Factor Models 101
Ross M. Miller
Miller Risk Advisors
October 25, 2004
My first exposure to a factor model came in late 1989 at
Kidder, Peabody. The model in question was Barra's Japan model. Kidder had
a test copy and I was the designated guinea pig who was charged with
finding a way to game it. Its central feature was a portfolio optimizer.
My first test for the optimizer was to construct the portfolio that would
most closely track the Nikkei 225 from a universe of stocks that included
the entire Nikkei 225. After crunching along on a PC for several minutes,
the model spit out a portfolio that looked nothing like the Nikkei 225 and
had loads of tracking error. Oh, well.
I think that it is fair to call William Sharpe the
father of factor models. When Sharpe was working on turning Harry
Markowitz's variance/covariance model of portfolio selection into a
practical tool for constructing real-world "optimal" portfolios,
he immediately ran into a problem. If you wanted to select from a large
universe of stocks, then you had to estimate the covariance between every
possible pair of stocks. For 1000 stocks, there are 499,500 covariances to
consider (along with the 1000 variances). Back in the early 1960s that was
too much to ask of even the largest mainframe computer and still takes a
good chunk of compute cycles.
William Sharpe came up with a clever shortcut that
effectively swept all the covariances under the rug. He assumed that
stocks were only correlated to the extent that they shared a common
factor. This factor, we'll call it the market factor, reflected the
well-known empirical observation that all stock prices were to varying
degrees correlated with one another. The variability of each stock was
determined by how much of this market factor it contained and how much of
a stock-specific factor that it contained. This stock-specific factor was
different for each stock and each had zero correlation with the market
factor and zero correlation with each other. Bill Sharpe looked deeply
inside his one-factor model of the stock market and what he discovered
there was the Capital Asset Pricing Model.
In an appropriate equilibrium setting, all the
stock-specific factors can be ignored and the market factor is all that
matters. If you want to know the equilibrium rate of return for a stock,
all you need to know is how much of the market factor it contains (and a
few other details, like the "risk-free" rate of return and the
"market" rate of return). This pioneering single-factor model
had some interesting consequence, like ownership of anything other that
the "market portfolio" (appropriate leveraged) was suboptimal.
It seemed like an odd result at the time (and still does for many), but it
was the idea that launched a thousand index funds.
The Barra model that I got to play with was a direct
descendent of Bill Sharpe's single-factor model. It had a market factor,
represented by the Nikkei 225, and a panoply of other "factors"
that seemed to be there as much because clients wanted them there as for
any fundamental economic reason. (Barr Rosenberg, the company's founder,
had already departed from his namesake firm to partner in an ill-fated
portfolio insurance venture. As a result, Barra shifted from a purist
academic orientation to a marketing orientation. Barra is now owned by
Morgan Stanley.) Whatever the reason, it seems clear that more than one
"systematic" factor is driving equity markets around the world,
even if no one seems to agree on what the factors other than Sharpe's
original market factor are.
To make matters concrete, what a factor model does is to
"decompose" each stock into a series of factor betas,
also known as factor loadings. The original Sharpe model had a
single beta that measured market risk; modern factor models have sixty or
more factors. Of course, each stock has its own stock-specific factor, but
for the holdings in a typical mutual fund, these factors effectively
cancel each other out in all but the most extreme cases.
At the time of my introduction to factor models, their
use was limited to the more quantitatively oriented shops, and the group
that I worked with at Kidder was arguably the most quantitative on the
Street back then. As the use of factor models to assess the performance of
investment managers has spread (even the Morningstar star system can be
said to be factor-based), so has the defensive deployment of these models
by investment managers. So, stocks are no longer treated as corporate
holdings, but as bundles of factors. In looking at a portfolio, one can
ignore the stocks and go straight to the factors.
The interesting thing about factor models is that their
use (though "abuse" may be a better word) results in a kind of
self-fulfilling prophecy. Like any output of a statistical model, factor
betas are not precise estimates, rather they have "error bands"
and these bands are typically quite large. The typical portfolio
management tool, including those bundling with factor models, ignores
these error bands. The neat thing is that if everyone ignores these error
bands, they will tend to shrink. For example, if everyone believes that
GE's beta relative to the S&P 500 Index is 1.1, then in the absence of
"news" that specifically concerns GE, every time the S&P 500
Index moves up by 1%, the minions of factor-model geeks (or, more
accurately, their computers) will reflexively push GE up by 1.1%. More
generally, factor models can convert any form of general news into precise
movements for individual stocks.
What would a market run by factor models look like?
You're looking it now. As the models purge themselves of noise by treating
point estimates as the truth, volatility will decline. In economist's
terms, the market no longer performs price discovery, the factor models
do. Less of a role for the markets in price discovery means less
volatility. It also means increased stock correlations as the noise of
price discovery is purged from the system. Bingo again.
The S&P 500 and the other major indexes do not
constitute a closed system, however, so it is reasonable to suspect that
all the noise will go somewhere. Where does all the volatility go? You
need look only as far as Google, which has yet to gain membership in any
index. My students face the daunting task of predicting what will happen
to Google during the month of November as well as constructing a hedge for
it from a list of fourteen stocks (including SPY, QQQ, and IWM, which are
technically ETFs). Next Monday, after all their predictions are due, I'll
post my predictions and attempt to justify them in "November
Copyright 2004 by Miller Risk Advisors. Permission
granted to forward by electronic means and to excerpt or broadcast 250
words or less provided a citation is made to www.millerrisk.com.