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Home/Financial Mathematics/Course 6

Course 6

Advanced Theory

5-6 hours

Arbitrage Pricing Theory

Multi-factor asset pricing models and the no-arbitrage approach

Learning Objectives

•Understand factor models and how they decompose returns into systematic and idiosyncratic components
•Derive APT from no-arbitrage arguments for well-diversified portfolios
•Compare APT with CAPM in terms of assumptions and predictions
•Learn the Fama-French three-factor and five-factor models
•Apply multi-factor models to estimate expected returns and evaluate performance

1. Factor Models of Asset Returns

Factor models express asset returns as linear functions of systematic risk factors plus idiosyncratic noise. They provide a parsimonious way to model return correlations and systematic risk.

Definition 1.1: Single-Factor Model

The return on asset $i$ is:

R_i = \mathbb{E}[R_i] + \beta_i F + \epsilon_i

where:

$F$ = systematic factor (mean zero, represents economy-wide shocks)
$\beta_i$ = factor loading (sensitivity to factor $F$ )
$\epsilon_i$ = idiosyncratic shock (mean zero, uncorrelated with $F$ and other $\epsilon_j$ )

Definition 1.2: Multi-Factor Model

With $K$ factors:

R_i = \mathbb{E}[R_i] + \beta_{i1}F_1 + \beta_{i2}F_2 + \cdots + \beta_{iK}F_K + \epsilon_i

In matrix notation:

R_i = \mathbb{E}[R_i] + \boldsymbol{\beta}_i^T \mathbf{F} + \epsilon_i

where $\boldsymbol{\beta}_i = (\beta_{i1}, \ldots, \beta_{iK})^T$ and $\mathbf{F} = (F_1, \ldots, F_K)^T$ .

Theorem 1.1: Covariance Structure in Factor Models

For a single-factor model, the covariance between assets $i$ and $j$ is:

\text{Cov}(R_i, R_j) = \beta_i \beta_j \sigma_F^2

The variance of asset $i$ is:

\text{Var}(R_i) = \beta_i^2 \sigma_F^2 + \sigma_{\epsilon_i}^2

Proof of Covariance Formula:

Since $\mathbb{E}[F] = 0$ , $\mathbb{E}[\epsilon_i] = 0$ , and $\epsilon_i$ is uncorrelated with $F$ and $\epsilon_j$ :

\text{Cov}(R_i, R_j) = \mathbb{E}[(\beta_i F + \epsilon_i)(\beta_j F + \epsilon_j)]

= \beta_i \beta_j \mathbb{E}[F^2] + \beta_i \mathbb{E}[F\epsilon_j] + \beta_j \mathbb{E}[\epsilon_i F] + \mathbb{E}[\epsilon_i \epsilon_j]

= \beta_i \beta_j \sigma_F^2

For variance ( $i = j$ ):

\text{Var}(R_i) = \beta_i^2 \sigma_F^2 + \sigma_{\epsilon_i}^2

∎

Example 1.1: Estimating Factor Model

To estimate a single-factor model, run regression:

R_i - \bar{R}_i = \beta_i F + \epsilon_i

Example: If factor is market excess return $R_M - R_f$ , this reduces to CAPM regression:

R_i - R_f = \alpha_i + \beta_i(R_M - R_f) + \epsilon_i

Under CAPM, $\alpha_i = 0$ . Non-zero alpha indicates mispricing or model mis-specification.

Remark 1.1: Identifying Factors

Common factor choices:

Macroeconomic factors: GDP growth, inflation, interest rates, oil prices
Statistical factors: Principal components of return covariance matrix
Fundamental factors: Industry returns, style factors (size, value, momentum)
Return-based factors: Portfolios formed on characteristics (Fama-French)

2. Arbitrage Pricing Theory Derivation

Definition 2.1: Well-Diversified Portfolio

A portfolio is well-diversified if its idiosyncratic risk is negligible. For $n$ assets with equal weights:

\text{Var}(\epsilon_p) = \frac{1}{n^2}\sum_{i=1}^n \sigma_{\epsilon_i}^2 \approx \frac{\bar{\sigma}_{\epsilon}^2}{n} \to 0 \text{ as } n \to \infty

As $n$ increases, only systematic (factor) risk remains.

Theorem 2.1: APT Pricing Formula

For well-diversified portfolios in the absence of arbitrage:

\mathbb{E}[R_i] = R_f + \beta_{i1}\lambda_1 + \beta_{i2}\lambda_2 + \cdots + \beta_{iK}\lambda_K

where $\lambda_k$ is the risk premium for factor $k$ .

Equivalently:

\mathbb{E}[R_i] = R_f + \sum_{k=1}^K \beta_{ik}\lambda_k

Proof of APT (Intuitive Argument):

Consider three well-diversified portfolios A, B, C with factor loadings and expected returns:

\text{Portfolio A: } \mathbb{E}[R_A], \beta_{A1}, \ldots, \beta_{AK}

\text{Portfolio B: } \mathbb{E}[R_B], \beta_{B1}, \ldots, \beta_{BK}

\text{Portfolio C: } \mathbb{E}[R_C], \beta_{C1}, \ldots, \beta_{CK}

Arbitrage argument: If we can form portfolio C as a combination of A and B with same factor loadings but different expected return, arbitrage exists.

Specifically, if $C = wA + (1-w)B$ replicates C's factor loadings:

w\beta_{Ak} + (1-w)\beta_{Bk} = \beta_{Ck} \text{ for all } k

Then no-arbitrage requires:

\mathbb{E}[R_C] = w\mathbb{E}[R_A] + (1-w)\mathbb{E}[R_B]

This implies expected returns are linear in factor loadings, yielding the APT formula.

∎

Proposition 2.1: APT as Generalization of CAPM

If there's only one factor (the market portfolio) and all portfolios are well-diversified:

\mathbb{E}[R_i] = R_f + \beta_i \lambda_M

Setting $\lambda_M = \mathbb{E}[R_M] - R_f$ gives CAPM. So CAPM is a special case of APT with one factor.

Remark 2.1: Key Differences: APT vs CAPM

Aspect	CAPM	APT
Factors	One (market portfolio)	Multiple
Key assumption	Mean-variance preferences	No arbitrage
Market portfolio	Must be mean-variance efficient	Not required
Factor specification	Clear (market)	Unspecified (weakness)
Applicability	All assets	Well-diversified portfolios

3. Fama-French Factor Models

Definition 3.1: Fama-French Three-Factor Model

Fama and French (1993) propose three factors:

R_i - R_f = \alpha_i + \beta_{iM}(R_M - R_f) + \beta_{iSMB}SMB + \beta_{iHML}HML + \epsilon_i

Factors:

$R_M - R_f$ : Market excess return (as in CAPM)
$SMB$ : Small Minus Big - return on small-cap stocks minus large-cap stocks (size factor)
$HML$ : High Minus Low - return on high book-to-market stocks minus low book-to-market stocks (value factor)

Example 3.1: Interpreting Factor Loadings

A stock has estimated coefficients:

$\beta_M = 1.2$ : 20% more volatile than market
$\beta_{SMB} = 0.8$ : positive exposure to small-cap premium
$\beta_{HML} = -0.3$ : negative exposure to value premium (growth stock)

If $R_M - R_f = 8\%$ , $SMB = 3\%$ , $HML = 4\%$ , $R_f = 2\%$ :

\mathbb{E}[R_i] = 2\% + 1.2(8\%) + 0.8(3\%) + (-0.3)(4\%)

= 2\% + 9.6\% + 2.4\% - 1.2\% = 12.8\%

Definition 3.2: Fama-French Five-Factor Model

Fama and French (2015) add two more factors:

R_i - R_f = \alpha_i + \beta_M MKT + \beta_{SMB}SMB + \beta_{HML}HML + \beta_{RMW}RMW + \beta_{CMA}CMA + \epsilon_i

Additional factors:

$RMW$ : Robust Minus Weak - profitability factor (high operating profitability minus low)
$CMA$ : Conservative Minus Aggressive - investment factor (low investment firms minus high investment)

Remark 3.1: Empirical Performance

Three-factor model:

Explains 90%+ of diversified portfolio returns
Size and value premiums well-documented historically
But doesn't capture momentum effect

Five-factor model:

HML becomes redundant when RMW and CMA are included
Better explains cross-section of expected returns
Still misses momentum and low-volatility anomalies

4. Applications and Limitations

Proposition 4.1: Expected Return Estimation

Using APT to estimate expected returns:

Identify relevant risk factors
Estimate factor loadings $\beta_{ik}$ via regression
Estimate factor risk premiums $\lambda_k$ from historical data or economic forecasts
Calculate: $\mathbb{E}[R_i] = R_f + \sum_k \beta_{ik}\lambda_k$

Example 4.1: Performance Attribution

A portfolio manager claims skill. Regression gives:

R_p - R_f = 2\% + 0.9(R_M - R_f) + 0.5 SMB + 0.3 HML

Interpretation:

Alpha = 2%: outperformance after controlling for factors
Lower market beta (0.9): defensive strategy
Small-cap tilt (0.5 SMB loading)
Value tilt (0.3 HML loading)

Returns may come from factor tilts rather than genuine skill. Need to test if alpha is statistically significant.

Remark 4.1: Limitations of APT

Factor specification: APT doesn't tell which factors matter - must be determined empirically
Well-diversified assumption: Strict APT applies only to well-diversified portfolios
Time-varying risk premiums: Factor premiums may change over time
Data mining risk: Easy to find spurious factors that fit historical data
Out-of-sample performance: Factors that worked historically may not work in the future

Key Takeaways

1.APT prices assets based on multiple risk factors using no-arbitrage arguments
2.Factor models decompose returns into systematic (factor) risk and idiosyncratic risk
3.APT more general than CAPM - weaker assumptions, multiple factors, but no factor specification
4.Fama-French models identify size, value, profitability, and investment as systematic risk factors
5.Well-diversified portfolios eliminate idiosyncratic risk, leaving only priced factor risk
6.Multi-factor models improve return explanation and performance attribution

Frequently Asked Questions

How is APT different from CAPM?

CAPM has one factor (market portfolio) and strict assumptions (mean-variance preferences, homogeneous expectations). APT allows multiple risk factors and relies only on no-arbitrage. APT is more general and flexible, but doesn't specify which factors matter.

What are the main factors in the Fama-French model?

The three-factor model includes: (1) Market factor (RM - Rf), (2) Size factor SMB (Small Minus Big - return difference between small and large cap stocks), (3) Value factor HML (High Minus Low - return difference between high and low book-to-market stocks).

Why doesn't APT require identifying the market portfolio?

APT works with any set of systematic risk factors. It doesn't need the market portfolio to be mean-variance efficient or even observable. This addresses Roll's critique of CAPM. However, APT doesn't tell us which factors to use.

What is factor loading (beta) in APT?

Factor loading measures an asset's sensitivity to a particular risk factor. If factor k is industrial production growth, a loading of 1.5 means a 1% change in IP growth associates with a 1.5% change in the asset's return (on average).

Can APT have arbitrage opportunities?

No - APT is derived from the absence of arbitrage. If APT pricing doesn't hold, there would be arbitrage opportunities that traders would exploit until prices adjust to satisfy APT.

Practice Quiz

Arbitrage Pricing Theory

Questions

Correct

Accuracy

In a single-factor model with risk-free rate 3%, factor risk premium 8%, and asset factor loading 1.2, what is the expected return?

Easy

Not attempted

APT relies primarily on:

Easy

Not attempted

If a two-factor model has Rf = 2%, λ1 = 5%, λ2 = 3%, and asset has β1 = 0.8, β2 = 1.4, what is E[R]?

Medium

Not attempted

In the Fama-French three-factor model, SMB represents:

Easy

Not attempted

Which is NOT an advantage of APT over CAPM?

Medium

Not attempted

A well-diversified portfolio in APT has:

Medium

Not attempted

If actual return is 15% and APT predicts 12%, the mispricing (alpha) is:

Easy

Not attempted

Factor risk premium is:

Medium

Not attempted

In APT, idiosyncratic risk is:

Easy

Not attempted

Which research identified the three-factor model?

Easy

Not attempted