The Law of Large Numbers in the Arbitrage Pricing Theory Derivation

# Law of large numbers necessary for APT derivation?


# 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.)

## Answer by Drew (score 1, accepted)

https://quant.stackexchange.com/a/16192

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.

EDIT: An interesting resource is also Jay Shankens paper.