The APT model

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Risky asset returns are said to follow a factor structure if they can be expressed as:

$r_j = a_j + b_{j1}F_1 + b_{j2}F_2 + \cdots + b_{jn}F_n + \epsilon_j$

where

a_j is a constant for asset j

F_k is a systematic factor

$b_{jk}$ is the sensitivity of the jth asset to factor , also called factor loading,

and $\epsilon_j$ is the risky asset's idiosyncratic random shock with mean zero.

Idiosyncratic shocks are assumed to be uncorrelated across assets and uncorrelated with the factors.

The APT states that if asset returns follow a factor structure then the following relation exists between expected returns and the factor sensitivities:

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

where

RP_k is the risk premium of the factor,

r_f is the risk-free rate,

That is, the expected return of an asset j is a linear function of the assets sensitivities to the n factors.

Note that there are some assumptions and requirements that have to be fulfilled for the latter to be correct: There must be perfect competition in the market, and the total number of factors may never surpass the total number of assets (in order to avoid the problem of matrix singularity),

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The APT model