The cross-sectional factor model

Module 7 ran one regression per stock, through time. This module turns the problem around and runs one regression per week, across stocks.

Most commercial equity risk models are built this way, so it also makes vendor documentation much easier to follow.

The idea

The question here is which characteristics were rewarded in a given week.

The exposures X are known in advance — you can look up a company's size and industry today. The factor returns f are what you solve for. Each week you regress that week's returns across all 81 stocks on their characteristics, and the coefficients tell you what the market paid, that week, for being large, or for having momentum, or for being a miner.

The roles are reversed compared with Module 7. There, the factor return (the market) was observed and beta was estimated. Here the exposure is observed and the factor return is estimated.

Can you actually observe exposures?

The cross-sectional model has one clear advantage: exposures are observable now. A company that listed three months ago has no usable return history, so the time-series model in Module 7 has nothing to say about it. But you know its market cap and its industry today, so a cross-sectional model can give it a covariance with everything else straight away. That is a large part of why commercial risk models are built this way.

It rests on a big assumption, though: that you can observe a company's exposures in the first place. The model takes a description of what each company is — its essence, if you like — and backs the factor returns out of that. If the description is wrong, so is everything downstream.

Descriptions go wrong more often than you might expect:

• When is a car maker actually a finance firm? A manufacturer with a large lending arm can trade more like a bank than an industrial company when credit markets get nervous.
• When is a commodity firm basically USD exposure in disguise? The FTSE miners and oil majors sell commodities priced in dollars and report in dollars. For a sterling investor, a good part of what looks like commodity exposure is currency exposure.
• 3i is classified as Financials, but much of its value is its stake in Action, a European discount retailer.
• Melrose buys industrial businesses, fixes them and sells them, so what it is exposed to changes with every deal, while its classification stays exactly where it was.

A time-series model does not need the description. It regresses returns on a factor and lets the data say how exposed each company is, which in effect regresses the essence out of the returns. If a car maker's shares move with credit spreads, a regression on credit spreads will find that, whatever the company calls itself. The price is that it needs a long return history, it is slow to notice when a business changes, and it can only find exposure to factors you thought to include.

The three models in this programme sit at different points on that line:

Opinions differ on which is the right answer, and experienced risk managers land on both sides. You will see two places later in this module where the cross-sectional assumption fails on our own data: a Financials factor that cannot tell banks from insurers, and a single industry label that covers both Burberry and Unilever. Keep this section in mind when you get there.

Step 1: build the exposures

We fix exposures as at Wednesday 27 September 2023 and regress every week after that date. Freezing them keeps the exercise reproducible, and measuring them before the returns they explain avoids look-ahead. A production model refreshes exposures monthly, but the method is the same.

Four factor groups, all buildable from your existing download:

Each style factor is z-scored across the universe and then clipped at ±3:

Z-scoring is needed because the raw units are not comparable: log market cap is around 23, while momentum is around 0.1. Standardising puts every factor return in the same unit, the return to a one-standard-deviation exposure.

Clipping matters because one silly value would otherwise become the factor.

raw       =LN(mcap)
z-score   =(LN(mcap) - AVERAGE(logcaps)) / STDEV.S(logcaps)
clipped   =MAX(-3, MIN(3, zscore))

=INDEX(B:B, n-21) / INDEX(B:B, n-252) - 1

=--($D2 = E$1)

Step 2: one regression per week

Regress that week's 81 returns on the 14 exposure columns, with no intercept.

Weight the regression by the square root of market cap. Residuals vary with size (small companies are noisier), and without weights the smallest names in the index have too much influence on the factor returns. Square-root-of-cap is the usual convention: it reduces the influence of small names without letting the largest ones take over.

y*  =return   * SQRT(mcap)
X*  =exposure * SQRT(mcap)

Drag that down for every week after 27 September 2023. You will end up with roughly 154 rows, which is a time series of factor returns.

Step 3: what the factor returns look like

Annualise the volatility of each factor return series. On this data you should see something close to:

Two things stand out.

Style factors are much less volatile than stocks. A typical FTSE 100 name runs at 25–35% annualised; the size factor at under 5%. Diversification across 81 names reduces the volatility of a common signal, so a style tilt has to be large to have much effect.

Industry factors are much more volatile than styles, because they include the market. Our specification has no separate market factor: with the dummies acting as the intercept, each industry factor return is roughly that industry's return, market included. Commercial models often extract a market factor explicitly so that the industry factors become market-relative, which is worth knowing when comparing vendor factor volatilities with ours.

Step 4: the covariance matrix

Now assemble it:

• X is 81 × 14 — the exposures.
• F is 14 × 14 — the covariance matrix of your factor returns.
• Δ is 81 × 81 diagonal — the variance of each stock's regression residuals.

F         =MMULT(TRANSPOSE(Fdev), Fdev) / (COUNT-1)      ' 14x14
XFX'      =MMULT(MMULT(X, F), TRANSPOSE(X))              ' 81x81
residual  =VAR.S(residuals for each stock)               ' down the diagonal

On this data the median specific volatility comes out around 21% annualised, with a range of roughly 5% to 40%. So a typical FTSE 100 stock has about 21 points of volatility that no factor in our model explains. That is expected: specific risk is real, and it is the part a stock-picker is paid to take.

Count the parameters

Fourteen factors use almost as few parameters as one, while capturing more of the correlation structure: two miners are now correlated through their shared industry as well as through beta.

What you should find

Compute your portfolio's volatility under this Σ and compare with Module 3. They will be close but not equal.

Then run the concentration test. Build four equally-weighted portfolios and price each under the sample, single-factor and cross-sectional matrices, comparing against what each actually realised over the same 154 weeks:

The single-factor model understates every concentrated portfolio, by between 3.7 and 7 points, as in Module 7. It only sees market exposure, so it misses co-movement that is not driven by the market.

The cross-sectional model improves the estimate for the miners and the utilities. It does not help for the banks, where it is slightly worse.

Why it does not work for banks

Compare the actual average pairwise correlation with what the model implies:

The model gets every group roughly right except the banks, whose correlation it understates by 0.12.

The reason is our industry classification. ind_Financials contains 27 names: five banks, but also life insurers, general insurers, asset managers, private equity vehicles and a couple of investment trusts. Barclays, Aviva and 3i all load 1.0 on the same factor. So the Financials factor return is an average across a heterogeneous set of businesses, which dilutes the common movement among the banks.

The miners do not have this problem: ind_Basic Materials is seven names, nearly all of them miners, so the factor is more specific.

A factor model depends heavily on how its factors are defined, and that is the main point of this module.

A missing factor: value

Value. Book-to-price needs book value, and there is no free point-in-time source for it. Dividend yield is often used instead, but it is a weak substitute: it mixes cheapness with payout policy and with quality, and it drops to zero for any company that suspends its dividend, which tends to happen when the stock has become cheap.

In practice, the quality of the exposures limits a cross-sectional model more than the regression does. The regressions are quick to set up; clean, point-in-time, survivorship-free fundamental data is much harder to obtain, and it is a large part of what vendors charge for.

What "stock-specific" really means

A factor model splits risk in two: the part its factors explain, and the rest, which it calls specific or idiosyncratic risk. Risk reports lean hard on that split. "Most of this manager's tracking error is stock-specific" sounds like a compliment: the manager is picking stocks rather than making large bets on sectors or styles.

Handle it with care. Specific risk is defined as whatever the model's factors failed to explain, so it describes the model as much as the manager. A missing factor, an industry that is too coarse, or a style measured on the wrong universe all get reported as stock selection.

Same portfolios, different models

Take the 300 portfolios built the same way as yours. Measure each one's active risk against a cap-weighted benchmark of all 81 names, over the same 154 weeks, and ask different models what share of that active variance is specific:

For a typical portfolio the answer moves across 57 percentage points depending on which model you ask. Nothing about the portfolios changed.

The single-factor model is the extreme case, and a useful warning. A long-only portfolio measured against its own benchmark has almost no net exposure to the market, so a model whose only factor is the market will call nearly all of its active risk specific, whatever the manager actually did.

When the model does not fit the universe

The same thing happens, less visibly, when a model is built for a different universe from the one the manager invests in.

Take a UK manager with a style bias, measured on a global risk model. A global model standardises size, momentum and value across the whole world's stocks. By global standards every FTSE 100 company is a large cap, so a UK manager tilting towards the smaller end of the FTSE barely registers in the global size exposure. The tilt is still there, and so are the returns it produces, but the model has no factor that sees it, so it lands in specific risk. The same goes for factor returns estimated mostly on US stocks: a style that paid off in the UK but not elsewhere has nothing in the model to explain it.

Removing the style factors from our own model is a crude stand-in for that situation, a model that cannot see UK style differences at all. The typical portfolio's specific share rises from 42% to 59%, and a manager who was a moderate stock-picker becomes a committed one on paper.

The banks, again

The five-bank portfolio from earlier in this module, measured against the same cap-weighted benchmark:

With one Financials factor covering banks, insurers and asset managers, more than a quarter of the banks' active risk is labelled stock-specific, and the model falls well short of the tracking error that actually happened. Give banks their own factor and the specific share more than halves, and the prediction comes close. (These are in-sample figures, so Module 10's warning applies. The gap is still telling.)

There is a check you can run yourself. If the specific returns really are specific, they should be roughly uncorrelated with each other. Under the 11-industry model the five banks' specific returns have an average pairwise correlation of +0.10. Pick five names at random from the universe and 99% of baskets come out below that. The banks' "specific" returns were moving together, which is the signature of a missing factor.

With banks split out the average becomes −0.17. A factor built from only five names absorbs their shared movement and then some, which is its own small warning about very narrow factors.

The check is useful but not foolproof. The twelve smallest names in the universe have a specific share of 31% under the full model and 46% without the size factor, yet their specific returns are barely correlated either way (+0.04 and +0.03). A missing factor with modest returns can shift the attribution a long way without leaving an obvious trace.