Asset Pricing John H. Cochrane June 12, 2000

CHAPTER 4 THE DISCOUNT FACTOR
many contingent claims economies consistent with our observations.
Finally, the absence of arbitrage is another very weak characterization of preferences. The
theorem tells us that this is enough to allow us to use the p = E(mx) formalism with m>0.
As usual, this theorem and proof do not require that the state space is RS . State spaces
generated by continuous random variables work just as well.
4.3 An alternative formula, and x ∗ in continuous time
In terms of the covariance matrix of payoffs,
Just like x ∗ in discrete time,
x ∗ = E(x ∗ )+[p−E(x ∗ )E(x)] 0 Σ −1 (x−E(x)).
dΛ ∗
Λ ∗ = −rf dt −
prices assets by construction in continuous time.
µ
µ + D
0
− r Σ
p −1 dz.
Being able to compute x∗ is useful in many circumstances. This section gives an alternative
formula in discrete time, and the continuous time counterpart.
A formula that uses covariance matrices
E(xx0 ) in our previous formula (4.51) is a second moment matrix. We typically summarize
data in terms of covariance matrices instead. Therefore, a convenient alternative formula
is
where
x ∗ = E(x ∗ )+[p−E(x ∗ )E(x)] 0 Σ −1 (x−E(x)) (52)
Σ ≡ E ¡ [x−E(x)] [x−E(x)] 0¢
denotes the covariance matrix of the x payoffs. (We could just substitute E(xx 0 )=Σ +
E(x)E(x 0 ), but the inverse of the sum is not very useful.) We can derive this formula by
postulating a discount factor that is a linear function of the shocks to the payoffs,
x ∗ = E(x ∗ )+(x−E(x)) 0 b,
and then finding b to ensure that x ∗ prices the assets x :
p = E(x ∗ )E(x)+E £ (x−Ex)x 0¤ b
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