The capital asset pricing model (CAPM) is the most widely used measure of risk, beta, and alpha; it implies that the market portfolio is mean–variance efficient and is thereby used to advocate for passive investment. Many academic studies empirically reject the CAPM. The authors illustrate through a reverse optimization method that the CAPM is consistent with the empirically observed return parameters and the market proxy portfolio weights. They conclude that the CAPM risk–return relationship is valid.

Investors and corporate managers can gain far-reaching information from the CAPM. But academics have found—and the financial community seems to believe—that the CAPM model cannot be justified with empirical evidence. Practitioners continue to use the model in their investment processes (to a varying degrees) because they lack a better alternative. The typical approach for testing the CAPM model involves empirically estimating the stock return parameters and examining whether these parameters satisfy the relationships implied by the model.

Using a different approach, the authors choose return parameters that ensure an efficient market proxy and are as close as possible to their sample counterparts. Given a market proxy, the authors’ optimization method looks for a set of mean return and standard deviation vectors that satisfy the condition of mean–variance efficiency for the proxy and are closest to the sample parameters. This approach does not assume the existence of a risk-free asset and considers simultaneous adjustments to average returns and standard deviations.

The authors show that the empirical proxy portfolio parameters are perfectly consistent with the CAPM, given an allowance for slight estimation errors in the return moments. Their analysis reveals that the CAPM delivers an improved estimate of expected return when they first calculate the adjusted mean return for the market index proxy and its corresponding zero-beta portfolio and then use those results along with the sample beta (which is close to the adjusted beta) in the model.

The authors collect a sample of 120 monthly return observations for each of the 100 largest stocks in the U.S. market for the period January 1997–December 2006. They implement MATLAB’s fmincon function to arrive at a set of adjusted expected returns and standard deviations and show that these adjusted parameters are not significantly different from the sample parameters used in t-tests.

They conduct two more tests: one with the assumption that individual stock returns are drawn from a multivariate normal distribution and the other with the assumption of a nonnormal distribution. They use the bootstrap method to show that the proxy portfolio is mean–variance efficient, even with some interdependence in estimation errors.

The authors show that the CAPM model cannot be empirically rejected because minor changes in return parameters lead to results that contradict previous negative and disappointing findings for the model. A comparative analysis of other asset pricing models and the CAPM, in which simultaneous corrections to the means and variations are considered, would aid practitioners in selecting an appropriate model for asset pricing.