A. 0.8 • Red dot indicates 0.6 proportion the proportion of A in the market 0.4 portfolio. 0.2 • Individuals are all over the place. 0 Allocations do not improve with time. 1 2 3 4 5 6 7 8 period 18 9 ¡

7/26/12 ¡ INDIVIDUAL Behavior 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930313233343536 − 1 0 1 8 7 6 5 4 3 2 1 subject period AD holdings: no improvement in poor rank correlations among final wealths 19 10 ¡