By Evgenia "Jenny" Nitishinskaya and Delaney Granizo-Mackenzie
Part of the Quantopian Lecture Series:
Notebook released under the Creative Commons Attribution 4.0 License.
Arbitrage pricing theory is a major asset pricing theory that relies on expressing the returns using a linear factor model:
$$R_i = a_i + b_{i1} F_1 + b_{i2} F_2 + \ldots + b_{iK} F_K + \epsilon_i$$This theory states that if we have modelled our rate of return as above, then the expected returns obey
$$ E(R_i) = R_F + b_{i1} \lambda_1 + b_{i2} \lambda_2 + \ldots + b_{iK} \lambda_K $$where $R_F$ is the risk-free rate, and $\lambda_j$ is the risk premium - the return in excess of the risk-free rate - for factor $j$. This premium arises because investors require higher returns to compensate them for incurring risk. This generalizes the capital asset pricing model (CAPM), which uses the return on the market as its only factor.
We can compute $\lambda_j$ by constructing a portfolio that has a sensitivity of 1 to factor $j$ and 0 to all others (called a pure factor portfolio for factor $j$), and measure its return in excess of the risk-free rate. Alternatively, we could compute the factor sensitivities for $K$ well-diversified (no asset-specific risk, i.e. $\epsilon_p = 0$) portfolios, and then solve the resulting system of linear equations.
There are generally many, many securities in our universe. If we use different ones to compute the $\lambda$s, will our results be consistent? If our results are inconsistent, there is an arbitrage opportunity (in expectation). Arbitrage is an operation that earns a profit without incurring risk and with no net investment of money, and an arbitrage opportunity is an opportunity to conduct such an operation. In this case, we mean that there is a risk-free operation with expected positive return that requires no net investment. It occurs when expectations of returns are inconsistent, i.e. risk is not priced consistently across securities.
For instance, there is an arbitrage opportunity in the following case: say there is an asset with expected rate of return 0.2 for the next year and a $\beta$ of 1.2 with the market, while the market is expected to have a rate of return of 0.1, and the risk-free rate on 1-year bonds is 0.05. Then the APT model tells us that the expected rate of return on the asset should be
$$ R_F + \beta \lambda = 0.05 + 1.2 (0.1 - 0.05) = 0.11$$This does not agree with the prediction that the asset will have a rate of return of 0.2. So, if we buy \$100 of our asset, short \$120 of the market, and buy \$20 of bonds, we will have invested no net money and are not exposed to any systematic risk (we are market-neutral), but we expect to earn $0.2 \cdot 100 - 0.1 \cdot 120 + 20 \cdot 0.05 = 9$ dollars at the end of the year.
The APT assumes that these opportunities will be taken advantage of until prices shift and the arbitrage opportunities disappear. That is, it assumes that there are arbitrageurs who have sufficient amounts of patience and capital. This provides a justification for the use of empirical factor models in pricing securities: if the model were inconsistent, there would be an arbitrage opportunity, and so the prices would adjust.