### CAPM vs. Arbitrage Pricing Theory: An Overview

In the 1960s, Jack Treynor, William F. Sharpe, John Lintner, and Jan Mossin developed the capital asset pricing model (CAPM) to determine the theoretical appropriate rate that an asset should return given the level of risk assumed. Thereafter, in 1976, economist Stephen Ross developed the arbitrage pricing theory (APT) as an alternative to the CAPM. The APT introduced a framework that explains the expected theoretical rate of return of an asset, or portfolio, in equilibrium as a linear function of the risk of the asset, or portfolio, with respect to a set of factors capturing systematic risk.

### Capital Asset Pricing Model

The CAPM allows investors to quantify the expected return on an investment given the investment risk, risk-free rate of return, expected market return, and the beta of an asset or portfolio. The risk-free rate of return that is used is typically the federal funds rate or the 10-year government bond yield.

An asset's or portfolio's beta measures the theoretical volatility in relation to the overall market. For example, if a portfolio has a beta of 1.25 in relation to the Standard & Poor's 500 Index (S&P 500), it is theoretically 25 percent more volatile than the S&P 500 Index. Therefore, if the index rises by 10 percent, the portfolio rises by 12.5 percent. If the index falls by 10 percent, the portfolio falls by 12.5 percent.

#### CAPM Formula

The formula used in CAPM is: E(ri) = rf + βi * (E(rM) - rf), where rf is the risk-free rate of return, βi is the asset's or portfolio's beta in relation to a benchmark index, E(rM) is the expected benchmark index's returns over a specified period, and E(ri) is the theoretical appropriate rate that an asset should return given the inputs.

### Arbitrage Pricing Theory

The APT serves as an alternative to the CAPM, and it uses fewer assumptions and may be harder to implement than the CAPM. Ross developed the APT on the basis that the prices of securities are driven by multiple factors, which could be grouped into macroeconomic or company-specific factors. Unlike the CAPM, the APT does not indicate the identity or even the number of risk factors. Instead, for any multifactor model assumed to generate returns, which follows a return-generating process, the theory gives the associated expression for the asset’s expected return. While the CAPM formula requires the input of the expected market return, the APT formula uses an asset's expected rate of return and the risk premium of multiple macroeconomic factors.

#### Arbitrage Pricing Theory Formula

In the APT model, an asset's or a portfolio's returns follow a factor intensity structure if the returns could be expressed using this formula: ri = ai + βi1 * F1 + βi2 * F2 + ... + βkn * Fn + εi, where ai is a constant for the asset; F is a systematic factor, such as a macroeconomic or company-specific factor; β is the sensitivity of the asset or portfolio in relation to the specified factor; and εi is the asset's idiosyncratic random shock with an expected mean of zero, also known as the error term.

The APT formula is E(ri) = rf + βi1 * RP1 + βi2 * RP2 + ... + βkn * RPn, where rf is the risk-free rate of return, β is the sensitivity of the asset or portfolio in relation to the specified factor and RP is the risk premium of the specified factor.

### Key Differences

At first glance, the CAPM and APT formulas look identical, but the CAPM has only one factor and one beta. Conversely, the APT formula has multiple factors that include non-company factors, which requires the asset's beta in relation to each separate factor. However, the APT does not provide insight into what these factors could be, so users of the APT model must analytically determine relevant factors that might affect the asset's returns. On the other hand, the factor used in the CAPM is the difference between the expected market rate of return and the risk-free rate of return.

Since the CAPM is a one-factor model and simpler to use, investors may want to use it to determine the expected theoretical appropriate rate of return rather than using APT, which requires users to quantify multiple factors.

### Key Takeaways

- The CAPM lets investors quantify the expected return on investment given the risk, risk-free rate of return, expected market return, and the beta of an asset or portfolio.
- The arbitrage pricing theory is an alternative to the CAPM that uses fewer assumptions and can be harder to implement than the CAPM.
- While both are useful, many investors prefer to use the CAPM, a one-factor model, over the more complicated APT, which requires users to quantify multiple factors.