Arbitrage Pricing Theory

Arbitrage Pricing Theory

Multi-factor model for asset pricing which relates various macro-economic (systematic) risk variables to the pricing of financial assets.
Proposed by economist Stephen Ross in 1976
APT is founded upon the law of one price, which suggests that within an equilibrium market, rational investors will implement arbitrage such that the equilibrium price is eventually realised.
APT assumes that there are no available arbitrage opportunities, and that if one does exist, it will very quickly evaporate due to the trading actions of market participants.
Ri = E(Ri ) + β1F1 + β2F2 + ... + βkFk + ei

where: Ri = the actual return on stock i E(Ri ) = the expected return on stock i
β1 = the beta (factor sensitivity) for factor 1
F1 = the first in a series of risk factors that could add return deviation from the expected return -systematic factor (macroeconomic or company-specific factor)
βk = the beta (factor sensitivity) for factor k
Fk = the last in a series of risk factors that could add return deviation from the expected return
ei = a random error term that accounts for company-specific (idiosyncratic) risk

While the CAPM formula requires the input of the expected market return, the APT formula uses an asset's expected rate of return and the risk premium of multiple macroeconomic factors.
APT does not specify the multiple factors to include in the analysis
core of the APT model is to find a combination of granular risk factors, such as those presented, that more closely predict the return of a financial asset.
An analyst would be wise to buy a security whose market price drifts lower than APT would suggest (due to unexpected factors) and to potentially short a stock whose price is too much higher than APT’s calculated return.
This logic introduces model risk and also the need to periodically update model coefficients to ensure robustness.
CAPM can be considered a "special case" of the APT in that the securities market line represents a single-factor model of the asset price, where beta is exposed to changes in value of the market.
APT can be seen as a "supply-side" model, since its beta coefficients reflect the sensitivity of the underlying asset to economic factors. On the other side, the capital asset pricing model is considered a "demand side" model. Its results, although similar to those of the APT, arise from a maximization problem of each investor's utility function, and from the resulting market equilibrium

Assumptions

Market participants are seeking to maximize their profits.
There are no arbitrage opportunities, and if any are uncovered, then they will be very quickly exploited by profit-maximizing investors.
Markets are frictionless (i.e., no barriers due to transaction costs, taxes, or lack of access to short selling).

Multifactor Model Inputs

first input is the expected return for the stock in question
A beta (factor sensitivity) is needed for each variable included in the model, and a value is needed for each factor as well
The error term (ei ) represents firm-specific return that is otherwise unexplained by the model. (Because firm-specific events are random, the expected (i.e., default) value for the error term is zero.)
multi factor model takes into account either company-specific risk or systematic risk exposure that is not captured by the single factor model.

Hedging Exposure to Multiple Factors

granular exposures captured by multifactor models enable a unique hedging opportunity
Using calculated factor sensitivities, an investor can build factor portfolios, which retain some exposures and intentionally mitigate others through targeted portfolio allocations
One challenge of hedging exposures when using multifactor models is the potential for error. Because this hedging process is based on the calculated model, there will always be an element of model risk
Another challenge arises if the hedging strategy is either rebalanced too infrequently or too often. Trading costs from frequent rebalancing could erode profits, and infrequent rebalancing could risk undesired exposures as relationships dynamically change in the markets (i.e., increase tracking error)
A third challenge results from assuming the underlying asset distribution is stationary over time.

The three main types of multi-factor models are Macroeconomic Factor Models, Fundamental Factor Models, and Statistical Factor Models.

Macroeconomic Factor Models

factors are associated with surprises in macroeconomic variables that help explain returns of asset classes.
The surprise or incremental return can be calculated as the actual value less the forecasted value, and the mean of the return is typically zero.
Chen, Roll and Ross introduced first macro economic model in 1980 and proposed the following four factors as one way to structure an APT model

The spread between short-term and long-term interest rates (i.e., the yield curve)
Expected versus unexpected inflation
Industrial production
spread between low-risk and high-risk corporate bond yields

Fundamental Factor Models

factors are characteristics of stocks or companies that can be used to explain the changes in stock prices. Examples of such factors are price-to-earnings ratio, market capitalization, and financial leverage.
In 1996, economists Eugene Fama and Kenneth French famously specified a multifactor model with three factors:

Small minus big (SMB) is the difference in returns between small firms and large firms. This factor adjusts for the size of the firm because smaller firms often have higher returns than larger firms
High minus low (HML) is the difference between the return on stocks with high book-to-market metrics and ones with low bookto-market values
A high book-to-market value means that the firm has a low price-tobook metric (book-to-market and price-to-book are inverses). This last factor basically means that firms with lower starting valuations are expected to potentially outperform those with higher starting valuations.

In 1997, Mark Carhart added a momentum factor to the Fama and French model to yield a four-factor model
In 2015, Fama and French themselves proposed adding factors for “robust minus weak” (RMW) that accounts for the strength of operating profitability and “conservative minus aggressive” (CMA) to adjust for the degree of conservatism in the way a firm invests

Statistical Factor Models

use various maximum likelihood and principal-components-based factor analysis procedures on cross-sectional/time-series samples of security returns to identify the pervasive factors in returns

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