Your factor model is 66° wrong

Plain-language explainer for doi:10.5281/zenodo.21284204 (#580)

The central observation

Every $k$-factor risk model — CAPM, Fama-French, Barra, APT — picks $k$ directions in the space of asset returns and says: “these are the systematic risks.” The $k$ directions span a $k$-dimensional subspace of the $n$-dimensional return space.

The space of all possible $k$-dimensional subspaces of $\mathbb{R}^n$ is the Grassmannian $\mathrm{Gr}(k, n)$. Every factor model is a point on this manifold.

The Grassmannian has a natural metric — the Fubini-Study distance — that measures the angle between two subspaces. Two factor models define two points on $\mathrm{Gr}(k, n)$, and the Fubini-Study distance between them is a computable, basis-independent number between 0° and 90°:

  • 0°: the two models identify identical systematic risks
  • 90°: the models are orthogonal — they share no systematic risk exposure at all

This gives something the financial industry has never had: a metric for model risk.

What the Grassmannian angle measures

Given two loading matrices $B_1, B_2$ (each $n \times k$, orthonormal columns), the Fubini-Study distance is:

\[d_\mathrm{FS}(V, W) = \arccos\!\bigl(\sigma_{\max}(B_1^T B_2)\bigr)\]

where $\sigma_{\max}$ is the largest singular value of the $k \times k$ inner product matrix. This is the leading principal angle — the smallest angle between any unit vector in $V$ and any unit vector in $W$.

This number is:

  • Computable: one SVD, available in any linear algebra library
  • Basis-independent: rotating either loading matrix doesn’t change it
  • Interpretable: it is literally an angle between risk directions

The model risk of using model $B$ instead of the truth is $d_\mathrm{FS}(p_B, p_\mathrm{true})$. If your model is 45° away from reality, you are capturing at most $\cos^2(45°) = 50\%$ of the systematic variance in the direction of largest overlap.

The 2008 diagnosis: 66°

The Gaussian copula (Li 2000) was the dominant pricing model for CDOs before 2008. It parametrises joint default risk with a single correlation parameter — a rank-1 model, corresponding to a point in $\mathrm{Gr}(1, n)$.

The actual 2008 crisis covariance structure required at minimum three independent systematic factors:

  1. A market factor (broad equity decline)
  2. A funding factor (interbank liquidity stress)
  3. A credit factor (structured-product spread widening)

The paper computes the Fubini-Study distance from the Gaussian copula’s rank-1 subspace to the nearest rank-3 approximation of the crisis covariance:

\[d_\mathrm{FS}(p_\mathrm{copula},\; p_\mathrm{true}^{(k=3)}) \approx 66°\]

66°. The model used by the entire industry was modelling a risk direction more than halfway to orthogonal from reality. No institution or regulator computed this number before the crisis. This paper argues they should compute it every quarter.

Three applications

1. The factor zoo

There are over 400 proposed risk factors in the academic literature (Hou-Xue-Zhang 2020). The Grassmannian gives a principled deduplication: cluster all proposed factors on $\mathrm{Gr}(1, n)$ using $d_\mathrm{FS}$ as the distance metric. Factors within $\theta < 15°$ of each other are effectively the same risk direction; the number of genuinely distinct clusters is the true factor count.

Experiment x580a finds 7–12 genuine clusters in US equity factors — consistent with theory (Kozak-Nagel-Santosh 2020), and far fewer than the 400+ named factors suggest.

2. Cointegration as Grassmannian geometry

A vector error-correction model (VECM) has a cointegrating matrix $\beta$ whose column space $p_\beta \in \mathrm{Gr}(k, n)$ is the long-run attractor of $n$ integrated time series. A structural break — a currency peg breaking, a basis trade unwinding — is a discontinuous jump of $p_\beta$ on the Grassmannian.

The paper defines a Grassmannian snap test: flag a structural break when

\[d_\mathrm{FS}(p_\beta(t),\; p_\beta(t-1)) > \theta^*\]

where $\theta^*$ is calibrated from the null distribution. Experiment x580b validates this against four known breaks: Lehman 2008, European sovereign 2011, CHF de-pegging 2015, UK gilt crisis 2022. The snap test correctly identifies all four.

The Johansen cointegration test is, in this language, an estimator of $\cos^2(d_\mathrm{FS})$ — the Grassmannian restatement makes its geometric meaning explicit.

3. Subspace velocity as a leading crisis indicator

Define the subspace velocity:

\[\theta_G(t) = d_\mathrm{FS}(p_t,\; p_{t-1})\]

This is the speed at which the factor structure is rotating on $\mathrm{Gr}(k, n)$, estimated from rolling PCA in successive windows. High $\theta_G$ signals that the dominant risk directions are changing rapidly.

Experiment x580c (S&P 500 sector returns 2004–2010) finds that $\theta_G(t)$ begins rising in 2007 Q3 — four quarters before the Lehman collapse — and peaks in 2008 Q4. VIX and credit spreads give no comparable advance signal. The covariance structure rotates before it snaps; $\theta_G$ measures the rotation.

The H^k hierarchy in finance

The paper connects the Grassmannian geometry to the ISA opcode hierarchy:

ISA tier Opcode Finance meaning Grassmannian object
H⁰ ORBIT Stable factor structure; regime Fixed point $p \in \mathrm{Gr}(k,n)$
H¹ TWIST Factor rotation; model risk accumulates Path $\gamma$ on $\mathrm{Gr}(k,n)$
H² BIND Systemic snap; no smooth recovery Holonomy of loop; $\theta_G \to 90°$

The 2008 crisis is an H² event: the factor subspace jumped discontinuously to near-orthogonal, and no smooth deformation connected the pre-crisis and crisis factor structures. The 66° distance is the metric complement to the topological obstruction identified in #397.

The same three tiers describe molecular bonding (#570), quantum error correction (#577), and nuclear structure (#575). Finance joins the same geometric family.

Practical implications

For portfolio managers: compute $\theta_G(t)$ alongside VaR and tracking error. A sustained increase warns that your factor model is drifting from reality before the crisis hits.

For risk managers: model risk capital should scale with $d_\mathrm{FS}(p_\mathrm{model}, p_\mathrm{estimated})$. A model 10° from the estimated true subspace needs less capital than one 45° away.

For regulators: if twenty systemically important institutions all have factor subspaces within 10° of each other, a shock to that shared direction is a systemic event. Grassmannian overlap across SIFIs is a natural stress test input.

What this paper does not claim

The 66° diagnosis assumes a three-factor true structure for the 2008 crisis. If the true number of crisis factors was higher than three, the actual distance was larger — the estimate is a lower bound. The subspace velocity $\theta_G(t)$ as a leading indicator is validated on one crisis episode (2004–2010); whether it leads reliably across different crisis types is an open empirical question requiring longer data.

See also:

For the full technical treatment, see doi:10.5281/zenodo.21284204