Interpreting a Normalized Pricing Kernel with a Risk-Free Component

Summary

The document asks how to reconcile a normalized pricing-kernel equation from a quality-investing paper with the consumption-based stochastic discount factor, which is usually expressed using a discount factor and the ratio of marginal utilities. The response interprets the paper’s equation as expressing the one-period change in the kernel relative to its current level as a risk-free discount component multiplied by a random innovation component.

In this interpretation, the risk-free rate sets the baseline scaling, while the zero-mean innovation represents uncertainty in the kernel’s evolution. The equation is therefore presented as a convenient return-like representation, rather than a competing statement about marginal utility. The response offers intuition but does not derive the equivalence from a consumption model, establish assumptions connecting the formulations, or discuss pricing implications. Readers should treat it as a brief explanation of notation and decomposition, not a complete account of the paper’s pricing-kernel framework.

Key ideas

• The paper writes the next-period kernel relative to its current value.
• The representation separates risk-free scaling from a random zero-mean innovation.
• The response interprets the expression as return-like evolution of the stochastic discount factor.
• The brief explanation does not derive the link to marginal utility or specify its assumptions.

Tags

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


# 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.

## Answer by KaiSqDist (score 0, accepted)

https://quant.stackexchange.com/a/78954

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.

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