APT and Multifactor Models Notes Summary & Study Notes

🪙 Arbitrage Pricing Theory (APT)

Arbitrage is a zero-investment portfolio yielding a sure profit; in efficient markets, profitable arbitrage opportunities vanish quickly.

In APT, the return on security $i$ is modeled as:

$r_i = E[r_i] + \beta_i F + e_i$

• E[r_i] is the expected return.
• F is a macroeconomic/systematic factor.
• \beta_i is the sensitivity of $i$ to $F$.
• e_i is the firm-specific surprise with Cov($e_i$, $e_j$)=0 for $i \neq j$, and Cov($F$, $e_i$)=0.

$F$ and $e_i$ are distributed with means 0 and standard deviations $\sigma_F$ and $\sigma_{e_i}$, respectively.

A portfolio $p$ with weights $w_i$ has:

$r_p = E[r_p] + \beta_p F + e_p$

$$E[r_p] = \sum_i w_i E[r_i]$$

$$\beta_p = \sum_i w_i \beta_i$$

$$e_p = \sum_i w_i e_i$$

The portfolio variance is:

$$\sigma_p^2 = \beta_p^2 \sigma_F^2 + \sigma_{e_p}^2$$

and \sigma_{e_p}^2 = \mathrm{Var}\left( \sum_i w_i e_i
ight); under broad diversification, $\sigma_{e_p}^2 \rightarrow 0$ as the number of assets grows.

APT: Returns as a Function of the Systematic Factor

Consider a well-diversified portfolio A with $\beta_A = 1$. If the expected return is 10% when $F=0$, then the return is

$r_A = E[r_A] + \beta_A F = 10\% + 1\cdot F$.

A single stock with $\beta_S = 1$ has

$r_S = E[r_S] + \beta_S F + e_S = 10\% + 1\cdot F + e_S$.

Consider two well-diversified portfolios A and B with $\beta_A = \beta_B = 1$, $E[r_A] = 10\%$, $E[r_B] = 8\%$. Their returns imply a potential arbitrage unless priced consistently.

No matter what the realization of $F$ is, portfolio A outperforms portfolio B; this is an arbitrage opportunity if it persists.

How to exploit:

• Buy $1$M of A and short $1$M of B: zero net investment. The payoff is $(10\%+F) - (8\%+F) = 2\%$.

To exclude arbitrage, prices must adjust so that all well-diversified portfolios lie on a common straight line through the risk-free asset.

One-Factor Security Market Line

Let $M$ denote the market portfolio, a well-diversified portfolio. The factor realized here is the risk premium on $M$. The SML in the APT context is

$$E[r_p] = r_f + (E[r_M] - r_f) \beta_p^F.$$

In words: the expected return of a well-diversified portfolio lies on a line defined by the risk-free rate and the market risk premium, proportional to the portfolio's systematic exposure.

APT vs CAPM

The APT does not require that the benchmark on the SML be the true market portfolio; any sufficiently diversified portfolio suffices. The SML in APT holds for well-diversified portfolios, not necessarily for individual assets. The APT cannot guarantee that the expected return–beta relation holds for a given single asset; CAPM would be needed to claim a beta-for-every-security relationship.

Multifactor Models

General form for security $i$ with $K$ factors:

$$r_i = E[r_i] + \sum_{k=1}^K \beta_{i,k} F_k + e_i.$$

Each factor $F_k$ denotes a systematic risk shock; $\beta_{i,k}$ is the sensitivity to factor k; $E[r_i]$ is the expected return; $e_i$ is idiosyncratic noise.

The expected return on security i is given by the SML analog for multifactor models:

$$E[r_i] = r_f + \sum_{k=1}^K \beta_{i,k} RP_k,$$

where the risk premium of factor $k$ is

$$RP_k = E[F_k] - r_f.$$

Multifactor Models: Example

Suppose a two-factor model with $E[F_1] = 10\%$, $E[F_2] = 12\%$, and $r_f = 4\%$. Consider a well-diversified portfolio A with $\beta_{A,1}=0.5$ and $\beta_{A,2}=0.75$. The risk premium from factor 1 is $0.5\cdot(E[F_1]-r_f) = 0.5\cdot 6\% = 3\%$. The risk premium from factor 2 is $0.75\cdot(E[F_2]-r_f) = 0.75\cdot 8\% = 6\%$. Therefore,

$$E[r_A] = r_f + 3\% + 6\% = 13\%.$$

Fama–French Three-Factor Model (FF3F)

The FF3F model is

$$r_i - r_f = \alpha_i + \beta_i (r_M - r_f) + s_i \mathrm{SMB} + h_i \mathrm{HML} + e_i,$$

where

• SMB: Small Minus Big, the return on a portfolio of small stocks in excess of the return on a portfolio of large stocks.
• HML: High Minus Low, the return on a portfolio of stocks with high book-to-market ratio in excess of the return on a portfolio of stocks with a low book-to-market ratio.
• $s_i$ and $h_i$ are factor loadings of security i on SMB and HML, respectively.

FF3F: Construction of SMB and HML

Ken French provides the returns on SMB and HML and data needed to construct them on his website. The construction steps are:

• Sort firms by market cap and by book-to-market.
• SMB is constructed as the difference in returns between the smallest and largest thirds of firms.
• HML is constructed as the difference in returns between high and low book-to-market firms.

FF3F: Empirical Tests (Davis, Fama, French, 2000)

For each of nine portfolios formed by three size groups and three book-to-market groups, estimate:

$$r_i - r_f = a_i + b_i (r_M - r_f) + s_i \mathrm{SMB} + h_i \mathrm{HML} + e_i.$$

Intercepts are typically not significantly different from zero, except for some small and high-B/M portfolios. Market betas are close to 1; SMB loads negatively on big firms and positively on small firms; HML loads positively on high-B/M firms and negatively on low-B/M firms; $R^2$ of FF3F regressions is very high (often above 0.91).

FF3F + Momentum

A fourth factor emerged—the momentum factor. The Carhart extension adds momentum (UMD):

$$r_i - r_f = a_i + b_i (r_M - r_f) + s_i \mathrm{SMB} + h_i \mathrm{HML} + u_i \mathrm{UMD} + e_i,$$

where UMD stands for Up Minus Down, i.e., the performance of stocks that did well recently versus those that did poorly.

Momentum evidence is strong across many asset classes, though the reason remains debated. It reflects a potential risk–return trade-off that is still under examination.

FF3F + Momentum + Liquidity

Pastor and Stambaugh (2003) propose liquidity as an additional factor. The model becomes

$$r_i - r_f = a_i + b_i (r_M - r_f) + s_i \mathrm{SMB} + h_i \mathrm{HML} + u_i \mathrm{UMD} + l_i \mathrm{Liq} + e_i.$$

FF5F

A five-factor extension adds two more firm characteristics: RMW and CMA. The model is

$$r_i - r_f = a_i + b_i (r_M - r_f) + s_i \mathrm{SMB} + h_i \mathrm{HML} + r_i \mathrm{RMW} + c_i \mathrm{CMA} + e_i,$$

where

• RMW: Robust Minus Weak, i.e., the return of firms with robust profitability minus those with weak profitability,
• CMA: Conservative Minus Aggressive, i.e., the return of firms with low investment (conservative) minus high investment (aggressive).

End notes

• The practical use of multifactor models is pricing and risk management for diversified portfolios.
• Always check factor significance and interpret intercepts with caution.