# Law of large numbers necessary for APT derivation?

The question refers to the well-known Ross (1976) paper with the derivation of the Asset Pricing Theory.

In the APT, the return of asset $i$ is driven by a linear factor model:

$$R_i = \alpha_i + \sum_{j=1}^m \beta_i^j \mathcal{F}_j + \varepsilon_i$$ where $\alpha_i$ is the intercept, $\beta_i^j$ is the sensitivity of asset $i$ to factor $j$ (the factor loading) and $\mathcal{F}_j$ is the value of factor $j$. $\varepsilon_i$ is the idiosyncratic risk of asset $i$.

Now what I want to derive is ($R_f$ is the risk free rate)

$$\pi_i := \mathbb{E} R_i - R_f = \sum_{j=1}^m\beta_i^j \pi(\mathcal{F}_j)$$

where $\pi(\mathcal{F}_j) = \mathbb{E}\mathcal{F}_j - R_f$. As the name suggests, this is done by a no arbitrage argument and the result means that the asset risk premia are determined by the factor risk premia via the factor loadings $\beta_i^j$.

In the paper the author assumes that for an arbitrage portfolio $x$ with asset weights $x_i$, $\sum_{i=1}^nx_i\varepsilon_i \approx 0$ by the law of large numbers if the $\varepsilon_i$ are "sufficiently independent for the law of large numbers to hold". Translated, this basically means that the arbitrage portfolio does not show any substantial idiosyncratic risk.

Then, the author proceeds that the net factor exposure of an arbitrage portfolio should be $0$: $\sum_{i=1}^n x_i \beta_i^j = 0$ and that the arbitrage portfolio does not use any capital $\sum_{i=1}^nx_i = 0$.

Then he continues with the derivation (which ends up in a linear algebra argument and finally the APT equations).

# Question

The question is why does the author need the law of large numbers? Doesn't this implicitly assume that the number of $n$ assets is large? Wouldn't it be better to just assume that for an arbitrage portfolio $\sum_{i=1}^n x_i\varepsilon_i=0$?

I think the answer is somehow tied to the question: If the linear relation between factor risk premia and asset risk premia does NOT hold, does this mean that there is an arbitrage portfolio? (in the sense that $\sum_{i=1}^nx_i\varepsilon_i=0$)

(The question arose from Appendix A1 of this document here, where the authors dont provide details about this.)

In small sample, there is no reason why $x' \epsilon$ will be 0. In fact, there is no real reason why $\epsilon$ should be independent. The fact that you are assuming a linear specification for the returns means you are to some extent making assumptions of linear regression. Justifying the errors being uncorrelated with the independent variables is justified through asymptotically diversifying idiosyncratic risk.
• I am sorry I have to downvote this answer. To get the APT relation you could simply assume $x^\prime \varepsilon = 0$ for an arbitrage portfolio... If you think its about the assumptions, feel free to be more specific about "some extent making assumptions of linear regression". Please also explain in detail where these assumptions are needed and how they imply the need for the law of large numbers with respect to the no-arbitrage argument! – vanguard2k Jan 14 '15 at 8:36
• To upvote and accept your answer I had to edit it first so I added the resource. Your explanation is not different from the paper but my question was aiming at something different. The shanken paper answered it though. What confused me was that the APT equation is only an approximate result! (which is unusual for an arbitrage argument and very seldomly stressed in the literature) Of course, $x^\prime \varepsilon=0$ can never hold for independent $\varepsilon_i$ and thus there will never be an arbitrage portfolio, even if arbitrage possibilities are present. – vanguard2k Jan 15 '15 at 7:33