FRM 6: The Arbitrage Pricing Theory and Multifactor Models of Risk and Return
Stephen Ross set out the Arbitrage Pricing Theory in 1976. Where the capital asset pricing model prices one source of risk, the expected risk premium on the market above a risk-free rate, APT makes the expected rate of return a linear function of several factors: the market among them, plus macroeconomic conditions and fundamental company attributes.
The theory will not name those factors, though. It shows several operate without saying which add explanatory power, so identifying them falls to economic logic and empirical analysis.
The return generating equation
Each bracket holds the difference between the observed and expected values in factor k, the surprise factor. The coefficient measures how strongly that shift reaches security i, and the last term is the noise factor, idiosyncratic to i alone.
Three assumptions
Systematic factors reaching every security explain asset returns. Diversification lets an investor strip specific risk out of a portfolio. And no arbitrage opportunities survive among well-diversified portfolios, since any that appeared would be traded away.
The arbitrage argument
Underneath sits a plain premise: an investor can assemble a portfolio carrying zero beta for zero net investment, and should it return anything positive, arbitrage delivers a sure profit. Ross proved that ruling arbitrage out forces every well-diversified portfolio to satisfy one condition.
The beta term is the factor loading for portfolio P on factor k, the zero-beta portfolio earns the risk-free rate with Cov(Ik, RZ) = 0 throughout, and each bracket is the premium factor k carries.
APT compared with the CAPM
Both models agree that specific risk earns nothing. APT goes further: it does not assume asset returns are normally distributed, and asks only that arbitrage opportunities are absent.
| CAPM | APT | |
|---|---|---|
| Factors | One, the market | Several, count not fixed |
| Foundation | Investor behaviour | No-arbitrage argument |
| Assumptions | Mean-variance efficiency, risk aversion | Neither required |
Source: the chapter.
With the market index as its only explanatory variable, the CAPM sits inside APT as a special case, and empirical work often prefers the multi-factor model for the way it decomposes what each factor contributes. One problem catches both: an expected rate of return cannot be observed, so testing uses the realised historical average.
Gregory Connor separates the factor models used in practice into three families.
Macroeconomic factor models
The first such model, from Chen, Roll and Ross in the 1980s, was tested for stocks listed on the New York Stock Exchange (NYSE). Four variables explained realised average rates of return: the spread of long-term over short-term interest rates, for time preferences; expected and unexpected inflation; industrial production, for cash flow expectations; and the yield difference between high-risk and low-risk corporate bonds, for risk preferences.
A later proprietary model from Roll and Ross with Burmeister and Ibbotson names five exposures: confidence risk from investor confidence, time horizon risk from interest rates, inflation risk, business cycle risk from real business activity, and market timing risk from the market index.
Fundamental factor models
Fundamental factors are attributes of a company or its industry: a price/earnings ratio, a book/price ratio, estimated revenue growth, market capitalisation. Fama and French work with two of them.
Statistical factor models
A statistical factor model works from historical, cross-sectional stock return data. Principal components analysis produces factors that are linear return combinations, uncorrelated with each other, explaining as much of the observed variance as they can. Compute monthly returns for 2,000 companies over 10 years and perhaps five factors capture most of the variation. Those five are statistical artifacts, and economic meaning is read into them later.
One factor reduces the calculation to the CAPM: the risk-free rate plus beta times the market risk premium.
An equity carries a beta of 1.20. The risk-free rate is 3.0 percent and the expected market risk premium is 5.5 percent.
The multifactor case
Three inputs drive a multifactor model: the return on the zero-beta portfolio, a risk premium per factor, and a factor beta measuring the sensitivity of the asset to each one. Factor betas come out of a time-series regression of asset excess returns against the factors.
State Street Global Advisors builds tradable baskets of equities from subsets of the S&P 500, gathered in the Standard and Poor’s Depository Receipts (SPDR) family of exchange traded funds. They are fairly liquid, and because the baskets themselves trade, an exposure the model measures can be acted on directly, by trading the relevant sector basket against the target equity.
| Symbol | Sector |
|---|---|
| XLB | Materials |
| XLE | Energy |
| XLF | Financials |
| XLI | Industrials |
| XLK | Technology |
| XLP | Consumer Staples |
| XLU | Utilities |
| XLV | Health Care |
| XLY | Consumer Discretionary |
Source: the chapter.
Fitted on five-day excess returns over the one-week US Treasury rate, December 1998 through June 2019, the model gives J.P. Morgan a beta of 1.000 on XLF, 0.223 on XLK and -0.212 on XLP.
An analyst forecasts weekly excess returns of XLF = 5%, XLK = -4.0% and XLP = 2.0%.
Factors also cut the estimation burden. Portfolio risk taken from every S&P 500 constituent would need roughly 125,000 different variances; through nine factors that count falls below 5,019, about as good given the error margins.
Eugene Fama and Kenneth French bolt two fundamental factors onto the CAPM. Small Minus Big (SMB) is the return on small stocks less that on large ones. High Minus Low (HML) sets stocks carrying high book-to-market values against those whose values are low.
Fama and French read the HML slope as a proxy for relative distress. A firm earning consistently high profits normally commands a market value well above book, and shows a negative HML slope; weak firms on consistently low earnings show positive ones. SMB captures the covariation in returns on small stocks, the small firm effect.
Once the model is estimated, signs and significance both matter. The alpha usually fails to stand apart from zero while the betas do, and a large company such as Coca-Cola carries a negative, significant SMB loading, its equity suffering when small stocks beat large ones.
A portfolio shows an alpha of zero, a market beta of 1.10, an SMB sensitivity of 0.70 and an HML sensitivity of -0.40. The risk-free interest rate is 3.0 percent, with premiums of 5.5 percent on the market, 2.4 on SMB and 3.0 on HML.
Two further fundamental factors arrived in 2015. Robust Minus Weak (RMW) is the return difference between companies whose operating profitability is robust and those where it is weak. Conservative Minus Aggressive (CMA) separates conservative investors of capital from aggressive ones. With both in the model, HML becomes redundant, and CMA carried a -0.7 correlation with it. Carhart (1997) added a momentum factor (MOM), the gap between stocks that rose in value over the prior month and those that fell.
Diversification disposes of idiosyncratic risk in theory, and does nothing about systematic risk. Factor betas make the systematic part hedgeable. Each factor can be treated as a fundamental security in its own right, so a countervailing exposure in a hedge portfolio H offsets a specific type of risk sitting in portfolio P.
Neutralising everything produces a zero-beta portfolio, opposite positions in each factor leaving no exposure at all. That is a choice: a manager who wants to keep certain systematic risks hedges only the rest.
A portfolio worth USD 10 million carries a beta of 0.60 to factor 1 and 0.90 to factor 2. Instrument A loads 1.00 on factor 1 alone. Instrument B loads 0.20 on factor 1 and 1.50 on factor 2.
The challenges of hedging with a factor model
Selecting the factors comes first, and a parsimonious choice matters, since each has to serve the risk-adjusted return objectives of the institution. Judgement enters, and no single set is perfect for all investors.
How often to adjust the hedge is the next question, a tradeoff between hedging cost and staying aligned with the portfolio. Run the strategy discontinuously and tracking errors appear; update it too frequently and trading costs mount and drag performance down.
Model risk covers factor model errors as well as errors in implementation. Mathematical mistakes and misleading assumptions both qualify, and a hedge built on linear factor models will be flawed wherever relationships among the factors are nonlinear. Assuming stationarity in the underlying asset distribution is another common error. Assumptions that hold in calm conditions fail under stress: in the 2007-2009 financial crisis many market-neutral hedge funds performed poorly.