Factor investing: the capital asset pricing model

This post is part of a series aimed at introducing factor investing. To understand what it is and whether it’s something you might consider for your own portfolio, we recommend first reading the introduction to the topic:

Factor investing: a gentle introduction

Factors help explain the sources of stock returns, and factor investing is a disciplined approach to capture these differences in order to achieve higher risk-adjusted returns.

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A fundamental question in finance is how the riskiness of an asset should affect its expected return. The Capital Asset Pricing Model (CAPM) provided the first coherent framework for answering this question. The CAPM was developed in the early 1960s by William Sharpe, Jack Treynor, John Lintner and Jan Mossin independently, building on the earlier work of Harry Markowitz on diversification and modern portfolio theory.

According to this research, an individual investment like a stock is characterized by two kinds of risk:

• Systematic risks are market-wide risks that even portfolio diversification cannot reduce. Examples include inflation, financial crises, wars, and geopolitical instability.
• Idiosyncratic risks are specific to a company, an industry, or an asset class. Examples include mismanagement, shortages of necessary materials, cybersecurity breaches, and scandals.

The CAPM is based on the idea that not all risks should affect asset prices. In particular, idiosyncratic risks can largely be reduced through diversification by holding other securities in the same portfolio. According to the CAPM, then, the formula for calculating the expected return of an asset given only the systematic risk [1] is:

\[R_i = R_f + \beta (R_m - R_f) \] Where: \[ \begin{align} R_i &= \text{expected return of investment}\\ R_f &= \text{risk-free rate}\\ \beta &= \text{systematic risk of the investment}\\ R_m &= \text{market return} \end{align} \]

Let's unpack it.

\(R_f\) is the risk-free rate, which is the return you'd get from a risk-free investment. No investment is truly risk-free, but short-term government bonds are typically used as a proxy. Such bonds would only fail to deliver if the government defaulted on its debt, which is considered highly unlikely in the very short term. The rationale for this term in the equation is that it represents the bare minimum you should accept for taking on systematic risk; if the expected return of your stock is lower, you’d be better off investing in the risk-free asset instead.

\(R_m\) is the return of the broad market, and the expression \((R_m - R_f)\) is often called the market risk premium. This is the extra return above the risk-free rate that investors are rewarded for taking on market risk.

\(\beta\) describes how much the asset tends to move relative to the market; you can think of it as a measure of sensitivity. A value of 1.0 means that if the market rises by 10%, the asset would also rise by about 10%. Values higher than 1.0 mean the asset amplifies the market’s movements (higher volatility), while values between 0 and 1 indicate the asset is less volatile than the market. A negative value, although rare, indicates that the asset acts as a hedge, moving in the opposite direction of the market.

For example, suppose a stock has \(\beta = 1.2\) (making it more sensitive to market risk), the expected return of the broad market is \(8\%\), and the risk-free rate is \(1\%\).

\[ R_i = R_f + \beta (R_m - R_f) = 1 + 1.2 \cdot (8 - 1) = 9.4\%\]

According to CAPM, then, you would expect this asset to yield an expected return of \(9.4\%\) as fair compensation for the risk. This example focused on a single asset, but the same formula applies to a portfolio as well; the only difference is that you would then consider the portfolio’s sensitivity, \(\beta\), to market risk.

If you invest in a diversified broad-market portfolio, then under CAPM the \(\beta\) of your investment is by definition 1.0, meaning your returns move in line with the overall market.

\[ R_i = R_f + 1.0 \cdot (R_m - R_f) = R_m \]

As hinted in the introduction, by holding a diversified portfolio you are, in effect, a factor investor. The first such factor is market risk, and investors are rewarded for bearing this risk.

In the years following the development of CAPM, however, several critiques were raised, particularly concerning its assumptions. The model underwent refinements; for example, Fischer Black introduced a version in 1972, now known as the Black CAPM, which removed the need to assume the existence of a truly risk-free asset. Despite such improvements, it became increasingly clear that CAPM did not fully capture the behavior of asset returns.

A major challenge came in 1981, when Rolf Banz published a seminal paper titled "The Relationship Between Return and Market Value of Common Stock" [2]. In it, he showed that stocks of smaller companies earned consistently higher average returns than the CAPM would predict. This size anomaly suggested that market risk alone was not sufficient to explain differences in returns.

Four years later, CAPM took another major blow. Barr Rosenberg , Kenneth Reid and Ronald Lanstein published "Persuasive Evidence of Market Inefficiency" [3], which showed that stocks with high book-value relative to their market price (i.e. high book-to-market) also earned higher average returns than could be explained by market risk alone.

The book-to-market ratio compares the accounting value of a company’s assets (its book value, essentially assets such as buildings, machines, and inventory minus all the liabilities they owe) with the value investors assign to the company in the stock market (its market value). A high book-to-market ratio means the stock is trading close to or even below its book value, suggesting it may be undervalued. These stocks, now commonly called value stocks, appeared to deliver excess returns in the historical data.

Together, these papers pointed to clear anomalies that market risk alone could not explain; CAPM was simply too limited as a model of portfolio returns. The work of Banz and Rosenberg, among others, increasingly hinted at the existence of additional drivers of performance, what we now call factors, and the search for them had only just begun.

[1] Sharpe, William F. (1964). "Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk." The Journal of Finance, https://doi.org/10.2307/2977928
[2] Banz, R. W. (1981). "The relationship between return and market value of common stocks". Journal of Financial Economics, https://doi.org/10.1016/0304-405X(81)90018-0
[3] Rosenberg, B., Reid, K. and Lanstein, R. (1985). "Persuasive Evidence of Market Inefficiency". Journal of Portfolio Management
http://dx.doi.org/10.3905/jpm.1985.409007