复旦大学：《投资学讲义》（英文版） Chapter 17: Conditioning information

Part 3: deep discussion: Estimating and evaluating sset pricing models Chapter 17 Conditioning information Fan longzhen

Chapter 17: Conditioning information Fan Longzhen Part 3: deep discussion: Estimating and evaluating asset pricing models

Discount model is in terms of conditional moments The first order condition is p,u(C (1)=BE[v(c1)x+1 The expectation is conditional expectation on investors time t information The basic pricing equation is P,=E,(mrD) If the payoff were independent and identically distributed over time then the conditional expectation is the same as unconditional expectation, we would not worried about the distinction Because the information may be not observable to economist describing the conditional distribution is not possible, we eventually have to think about unconditional information Unconditional moments are interesting themselves, for example, we may concern why average stock return are high than risk free rate

Discount model is in terms of conditional moments • The first order condition is • The expectation is conditional expectation on investor’s time t information; • The basic pricing equation is • If the payoff were independent and identically distributed over time, then the conditional expectation is the same as unconditional expectation, we would not worried about the distinction; • Because the information may be not observable to economist, describing the conditional distribution is not possible, we eventually have to think about unconditional information; • Unconditional moments are interesting themselves, for example, we may concern why average stock return are high than risk free rate. '( ) [ '( ) ] t t = t t+1 t+1 p u c βE u c x ( ) t = t t+1 t+1 p E m x

Scaled payoffs From the pricing model p,=E,(m,,,1 Take unconditional expectations to obtain E(p,)=E(m1+x) Multiply the payoff and price by any variable or instruments -t observed at time t, and take expectations E(p, -)=E(m,* 21) Sufficiency of adding scaled returns ⅡB()=B0mx)E1→月=B(mx1)

Scaled payoffs • From the pricing model • Take unconditional expectations to obtain • Multiply the payoff and price by any variable or instruments observed at time t, and take expectations • Sufficiency of adding scaled returns: • If ( ) t = t t+1 t+1 p E m x ( ) ( ) t = t+1 t+1 E p E m x t z ( ) ( ) t t t 1 t 1 t E p z E m x z = + + ( ) ( ) ( ) t t t 1 t 1 t t t t t 1 t 1 t E z p E m x z z I p E m x I = + + ∀ ∈ ⇒ = + +

Conditional and unconditional models in discount factor language As an example, consider the Capm. m=a-bR Where rl is the return on the market or wealth portfolio · From equation =n+E, (RD 1=E(m,, 1=(m )R b E, (R4D-R/ RO(R4) Above equation shows explicitly that a and b must vary over time, as E, (R), 0, (R +D, R vary over time; model with time-varying weights of the form m+=a-b, K If it is to price asset conditionally the capm must be a linear facto Form the conditional pricing model 1=El(- RDR And take unconditional expectation, we have

Conditional and unconditional models in discount factor language • As an example, consider the CAPM, • Where is the return on the market or wealth portfolio • From equation • Above equation shows explicitly that a and b must vary over time, as • vary over time; • If it is to price asset conditionally, the CAPM must be a linear factor model with time-varying weights of the form • Form the conditional pricing model • And take unconditional expectation, we have W m = a − bR W R ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ − = = + ⇒ ⎪⎩ ⎪ ⎨ ⎧ = = + + + + + + ( ) ( ) ( ) 1 1 ( ) 1 ( ) 1 2 1 1 1 1 1 W t t f t f t W t t W t t f t f t t t W t t t R R E R R b bE R R a E m R E m R σ f t W t t W Et( Rt ), ( R 1), R 2 +1 σ + W t t tRt m a b +1 = − +1 1 [( ) ] = − +1 t+1 W Et at btRt R

continued =Ela, +6, RMR E(a, E(R D-E(b E(R+R+)+ cov(a, R-1)-COV(b, RmR The conditional model 1=e[(e(a, )-e(b r+ir-1 only holds if the covariance terms happen to be zero, this is generally not right On the other hand, suppose that it is true that a, b, are constant over time, even if R, R are not i.i. d, a, can be constant over time Then [a-bRAdR es implies =Ela-bRIMR

continued • The conditional model • only holds if the covariance terms happen to be zero, this is generally not right. • On the other hand, suppose that it is true that are constant over time, even if are not i.i.d., can be constant over time. Then • Does implies ( ) ( ) ( ) ( ) cov( , ) cov( , ) 1 [( ) ] 1 1 1 1 1 1 1 1 + + + + + + + + = − + − = + t W t t t t t W t t t t t W t t t E a E R E b E R R a R b R R E a b R R 1 [( ( ) ( ) ) ] = − +1 t+1 W E E at E bt Rt R t t a ,b W t f Rt , R t t a ,b 1 [( ) ] = − +1 t+1 W Et a bRt R 1 [( ) ] = − +1 t+1 W E a bRt R

Conditional and unconditional in an expected return -beta model To put the observation into beta pricing language E, (RA D=R+B h Does not imply iply E(R+D=y+B2 Even if i=h is a constant, you cannot derive the pricing model also because B=cov(R I,f )/var()+ e(cov, r ,)/var, (iD)

Conditional and unconditional in an expected return-beta model • To put the observation into beta pricing language • Does not imply • Even if is a constant, you cannot derive the pricing model also, because t i t f t i Et( Rt+1) = R + β λ γ β λ i i Et( Rt+1) = + λt = λ cov( , )/ var( ) [cov ( , )/ var ( )] = +1 +1 +1 ≠ +1 t+1 t t+1 i t t t t i t i β R f f E R f f

a partial solution: scaled factors How to handle the parameters a, b, change with time a partial solution is to model the dependence of parameters a, b on variables in the time t information set et -t be a vector of variables observed at time t In particular, a linear function of zt The discount factor is now as follows +1=a(-1)+b(=,)f a0+a2+(b+b2 0+a-,+ bf(1+b(-,f1) In place of the one-factor model with time-varying coefficients, we have a three-factor(=,f 1, -f) with fixed coefficients

A partial solution: scaled factors • How to handle the parameters change with time ; • A partial solution is to model the dependence of parameters on variables in the time t information set; • Let be a vector of variables observed at time t; • In particular, a linear function of ; • The discount factor is now as follows: • • In place of the one-factor model with time-varying coefficients, we have a three-factor with fixed coefficients. t t a ,b t t a ,b t z t z ( ) ( ) ( ) ( ) 0 1 0 1 1 1 0 1 0 1 1 1 1 + + + + + = + + + = + + + = + t t t t t t t t t t t a a z b f b z f a a z b b z f m a z b z f ( , , ) t t+1 t t+1 z f z f

continued Since the coefficients are now fixed we can use the scale factor model with unconditional moments P=E[(0+a121+bf1+b(,f1)x1]→ E(p)=E[(an+a121+bf1+b()

continued • Since the coefficients are now fixed, we can use the scale-factor model with unconditional moments: ( ) [( ( )) ] [( ( )) ] 0 1 0 1 1 1 1 0 1 0 1 1 1 1 + + + + + + = + + + = + + + ⇒ t t t t t t t t t t t t t E p E a a z b f b z f x p E a a z b f b z f x