The statistical risk model

In Modules 7 and 8 the factors were chosen in advance: the market in Module 7, and size, momentum, volatility and industry in Module 8. The table near the start of Module 8 compares all three approaches by what each one assumes.

What principal components are

Principal components analysis finds the directions of greatest variance in your covariance matrix.

• PC1 is the portfolio of your 81 stocks with the highest variance.
• PC2 is the highest-variance portfolio uncorrelated with PC1.
• And so on, each orthogonal to all the ones before it.

Formally these are the eigenvectors of Σ, ordered by eigenvalue. The eigenvalue is the variance of that portfolio, so the share of variance explained is simple to calculate:

The trace (the sum of the diagonal) is the total variance across all your stocks, because eigenvalues sum to the trace. So you can compute the share explained without finding all 81 components.

Doing eigenvectors in Excel

Excel has no EIGEN() function. You do not need one: power iteration is a few cells and a drag. Multiply any starting vector by Σ, normalise it, and repeat. It converges to the dominant eigenvector, because each multiplication stretches the vector further along the direction of greatest variance:

1. Start with v = a column of 81 ones.
2. Compute w = Σv.
3. Normalise: v = w / √(sum of w²).
4. Go back to step 2.

=MMULT(Cov!$B$2:$CC$82, B2:B82) / SQRT(SUMSQ(MMULT(Cov!$B$2:$CC$82, B2:B82)))

=MMULT(TRANSPOSE(v), MMULT(Cov!$B$2:$CC$82, v))

For the next component, deflate, by subtracting the one you just found:

Then run the same routine on Σ′. Because PC1's contribution has been removed, power iteration now converges to PC2. Repeat for as many as you want.

=Cov!B2:CC82 - lambda1 * MMULT(v1, TRANSPOSE(v1))

What you will find

On the 81-name weekly covariance since 2005:

PC1 is the market, and nobody told it to be

Look at the PC1 loadings. All 81 have the same sign, so PC1 is a long-only portfolio of the whole index.

Now score it: for each week, compute the loadings dotted with that week's returns, giving a time series. Correlate that against the simple equal-weighted average of all 81 stocks.

You should get about 0.991.

The algorithm was given only the covariance matrix, with no index or benchmark, and it still produced something very close to the market. The market is the largest single source of common variation in the data, so methods like this one find it.

Look at the size of the loadings, too:

• Largest: Barclays, NatWest, Barratt, Lloyds — banks and a housebuilder.
• Smallest: Reckitt, Unilever, National Grid, Severn Trent, United Utilities.

This ordering is close to a ranking by beta.

PC2: commodities versus domestic UK

PC2 has loadings of both signs. Average the PC2 loading within each ICB industry:

Global commodities are at one end, and domestic UK financials and property at the other. This reflects the make-up of the FTSE 100, which staples a set of enormous global miners and oil majors onto a set of domestic British banks and REITs. The difference between those groups is the second-largest source of common variation. Nobody specified that factor; it fell out of the data.

Check the orthogonality while you are here: correlate the PC2 score against the market. You should get roughly −0.08, which is close to zero. The components are uncorrelated by construction, so this is a good test that your deflation worked.

Building a risk model from it

Keep the first k components and rebuild:

where V holds the k eigenvectors and L the eigenvalues on its diagonal. The diagonal term keeps each stock's total variance. Without it, a stock that is poorly captured by the first five components would appear almost riskless.

systematic =MMULT(MMULT(V, L), TRANSPOSE(V))
residual   =MAX(0, original diagonal - systematic diagonal)
Sigma_k    =systematic + diagonal of residuals

How many components?

On this data it takes 30 components to reach 80% of total variance.

PC1 accounts for about a third, the next four add about 16 points between them, and then there is a long, flat tail of components each explaining 1–2%. That reflects how much of each FTSE 100 stock's movement is specific to that company.

The usual selection methods and what they suggest:

• Scree plot. Chart the eigenvalues and look for the elbow. Here it is clear: after PC1, everything is small. That argues for very few factors.
• Variance target. Retain enough for 80% or 90%. That argues for 30 or more.
• Marchenko–Pastur / random matrix theory. Components whose eigenvalues fall inside the range you would expect from pure noise, given your ratio of stocks to observations, are indistinguishable from noise and can be dropped. This has a stronger theoretical basis and usually keeps fewer than ten components.

People reasonably choose different numbers. Five is a defensible working choice, and it is what the self-check uses.

The catch

Principal components are statistical constructs without a direct economic meaning. This has three consequences:

1. They have no names. PC1 is the market and PC2 is arguably commodities-versus-domestic, but PC4 is whatever it is, and you cannot walk into a meeting and attribute a portfolio's risk to PC4.
2. They change. Re-estimate on a different window and the components move, sometimes swapping order when two eigenvalues are close. A fundamental factor called "Financials" means the same thing every month; PC3 does not.
3. The sign is arbitrary. An eigenvector multiplied by −1 is still an eigenvector. Any convention will do (we force the largest loading to be positive), but the sign itself has no meaning.

Their strength is capturing correlation structure that nobody specified in advance, including unusual changes in correlation.

For this reason many risk teams run both a fundamental and a statistical model, and investigate cases where the two disagree. Module 10 compares the models.