The problem factor models solve
Mean-variance portfolio theory needs every variance and every covariance. For N assets that is N(N+1)/2 numbers — 5,050 for a hundred assets, and over half a million for a thousand. Three difficulties follow, and they are fatal rather than inconvenient.
There is rarely enough data: estimating 5,050 parameters from a few years of monthly returns means more parameters than observations. The estimates are noisy, and an optimiser fed noisy covariances produces confident nonsense. And the result is not interpretable — nobody can look at a 100-by-100 matrix and say whether it is sensible.
A factor model's answer is to assume that assets move together only because they respond to a small number of common influences. Model those, and every covariance follows.
The single-index model
In words: an asset's return is a constant, plus a multiple of the market's return, plus something specific to that asset.
Two assumptions make it useful, and both are assumptions rather than facts:
- the specific terms have mean zero and are uncorrelated with the index;
- the specific terms are uncorrelated with each other.
Taking variances and covariances, and using them:
σij = βiβjσM2
The second line is the whole point. Every covariance comes from one multiplication of two betas. The parameter count collapses from N(N+1)/2 to 2N + 1: a beta and a specific variance for each asset, plus the index variance.
And notice what the model asserts by what it leaves out. Two assets are correlated only through the index. Two banks in the same city with the same regulator and the same borrowers are assumed to have nothing in common beyond their market exposure, which is plainly false. That falsehood is the reason for more factors.
This is not the Capital Asset Pricing Model
They look similar and do different jobs, and a question will test the distinction.
- The single-index model is a statement about how returns are generated. It is an empirical decomposition and it predicts nothing about what αi should be.
- The Capital Asset Pricing Model is a statement about equilibrium. It says what the expected return must be if markets clear, namely
rf + βi(E[RM] − rf), which pins α down.
So the index model is the measurement and the Capital Asset Pricing Model is the theory about what the measurement should show. One can hold without the other: returns can be well described by one factor while the pricing relationship fails.
More than one factor
With uncorrelated factors the formulae extend in the obvious way:
σij = ∑k bikbjkσFk2
In words: each factor contributes to a variance in proportion to the square of the sensitivity, and to a covariance in proportion to the product of the two assets' sensitivities.
If the factors are correlated the covariance formula gains cross terms in bikbjlσFkl. Factors are usually constructed to be uncorrelated precisely to avoid that, which is worth saying when a question asks why.
A crucial sign point: a covariance contribution is negative when the two assets have opposite sensitivities to a factor. Two assets that respond in opposite directions to interest rates become less correlated through that factor, which is exactly the effect a single-index model cannot represent.
The three kinds of multifactor model
They differ in where the factors come from, and the trade-off is the same each time: interpretability against fit.
Macroeconomic. The factors are observable economic series — interest rates, inflation, industrial production, oil, the exchange rate. Easy to explain to anyone, easy to stress test ("what if rates rise two points"), and they often fit returns less well than the alternatives.
Fundamental. The factors are characteristics of the companies themselves — size, price-to-book, sector, leverage, dividend yield. These fit well, they are the standard in commercial risk systems, and they are harder to tell an economic story about: it is not obvious why small companies should move together beyond the fact that they do.
Statistical. The factors are extracted from the return data itself, by principal components or factor analysis. The best fit of the three by construction, and the factors have no meaning at all — they are whatever linear combinations happen to explain the variance. Fine for building a covariance matrix, useless for explaining a result to a client, and they can be unstable from one estimation period to the next.
Diversification, in the model's own terms
Hold N assets in equal weights. The systematic part of the portfolio's risk is driven by the average of the betas and does not shrink. The specific part does:
So specific risk falls like 1/N while systematic risk does not fall at all. That one line is the formal version of everything said about diversification earlier in the syllabus: there is a floor, the floor is the factor exposure, and the Capital Asset Pricing Model prices what is left after the rest has been diversified away.
It also says how fast. Going from one asset to ten removes 90% of the specific variance, and from ten to a hundred removes 90% of the remainder. Most of the benefit arrives early, which is why a portfolio of twenty or thirty holdings is already close to the floor.
Where marks are lost
Forgetting to square the sensitivities in the variance. It is b2σF2 in a variance and bibjσF2 in a covariance. One is squared, the other is a product of two different numbers.
Adding a specific-risk term to a covariance. Specific terms are uncorrelated across assets, so they contribute nothing to a covariance. If a covariance calculation contains a specific variance, it is wrong.
Multiplying by the factor standard deviation instead of its variance. Both formulae carry σF2.
Weighting a portfolio's specific variance linearly. It is ∑wi2σεi2, with the weights squared. This is the error that makes diversification vanish from the answer.
Confusing the index model with the Capital Asset Pricing Model. Covered above, and worth checking before writing a sentence about either.
Questions students ask
How many factors should a model have?
Enough to capture the ways assets genuinely move together, and no more. Each extra factor fits the historical data better by construction and adds N parameters to estimate, so beyond a few the gain is fitting noise. Commercial equity models typically use somewhere between about five and a few dozen, and the number is chosen by how well the model predicts out of sample rather than by how well it fits.
Is the specific risk really uncorrelated between assets?
No, and that is the assumption to challenge when a factor model misbehaves. Two companies in the same industry share influences the factors have not captured, so their residuals move together, and the model then understates the risk of a portfolio concentrated in that industry. Adding an industry factor is the standard repair, and it is why commercial models carry so many of them.
Why go to the trouble when the full covariance matrix could just be estimated?
Because the full matrix estimated from limited data is usually worse, not better. With fewer observations than assets it is not even invertible, so it cannot be used for optimisation at all; and even when it is, its errors are large enough that the resulting portfolio is driven by estimation noise. A factor model imposes structure, and the structure — although wrong in detail — removes far more error than it introduces. That is the real argument, and it is a statistical one rather than a computational one.
Where does arbitrage pricing theory fit in?
It is the equilibrium theory built on top of a multifactor model, as the Capital Asset Pricing Model is built on the single-index one. It says that expected return must be the risk-free rate plus a sum of factor sensitivities multiplied by factor risk premiums, or an arbitrage would be available. It requires far weaker assumptions than the Capital Asset Pricing Model and pays for that by not saying what the factors are, which is both its strength and the reason it is harder to test.
A worked example
Two assets, P and Q, described by a two-factor model with uncorrelated factors.
| Asset | Sensitivity to factor 1 | Sensitivity to factor 2 | Specific risk |
|---|---|---|---|
| P | 0.8 | 0.4 | 9% |
| Q | 1.2 | −0.3 | 12% |
The factors have standard deviations of 15% and 8%. All seven figures are invented for this example. Work in percentage points, so variances are in squared percentage points.
Step 1: the two variances
= 144.00 + 10.24 + 81.00 = 235.24, so σP = 15.3375%
= 324.00 + 5.76 + 144.00 = 473.76, so σQ = 21.7660%
Note that the negative sensitivity was squared and contributed positively. A variance cannot be reduced by a negative exposure; only a covariance can.
The split is informative on its own:
| Asset | Systematic variance | Specific variance | Proportion systematic |
|---|---|---|---|
| P | 154.24 | 81.00 | 65.6% |
| Q | 329.76 | 144.00 | 69.6% |
About a third of each asset's risk is specific, earns nothing in equilibrium, and can be diversified away for free.
Step 2: the covariance, which is the point of the model
Only the shared factors contribute. The specific terms are uncorrelated and add nothing.
= 216.00 − 7.68 = 208.32
Two things worth reading there. The second factor made a negative contribution of 7.68, because P and Q respond to it in opposite directions — so that factor pushes them apart while the first pushes them together. A single-index model could not have produced that at all. And the correlation of 0.624 came out of seven parameters rather than being estimated directly, which is the whole economy of a factor model.
A free check: a covariance cannot exceed the product of the standard deviations, which is 333.84. At 208.32 it is comfortably inside, as a legitimate covariance must be.
Step 3: a portfolio, computed two ways
Hold 60% in P and 40% in Q. First work through the factors, by finding the portfolio's own sensitivities:
b2 = 0.6 × 0.4 + 0.4 × (−0.3) = 0.24 − 0.12 = 0.12
= 207.3600 + 0.9216 + 52.2000 = 260.4816
Now the same portfolio through the covariance found in step 2, which uses none of the factor arithmetic above:
= 84.6864 + 75.8016 + 99.9936 = 260.4816
260.4816 both ways, identical to ten decimal places. The two routes share no intermediate figure, so the agreement confirms the factor sensitivities, the specific variances and the covariance all at once. This is the check to run on any factor-model question that reaches a portfolio.
What the diversification was worth
The weighted average of the two standard deviations is 0.6 × 15.3375 + 0.4 × 21.7660 = 17.9089%. The portfolio's actual standard deviation is 16.1394%, so 1.77 percentage points were saved by holding both rather than a notional blend of the two risks.
And look where the saving came from. The second factor's contribution to the portfolio's variance has nearly vanished — 0.122 × 64 = 0.92, against 10.24 for P alone and 5.76 for Q alone — because the opposite sensitivities almost cancelled. The portfolio is close to immune to factor 2, and nobody chose that; it fell out of holding the two assets together.
And with more assets
The two specific variances average 112.5. For an equally weighted portfolio of N similar assets the specific variance is 112.5/N:
| N | Specific variance | Specific standard deviation |
|---|---|---|
| 1 | 112.50 | 10.61% |
| 2 | 56.25 | 7.50% |
| 5 | 22.50 | 4.74% |
| 20 | 5.63 | 2.37% |
| 100 | 1.13 | 1.06% |
Against the 60/40 portfolio's systematic variance of 207.36 + 0.92 = 208.28, which does not fall at all. At twenty assets the specific part is 5.63 against 208.28 — under 3% of the total — and at a hundred it has effectively gone. The floor is the factor exposure, and reaching it takes a few dozen holdings rather than hundreds.
The count that justifies the whole approach
| Approach, 100 assets | Parameters to estimate |
|---|---|
| Full covariance matrix | 5,050 |
| Two-factor model | 303 |
Two sensitivities and one specific variance for each of the hundred assets is 300, plus two factor variances and their covariance, which gives 303 against 5,050. That is a sixteen-fold reduction, and it is the difference between a model that can be estimated from a few years of monthly data and one that cannot be estimated at all.
Where this fits in CM2
This page explains one idea from CM2 Financial Engineering & Loss Reserving. The full CM2 syllabus, with what each part is examined on, sits on the course page, and past-paper style mock exams with full solutions are where you find out whether the idea has stuck under time pressure.
Other CM2 topics explained
- Utility theory and risk aversion: non-satiation, A(w) and R(w)
- Measures of investment risk: variance, semi-variance, VaR and Tail VaR
- CAPM and the security market line: beta and the market price of risk
- Ito's lemma explained term by term, and what it does to a share price
- The chain ladder method: estimating outstanding claims from a run-off triangle
- The Bornhuetter-Ferguson method: the chain ladder with a second opinion
- Binomial option pricing: where the risk-neutral probability comes from
- The Black-Scholes formula and put-call parity, read term by term
- Ruin theory: the adjustment coefficient and Lundberg bound explained
- Mean-variance portfolio theory: the efficient frontier and why diversification works
- The term structure: spot rates, forward rates and par yields
- Short-rate models: Vasicek, Cox-Ingersoll-Ross and Hull-White compared
- The Merton model: company equity as an option, and where a credit spread comes from
- The Greeks: delta, gamma, vega, theta and rho, and how a hedge decays
- The inflation-adjusted chain ladder: taking past inflation out and putting a future assumption in
- The average cost per claim method: developing numbers and severity separately
- Valuing a maturity guarantee by simulation, and checking it against the formula
- The Efficient Markets Hypothesis: the three forms, and how each one is tested
- Excess-of-loss reinsurance: the retained claim, and choosing the retention
- State price deflators: pricing with real-world probabilities
- What makes a risk insurable, and what to do when it almost is
- Two-state and intensity credit models: backing a default rate out of a bond price
- The martingale representation theorem, and why hedging is possible at all
Stuck on a term rather than the idea? The actuarial glossary defines the notation used above. If something here is wrong or unclear, tell us — these pages get corrected.