复旦大学：《投资学讲义》（英文版） Chapter 12.2: General asset pricing model

Chapter 12.2: General asset pricing mode」 Fan longzhen July. 2003

Chapter 12.2: General asset pricing model Fan Longzhen July, 2003

Consumption-Based model and Basic Pricing model Basic question to decide for an investor ·(1) how much to save; (2) how much to consume ()what portfolio of assets to hold Pricing equation come from the first order condition for this decision

Consumption-Based Model and Basic Pricing Model • Basic question to decide for an investor: • (1) how much to save; • (2) how much to consume; • (3)what portfolio of assets to hold. • Pricing equation come from the first order condition for this decision

Marginal utility and its indicator Marginal utility, not consumption, is the fundamental measure of how you feel Theory of asset pricing is about how to go from marginal utility to observable indicators Consumption is an indicator of marginal utility When consumption is low, marginal utility is high a large index may be an indicator. a large index goes down, the wealth of investors goes down consumption goes down, marginal utility becomes

Marginal utility and its indicator • Marginal utility, not consumption, is the fundamental ,measure of how you feel. • Theory of asset pricing is about how to go from marginal utility to observable indicators. • Consumption is an indicator of marginal utility. When consumption is low, marginal utility is high. • A large index may be an indicator. A large index goes down, the wealth of investors goes down, consumption goes down, marginal utility becomes high

Basic Pricing equation Basic pricing problem is to price stream of any uncertainty cash flows; Any asset, for example, a stock, with price p and dividend d, next period. The payoff next period is xu 1 =p+d What is value of the payoff today for a typical investor We need a model to model typical investors utility happiness)of consumption +1=0(c)+ BE JU(C DI The utility functions may be a power utility form

Basic Pricing Equation • Basic pricing problem is to price stream of any uncertainty cash flows; • Any asset, for example, a stock, with price and dividend next period. The payoff next period is • What is value of the payoff today for a typical investor. • We need a model to model typical investor’s utility ( happiness) of consumption • • The utility functions may be a power utility form ( , ) ( ) [ ( ) ] t t+1 = t + t t+1 U c c U c βE U c γ γ − − = 1 1 1 ( )t t U c c t+1 p dt+1 t+1 = pt+1 + dt+1 x

Utility Utility function satisfy conditions (1)desire for more consumption, increasing function; (2)marginal utility is decreasing: ()impatience for time: B--subjective discount factor (4)aversion for risk curvature of utility function

utility • Utility function satisfy conditions: • (1) desire for more consumption, increasing function; • (2)marginal utility is decreasing; • (3) impatience for time: β--subjective discount factor; • (4) aversion for risk:curvature of utility function

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