APT and Multifactor Models Notes Flashcards

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Arbitrage Pricing Theory

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Arbitrage Pricing Theory (APT) is a multifactor asset-pricing framework in which returns are driven by multiple $K$ macroeconomic factors. The return on asset $i$ is modeled as $r_i = E[r_i] + \sum_{k=1}^K \beta_{i,k} F_k + e_i$, where $e_i$ is idiosyncratic risk and is uncorrelated with other assets. Arbitrage opportunities imply zero‑investment portfolios with a sure profit, and in efficient markets such opportunities should disappear quickly.

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Arbitrage

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Arbitrage is the construction of a zero‑investment portfolio that yields a sure profit with no risk. In efficient markets any profitable arbitrage opportunity will be quickly exploited away. The concept underpins APT as a pricing reason: risk is rewarded through factors, not through mispricings.

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Single-factor model

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The APT is built as a single-factor model where asset returns are determined by a primary systematic factor. The return on asset $i$ is $r_i = E[r_i] + \beta_i F + e_i$, with $F$ representing macro shocks and $e_i$ idiosyncratic risk. Diversification reduces the idiosyncratic risk term $e_i$.

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Systematic factor

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A systematic factor $F$ represents macroeconomic shocks that are unanticipated and affect many assets. The factor is assumed to be uncorrelated with firm-specific surprises: $\mathrm{Cov}(F, e_i) = 0$ for all $i$. $F$ and $e_i$ are typically modeled as random with mean zero and variances $\sigma_F^2$ and $\sigma_{e_i}^2$ respectively.

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Idiosyncratic risk

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$e_i$ captures firm-specific surprises and is uncorrelated across firms: $\mathrm{Cov}(e_i, e_j) = 0$ for $i \neq j$. In a well-diversified portfolio the idiosyncratic risk component shrinks, i.e., $\lim_{n \to \infty} \sigma_{e,p}^2 = 0$. The portfolio risk is then dominated by the systematic factor risk.

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Beta

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$\beta_i$ measures the sensitivity of security $i$ to the factor $F$ in the one‑factor model. For a portfolio, $\beta_p = \sum_i w_i \beta_i$. It captures how much the asset’s return moves with the systematic factor.

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Factor

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A factor $F_k$ denotes a macroeconomic risk source. For security $i$, sensitivities to factors are $\beta_{i,k}$. The return is modeled as $r_i = E[r_i] + \sum_k \beta_{i,k} F_k + e_i$.

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Portfolio return

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For portfolio $p$, $r_p = E[r_p] + \beta_p F + e_p$, with $E[r_p] = \sum_i w_i E[r_i]$, $\beta_p = \sum_i w_i \beta_i$, and $e_p = \sum_i w_i e_i$.

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Risk premium

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Each factor $k$ has a risk premium $RP_k = E[r_{F_k}] - r_f$. The expected return on asset $i$ can be written as $E[r_i] = r_f + \sum_k \beta_{i,k} RP_k$. This links factor exposures to expected returns.

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SML (One-Factor)

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One‑Factor Security Market Line: $E[r_p] = r_f + [E(r_M) - r_f] \beta_p^F$ for well-diversified portfolios. It mirrors the CAPM idea but applies to the APT framework for portfolios with a single systematic factor.

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FF3F

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Fama-French Three-Factor model: $r_i - r_f = \alpha_i + \beta_i (r_M - r_f) + s_i \mathrm{SMB} + h_i \mathrm{HML} + e_i$. SMB captures size, HML captures value via book-to-market; $s_i$ and $h_i$ are factor loadings.

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SMB

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Small Minus Big (SMB) is the FF3F factor representing the excess return of small-cap over large-cap portfolios. It captures the size effect and contributes to explaining cross‑sectional returns.

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HML

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High Minus Low (HML) is the FF3F factor representing the excess return of high book‑to‑market stocks over low book‑to‑market stocks. It captures the value effect in asset pricing.

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Momentum

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Momentum (UMD) is a factor capturing persistence in stock performance: stocks that performed well recently tend to continue performing well. Carhart extended FF3F by adding this momentum factor to form a four-factor model.

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Liquidity

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Liquidity (Liq) is a factor capturing liquidity risk premia; high‑liquidity risk premium compensates investors for holding less liquid assets. Pastor and Stambaugh proposed including liquidity in factor models.

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RMW

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Robust Minus Weak (RMW) is a profitability factor: firms with robust profitability outperform those with weak profitability. It is included in the five‑factor model (FF5F) as one of the additional factors.

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CMA

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Conservative Minus Aggressive (CMA) is an investment factor: firms with conservative (low investment) policies tend to outperform aggressive (high investment) firms. Included in FF5F as another additional factor.

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Carhart Model

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Carhart’s four-factor model adds Momentum to FF3F: $r_i - r_f = \alpha_i + \beta_i (r_M - r_f) + s_i \mathrm{SMB} + h_i \mathrm{HML} + u_i \mathrm{UMD} + e_i$, where UMD is the momentum factor.

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FF5F

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Five-factor model extends FF3F with RMW and CMA: $r_i - r_f = \alpha_i + \beta_i (r_M - r_f) + s_i \mathrm{SMB} + h_i \mathrm{HML} + r_i \mathrm{RMW} + c_i \mathrm{CMA} + e_i$. It combines profitability and investment factors with size and value effects.

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Multifactor Model

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Multifactor models generalize APT to multiple factors: $r_i = E[r_i] + \sum_{k=1}^K \beta_{i,k} F_k + e_i$. The expected return is $E[r_i] = r_f + \sum_{k=1}^K \beta_{i,k} RP_k$ with $RP_k = E[r_{F_k}] - r_f$. These models encompass FF3F, Carhart, and FF5F as special cases.