Question about pricing kernel definition in "Quality minus junk" paper

I'm reading the paper "Quality minus junk" by Asness et al. published in Review of Accounting Studies (2019). The authors present the following definition of the pricing kernel on page 2:

$$\frac{M_{t+1}}{M_t} = \frac{1}{1+r^f} \left(1 + e^M_{t+1}\right)$$

where $$r^f$$ is the risk-free rate and $$e^M_{t+1}$$ is the zero-mean innovation to the pricing kernel.

This doesn't match the more standard definition of the pricing kernel I'm familiar with:

$$M_{t+1} = \beta \frac{u'(c_{t+1})}{u'(c_t)}$$

where $$\beta$$ is the discount factor and $$u'(c_t)$$ is the marginal utility of consumption at time $$t$$.

Can someone help explain the difference between these two definitions and provide some intuition for the equation used in the "Quality minus junk" paper?

Any insights would be greatly appreciated. Please let me know if you need me to provide any additional context from the paper.

• What you describe is a consumption based pricing kernel, but that is only a special case of pricing kernel. A more general definition is given here en.wikipedia.org/wiki/Stochastic_discount_factor It is really just a state variable that measure the value to you of receiving one dollar in a particular future state. Commented Apr 12 at 8:22
• @nbbo2 Oh, I've seen this, but I still fail to link the equation '$$\frac{M_{t+1}}{M_t} = \frac{1}{1+r^f} \left(1 + e^M_{t+1}\right)$$' to the more general model. Could you shed some light on this for me? Commented Apr 12 at 10:19

I believe a good analogy to understand it is - the change/evolution in the stochastic discount factor throughout time can be attributed to a riskless $$1+r_f$$ and a random $$1+e^M_{t+1}$$ component, which at least to me, makes a lot of sense.
• Thanks @KaiSqDist!, I saw you mentioned that the innovation term $e^M_{t+1}$ captures the random component in the change/evolution of the stochastic discount factor over time. I'm still a bit unclear on how this innovation term directly relates to the CAPM model specification shown in This euqation. Could you provide some intuition on why and how the random fluctuations in the stochastic discount factor dynamics are specifically expressed in terms of the market risk premium and deviations of the market return, as per the CAPM equation? Commented Apr 12 at 16:41