Sunday, July 26, 2009

Aaron Brown Reviews Finding Alpha

Not quite Michiko Kakutani, but someone finally thoughtfully reviewed my book. Aaron Brown blogs a lot over at the Wilmott general forum, and he's the kind of practitioner I was hoping to reach: a practitioner who understands financial mathematics. Given his other reviews on Amazon, we don't have a lot in common: he likes poker (which I find boring), and he likes a lot of books I don't. He wrote a neat book, the Poker Face of Wall Street, about how our innate desire to gamble leads to all sorts of markets, even financing. He has a rather long review over at Amazon. While he gave it a mixed review (3 out of 5 stars), I take pride in that all his other reviews are "5" or "1", which means I made him think. I think the positives he mentions are objectively more positive than the negatives--I'd rather be sloppy but make an important novel point, than not have such a point but write with precision and verve. As my book doesn't get a thoughtful review very often, I want to address his criticisms:

First, he notes the following typo:

An example of the sloppiness (and there are many) is on page 21, "The key to the portfolio approach is the variance of two random variables is less than the sum of their variance." This makes no sense. He might mean "The variance of THE SUM of two random variables is less than the sum of their variances,"

Alas, he is exactly right. I meant 'variance of the sum' but wrote 'sum of the variances'. A classic brain fart, like writing 'the wet was water', a phrase one's brain turns into 'the water was wet' when you read it 50 times because that's what brains do, and reading it over and over makes it even harder to see.

Fundamentally, there was only light editing of my work. I recognize that meticulousness is not one of my strong points, and should have paid for extra editing, but I was grateful to Wiley for publishing my book, and so took what I got. As the list of acknowledgements will attests (ie, none), this book did not benefit from extensive vetting. There are about 5 such mistakes, and as to whether this is 'many' is a matter of perspective. Yet in context generally it was obvious as to what is meant. For example, in this case it was right above a graph that shows that portfolio variance declines as a function of portfolio size, and talk about how diversification lowers risk. But I can sympathize that such mistakes are distracting.

Brown also bemoans the $95 price. Gee, I didn't know anyone would pay that much. Amazon sells it for about $60, and the 'used' copies are basically new copies released to small stores that are selling it, and these go down to $52. Nevertheless, it is above the usual $20 to $30 one is used to paying for books, but it's a niche, and books with math usually have sufficiently small audiences that such is the price. I would have liked to sell a $30 book so it could be at Barnes and Noble, but it's a rather specific, technical arguement with a limited audience.

A final criticism was that I "Lack of appreciation for other people's thought", and he specifically mentions Taleb. Alas, I have Taleb fatigue and consider him a lightweight blowhard, not The Establishment I am criticizing (represented by, say, Fama, French, Cochrane, Campbell, Harvey, Schwert, etc.). In a sense, like Taleb, I am criticizing the Establishment, but we have as much in common as typical third party candidates. Considering that Taleb's main empirical points are about fat tails and peso problems, these are standard problems and are addressed through the extant, large literature, that exists independent of Taleb.

But in general, I think I amply provide references to earlier research, and for almost every empirical point provide specific papers. My novelty, empirically, is highlighting that the scope of data suggest a flat risk-return relation in general; each individual finding, such as within equities, or currencies, points to literature in that domain, literature that is often quite deep.

Brown states:
The book's summary of quantitative finance is backed up by lots of references, but I would bet that the author has not read all the references. He doesn't even seem to be familiar with Hyman Minksky's work, and he was a graduate assistant for Minsky.

Actually, I have thoroughly read all my references, and I do appreciate Minsky's work (I was his TA as an undergrad, not grad), and noted the Minskian notion of Keynesian uncertainty was not fruitful because metrics of uncertainty, emprically, are not positively related to returns (eg, IPOs, or firms with relatively high amounts of analyst forecast dispersion, have lower returns than average).

Brown then states

And "expected return" only makes sense in a rigorous context (who does the "expecting," and when?). The author scorns rigor, but then uses the concept of expected return in a model with divergent expectations among investors, without discussion of whose beliefs define the expected return, and fails to distinguish that concept clearly from ex post average return and market-clearing return.

I don't make a distinction between long and short term expectations, and most asset pricing models are simple one-period models. In many cases, one can find short and long term expectations have different properties unless returns have certain distributions, or utility functions have specific functional forms, and generally these discussions lead to hair splitting, and generally add little insight. Thus, like almost everyone in this space I found this distinction an uninteresting complication. More fundamentally, expected returns should equal average returns over large enough sample, just as a sample mean converges to the population mean over time. That so many assets show sample means in direct contrast to theory implies to many of Finance's founding fathers that 'expected returns' are hard to measure, but after 50 years I find this rather uncompelling.

But let's get on to the good things he had to say, after all, it was 3 stars, not 1!

What I like about this book:
* It contains important new ideas that can help any risk-taker with quantitative skills succeed
* It challenges conventional wisdom
* The meat of the book is based on practical experience, not just things that seem right to the author, but things he has tried, and generally with success

Those are good things, I think, and highlight in make a new point about something important, with a practical slant. I think for $52, even with typos, that makes it a good buy.

Brown ends with
This idea is integrated into a reasonably complete financial theory. The foundation seems solid but, as described above, its superstructure is jumbled and ugly.

The ugliness? I too dislike ugliness. I did not want the book to be too technical, however, so my fundamental model is a bit incomplete in the book, but the gist is rather simple:

Payoffs to Assets X and Y in States 1 and 2
Total Return
Relative Return
state 1
state 2

As shown in the table above, Y is usually considered riskier, with a 60 point range in payoffs versus a 20 point range for X. Yet on a relative basis, each asset generates identical risk. Everything follows from that. The proof of this is rather straightforward, and I outline several models of varying degrees of elegance in this SSRN paper here. But that's the big idea in a simple nutshell, and I don't find such a simple model ugly.


Michael Stack said...

"this book did not benefits from extensive vetting."

Nor the blog! :)

That's OK though - I'd rather see a good blogger spend more time writing new posts than proof-reading existing blog posts.

Aaron Brown said...

Thanks for a graceful response to a mixed review.

I am puzzled by one point, you say that the sum of the variances of two random variables is less than the sum of their variances. But that is only true for assets with negative covariance. The formula is:

Variance(A + B) = Variance(A) + Variance(B) + 2*Covariance(A,B)

I would not have singled out the omission of "sum of" by itself, but in this case it left me guessing about your meaning. I'm still guessing.

I disagre with, "expected returns should equal average returns over large enough sample, just as a sample mean converges to the population mean over time." The trouble is parameters are not constant over a sufficiently long period to make this true in practice. Expected return of individual assets has to be a matter of faith, not statistical proof. That doesn't mean you can't get reliable empirical results about strategies.

I agree wholeheartedly that the deviation of ex post asset returns from simple theory means the simple theory is wrong. There is plenty of evidence for that. But I think you and I are both really interested in future returns, not past data or anyone's expectations. If you expressed your theory in terms of the best possible current prediction of future returns it would retain all its practical value, and would fit in quite well with modern academic theories.

The past is for archeologists. A dead record of historical prices is problematic to analyze, because you don't really know what you could have executed at what price. And even if you explain it correctly, you have only explained how things used to be. Finance is much to dynamic for that to have much value.

I love your book because it deals with living issues in useful and important ways, but I still consider it an ugly description of the dead past.

I concede Taleb is an acquired taste, although I found much more in his books than you did. I did not cite him as someone so authoritative you had to mention him, but as the most obvious source on the subject.

Eric Falkenstein said...

Right again! I mean volatility. The basic point is that diversification lowers risk, and this leads to idiosyncratic risk going to zero, leaving only covariance as the determinate of portfolio volatility. This is the essence of the Markowitz's point, and the surprising thing was that this is not zero (surprising to the conventional wisdom back then). But you are correct, I regret the error.

As to expected returns changing over time, this is possible, but with junk bonds vs. inv. grad bonds, high beta stocks vs. low beta stocks, out-of-the-money calls vs. in the money, the changes in their expected returns due to stochastic discount factors, and changes in their 'betas' (to the SDF of choice), are insufficient to explain the fact that the sample averages over all the time we have show no relation, if not a negative relation, between groupings of low and high risk. For example, a beta 2.0 stock will change its beta over time, usually towards 1, but on average the betas are well above 1, and the risk premiums should be above those stocks with betas around 0.5 (whose betas may move up to 0.7). Over long periods of time, assets with higher volatility, default risk, uncertainty, and covariance of the market have at best equaled the return of their asset class counterparts. I don't think 'expected returns' can save them. I think going forward, avoiding these assets, for someone looking to invest in size, is I think optimal. That is, if you just want to maximize your Sharpe or Information Ratio, it seems a very good strategy.

Aaron Brown said...

You get no argument from me, or anyone with a brain who has looked at the data, that the lack of relation between sample averages and risk groupings means the simple model cannot be right. I prefer to speak in terms of strategies (e.g. "Buying a diversified portfolio of stocks with measured Betas over 1.5 over the prior three years and shorting a matched diversified portfolio of stocks with measured Betas under 0.75 does not produce consistently positive returns over long periods") because they are precise and testable, with plenty of data. I understand it's simpler to speak of assets, e.g. "high Beta stocks do not have carry a risk premium over low Beta stocks," but there are never enough data to form a conclusion about any individual asset, or even large portfolio of assets. It's even problematic to prove that the S&P500 outperforms T-bills on average. But that's okay because it's strategies that matter, not assets.

Now I understand what you meant by that sentence, but I think you're jumping over some basic issues. If you think about one period, then Sharpe ratio (mean excess return / standard deviation) or something else in the same units of dollars / dollars is the natural way to rank portfolios. At the other extreme, if you think about a long sequence of investments, then Kelly (mean excess return / variance) or something else in the units $ / $^2 is natural.

This is the difference between portfolio management and risk management. A portfolio manager wants to know the beneficial investor's time horizon, but the size of the investment is a relatively minor consideration. Sharpe ratio declines with time horizon but is independent of size. The risk manager wants to know the size, but time horizon is not important. Kelly increases with bet size, but is independent of time horizon.

Therefore, using the diversification argument as you do carries a lot of baggage with it. From a Kelly perspective, the risk of the sum of two uncorrelated gambles is equal to the sum of the risks.

In your terms, once someone has found alpha they might choose to run out and raise a lot of money from others to make a one-time killing, and then worry about tomorrow; or they might choose to raise the minimum necessary capital in order to keep as much equity as possible, and grow it exponentially. The first makes sense if you think of standard deviation as risk, if you're a portfolio manager at heart, if you work for a big institution. The second makes sense if you think of variance as risk, if you're a risk manager at heart, if you work for yourself. The first guy loves diversification, both lots of bets and long time horizon, and the CAPM speaks to him. If the CAPM is wrong, he needs a replacement in the same units. The second gal has no use for diversification or the CAPM, however reformulated. She doesn't have a problem that can be profitably analyzed in a single-period model, or even a multi-period extension of a single-period model. She speaks a different language.

After reading your book, I think your biggest issue with CAPM would still apply if it were true. In order to write the equation down you have to make assumptions that are incompatible with alpha. I don't mean assumptions that require alpha = 0, I mean assumptions that make the concept of alpha impossible. You don't want a replacement CAPM, you want to work in another dimension.

Eric Falkenstein said...

Aaron: I think the Siegel's finding that volatility decreases by time horizon is interesting, and a recent paper by Pastor and Stambaugh suggests this is crucially dependent on the fact that this is a sample estimate. If one assumes the mean return has a prior distribution, stocks are actually more volatile over long horizons. See Are Stocks Really Less volatile in the Long Run?.

I think if the CAPM were true, it would not be obvious that low beta investing is better. Given the CAPM is not true, it is. This 'arbitrage' is small, not hedge-fund worthy, but rather, like index investing, a good idea for most investors. Basically, quit chasing last year's big winners, accept a moderate return, and you can get a 1% return premium with lower intuitive measures of risk (beta, volatility). I think an institution like AQR would be in a position to provide this, because it needs to be sold, just as early index funds did. That is, it's obvious based on the math, but counterintuitive to many investors. You need data, theory, and a brand, to really sell this.