# Covariance, stochastic discount factor (SDF) and risk aversion

John Cochrane states, that if the covariance between the stochastic discount factor and the payoff is zero - then risk aversion should have no impact on the pricing. I do not fully understand why this is the case. Since in the pricing formula P = E(Mx) with M as the SDF and x as they payoff if I have a different risk aversion should that not still change the price? Or is this statement with respect that there is just no risk premium in this case even with different risk aversion?

• The question shouldn’t have been edited this way Oct 15 '20 at 18:06
• Please don't destroy your question. @KeSchn spent time on helping not just you but also other users. Don't let that time go to waste. Oct 15 '20 at 18:14

The Euler equation is $$p=\mathbb{E}[MX]=\mathbb{C}\text{ov}(M,X)+\mathbb{E}[M]\mathbb{E}[X].$$ If the payoff $$X$$ doesn't covary with the stochastic discount factor (or pricing kernel), then it does not have systematic risk. Because $$\mathbb{E}[M]=\frac{1}{R_f}$$, where $$R_f$$ is the risk-free rate, you indeed obtain $$p=\frac{\mathbb{E}[X]}{R_f}.$$ As you see, you compute the (real world) expected payoff and simply discount at the risk-free rate. The agent's utility (be it time separable, recursive or whatever) and risk aversion does not enter the pricing formula.
Remember that in the CAPM only comovement with the market (as measured by market beta) is priced. Well, in the CAPM, the SDF is a linear function of the market returns. Thus, the same intuition applies here: if the payoff of your asset doesn't covary with the market ($$\beta=0$$), then the risk-free rate is the appropriate discount rate -- regardless of potential idiosyncratic risk. Only covariance with the SDF (market) is priced.
• But Cochrane defines $R_f$as 1/E(M), so the ratio of marginal utilities and hence, the risk aversion still enters the pricing formula no? Oct 15 '20 at 16:19
• @Kermittfrog Interesting point. Do you mean recursive Epstein Zin with the elasticity of intertemporal substitution being disentangled from the risk aversion coefficient? These models also have $\frac{1}{\mathbb{E}[M]}$. By definition, the risk-free asset has $p=1$ and returns $r_f$. The Euler equation then immediately gives you $r_f=\frac{1}{\mathbb{E}[M]}$. So, for $r_f\neq\frac{1}{\mathbb{E}[M]}$, you’d need the Euler equation to be violated (i.e. consider models with arbitrage?) Also $M>0$ exists and $p=\mathbb{E}[MX]$ holds, even if there’s no equilibrium for an investor. Oct 15 '20 at 17:06