Is CAPM dead? Beta, multi-factor models and what to keep

Suppose you are the finance director of a distribution company and the board asks which discount rate to use for a second warehouse. The textbook gives a formula: the risk-free rate plus beta times the market risk premium. A colleague shakes his head: "CAPM died years ago." This post reads three original papers to see who is right about what: Sharpe (1964) built the model, Roll (1977) asked whether it can be tested, and Fama and French (1993) looked at what the data say. See also Capital asset pricing model (CAPM) and From CAPM to multi-factor models.

What does CAPM say, and how does Sharpe describe his own assumptions?

Sharpe says the price of risk sits in the part that diversification cannot remove, and he knew his assumptions were unrealistic. The market, he writes, offers an investor two prices: the price of time (the pure interest rate) and the price of risk, the extra expected return for each unit of risk borne (Sharpe, 1964, p. 425).

To derive an equilibrium he assumes that all investors borrow and lend at one pure rate and agree on the prospects of every investment (Sharpe, 1964, pp. 433–434). He says plainly that these are "highly restrictive and undoubtedly unrealistic", and then defends them: a theory should be judged by its implications, not by the realism of its assumptions (Sharpe, 1964, p. 434). In my reading, that defence puts the whole later argument on the implications: if the data reject them, nothing is left to hide behind.

The main implication: an asset's return has a part that moves with the efficient combination and a part that does not. The first is systematic risk (systematic risk), the second is unsystematic, and only the systematic part survives when the asset is added to a portfolio (Sharpe, 1964, pp. 439–440). Sharpe writes this sensitivity as the coefficient B_ig; textbooks later call it beta. He then offers an interpretation: if the common risk is mainly swings in the economy, an asset unaffected by the economy earns the pure interest rate, and one that moves with the economy promises a higher expected return (Sharpe, 1964, pp. 441–442).

What hole did Roll point to?

Roll does not say CAPM is wrong. He says nobody has tested it correctly. The only testable proposition is that the market portfolio is mean-variance efficient; the linear relation between expected return and beta follows from it and cannot be tested on its own (Roll, 1977, p. 130).

From this he draws, in the paper's own summary:

1. A correct test needs the exact composition of the true market portfolio, meaning every asset must be in the sample (Roll, 1977, p. 130).
2. A stand-in portfolio (a proxy) can be efficient when the true market is not, or the reverse; every sample contains efficient portfolios that satisfy all the theory's implications (Roll, 1977, p. 130).
3. Two researchers who disagree about what the market contains cannot be brought to agree on the test results (Roll, 1977, p. 131).

My reading for anyone in corporate finance: the beta on a data screen is a beta against a stock index, not against the market in the theory's sense.

What did Fama and French find in the data?

They found that beta explains very little of the differences in average returns across US stocks, and that two added factors make the model fit. The opening of the 1993 paper says that average returns on US common stocks show "little relation" to the market beta of the Sharpe–Lintner model (Fama and French, 1993, p. 3). Their earlier paper (1992) had found that beta, alone or with other variables, carries very little information about average returns, and that size and the ratio of book to market equity (BE/ME) do better (Fama and French, 1993, p. 4).

The 1993 paper uses 25 portfolios of NYSE, Amex and NASDAQ stocks sorted by size and BE/ME, with monthly returns from July 1963 to December 1991 (Fama and French, 1993, p. 10). Each portfolio's return above the Treasury bill rate is regressed on three factors: the market, the gap between small and big stocks (SMB), and the gap between high and low BE/ME stocks (HML).

The two sentences worth remembering are on page 38: the size and BE/ME factors explain why average returns differ across stocks, while the market factor is needed to explain why stocks on average earn more than the one-month bill (Fama and French, 1993, p. 38). Beta was not thrown out; it was judged insufficient.

How much do Fama and French doubt their own model?

Quite a lot. They write that the joint test that all the leftovers for 25 stock portfolios and 7 bond portfolios are zero is still rejected at about the 0.95 level. Because three factors explain nearly all the variation (typical R² above 0.93), even a small leftover is enough to show that the model is just a model, which is to say, false. They still conclude it does "surprisingly well": only one of the 25 stock leftovers differs from zero in any practical sense (Fama and French, 1993, p. 41).

They name one weakness themselves. The choice of factors, size and BE/ME above all, rests on empirical experience. No theory gives the exact form of the common factors, so any choice is "somewhat arbitrary", and detailed stories about the coefficients are suggestive rather than definitive (Fama and French, 1993, p. 53). A second weakness is mine, not theirs: the data are listed US stocks over 29 years, and nothing guarantees the same holds for Vietnamese firms. They also write that many people still use the one-factor Sharpe–Lintner model to estimate the cost of capital although there is no evidence that it fits, and they hope their results will help end that habit (Fama and French, 1993, p. 54).

What should a finance director keep from these three papers?

Keep the argument, drop the false precision. This is my reading.

Keep three things. First, only risk that diversification cannot remove earns a premium (Sharpe, 1964, pp. 440–442). Second, the risk-free rate plus a premium for sensitivity to the market is still a starting point that is easy to explain to a board. Third, Roll's question: against what was this beta measured (Roll, 1977, p. 131)?

Drop three things. The belief that one beta gives one exact discount rate. The idea that CAPM has been verified. And building a three-factor model for a single project without a reason: Fama and French expect sampling error to be serious for individual securities, although they say so of their five-factor model (Fama and French, 1993, p. 54).

The example below is hypothetical and every number is mine. A second warehouse costs 10 billion dong and returns 2.0 billion a year for seven years, an IRR of about 9.2%. Suppose the risk-free rate is 3% and the market premium 5%, so the cost of equity is 3% plus beta times 5%.

Suppose you can only estimate beta somewhere between 0.8 and 1.4, because only a few listed firms serve as references. Across that range the decision flips. The lesson is not to pick the right beta but to ask where the project survives: here NPV stays positive if beta is below about 1.24. One sentence like that tells a board more than a single figure such as 8%. For NPV and IRR side by side, see NPV or IRR.

Going deeper: what I read and what I did not

My copies are not the publishers' pages, which charge for access. I read Sharpe (1964), Roll (1977) and Fama and French (1993) through publicly posted PDF copies, scanned images that I ran through optical character recognition; I rechecked every number I quote, and page numbers are the printed ones on the scans.

What I read: Sharpe's introduction, section III and the end, without following each step of the proof; Roll, pages 129 to 131, but not the detailed analysis of individual tests that follows; Fama and French, the introduction, the data section, the main results and section 7, but not the tables closely, the bootstrap simulations or section 6. I did not read Lintner (1965), Mossin (1966), Ross (1976), Fama and French (1992, 2015) or Carhart (1997); what this post says about the 1992 paper is the authors' own summary in the 1993 one. It does not ask whether the factors still hold in later data.

The "keep" and "drop" lists are my reading, not any paper's conclusion. The warehouse is a paper exercise with no Vietnamese market data behind it.