How SDF Covariance Determines an Asset’s Risk Premium
Summary
The document explains how the stochastic discount factor prices a payoff through its expected value and its covariance with that payoff. When covariance is zero, the payoff has no exposure to the priced source of risk represented by the discount factor. Its price is then its expected payoff discounted at the risk-free rate, so changing risk aversion does not add a risk premium through this channel.
The CAPM provides an analogy: when the discount factor is a linear function of market returns, an asset with zero market beta has no market-related premium, even if it retains idiosyncratic uncertainty. This conclusion concerns pricing under the stated covariance condition and the given pricing framework. It does not imply that all risk is absent, or that risk aversion never matters; it says that risk uncorrelated with the SDF does not affect the price through a covariance premium.
Key ideas
- The pricing equation separates the expected payoff component from the covariance with the SDF.
- A payoff uncorrelated with the SDF is discounted at the risk-free rate in this framework.
- Only risk that covaries with the pricing kernel contributes a risk premium here.
- Zero market beta is analogous when the SDF is a linear function of market returns.
- Idiosyncratic risk may remain even when it does not affect price through SDF covariance.
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# Covariance, stochastic discount factor (SDF) and risk aversion
# 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?
## Answer by Kevin (score 4)
https://quant.stackexchange.com/a/58712
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.