# Arbitrage Pricing Theory

Arbitrage pricing theory (APT) is an asset pricing model which builds upon the capital asset pricing model (CAPM) but defines expected return on a security as a linear sum of several systematic risk premia instead of a single market risk premium. While the CAPM is a single-factor model, APT allows for multi-factor models to describe risk and return relationship of a stock.

Arbitrage pricing theory is based on the argument that there can be no arbitrage, i.e. no one can earn any profit without undertaking any risk. Based on the capital asset pricing model, stocks must fall on the security market line. If any stock is plotted above SML, i.e. it has higher expected return per unit of systematic risk i.e. beta, it is underpriced and vice versa. Arbitrage pricing theory states that any portfolio can deviate from the SML because it is exposed to a different systematic risk factors and such deviation doesn’t necessarily mean that the security is underpriced. Because if it was, someone can easily exploit the arbitrage opportunity by buying the portfolio/stock and short selling another stock or portfolio with the same beta thereby having zero systematic risk exposure, i.e. zero net-beta.

Arbitrage pricing theory is the foundation of multi-factor models, models which attempt to explain the expected return as a function of the risk-free rate plus the product of different components of systematic risk such as inflation rate, business cycle stage, central bank discount rate, etc. APT doesn’t define the risk factors nor it specifies any number. It just offers the framework to tie required return to multiple systematic risk components.

## Formula

Following is the general expression for required return under APT:

$$ E(R)=r_f+\beta_1\times{FP}_1+\beta_2\times{FP}_2+...++\beta_n\times{FP}_n $$

Where r_{f} is the risk-free rate, β_{1} is the beta for the first factor, FP_{1} is the factor risk premium associated with the first factor i.e. the additional return that is expected for taking on the associated risk; β_{2} is the beta for the second, FP_{2} is the risk premium associated with the second factor, and so on.

**Factor beta** is theoretically similar to the beta coefficient used in capital asset pricing model. It measures the sensitivity of expected return to a 1% change in the additional return required for taking on the associated risk.

**Factor (risk) premium** is the additional return that must be offered to the investor for him to take on the additional factor risk. It equals the expected return on the pure factor portfolio i.e. a portfolio that is only sensitive to that risk factor minus the risk-free rate.

We can define any number of risk factors having any plausible relationship to the expected return. However, the factors must be systematic in nature because any unique risk can be diversified away and isn’t compensated by an efficient market.

## Example: Fama-French Three-Factor Model

One of the most common multi-factor models is the Fama-French three-factor model which links expected return of a security to (a) the market risk premium, (b) a factor representing company size and (c) a factor representing whether the stock is a value stock or a growth stock.

The model can be expressed mathematically as follows:

$$ E(R)=r_f+\beta_m\times(r_m-r_f)+\beta_{SMB}\times{FP}_{SMB}+\beta_{HML}\times{FP}_{HML} $$

Where r_{f} is the risk free rate, the interest rate on long-term government Treasury bonds; β_{m} is the beta coefficient as defined in the CAPM; r_{m} is the expected return on the broad equity market index such as S&P 500; β_{SMB} (i.e. beta-small minus big) is the factor beta i.e. a measure of how sensitive return of a stock is to the factor risk premium FP_{SMB} which measures the difference in return between small-cap companies and large-cap companies; and β_{HML} (i.e. beta high minus low) is the factor beta related to factor risk premium FP_{HML} which equals the average return difference between stocks with high book value to market value minus stocks with low book value to market value ratios.

Let's say you have risk-free rate of 3.5%, expected return on the S&P 500 of 8%, expected return on the small-cap market index of 9.5% and large-cap market index of 7.5%; and expected return on value-style index of 8.3% and growth-style index of 7.9%.

Estimate required return on equity for two companies: Company A has CAPM beta of 1.2; beta to SMB factor of -0.5 and beta to HML factor of 0.3; while Company B has CAPM beta of 0.9, beta to SMB factor of 0.3 and beta to HML factor of -0.2

The required return on equity using the capital asset pricing model is calculated as follows:

$$ E(R_A)=r_f+\beta_m\times(r_m-r_f)=3.5\%+1.2\times(8\%-3.5\%)=8.5\% $$

$$ E(R_B)=r_f+\beta_m\times(r_m-r_f)=3.5\%+0.9\times(8\%-3.5\%)=7.55\% $$

Using the multi-factor model, required return on equity for Company A and B works out to 8.2% and 8.07% respectively:

$$ E(R_A)=3.5\%+1.2\times(8\%-3.5\%)+-0.5\times(9.5\%-7.5\%)+0.3\times(8.3\%-7.9\%)=8.2\% $$

$$ E(R_A)=3.5\%+0.9\times(8\%-3.5\%)+0.3\times(9.5\%-7.5\%)+-0.2\times(8.3\%-7.9\%)=8.07\% $$

It follows that the capital asset pricing model didn’t completely capture the associated risks correctly. It overstated the risk in Stock A and understated the risk in Stock B. Using multi-factor models in the arbitrage-pricing theory framework helps in pricing the different risks correctly.