BUSFIN4221 – Investments (Module 3 – Factor Models)

Module 3: Factor Models

(BUSFIN 4221 – Investments)

Instructor:	Andrei S. Gonçalves
E-mail:	Andrei_Goncalves@kenan-flagler.unc.edu
Website:	andreigoncalves.com/BUSFIN4221

Welcome to Module 3!

Download the class slides here and the “to print” version here.

While Portfolio Theory approaches the question of “How should investors construct a portfolio”, it is silent about what would be the implications if all investors started to use Portfolio Theory to construct their portfolios. This Module is precisely about such implications. It turns out that if investors use Portfolio Theory to construct their portfolios, then the return process of any security/portfolio can be described in a simple way:

$r_{i,t}=r_{f}+\beta_{i}\cdot\left(r_{o,t}-r_{f}\right)+\epsilon_{i,t}$

$\mathbb{E}\left[r_{i}\right]=r_{f}+\beta_{i}\cdot\mathbb{E}\left[r_{o}-r_{f}\right]$

where

$r_{i,t}$ represents the return on security $i$ at time $t$

$r_{f}$ represents the risk-free rate

$r_{o,t}$ represents the return on the tangent portfolio (or optimal risky portfolio) at time $t$

$\beta_{i}$ represents the exposure of security $i$ to the systematic risk embedded into $r_{o}$

In words, if investors use Portfolio Theory, then the returns of any security/portfolio can be decomposed into three pieces: (i) a risk-free component, $r_{f}$ ; (ii) a component associated with systematic risk, $\beta_{i}\cdot\left(r_{o,t}-r_{f}\right)$ ; and (iii) a component capturing “firm-specific” risk, $\epsilon_{i,t}$ . Moreover, investors are only compensated for systematic risk, $\beta_{i}$ , in the sense that the expected return of the respective security/portfolio only depends on $\beta_{i}$ .

All factor models do is to provide a description of the tangent portfolio, $o$ , so that we can measure systematic risk. The famous Capital Asset Pricing Model (CAPM) imposes assumptions in the economy such that the market portfolio is the tangent portfolio and we have:

$\mathbb{E}\left[r_{i}\right]=r_{f}+\beta_{i}\cdot\mathbb{E}\left[r_{M}-r_{f}\right]$

where $\beta_{i}$ is commonly referred to as “the market beta” and measures the exposure to market risk.

Similarly, if the basic assumptions of the CAPM do not hold, the tangent portfolio will typically be different from the market portfolio. We often think of it as a passive position on the market portfolio added to several active Long-Short positions. If we understand what are such active positions, then we have a “multi-factor model”:

$\mathbb{E}\left[r_{i}\right]=r_{f}+\beta_{i}\cdot\mathbb{E}\left[r_{M}-r_{f}\right]+\beta_{i,A}\cdot\mathbb{E}\left[r_{A}-r_{a}\right]+\beta_{i,B}\cdot\mathbb{E}\left[r_{B}-r_{b}\right]+...$

with $\left(r_{A}-r_{a}\right)$ , $\left(r_{B}-r_{b}\right)$ , … representing the long-short positions.

It turns out that that there are many alternative patterns in investor behavior (different from the use of Portfolio Theory) that would lead to the same general implications for expected returns. One such behavior is described by the “Arbitrage Pricing Theory” and many others exist. The key point is that factor models seem to be a robust way to describe expected returns (certainly the best we currently have) and we use it both in academia and in practice.

Finally, if we understand what factors should be used, then we can better understand expected returns, which can improve applications such as capital budgeting (in corporate finance) and active management evaluation.