The price of a generic asset can be written as
pt=ℂ𝕆𝕍t(mt+1,xt+1)+
1

1+Rf
𝔼t(xt+1)
where pt is the price of the asset at time t; xt+1 is the payoff of the asset at time t+1; Rf indicates the return on the risk-free asset; mt+1 is the stochastic discount factor; and 𝔼t and ℂ𝕆𝕍t denote the conditional expectation and covariance at time t, respectively.

Assume that the stochastic discount factor is a linear function of the market returns R
M
t+1
, that is
mt+1=θ1−θ2R
M
t+1
where both parameters are positive, θ10,θ20. If the market returns R
M
t+1
are uncorrelated with the asset payoff xt+1, then:

Question

The price of a generic asset can be written as
pt=ℂ𝕆𝕍t(mt+1,xt+1)+
1

1+Rf
𝔼t(xt+1)
where pt is the price of the asset at time t; xt+1 is the payoff of the asset at time t+1; Rf indicates the return on the risk-free asset; mt+1 is the stochastic discount factor; and 𝔼t and ℂ𝕆𝕍t denote the conditional expectation and covariance at time t, respectively.

Assume that the stochastic discount factor is a linear function of the market returns R
M
t+1
, that is
mt+1=θ1−θ2R
M
t+1
where both parameters are positive, θ1>0,θ2>0. If the market returns R
M
t+1
 are uncorrelated with the asset payoff xt+1, then:

BlackTom AI · Accepted Answer

The question asks us to assess what happens to the asset price when the stochastic discount factor mt+1 is modeled as a linear function of the market return RM,t+1 and the market return is uncorrelated with the asset payoff xt+1.

First, recall the pricing relation provided: pt = Covt(mt+1, xt+1) + (1 / (1 + Rf)) E_t(xt+1).

Next, compute Covt(mt+1, xt+1) given mt+1 = θ1 − θ2 RM,t+1. Since Covt(θ1, xt+1) = 0 (θ1 is a constant) and Covt(−θ2 RM,t+1, xt+1) = −θ2 Covt(RM,t+1, xt+1), we get Covt(mt+1, xt+1) = −θ2 Covt(RM,t+1, xt+1).

Now apply the assumption: RM,t+1 is uncorrelated with the asset payoff xt+1, which implies Covt(RM,t+1, xt+1) = 0. Therefore Covt(mt+1, xt+1) = 0.

With the covariance term vanishing, the pricing formula reduces to pt = (1 / (1 + Rf)) E_t(xt+1).

Thus, the asset price equals the discounted expected payoff, with no additional premium or discount arising from covariance with the stochastic discount factor. In other words, the pricing relation implies no risk premium beyond the risk-free discount when the payoff is uncorrelated with the market return that enters the SDF.

Why the other possibilities (if any) would be incorrect: since the only other term in the price expression is the covariance term, any nonzero covariance would introduce a premium or discount beyond the simple discounted expected payoff. However, the given assumption eliminates that term, so the statement that the price is the discounted expected payoff with no extra premium is correct under these conditions.

BU.232.630.W1.SP25 Quiz 2 solutions

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