Asset Pricing John H. Cochrane June 12, 2000

so
SECTION 4.3 AN ALTERNATIVE FORMULA, AND X ∗ IN CONTINUOUS TIME
b = Σ −1 [p−E(x ∗ )E(x)] .
Ifariskfreerateistraded,thenweknowE(x ∗ )=1/R f . If a riskfree rate is not traded –
if 1 is not in X – then this formula does not necessarily produce a discount factor x ∗ that is
in X. In many applications, however, all that matters is producing some discount factor, and
the arbitrariness of the risk-free or zero beta rate is not a problem.
This formula is particularly useful when the payoff space consists solely of excess returns
or price-zero payoffs. In that case, x ∗ = p 0 E(xx 0 ) −1 x gives x ∗ =0. x ∗ =0is in fact the
only discount factor in X that prices all the assets, but in this case it’s more interesting (and
avoids 1/0 difficulties when we want to transform to expected return/beta or other representations)
to pick a discount factor not in X by picking a zero-beta rate or price of the riskfree
payoff. In the case of excess returns, for arbitrarily chosen R f , then, (4.52) gives us
x ∗ = 1 1
−
Rf Rf E(Re ) 0 Σ −1 (R e −E(R e )); Σ ≡ cov(R e )
Continuous time
The law of one price implies the existence of a discount factor process, and absence of
arbitrage a positive discount factor process in continuous time as well as discrete time. At
one level, this statement requires no new mathematics. If we reinvest dividends for simplicity,
then a discount factor must satisfy
ptΛt = Et (Λt+spt+s) .
Calling pt+s = xt+s, this is precisely the discrete time p = E(mx) that we have studied all
along. Thus, the law of one price or absence of arbitrage are equivalent to the existence of a
or a positive Λt+s. The same conditions at all horizons s are thus equivalent to the existence
of a discount factor process, or a positive discount factor process Λt for all time t.
For calculations it is useful to find explicit formulas for a discount factors. Suppose a set
of securities pays dividends
and their prices follow
Dtdt
dpt
= µ tdt + σtdzt
pt
where p and z are N × 1 vectors, µ t and σtmay vary over time, µ(pt ,t,other variables),
E (dztdz0 t)=I and the division on the left hand side is element-by element. (As usual, I’ll
drop the t subscripts when not necessary for clarity, but everything can vary over time.)
We can form a discount factor that prices these assets from a linear combination of the
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