Consumption and investment decision If today's price of the payoff Pt. how much will the typical investor buy and sell? Denote by e the original consumption level(if the investor buy none of the asset), and denote by a the amount of the asset he choose to buy The problems is maxu(c,)+E, LSu(ci+D)] s.t. c=e-ps C,1=e,,+x The first-order condition for an optimal consumption and portfolio choice p, u'(c)=E,lBu(cx .2)
Consumption and investment decision • If today's price of the payoff Pt, how much will the typical investor buy and sell? • Denote by e the original consumption level(if the investor buy none of the asset), and denote by the amount of the asset he choose to buy. The problems is • The first-order condition for an optimal consumption and portfolio choice • Or ξ { } ξ ξ β ξ 1 1 1 1 . . max ( ) [ ( )] + + + + = + = − + t t t t t t t t t c e x s t c e p u c E u c '( ) [ '( ) ] ( 1.1 ) t t = t t+1 t+1 p u c E βu c x [ '( ) / '( )] ( 1.2 ) t t t 1 t 1 t p E u c x u c = β + +
Pricing formula p,=ELBu'(cm/u'(c,) Equation (1. 2 )is the central pricing formula It relate one endogenous variable, price, to the two other two exogenous variables, consumption and payoff. You can interpret consumption in terms of more fundamental, but we stop now
Pricing formula • Equation (1.2) is the central pricing formula. • It relate one endogenous variable, price, to the two other two exogenous variables, consumption and payoff. • You can interpret consumption in terms of more fundamental, but we stop now. [ '( ) / '( )] ( 1.2 ) t t t 1 t 1 t p E u c x u c = β + +
Stochastic discount factor If we define the stochastic discount factor m, as m,41 β u(cD The basic pricing formula can simply expressed as uIc P=E,(m21x1) Why called discount factor. if no uncertainty we can express price via the standard present value formula P1=x/(1+R,) m, is just like 1/(1+Rr-traditional discount factor It says something deep, all assets use a stochastic factor to adjust risk m,l are also called marginal rate of substitution, pricing kernel change of measure, or state-price density
Stochastic discount factor • If we define the stochastic discount factor as • The basic pricing formula can simply expressed as • Why called discount factor, if no uncertainty, we can express price via the standard present value formula • is just like 1/(1+R f)—traditional discount factor. • It says something deep, all assets use a stochastic factor to adjust risk; • are also called marginal rate of substitution, pricing kernel, change of measure, or state-price density. mt '( ) '( )1 1 t t t u c u c m + + = β ( ) t = t t+1 t+1 p E m x /( 1 ) t t R f p = x + mt+1 mt+1
Many other names of mt+ Marginal rate of substitute · Pricing kernel Change of measure · State-price density For gross return R,+=x+ p,, the pricing formula becomes 1=E、m2:3R)
Many other names of mt+1 • Marginal rate of substitute; • Pricing kernel; • Change of measure; • State-price density • For gross return , the pricing formula becomes t t pt R x / +1 = +1 1 ( ) = Et mt+1Rt+1
Prices, payoff, and notation This pricing formula is for any financial asset, stock, bond, option, and future. swap price p(t payoff p(t+1) stock p(t p(t+1-d(t+1) return Rt+1) price-dividend ratio p excess return one-period bond(t) c max(S(T)-1 Prices or returns can also be real or nominal The only difference is we use a real or nominal discount factor If p and X are nominal, we can create real pricing formula as uc u'( ∏ p,/I =E, C B P,=ECB (c))∏ u(c
Prices, payoff, and notation • This pricing formula is for any financial asset, stock, bond, option, and future, swap. • Prices or returns can also be real or nominal. The only difference is we use a real or nominal discount factor. • If P and X are nominal, we can create real pricing formula as: price p(t) payoff p(t+1) stock p(t) p(t+1)+d(t+1) return 1 R(t+1) price-dividend ratio p(t)/d(t) (p(t+1)/d(t+1)+1)d(t+1)/d（t） excess return 0 one-period bond p（t） 1 option c max(S（T）-1) ] '( ) '( ) / [ 1 1 1 + + + ⎟ Π ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Π = t t t t t t t x u c u c p E β ] '( ) '( ) [ 1 1 1 + + + Π Π ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = t t t t t t t x u c u c p E β