# Factor Model

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• Factor Model

### Factor Model

The central idea of modern financial economics: The average return of a stock is the payoff for taking risk.

### Factor Premium & Factor Exposure

Given a stock and a risk factor,Factor Premium quantifies the payoff to an investor who takes on this risk by buying the stock.Factor Exposure quantifies the exposure of the stock to this risk.Average stock return per this risk[box]average stock return = factor exposure X factor premium

### Mathematics

Suppose we have N stocks whose returns depend on K factors.

• the factor premiums for the K factors: $$f_1, cdots , f_K$$
• the factor exposures of stock i: $$beta_{i1}, cdots, beta_{iK}$$

The return on stock i, $$r_i$$” , can be written as $$r_{i} = alpha_{i} + beta_{i1}f_1 + cdots + beta_{iK}f_K + epsilon_{i}$$

The average stock return is $$E(r_i) = mathbf{beta_i}’E(mathbf{f})$$

where, $$mathbf{f} = {left( alpha_i, f_1, cdots,f_K right)}’$$, $$mathbf{beta_i} = {left( 1, beta_{i1}, cdots,beta_{iK} right)}’$$

Since return ($$r_i$$) and factor exposures ($$beta_{i1}, cdots, beta_{iK}$$) of the N stocks are known, we can compute, at any time point, the factor premiums ($$f_{1}, cdots, f_{k}$$) from the stock return equation using the ​OLS (Ordinary Least Square) ​regression.

$$r_{i} = alpha_{i} + beta_{i1}f_1 + cdots + beta_{iK}f_K + epsilon_{i}$$

### Panel Regression

However, knowing the factor premiums on separate time points (e.g. each month or season) would not be helpful for prediction and analysis. We would like to know whether there exist stable factor premiums for a longer time period (e.g. 2~3 years).

Given there are N stocks over T time periods, $$left { (r_{11}, cdots, r_{N1}), cdots,(r_{1T}, cdots,r_{NT}) right }$$

Factor exposures of N stocks over T time periods, $$left { (beta_{11}, cdots, beta_{N1}), cdots, (beta_{1T}, cdots,beta_{NT}) right }$$

We can estimate the factor premiums (f) from the following equation using OLS regression,

$$r_{it} = beta’_{it}f + epsilon_{it}$$

### Our Implementation in Algoquant

AlgoQuant has a package on QEPM to model factors, exposure and premium. We currently support the computation of premiums using cross-sectional or panel regressions (factor premiums on separate time points).

### Reference

​Ludwig B Chincarini and Daehwan Kim, Quantitative Equity Portfolio Management: An Active Approach to Portfolio Construction and Management, McGraw-Hill Library of Investment and Finance, 2006.