Monday, September 07, 2009

Are Expected Returns Unmeasurable?

Aaron Brown writes over at Willmott Forum (need to register):
I treasure the perfection in the Capital Asset Pricing Model, a necessary advance, even if expected return is nonmeasurable so the model cannot be tested.

Alas, I see often in responses to my critique of the standard theory that I am only using historical returns, not expected returns, so my take is invalid. Now, it is one things to say expected returns are difficult to measure, quite another to say they are unmeasurable. If large sample averages are not vaguely correlated with population averages, what does 'expected return' mean? It must mean one rationally expects randomness, making such a theory rather trivial. If a theory is by definition untestable, that is not merely 'imperfection', but rather, a bad theory (like asserting there are unmeasurable ghosts in my garage).

Nobel Prize Winner William Sharpe mentioned that expected returns are much more difficult to measure than anticipated in the 1960s, and I will admit these early tests contained several errors that needed fixing. For example, there's the 'two pass' sort, an adjustment for the 'errors in variables' such that high betas tend to be overestimated and low betas underestimated. But it has been 40 years, and we have data in the US back to 1927, and broader data, and only a small, ungeneralizabe fraction of it generates an intuitive scatter plot where some metric of risk is positively correlated with average returns. At what point does one say, this theory isn't untestable, it's wrong?

Currently, in empirical finance we know that size, value and momentum are related to returns, but it's not clear why. In the nineties most thought value reflected distress risk, but when measured financial distress actually is inversely related to future returns. One thing that is clear is beta and volatility are not positively correlated with returns. To say that financial theory is successful while the main facts were discovered via simple sorts, and sophisticated tests like GMM have uncovered nothing interesting to an investor, suggests current theory is pretty and consistent to be sure, yet I would only call a theory beautiful if it's true, because it in non-empirical, it's simply mathematical masturbation.


Anonymous said...

Good post. I think that Aaron Brown is an entertaining and thoughtful guy. But he is a guy with a schtick.

We have all probably peddled dubious empirical results for financial gain. If clients want to believe that they know what an asset's expected return happens to be then it is the job for us peddlers to apply marketing jujitsu to use the force of their beliefs to further our interests. If one has to invent a new set of supposed relationships to assert some set of expected returns, then make it up (won't be the first time). At some time in the future, the current generation of financial economists/theorists will be forgotten relics of the past along with other forgotten great thinkers such as the physiocrats and the mercantilists. Not just the names of individuals will be forgotten (Fama who, Cochrane who, anyone who), but the ideas put forward by these folks will be forgotten.

Aaron Brown is an interesting thinker. His day job is risk management. His boss, Cliff Asness, is a different sort of entertainer. Cliff tells stories and pulls in client assets. Aaron is part of the effort directed at fleshing out the AQR marketing materials so it looks like AQR is responsibly managing client assets. Aaron may be a seeker of truth. Cliff may be a seeker of a larger place in Greenwich. Derman may be a seeker of truth. Truth seekers rarely get what they seek, and maybe those focused on money and trinkets are more likely to succeed.

Walt French said...

Expected returns are indeed easy to measure when one plays dice or roulette. But what does one "measure" when talking about investors' beliefs about the unknowable future?

The post suggests that sufficient analysis of historical returns would suffice. What weight will you give to pre-Bolshevik Russia? ... to more prosperous, well-developed nations like Argentina or Japan?

OK, a survey of US investors. Maybe with an average weighted by wealth so it reflects the average dollar's expectations.

But of course, many individuals, when surveyed, will give offhand or inconsistent answers, and these expectations are quite volatile trend-followers over time, suggesting rather less than a well-defined notion that is being sampled. For example, try to make sense of high-net-worth investors' responses to simple stocks-vs-bonds questions in nominal and real terms, this year versus last.

Finally, my return expectations over the next 3 months need only be relative to one another to be useful inputs into the Markowitz formula; it's enough to know that I expect stocks will gain 2% more than cash, without caring what cash will return. Some of your fairly sophisticated respondents, therefore, may not even know how to give absolute answers that they would stand behind.

Net net: for me, expected returns are merely the numbers that I assign to assets when I'm doing asset allocation, and are my best swag at what, if I were to arithmetically average the independent results of a billion parallel universes that might happen in the future; the standard deviation is the same (history? pfft!) and the covariance also the product of individual scenarios' assets returns that are above or below the subsequent average over all the scenarios.

This rather dry view is perfectly consistent with all the AA and theoretical tools, and I find it very helpful in my work: it's all a forecast. Yet it says nothing about there being an "expected return" that means anything to anybody other than me.

Eric Falkenstein said...

Walt: I think one must abstract from individual expectations, which are very subjective and idiosyncratic, and the 'market', which is a 'rational' expectation.

But in any case, as you mention, ordinal rankings are definitely implied. It should be that even though we can't measure expectations over time in various asset classes, we should be able to say that 1931 was a riskier year to invest than 1926, or that Coke is a riskier stock than GM. When we look at various metrics of risk--over time, cross sectionall--we don't see any ordinal difference in average returns consistent with a risk premium. If the ordinal rankings don't show up over all these sorts, something is wrong with the basic theory.

Anonymous said...

Even if expected returns are measurable, who is to say that they will correlate with reality. The idea that basing your risk/reward situations off an aggregation of the market's sentiments still doesn't strike me as a good way to develop your own required rate of return. I think true investors have an inherent rate they want to attain regardless of the market's bearings.