asset pricing kernels

Lecture 4Asset Pricing Kernels From last lecture:1111'()11()0();'()'()()'()'()'()i e e i t t ft t t t t t t t u C E R E mR E mR R R R u C u C p E x p E mx u C u C m u C βββ++++ ==⇒==− ===m is a measure of aggregate discomfort[]t+1t+1f f1i 1t 11x x If asset is riskless, its price should be R 1+r ()()1() first restriction on m ()For any risky asset, p means all assets can be priced by defining t fi t t i t t t t p E mx x E m soE m R E x R p E m x ++++====←==t+1a single mEach asset-pricing model corresponds to different choices of m.- We have not assumed complete markets.- We have not assumed representative investor.- We have not restricted the payoff distributions or the form of the utility function. - We are not restricted to a one-period model t 0t t=0E [βu(c )]∞∑- Investors can have non-marketable assets (e.g. human capital).Appropriate risk-adjusted rate1()f R E m = What affects how R f is set in equilibrium?1t+1t t+111t 2111,11Assume: u(c) = ,'()where 0()1-c consumption growth is log normally distributed c c ln ln ln ln c ln ln ,ln 11()f t t t t t t t t f t t t t t c u c c R c c c c c n E c c R E m c E c γγγγγσβ−−++++++=>=∆==−∆∆∆== ∼2211,,ln ln 22211f t 1f f define :ln ,1ln ln 2more impatient ( increases)r increasesE ln increases r increases(high r induces savi t t t t f t f t f t E C C f t t t t t r R e R e e r E c c c γδγγσδβγδγσδ++−−−∆+∆−+++===⇒=+∆−∆⇒∆⇒2t 1f ngs higher consumption growth)ln increases r decreasesIf is high, then investor prefers smooth consumption more. Whenconsumption growth is volatile, investor wants to smooth consumption so t c σγ+⇒∆⇒ he is more willing to save (precautionary savings).Risk correction to price[]11t+1t+1()()cov()()()()cov(If cov()0high (low ()like insurance)cov '(),cov()'()'() is inversely related to c, so m is inversely related to c .If an asset's t t t p E mx E mx mx E m E x E x p mx R mx p E r u c x mx u c u c β++==+=+>⇒−=payoff covaries positively with consumption low p.This asset won't help you smooth consumption, so it looks risky.Risk correction to expected returns1()1()()cov(,)1cov(,)()()()E mR E m E R m R m R E R E m E m E ⇒==+=−f ()cov(,)If an asset's return covaries positively with consumption, itcovaries negatively with m, so E(R) > R .f f R R R m R =−Risk adjustmentIdiosyncratic riskLet R be very volatile but still have cov(m,R) = 0.Then E(R) = R fWhat matters is the projection of R onto m.ˆˆi i i i s s sR m e αβ=++i cov(,)()()cov(,)var()()var()()the factor is the pricing kernelis the price of riskIs + or - ? What about ?Mean-variance (efficient) frontier1(f f i mm m i CAPMm R E R R E m m R m E R R SMLm E m s E mR βλλβλβ=− −=+← =,,)1()()()()()()()()()()()()()()()()()1()second restriction on m ()i i i i m R i i f m R i i f e e i i e i e f i E m E R R m m E R R R E m m E R R R E m m E R R E m E R m R R ρσσσρσσσσσσσ=+=−∗−≤≤ ≥←()()e i e i E R R σ is asset i’s Sharpe ratio. This relation is used to construct the Hansen-Jaganathan bound, a region of permissible values for the moments of m.• gives a region where stocks can beE(r)R fσ(R)ρm,R i captures the degree of systematic risk.If ρm,R i = -1 , asset is perfectly negatively correlated with m and perfectly positively correlated with consumption (no idiosyncratic risk). These assets receive the highest expected return.()()()()i i f m E R R R CML E m σσ=+←You can reach any position on the efficient frontier if you have two assets on it.Any efficient portfolio carries all pricing information!,()() with respect to any efficient portfolio i f mv i mv f E R R E R R SML β =+−←()()() slope of CML ()()i f f i E R R m R m R E m σσσ−==←Again, let u’(c) = c -γ, lognormal consumption growth.big γSteep slope if economy is risky (consumption growth is volatile) and if consumers are very risk averse.Time-varying expected returns[]111,1cov ,()()t t t t t f t t m R E R R E m ++++=−Can have predictable returns as long as it is explained by changing expected consumption growth, changing covariance of return with consumption growth, or changing risk aversion.Present Value,'()'()t j j t t t j j t t t j u c P E d u c m β∞++=+=∑。