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

Question

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.

Answer 1

It seems like what the first equation is saying is - the return on the stochastic discount factor is equal to the zero-mean innovation to the pricing kernel discounted at the riskless rate.

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.