How APT Factor Returns Imply a Linear Pricing Kernel

Summary

The document asks how a linear multifactor model for asset returns under Arbitrage Pricing Theory can imply a pricing kernel, or stochastic discount factor, that is linear in the same factors. It defines asset factor loadings, factor returns, residuals, and the pricing relation between an asset’s payoff and the pricing kernel, then poses the derivation as an open question.

No derivation or supporting evidence is provided, so the proposed implication should not be treated as established by this text alone. The relationship depends on the precise APT assumptions and on how factors, residual risks, and pricing are defined; the document does not specify those conditions. It is useful as a prompt to examine the assumptions required to connect a return-generating model with a pricing representation.

Key ideas

• The document presents asset returns as a risk-free return plus factor exposures and an idiosyncratic residual.
• It asks whether the pricing kernel can be represented as a linear combination of those factors.
• It provides no derivation, so the claimed implication requires additional assumptions and argument.

Tags

# Linear factor representation Pricing kernel APT


# Linear factor representation Pricing kernel APT












following Cochrane (2005) and other insights, we know that under Arbitrage Pricing Theory (Ross, 1976), if investors believe returns follow a linear multifactor structure of the form

$x^i=r^f+\sum_{j=1}^{M}\beta_{ij}f_j+\epsilon_i$

where $x^i$ is a the asset $i$ return, $r^f$ is the return on a risk-free asset, $\beta_{ij}$ is the factor loading of asset $i$ with respect to factor $j$ and $f_j$ is the $j$-th factor; then we have that the pricing kernel or stochastic discount factor $m$ which prices assets by defintion according to $p(x^i)=E[mx^i]$, also satisfies

$m=a+\sum_{j=1}^{M}b_{j}f_j$

for some arbitrary $a$ and $b$ i.e. it satisfies a linear multifactor structure. Is anyone able to derive the second equation starting from the first one using the classical assumptions in the APT or at least give me some hints about it?

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.