Free Custom «Factor Models» Essay Paper

Generally factor models are used for describing and quantifying miscellaneous factors, affecting the return percentage for securities in the provided time frame. The range of use of the factor models is limited to making measurement and managerially supervising of different factors of economy, especially risks (energy price risk, interest risk etc.). A factor model could be also defined as a set of the security returns, separated into two categories. First one presents the security-correlated components and the second one reflects the components, not correlated across the securities.

Let us put the relevant economic factors together based on the common macroeconomic indicator. Supposedly all the factors are also firm-specific. They all come within a single company and are tightly related to it. Such way we get the simplest factor model – a single-factor model:

r_i = α_i + β_i F + e_i,

where r_i is return on stock, b_i – sensitivity of stock’s return to macroeconomic events, f is return to market index and e_i is a random stock to return. The market return-generating factor is the only one in this model.

 

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Provided there are more than one common factors, as it occurs in real-life situations, we get a different factor model:

r_i = α_i + β_1i F₁ + β_2i F₂ + … + β_ki F_k + e_i.

By means of the multifactor modeling we define the exposure of business to different macroeconomic risks. Security returns depend on a small number of common factors with different sensitivity of stocks. Portfolios are created for protection from possible risks.

To denote the risk premiums in the k-factors, λ₁, λ₂,…, λ_k are used. In other words, the expected return of factor portfolio 1 can be viewed as (r_f + λ₁), where r_f is the risk-free rate. Let us assume the world is generated by a two-factor model and assume that we wish to replicate asset X, which has sensitivity β_1x to the first factor and β_2x to the second factor. Also we will assume that the primary securities being worked with are well-diversified portfolios themselves. That is why any idiosyncratic risk in this derivation will be ignored.

To replicate asset X`s factor sensitivities, we construct a portfolio with weight β_1x on the first factor-replicating portfolio, weight β_2x on the second factor-replicating portfolio and weight 1 - β_1x - β_2x on the risk-free asset. Hence, the expected return of the replicating portfolio is:

β_1x (r_f + λ₁) + β_2x (r_f + λ₂) + (1 - β_1x - β_2x ) r_f = r_f + β_1x λ₁ + β_2x λ₂.

By doing this for the k-th factor criteria, we get:

E(r_i) = r_f + β_1i λ₁ + β_2i λ₂ + ... + β_ki λ_k.

In case we require a two-factor model and the expected returns on assets will adjust such that there are no arbitrage profits (pair of portfolios must have identical expected return in equilibrium) to be made between broad portfolios, portfolios that have loadings on two factors must have expected return equal to the risk free plus the sum of the beta-weighted risk premiums for these factors. Formally, E(r_i) = r_f + β_1i λ₁ + β_2i λ₂, which is the statement of APT.

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The APT is based on returns of securities, described by the factor models, and non-existence of the arbitrage opportunities. Based on these ideas, one might derive the multifactor version of CAPM risk-expected return equation, expressing the expected returns of every single security in terms of factor beta function. The pros of APM are using of simple market portfolios, and the cons are the lack of the theoretical approach, found in CAPM.