7. Implications of Existence and Equivalence Theorems

\(p=\mathbb E(mx)\) is Innocuous

Before Roll (1977), expected return-beta representations were derived inside specific explicit economic models (especially the CAPM), and any beta model's empirical success seemed to vindicate the whole structure. After Roll showed that mean-variance efficiency implies a single-beta representation, that changed: since some mean-variance efficient return always exists, some single-beta representation always exists. The asset pricing model only really claims that one particular return (say "the market") is mean-variance efficient. So "testing the CAPM" becomes a matter of being very choosy about the reference portfolio, to guard against stumbling on something that happens to be efficient and hence prices assets by construction.

Roll's insight led naturally to broader wealth indices (Stambaugh 1982) as the reference portfolio, but that route never caught on: stocks are priced by stock factors, bonds by bond factors; stocks sorted on size, book/market, and past performance are priced by portfolios sorted on those characteristics (Fama and French 1993, 1996). Part of the reason is that asset classes are not highly correlated and betas are small, so risk premia from one source barely affect another set of average returns; more comprehensive wealth measures (human capital, real estate) lack high-frequency price data. This leaves a nagging (and, to a researcher, exciting) suspicion that markets may be segmented, especially at high frequency.

Ex Ante and Ex Post

The theorems hold for any set of probabilities — agents' subjective probabilities, objective population probabilities, or the moments realized in a given sample. So they work ex ante and ex post alike. This means: if the law of one price holds in a sample, one can form an \(x^*\) from sample moments such that \(p(x)=E(x^*x)\) holds exactly in that sample (\(p\) is observed prices, \(E\) is the sample average); equivalently, if the sample covariance matrix of returns is nonsingular, there exists an ex-post mean-variance efficient portfolio whose sample average returns line up exactly with sample regression betas.

Discipline

So what is wrong with using an ex-post efficient portfolio or \(x^*\) that prices by construction? The mistake: a portfolio ex-post efficient in this sample, and so pricing all assets in it, is unlikely to be efficient in the next sample, and so prices the future poorly. Likewise the \(x^*=p'E(xx')^{-1}x\) built from the sample second-moment matrix has weights \(p'E(xx')^{-1}\) that swing wildly from sample to sample. Even if the CAPM is true and the market is ex-ante efficient, it is unlikely to be ex-post efficient in any given sample — an ex-post efficient portfolio is a "Monday-morning quarterback," loading up on whatever was lucky in-sample. "If only I'd bought Microsoft in 1982…" is no guide to forming an efficient portfolio today (indeed, mean-reversion and book/market effects suggest past winners tend to do poorly going forward!).

The only way out is to impose some discipline against dredging up spuriously good in-sample pricing. It is the same as in regression \(y=x'\beta+\varepsilon\): blindly adding right-hand variables yields arbitrarily good fit but is unstable and useless out of sample. Econometricians have wrestled with this for ~50 years, and the best answers are: (1) use economic theory to specify the right-hand side carefully; (2) use a battery of cross-sample and out-of-sample stability checks. Alas, both are hard: economic theory is usually silent on which variables to include, or allows a huge range — Fama (1991) calls APT and ICAPM a "fishing license" for almost any desired factor. Out-of-sample tests often reveal instability and force changes; once changed, there is no out-of-sample left. And even a researcher pure enough to wait 50 years for a fresh sample has impatient competitors and editors.

In the author's view, the best hope for pricing factors robust out-of-sample and across markets is to understand the macroeconomic sources of risk — tying asset prices to macro events the way the ill-fated consumption model does via \(m_{t+1}=\beta u'(c_{t+1})/u'(c_t)\). The consumption model has lost favor, but empirically determined factors are unstable too (since Merton 1973a and Ross 1976a launched multifactor models, the standard set has changed about every two years). Efforts like Lettau and Ludvigson (2001a) to find macro explanations for empirical factors may be a useful compromise.

Our current statistical methodology for evaluating models and papers naturally resists such discipline: when the last author fished up an ad hoc model with 1% average pricing errors, it is hard to persuade readers, referees, editors, and clients that your economically motivated model with 2% errors is more interesting. Your model may truly be better and keep performing out-of-sample after the fished one falls by the wayside of financial fashion — but it is hard to get past in-sample statistical fit. One hungers for a formal count of the hurdles imposed on a fishing expedition, like the degrees-of-freedom correction in \(R^2\). Absent a numerical correction, we must use judgment to discount apparent statistical successes by the amount of economic and statistical fishing that produced them.

Mimicking Portfolios

\(x^*=\operatorname{proj}(m|X)\) also has empirical bite: any model's pricing implications are equivalently captured by its factor-mimicking portfolio. If the economic variables driving \(m\) have measurement error, the mimicking portfolio for the true \(m\) will price assets better than an estimate of \(m\) using the measured macro variables. So it is usually unwise to run a statistical horse race of an economically interesting model against models that use portfolio returns as factors: even if the economic model is true and perfectly measured, in large samples it just equals its own factor-mimicking portfolio; with any measurement error it underperforms it; and both always lose in-sample to ad hoc factor models that find nearly ex-post efficient portfolios.

This does not say to bypass understanding the true macro factors and just fish for mimicking portfolios. Practitioner experience bears this out: large commercial factor models from extensive statistical analysis (a.k.a. fishing) perform poorly out-of-sample, as shown by factors and loadings \(\beta\) that change all the time. But there is an important place for return-factor models: once the underlying macro factors are found, practitioners are well advised to watch the mimicking portfolio day-by-day (even minute-by-minute), since good high-frequency data on it are available and many purposes do not require understanding the economics.

Irrationality and the Joint Hypothesis

Finance has a long history of fighting over "rationality vs. irrationality" and "efficiency vs. inefficiency." Many empirical papers are sold as evidence of "inefficiency" or "irrationality" (the October 1987 crash; small-firm, book/market, seasonal effects; long-term predictability). But none of these puzzles documents an exploitable arbitrage opportunity. So we know there is a "rational model" — a stochastic discount factor, an efficient portfolio for a single-beta representation — that rationalizes them all. And we can confidently predict this will continue: real arbitrage opportunities do not last long!

The Number of Factors

Many tests obsess over "how many factors are needed to price a cross-section." The equivalence theorems show this is a non-question. A linear factor model \(m=b'f\) (and its equivalent beta model) is not a unique representation: given any multifactor representation, one can always reduce it to a single-beta representation — \(m=b'f\), \(x^*=\operatorname{proj}(b'f|X)\), or the corresponding \(R^*\) all give single-beta models with exactly the multifactor model's pricing ability. One can equally rewrite it with a different number (greater than one) of factors, e.g. collapsing a three-factor \(m=a+b_1f_1+b_2f_2+b_3f_3\) into two factors \(a+b_1f_1+b_2\hat f_2\). Sometimes a multifactor representation is of interest (factors carry economic meaning lost on combination), but the pure number of factors is not meaningful.

Discount Factors vs. Mean, Variance, and Beta

Why did mean-variance and beta language develop first, while discount-factor language seems to be taking over? Asset pricing began by putting the mean and variance of returns on the axes, rather than payoff-in-state-1, payoff-in-state-2 as we do now. Markowitz (1952) posed it just right: treat assets in the apples-and-oranges, indifference-curve-and-budget-set framework of microeconomics — the trouble was what to label the axes. "IBM stock" and "GM stock" won't do; investors value not securities per se but aspects of their random cash flows. The brilliant move was to put portfolio mean and variance on the axes as the "hedonics" by which investors value portfolios (more mean, less variance), with "utility functions" over mean and variance and the mean-variance frontier as the "budget set." Then naturally: each security's mean return measures its contribution to portfolio mean, and its regression beta on the portfolio measures its contribution to portfolio variance — the mean-return-versus-beta description for each security followed (Sharpe 1964).

Summary

The existence theorems mean there are no quick proofs of "rationality" or "irrationality": the only game in town for explaining asset prices is thinking about economic models of the discount factor. \(p=\mathbb E(mx)\) is innocuous, the beta representation is innocuous — all content is in the specification and restrictions of \(m=f(\text{data})\). The real empirical challenge is not fitting a sample (that succeeds "by construction") but imposing economically motivated, out-of-sample-robust discipline. "The number of factors" is a non-question; discount-factor language wins for its linear budget constraints, generality, and natural handling of no-arbitrage and multiple horizons.