Pillar 06 · Micro Equity Execution
How much of a portfolio's total risk does each holding really add? Not its standalone volatility — what matters is how it co-moves with everything else. This tool shows that visually as a triangle, and computes it on the live S&P 500.
A portfolio's total risk is usually measured as its volatility (σp) — how much its value swings around. A natural question is: which holdings are responsible for that risk? The naïve answer — "the most volatile ones" — is wrong. What actually matters is how each holding moves together with the rest of the portfolio.
A wild, volatile stock that zigs when the portfolio zags can reduce total risk (a hedge). A calm, low-volatility stock that moves in lockstep with everything else can add more risk than its own volatility suggests. Risk contribution (RC) splits the portfolio's total volatility into a piece for each holding — and, by a result called Euler's theorem, those pieces add up exactly to σp.
Risk adds up like vectors, not like numbers — and vectors form triangles. Draw the portfolio's risk as the base of a triangle (σp), and a holding's weighted risk (wi·σi) as one of the other sides. The angle θ between them is set entirely by the correlation: θ = arccos(ρ).
Drop a perpendicular from the apex onto the base: the shadow it casts is the risk contribution. That single picture explains everything:
• ρ near +1 → θ near 0°: the side lies almost flat along the base; the holding contributes nearly its full standalone risk (no diversification).
• ρ = 0 → θ = 90°: the side stands straight up; its shadow on the base is zero — it adds no risk at all.
• ρ negative → θ obtuse: the shadow points backwards; the contribution is negative — the holding is a hedge that cancels risk elsewhere.
Risk contribution is a projection, not a property of the asset alone. The same stock can be a big risk adder in one portfolio and a hedge in another.
Risk budgeting & portfolio construction. Instead of choosing position sizes, managers choose how much risk each position is allowed to contribute. "Risk parity" portfolios equalize RC across holdings so no single name dominates.
Spotting hidden concentration. A book can look diversified by weight yet have most of its risk coming from a handful of high-beta names. RC reveals that — it's where the risk actually is.
Hedge sizing. RC tells you how much of a negative-correlation position you need to neutralize a given amount of portfolio risk.
Stress & crisis analysis. In a crash, correlations across stocks lurch toward +1 — every triangle flattens, every angle collapses, and diversification you thought you had simply disappears. RC makes that visible (see the Regime Comparison tab).
Drag the sliders to see how standalone risk, weight and correlation set the triangle — and how the contribution (the shadow DC on the base) responds. Or load a real S&P 500 stock to fill in its actual σi and ρ.
Real risk geometry for every S&P 500 member, from the dashboard's own data — annualized vol (σi), correlation to the index (ρ), CAPM beta (β), the triangle angle (θ), and the marginal contribution to index risk (MCR = β·σp). Click any row to load it into the simulator.
| Ticker | Sector | β | ρ | σi | θ | MCR | Δ10Y | RC share |
|---|
The same holding behaves very differently depending on the correlation regime. The table sweeps ρ from crisis (≈ +1) to deep hedge (≈ −1) for one representative position — exactly the shift a rolling-window analysis reveals through market stress.
| Regime | ρ | θ | RCi | RC share | Visual | Interpretation |
|---|
When ρ → 1 the angle θ → 0°, the apex lies over the right vertex, and the asset contributes nearly its full standalone risk. When ρ → −1, θ → 180°, the contribution is maximally negative and the hedge fully offsets. The triangle makes visible what the formula hides: risk contribution is a projection, not a property of the asset alone — which is why a crisis that pushes every ρ toward 1 erases diversification across the whole book at once.