Topic 4: Asset Pricing Models II (Arbitrage Pricing Theory)

Arbitrage Pricing Theory (APT) Fundamentals

• Arbitrage Pricing Theory (APT) was developed as an alternative to the Capital Asset Pricing Model (CAPM) to address significant criticisms of the latter. CAPM is often criticized for its reliance on unrealistic assumptions, the practical difficulties in selecting an appropriate proxy for the market portfolio to serve as a benchmark, and empirical evidence indicating that firm-specific characteristics other than beta possess explanatory power for cross-sectional stock returns.

• The APT was originally put forward by Stephen Ross in 1976.

• The theory is based on three major assumptions:

  • Capital markets are perfectly competitive.

  • Investors always prefer more wealth to less wealth with certainty.

  • The stochastic process generating asset returns can be expressed as a linear function of a set of $K$ identified factors or indexes, and unsystematic risk can be diversified away.

• In contrast to the CAPM, the APT does not require the following assumptions:

  • Normally distributed security returns.

  • A quadratic utility function.

  • The existence of a mean-variance efficient market portfolio.

The APT Model Equation and Variables

• The APT model states that the expected return on any Asset $i$ can be expressed as a linear $k$ factor model:

  • $E(R_i) = \lambda_0 + \lambda_1 b_{i1} + \lambda_2 b_{i2} + \dots + \lambda_k b_{ik}$

• Definitions of variables within the equation:

  • $E(R_i)$: The expected return on Asset $i$.

  • $\lambda_0$: The expected return on an asset with zero systematic risk (the zero-beta return).

  • $\lambda_j$: The risk premium related to the $j^{th}$ common risk factor.

  • $b_{ij}$: The pricing relationship between the risk premium and the asset, representing how responsive Asset $i$ is to the $j^{th}$ common factor (factor sensitivity).

Comparative Analysis: CAPM vs. APT

• Equation Structure:

  • CAPM: $E(R_i) = RFR + \beta_i [E(R_M) - RFR]$

  • APT: $E(R_i) = \lambda_0 + \lambda_1 b_{i1} + \lambda_2 b_{i2} + \dots + \lambda_k b_{ik}$

• Form of Equation: Both models are linear in nature.

• Number of Risk Factors:

  • CAPM is a single-factor model ($1$ factor).

  • APT is a multi-factor model ($K$ factors where K > 1).

• Factor Risk Premium Term:

  • CAPM uses the market risk premium: $[E(R_M) - RFR]$.

  • APT uses various risk premiums: $\{\lambda_j\}$.

• Factor Risk Sensitivity Term:

  • CAPM uses beta: $\beta_i$.

  • APT uses factor loadings: $\{b_{ij}\}$.

• Zero-Beta Return Variable:

  • CAPM uses the risk-free rate: $RFR$.

  • APT uses a zero-systematic risk asset return: $\lambda_0$.

Identification of Risk Factors

• While CAPM identifies the market portfolio return as the sole factor, APT does not specifically identify which risk factors should be used in application. The primary challenge of using APT for security valuation is the identification of these factors.

• Commonly suggested risk factors include:

  • Inflation (unanticipated changes in the rate of inflation).

  • Growth in Gross National Product (GNP) or unexpected changes in the growth rate of real GDP.

  • Major political upheavals.

  • Changes in interest rates.

• Successfully using APT requires identifying measurable risk factors and correctly interpreting their statistical significance via methods such as multivariate regression. APT allows for risk isolation, including macroeconomic factors, which enables the customization of portfolios to specific investor mandates (the foundation of "alpha" funds).

Security Valuation with the APT

Calculation Example

• Assume a model with two factors: Inflation ($\lambda_1$) and GDP Growth ($\lambda_2$).

• $\lambda_0 = 4\,\%$ (0.04)

• $\lambda_1 = 2\,\%$ (0.02) - Risk premium of 2% for every 1% change in inflation.

• $\lambda_2 = 3\,\%$ (0.03) - Risk premium of 3% for every 1% change in growth rate.

• Asset X Responsiveness:

  • $b_{x1} = 0.50$

  • $b_{x2} = 1.50$

  • $E(R_x) = 0.04 + (0.02)(0.50) + (0.03)(1.50) = 0.095 = 9.5\,\%$

• Asset Y Responsiveness:

  • $b_{y1} = 2.00$

  • $b_{y2} = 1.75$

  • $E(R_y) = 0.04 + (0.02)(2.00) + (0.03)(1.75) = 0.1325 = 13.25\,\%$

Price Estimation and Arbitrage

• If Stock A, B, and C are priced at $35\,USD$ today and pay no dividends, the estimated price one year later is derived from expected returns.

• Given: $\lambda_1 = 4\,\%$, $\lambda_2 = 5\,\%$, and $\lambda_0 = 0$.

  • $E(R_A) = (0.8)(4\,\%) + (0.9)(5\,\%) = 7.7\,\%$

  • $E(R_B) = (-0.2)(4\,\%) + (1.3)(5\,\%) = 5.7\,\%$

  • $E(R_C) = (1.8)(4\,\%) + (0.5)(5\,\%) = 9.7\,\%$

• Market Expected Prices (Calculated Value):

  • $E(P_A) = 35 \times (1.077) = 37.70\,USD$

  • $E(P_B) = 35 \times (1.057) = 37.00\,USD$

  • $E(P_C) = 35 \times (1.097) = 38.40\,USD$

• Identifying Mispricing:

  • If an investor believes actual future prices will be: $P_A = 37.20\,USD$, $P_B = 37.80\,USD$, and $P_C = 38.50\,USD$.

  • Stock A is overpriced (Investor expects 37.20 vs Market value 37.70).

  • Stock B is underpriced (Investor expects 37.80 vs Market value 37.00).

  • Stock C is underpriced (Investor expects 38.50 vs Market value 38.40).

• Arbitrage Execution:

  • Riskless arbitrage involves selling short the overpriced asset and buying the underpriced assets.

  • Portfolio Weights: $w_a = -1.0$, $w_b = +0.5$, $w_c = +0.5$.

  • Conditions for arbitrage:

    • No net wealth invested ($\sum w_i = 0$).

    • No systematic risk exposure ($\sum w_i b_{ij} = 0$).

    • Positive actual portfolio return (\sum w_i R_i > 0).

Empirical Tests of the APT

Roll-Ross Study (1980)

• Methodology:

  1. Used time-series data to estimate expected returns and factor loadings.

  2. Used these estimates to test cross-sectional pricing conclusions.

• Sample: Daily returns from 1962 to 1972. 1,260 stocks divided into 42 portfolios by alphabetical order (30 stocks per portfolio).

• Null Hypothesis: There exist non-zero constants $(\lambda_0, \dots, \lambda_k)$ such that $[E(R_i) - \lambda_0] = \lambda_1 b_{i1} + \dots + \lambda_k b_{ik}$.

• Findings:

  • Found a maximum of 5 reasonable factors using factor analysis.

  • With an assumed risk-free rate of $6\,\%$, 3 or 4 factors meaningful factors existed.

  • When $\lambda_0$ was estimated by the model, only 2 factors were consistently significant.

  • Inter-group testing (Group A vs Group B) showed similar $\lambda_0$ terms, suggesting consistent factors across groups.

Other Empirical Studies

• Cho, Elton, and Gruber (1984): Examined the number of factors in the return-generating process.

• Dhrymes, Friend, and Gultekin (1984/1985): Determined the number of factors varies based on portfolio size (number of stocks). They also found that unique risk could predict future returns.

• Roll and Ross (1984): Argued that the number of factors is secondary to the model's explanatory power.

• Connor and Korajczyk (1993): Developed a test for factor counts that allows for correlation among unsystematic risk components.

• Harding (2008): Investigated connections between systematic and unsystematic risk factors.

APT and Stock Market Anomalies

• Small-Firm Effect:

  • Reinganum (1981): Found results inconsistent with APT; small firms had positive risk-adjusted returns, large firms had negative.

  • Chen (1983): Supported APT over CAPM, arguing APT explains residual returns better.

• January Anomaly:

  • Gultekin and Gultekin (1987): APT did not perform better than CAPM in explaining this effect.

  • Burmeister and McElroy (1988): While APT didn't capture the effect better, their analysis still favored APT over CAPM.

Challenges to APT

• Shanken’s Challenge (1982): Argued that APT lacks testability because factors are not explicitly defined. Different factor structures might explain the same return sets. If a model fails, it might simply be due to the use of incorrect factors rather than a failure of the theory itself.

• Alternative Testing Techniques:

  • Jobson (1982): Multivariate linear regression.

  • Brown and Weinstein (1983): Bilinear paradigm.

  • Geweke and Zhou (1996): Exact Bayesian framework.

Multifactor Models in Theory and Practice

• A multifactor model allows investors to choose the exact number and identity of risk factors.

• Theoretical Equation: $R_{it} = a_i + [b_{i1} F_{1t} + b_{i2} F_{2t} + \dots + b_{iK} F_{Kt}] + e_{it}$

  • $F_{jt}$: Period $t$ return to the $j^{th}$ designated risk factor.

  • $R_{it}$: Nominal or excess return of Security $i$.

Macroeconomic-Based Risk Factor Models

• Chen, Roll, and Ross (1986): Proposed these factors:

  1. $R_m$: Return on value-weighted index of NYSE-listed stocks.

  2. $MP$: Monthly growth rate in US industrial production.

  3. $DEI$: Change in inflation (US CPI).

  4. $UI$: Difference between actual and expected inflation.

  5. $UPR$: Unanticipated change in bond credit spread.

  6. $UTS$: Unanticipated term structure shift (long-term minus short-term RFR).

• Burmeister, Roll, and Ross (1994): Analyzed models based on:

  • Confidence Risk: Changes in willingness to take investment risk.

  • Time Horizon Risk: Changes in desire for payouts.

  • Inflation Risk: Unexpected short and long-term components.

  • Business Cycle Risk: Changes in overall business activity.

  • Market Timing Risk: S&P 500 returns not explained by the other four factors.

• Observation: Smaller firms are more exposed to business cycle and confidence risk, but less to horizon risk.

Microeconomic-Based (Characteristic-Based) Models

• Fama-French Three-Factor Model (1993):

  • Equation: $(R_{it} - RFR_t) = a_i + b_{i1}(R_{mt} - RFR_t) + b_{i2} SMB_t + b_{i3} HML_t + e_{it}$

  • $SMB$: Small Minus Big (return on small-cap stocks less return on large-cap stocks).

  • $HML$: High Minus Low (return on stocks with high book-to-market ratios [Value] less low [Growth]).

• Fama-French Five-Factor Model (2015):

  • Equation: $(R_{it} - RFR_t) = a_i + b_1(R_m - RFR) + b_2(SMB) + b_3(HML) + b_4(RMW) + b_5(CMA) + e_i$

  • $RMW$: Robust Minus Weak (high profitability minus low profitability firms).

  • $CMA$: Conservative Minus Aggressive (firms with steady capital investment minus those with low investment).

  • Exclusion: Momentum (MOM) is excluded because it is viewed as a short-term market phenomenon rather than a long-term risk factor.

Estimating Expected Returns for Individual Stocks

Successful estimation requires four steps:

1. Identify a specific set of $K$ common risk factors.

2. Estimate the risk premia for those factors.

3. Estimate the sensitivities ($b_{ij}$) of the stock to each factor.

4. Compute the expected returns by combining these estimates within the APT framework.

Checkpoint Questions and Discussion

• Checkpoint 1:

  1. A similarity between APT and CAPM is the linear relationship between risk and return.

  2. A key attribute APT seeks to provide price outcomes for is mispricing identification (arbitrage).

  3. To calculate expected returns with two factors, you also need the zero-beta asset return ($\lambda_0$) and the risk premiums for each factor ($\lambda_1, \lambda_2$).

• Checkpoint 2:

  1. If Stock A has an SMB coefficient of $-0.22$ and an HML coefficient of $0.89$, it signifies Stock A is a Large-Cap Value stock (negative SMB targets large, high HML targets value).

  2. Fama-French factor risk premiums are sometimes at odds with high betas in CAPM, particularly regarding firm size and value characteristics that CAPM's beta may not capture.