复旦大学：《投资学讲义》（英文版） Chapter 13 Factor pricing model

Chapter 13 Factor pricing model Fan longzhen

Chapter 13 Factor pricing model Fan Longzhen

Introduction The consumption-based model as a complete answer to most asset pricing question in principle, does not work well in practice This observation motivates effects to tie the discount factor m to other data Linear factor pricing models are most popular models of this sort in finance They dominate discrete-time empirical work

Introduction • The consumption-based model as a complete answer to most asset pricing question in principle, does not work well in practice; • This observation motivates effects to tie the discount factor m to other data; • Linear factor pricing models are most popular models of this sort in finance; • They dominate discrete-time empirical work

Factor pricing models Factor pricing models replace the consumption-based expression for marginal utility growth with a linear model of the form m=a+b'f+ The key question: what should one use for factors

Factor pricing models • Factor pricing models replace the consumption-based expression for marginal utility growth with a linear model of the form • The key question: what should one use for factors 1 1 ' t+ = + t+ m a b f t+1 f

Capital asset pricing model ( capm CAPM is the model m=a+bR, rw is the wealth portfolio return Credited Sharpe(1964)and Interner(1965), is the first, most famous and so far widely used model in asset pricing Theoretically, a and b are determined to price any two assets, such as market portfolio and risk free asset Empirically, we pick a, b to best price larger cross section of assets We dont have good data, even a good empirical definition for wealth portfolio, it is often deputed by a stock index We derive it from discount factor model by (1 )two-periods, exponential utility, and normal returns (2 )infinite horizon, quadratic utility, and normal returns ()log utility (4) by seeing several derivations, you can see how one assumption can be traded for another. For example, the Capm does not require normal distributions, if one is willing to swallow quadratic utility instead

Capital asset pricing model (CAPM) • CAPM is the model , is the wealth portfolio return. • Credited Sharpe (1964) and Linterner (1965), is the first, most famous, and so far widely used model in asset pricing. • Theoretically, a and b are determined to price any two assets, such as market portfolio and risk free asset. • Empirically, we pick a,b to best price larger cross section of assets; • We don’t have good data, even a good empirical definition for wealth portfolio, it is often deputed by a stock index; • We derive it from discount factor model by • (1)two-periods, exponential utility, and normal returns; • (2) infinite horizon, quadratic utility, and normal returns; • (3) log utility • (4) by seeing several derivations, you can see how one assumption can be traded for another. For example, the CAPM does not require normal distributions, if one is willing to swallow quadratic utility instead. w m = a + bR w R

Two-period quadratic utility Investor have a quadratic preferences and live only for two periods, U(c,c)=-( C C,-1-C R:(1-c) B*,B(w,-c,) C-C C-C b, R

Two-period quadratic utility • Investor have a quadratic preferences and live only for two periods; [( *) ] 2 1 ( *) 2 1 ( , ) 2 1 2 1 U c c c c E c c t t+ = − t − − β t+ − w t t t w t t t t t t t t w t t t t t t a b R R c c w c c c c c c R w c c c c c c U c U c m 1 1 1 1 1 1 * * ( ) * ( ) * * * '( ) '( ) + + + + + + = + − − + − − = − − − = − − = = β β β β β

Exponential utility, Normal distributions We present a model with consumption only in the last period, utility is EU(C=E-e -ac If consumption is normally distributed, we haveE(U(c)=-e- a(cha o'lc) Investor has initial wealth w. which invest in a set of risk-free assets with return Rand a set of risky assets paying return r Let y denote the mount of wealth w invested in each asset, the budget constraint is c=y'R+yR w=y+yf Plugging the first constraint into the utility function, we obtain E(U(c) ce aly'R+yE(R)I+a/2y2y

Exponential utility, Normal distributions • We present a model with consumption only in the last period, utility is • If consumption is normally distributed, we have • Investor has initial wealth w, which invest in a set of risk-free assets with return and a set of risky assets paying return R. • Let y denote the mount of wealth w invested in each asset, the budget constraint is • Plugging the first constraint into the utility function, we obtain [ ( )] [ ] c E U c E e − α = − ( ) ( )/ 2 2 2 ( ( )) E c c E U c e − α + α σ = − f R w y y f c y R y R f f f ' ' = + = + y R y E R y y f f E U c e − + + Σ = − [ ' ( )] / 2 ' 2 ( ( )) α α

Exponential utility, Normal distributions--continued The optimal amount invested in risky asset IS y=Dy E(R-P Sensibly the investor invest more in risky assets if their expected returns are higher ess if his risky aversion coefficient is higher Less if assets are more risky The amount invested in risky assets is independent of level of wealth so we say absolute rather than relative( to wealth risk aversion Note also that these demand for risky assets are linear in expected returns Inverting the first-order conditions we obtain e(r)-R=a y=acoV(rr") If all investors are identical, then the market portfolio is the same as the individual portfolio, and also Ey gives the correlation of each return with pm R"=yR+yR

Exponential utility, Normal distributions--continued • The optimal amount invested in risky asset is • Sensibly, the investor invest more in risky assets if their expected returns are higher; • Less if his risky aversion coefficient is higher; • Less if assets are more risky. • The amount invested in risky assets is independent of level of wealth, so we say absolute rather than relative( to wealth risk) aversion; • Note also that these demand for risky assets are linear in expected returns. • Inverting the first-order conditions, we obtain • If all investors are identical, then the market portfolio is the same as the individual portfolio, and also gives the correlation of each return with α f E R R y − = Σ − ( ) 1 ( ) cov( , ) f w E R − R = αΣy = α R R Σy R y R y R m f f = +

Exponential utility, Normal distributions--continued Applying the formula to market return itself, we have E(R-R=aO(R) The model ties price of market risk to the risk aversion coefficient

Exponential utility, Normal distributions--continued • Applying the formula to market return itself, we have • The model ties price of market risk to the risk aversion coefficient. ( ) ( ) w f 2 w E R − R =ασ R

Quadratic value function Dynamic programming Since investor like for more than two periods, we have to use multi period assumptions, et us start by writing the utility function as this period consumption and next periods wealth: U=u(c)+BE Y(W+) His first-order condition is pr BE, LV W, + The discount factor is m1:=B V"(W) Suppose the value function is quadratic u'(c W-W* Then, we would have W,- -Bn 2(7-c;)-W* l'(c) u(Cr Bnw*. Bn(w-c once again u(c,) u'(c,) W 1/a.+ b, r

Quadratic value function, Dynamic programming • Since investor like for more than two periods, we have to use multi period assumptions; • Let us start by writing the utility function as this period consumption and next period’s wealth: • His first-order condition is • The discount factor is • Suppose the value function is quadratic • Then, we would have • Or , once again • U = u ( ct) + βEt V ( Wt+1 ） [ '( ) ] t = t t+1 t+1 p βE V W x '( ) '( )1 1 t t t u c V W m + + = β 2 1 1 ( *) 2 V (Wt+ ) = − Wt+ − W η W t t t t t t t t W t t t t R u c W c u c W u c R W c W u c W W m 1 1 * 1 1 '( ) ( ) '( ) * '( ) ( ) * '( ) + + + + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − ⎥ + − ⎦ ⎤ ⎢ ⎣ ⎡ = − − = − − = − βη βη βη βη W t t t R t m = a + b + 1

Quadratic value function, Dynamic programming-continued (1 ) the value function only depends on wealth. If other variables enter the value function, m would depend on other variables. The Icapm allows other variables in the value function and obtain more factors (other variable can enter the function, so long as they do not affect marginal utility value of wealth 2)the value function is quadratic, we wanted the marginal value function is linear

Quadratic value function, Dynamic programming-continued • (1) the value function only depends on wealth. If other variables enter the value function, m would depend on other variables. The ICAPM, allows other variables in the value function, and obtain more factors. • (other variable can enter the function, so long as they do not affect marginal utility value of wealth.) • (2) the value function is quadratic, we wanted the marginal value function is linear