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書名： Mastering R for Quantitative Finance
作者名： Edina Berlinger Ferenc Illés Milán Badics ?dám Banai Gergely Daróczi Barbara D?m?t?r Gergely Gabler Dániel Havran Péter Juhász
本章字數： 1001字
更新時間： 2021-07-23 20:10:43

Arbitrage pricing theory

APT relies on the assumption that asset returns in the market are determined by macroeconomic and firm-specific factors, and asset returns are generated by the following linear factor model:

Equation 1

Here, E(r_i) is the expected return of asset i, F_j stands for the unexpected change of the jth factor, and β_ij shows the ith security's sensitivity for that factor, while e_i is the return caused by unexpected firm-specific events. So Arbitrage pricing theory represents the random systemic effect, and e_i represents the non-systemic (that is idiosyncratic) effect, which is not captured by the market factors. Being unexpected, both and e_i have a zero mean. In this model, factors are independent of each other and the firm-specific risk. Thus, asset returns are derived from two sources: the systemic risk of the factors that affect all assets in the market and the non-systematic risk that impacts only that special firm. A non-systemic risk can be diversified by holding more assets in the portfolio. In contrast, a systemic risk cannot be diversified, as it is caused by economy-wide sources of risks that affect the overall stock market (Brealey-Myers, 2005).

As a consequence of the model, the realized return of an asset is the linear combination of multiple random factors (Wilmott, 2007).

Other important assumptions of APT are as follows:

There are a finite number of investors on the market who optimize their portfolio for the next period. They are equally informed and have no market power.
There is a riskless asset and an infinite number of risky assets traded continuously; thus, firm-specific risks can be totally eliminated by diversification. A portfolio that has zero firm-specific risks is called a well-diversified portfolio.
Investors are rational in the sense that if an arbitrage opportunity occurs (financial assets are mispriced relative to each other), then investors immediately buy the underpriced security/securities and sell the overpriced one(s), and they will take an infinitely large position in order to earn as much riskless profit as possible. Consequently, any mispricing will disappear on the spot.
Factor portfolios exist, and they are continuously tradable. A factor portfolio is a well-diversified portfolio that reacts only to one of the factors; specifically, it has a beta of 1 for that specified factor and a beta of 0 for all other factors.

From the preceding assumptions, it can be shown that any portfolio's risk premium equals the weighted sum of the factor portfolios' risk premium (Medvegyev-Száz, 2010). The following pricing formula can be derived in the case of a two-factor model:

Equation 2

Here, r_i is the return of the ith asset, r_f is the risk-free return, β_i1 is the sensitivity of the ith stock's risk premium to the first systemic factor, and (r₁-r_f) is the risk premium of this factor. Similarly, β_i2 is the sensitivity of the ith stock's risk premium to the second factor's excess return (r₂-r_f).

When we implement APT, we perform a linear regression in the following form:

Equation 3

Here, α_i stands for a constant and ε_i is the asset's non-systemic, firm-specific risk. All other variables are the same as mentioned previously.

If there is only one factor in the model, and it is the return of the market portfolio, the pricing equation of the CAPM model and APT model will coincide:

Equation 4

In this case, the formula to be tested on real market data is as follows:

Equation 5

Here, Arbitrage pricing theory is the return of a market portfolio represented by a market index (like the S&P 500). This is why we call Equation (5) the index model.

Implementation of APT

The implementation of APT can be split into four steps: identifying the factors, estimating the factor coefficient, estimating the factor premiums, and pricing with APT (Bodie et al. 2008):

Identifying the factors: As APT mentions nothing about the factors, they have to be identified empirically. These factors are usually macroeconomic factors, like stock market return, inflation, business cycle, and so on. The main problem in using macroeconomic factors is that factors are usually not independent of each other. The identification of the factors is often carried out by factor analysis. However, factors identified by factor analysis cannot necessarily be interpreted in an economically meaningful way.
Estimating factor coefficients: In order to estimate the coefficients in a multivariate linear regression model, a general version of Equation (3) is used.
Estimating the factor premiums: The estimation of the factor premiums is based on historical data, taking the average of the historical time-series data of the premiums of the factor portfolios.
Pricing with APT: Equation (2) is used for calculating the expected return of any asset by substituting the appropriate variables into the equation.

Fama-French three-factor model

Fama and French proposed a multifactor model in 1996, in which they used corporate indicators as factors instead of macroeconomic factors, since they found that these factors better describe the systemic risk of assets. Fama and French (1996) extended the index model by adding the firm size and the book-to-market ratio as return-generating factors to the market portfolio returns (Fama and French, 1996).

The firm size factor was constructed by taking the difference between the returns of small and large firms (r_SMB). The name of the variable was SMB, which is derived from "small minus big". The book-to-market factor was calculated by taking the difference between firms' returns that have a high and low book-to-market ratio (r_HML). The name of the variable was HML, which is derived from "high minus low".

Their model was the following:

Equation 6

Here, α_i is a constant, which shows the abnormal rate of return, r_f is the risk-free return, and β_iHML is the ith asset's sensitivity to the book-to-market factor, while β_iSMB is the ith asset's sensitivity to the factor of size, β_iM is the sensitivity of the ith stock's risk premium to the market index factor, (r_M-r_f) is the risk premium of this factor, and e_i is the asset's non-systemic, firm-specific risk with zero mean.

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Mastering R for Quantitative Finance

Arbitrage pricing theory

Implementation of APT

Fama-French three-factor model