In the previous lesson, we learned about the capital asset pricing model (CAPM). According to the CAPM, any risky asset’s expected return depends on its exposure to market risk. So, CAPM is based on the idea that market risk is the sole systematic risk factor. Because systematic risk can’t be eliminated via diversification, risk-averse investors demand a risk premium based on the amount of market risk they bear in their portfolios. But, what if there is more than one systematic risk factor? The arbitrage pricing theory offers a robust solution to calculate expected returns in the presence of systematic risk factors other than market risk.

Learning objectives

- Understand the APT formula.
- Evaluate different factor models.

## Arbitrage pricing theory formula

The capital asset pricing model posits a linear relationship between (market) risk and (expected) return as follows:

*E[R _{i}] = R_{f} + β_{i} (E[R_{m}] − R_{f})*

In this equation, *E[R _{i}]* is the expected return on asset

*i*and

*R*is the return on the risk-free asset.

_{f}*β*captures asset

_{i}*i*‘s exposure to the market risk, such that a higher

*β*implies higher exposure. Finally,

_{i}*E[R*

_{m}]*− R*is the market risk premium.

_{f }In comparison, the APT formula with *n* systematic risk factors is as follows:

*E[R _{i}] = R_{f} + β_{i1} λ_{1}*

*+ β*

_{i2}λ_{2}*+ … + β*_{in}λ_{n}where *β _{ij}* measures asset

*i*‘s exposure to the

*j*th systematic risk factor and

*λ*is the risk premium (i.e.,

_{j}*E[R*) on the same factor.

_{j}] − R_{f}*β*s are commonly referred to as

_{ij}**factor betas**or

**factor loadings**.

There are obvious similarities between the APT formula and the CAPM formula. In both formulas, the expected return on a risky asset depends on (1) the risk-free rate of return, (2) the risk premium on systematic risk factor(s), and (3) the asset’s exposure to the systematic risk factor(s).

## APT versus CAPM

A key advantage of the APT is that it accommodates multiple sources of systematic risk. In comparison, CAPM relies on a single systematic risk factor, which is the market risk. On the other hand, APT does not tell us what the systematic risk factors are (or how many of them exist). While CAPM offers a theoretical foundation for considering market risk as a systematic risk factor.

CAPM has a restrictive assumption that investors are mean-variance optimizers. This means that they choose portfolios based on their expected return and variance only. In reality, investors may care about other features of portfolios as well (e.g., skewness, kurtosis, etc.). APT has the advantage that it does not rely on this assumption of mean-variance optimization. It mainly requires that there are a sufficient number of sophisticated investors who monitor markets to exploit arbitrage opportunities. Their presence ensures that asset mispricings are short-lived, and prices satisfy the no-arbitrage condition most of the time.

One drawback of APT compared to CAPM is that APT may not hold as well for individual assets (e.g., stocks) as well-diversified portfolios. This is because individual assets carry a large amount of unsystematic risk, in which case the no-arbitrage condition, which APT relies on, is harder to satisfy.

## Factor models: Fama-French, Carhart, …

Following the theoretical development of the arbitrage pricing theory (see e.g., Ross, 1976 – full reference at the bottom), both academics and practitioners embraced the idea of multiple systematic risk factors. This has led to the development of so-called **factor models**.

One of the early factor models that gained prominence in both academia and industry is the renowned **Fama-French 3-factor model**. This model is developed jointly by the Nobel laureate Eugene Fama and Kenneth French. Apart from market risk, which is present in the CAPM, the 3-factor model includes the **size factor** and **value factor**. According to the authors, like market risk, these additional factors represent systematic risk factors. In particular, they argue that (i) small firms tend to be riskier than larger firms and (ii) value firms are generally riskier than growth firms. Their measure of size is market capitalization. They use the book-to-market ratio to capture value.

In their tests, Fama and French construct the following **factor-mimicking portfolios**: SMB and HML. SMB stands for small minus big. It is a portfolio long in small stocks and short in big ones. Consequently, SMB is exposed to the size factor. Similarly, HML stands for high minus low. It is long in high book-to-market value stocks and short in low book-to-market value ones. This makes HML exposed to the value factor. The analysis of Fama and French shows that both the size factor and value factor have significant explanatory power on the cross-section of stock returns. This is beyond what is explained by the market factor. Of course, this is a headache for CAPM as it claims the market factor to be the only systematic risk factor.

Another popular factor model is the **Carhart 4-factor model** (see Carhart, 1997 – reference at the end). It considers the **momentum factor **as an additional systematic risk factor. The momentum factor is based on stocks that exhibited significant price runups and those that experienced strong price rundowns. It can be captured by a factor-mimicking portfolio that is typically denoted as UMD. UMD stands for up minus down such that the portfolio is long in stocks whose prices went up the most and short in stocks whose prices went down the furthest. Carhart found that adding the momentum factor to the Fama-French 3 factors improves the power of the model in explaining asset returns.

More recently, Fama and French came up with a 5-factor model whereby they add the **profitability factor** and **investment factor **to their original 3-factor model (read Fama and French, 2015 under the further reading section at the bottom). The profitability factor is captured by the portfolio RMW, which deals with stocks that have robust versus weak profitability. The investment factor is captured by the portfolio CMA. CMA is long in stocks with conservative investments and short in stocks with aggressive investments.

Of course, there are many other factor models out there using different combinations of factors such as size, value, momentum, quality, profitability, liquidity, volatility, and so on. In fact, there are many funds out there that help investors to gain exposure to specific factors.

##### What is next?

This post is part of our free course on investments. In the previous post, we studied the capital asset pricing model (CAPM) in detail. In the next post, we will introduce the Sharpe ratio, which is a reward-to-risk ratio, as a risk-adjusted performance measure.

Further reading

- Carhart (1997), ‘On Persistence in Mutual Fund Performance‘,
*Journal of Finance*, Vol. 52(1), 57-82. - Fama and French (1993), ‘Common risk factors in the returns on stocks and bonds‘,
*Journal of Financial Economics*, Vol. 33(1), 3-56. - Fama and French (2015) ‘A five-factor asset pricing model‘,
*Journal of Financial Economics*, Vol. 116(1), 1-22. - Ross (1976) ‘The arbitrage theory of capital asset pricing‘
*Journal of Economic Theory*, Vol. 13(3), 341-360.

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