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