Asset Pricing John H. Cochrane June 12, 2000

SECTION 6.1 FROM DISCOUNT FACTORS TO BETA REPRESENTATIONS
f = m, x*, R*
E(R i ) = α + β i’λ
m = b’f
proj(f|R) on m.v.f.
f = R mv
LOOP m exists
p = E(mx)
m = a + bR
R* is on m.v.f.
mv
R mv on m.v.f.
E(RR’) nonsingular R mv exists
Figure 18. Relation between three views of asset pricing.
6.1.1 Beta representation using m
p = E(mx) implies E(R i )=α + β i,mλm. Startwith
Thus,
1=E(mR i )=E(m)E(R i )+cov(m, R i ).
E(R i )= 1
E(m) − cov(m, Ri )
.
E(m)
Multiply and divide by var(m),define α ≡ 1/E(m) to get
E(R i µ µ i cov(m, R )
)=α +
−
var(m)
var(m)
= α + β
E(m)
i,mλm.
As advertised, we have a single beta representation.
For example, we can equivalently state the consumption-based model as: mean asset
returns should be linear in the regression betas of asset returns on (ct+1/ct) −γ .Furthermore,
the slope of this cross-sectional relationship λm is not a free parameter, though it is usually
treated as such in empirical evaluation of factor pricing models. λm should equal the ratio of
99