Talk:Arbitrage pricing theory

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[edit] request for clarification

This is a very nice article, but I'm confused by the sentence that starts "If APT holds, ...". It sounds like an assumption of APT is that risky assests satisfy the equation $E\left(r_j\right) = r_f + b_{j1}RP_1 + b_{j2}RP_2 + ... + b_{jn}RP_n r_j = E\left(r_j\right) + b_{j1}F_1 + b_{j2}F_2 + ... + b_{jn}F_n + \epsilon_j.$ Is this an assumption or a consequence of the theory? For example, does this linear relationship follow from an another assumption such as lognormality of returns or is it a fundamental assumption?

Thanks.

$E\left(r_j\right) = r_f + b_{j1}RP_1 + b_{j2}RP_2 + \cdots + b_{jn}RM_n$

$r_j = E\left(r_j\right) + b_{j1}F_1 + b_{j2}F_2 + \cdots + b_{jn}F_n + \epsilon_j$

where

$E (r j)$ is the risky asset's expected return,
$R P k$ is the risk premium of the factor,
$r f$ is the risk-free rate,
$F k$ is the macroeconomic factor,
$b j k$ is the sensitivity of the asset to factor $k$ , also called factor loading,
and $ε j$ is the risky asset's idiosyncratic random shock with mean zero.