Question

What is the exact difference between the single index model, arbitrage pricing model and linear regression model? Based on the residual value's differentiation, will the single index model be included in the linear regression model rather than the arbitrage pricing model? Thanks.

Answer 1

The Single Index Model (SIM) is an asset pricing model, according to which the returns on a security can be represented as a linear relationship with any economic variable relevant to the security.

In case of stocks, this single factor is the market return.

The SIM for stock returns can be represented as follows:

Where:

• Alpha (α) represents the abnormal returns for the stock
• β(r_m − r_f) represents the movement of the market modified by the stock’s beta
• ε represents the unsystematic risk of the security due to firm-specific factors.

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 betathereby 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)=rf+β1×FP1+β2×FP2+...++βn×FPn

Where r_f is the risk-free rate, β₁ is the beta for the first factor, FP₁ is the factor risk premium associated with the first factor i.e. the additional return that is expected for taking on the associated risk; β₂ is the beta for the second, FP₂ 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.

Regression modelling / analysis is basically a study to establish relation between 2 entities - if any.

Following places where one can use regression analysis:

1) To find out correlation between stock prices of 2 companies (this can be extended to find correlation between 2 competing companies, 2 companies operating in an unrelated industry etc)

2) The financial performance of a company with the performance of a particular industry

3) To determine correlation between indices of 2 countries (for ex Nasdaq vs FTSE) - to see if there is any correlation between the performance of them - if any.