Which holding is actually contributing your risk?

A portfolio's weights tell you where the money is. They do not tell you where the risk is, and the two are often nowhere near each other.

This lesson makes that difference numerical, using arithmetic you can do on paper.

The intuition first

A holding's contribution to portfolio risk depends on three things, not one:

1. How much you own (its weight)
2. How volatile it is (its own standard deviation)
3. How it moves with everything else (its correlation with the rest)

The third factor is what makes this non-obvious. A volatile asset that moves against the portfolio can contribute very little risk — sometimes it reduces it. A moderate asset that moves in perfect step with the portfolio contributes its full weight and more.

The arithmetic

Two-asset portfolio: 60% equities (σ = 16%) and 40% bonds (σ = 6%), correlation 0.10.

Portfolio variance:

σ²ₚ = (0.6² × 0.16²) + (0.4² × 0.06²) + (2 × 0.6 × 0.4 × 0.10 × 0.16 × 0.06) = 0.009216 + 0.000576 + 0.000461 = 0.010253

So σₚ = 10.1%. Already interesting: the weighted average of the two volatilities is 0.6 × 16% + 0.4 × 6% = 12.0%. Diversification bought you about 1.9 percentage points.

Now split that 10.1% between the two holdings. Each asset's risk contribution is its weight times its covariance with the portfolio, divided by portfolio volatility:

Equities: covariance with portfolio = (0.6 × 0.16²) + (0.4 × 0.10 × 0.16 × 0.06) = 0.01536 + 0.000384 = 0.015744 Contribution = 0.6 × 0.015744 ÷ 0.101256 = 0.0933 → 9.33%

Bonds: covariance with portfolio = (0.4 × 0.06²) + (0.6 × 0.10 × 0.16 × 0.06) = 0.00144 + 0.000576 = 0.002016 Contribution = 0.4 × 0.002016 ÷ 0.101256 = 0.0080 → 0.80%

Check: 9.33% + 0.80% = 10.13% = σₚ. The contributions add up to total portfolio risk exactly — that's the property that makes this decomposition useful.

The result, in percentages

Sixty percent of the money is producing ninety-two percent of the risk. The classic balanced portfolio, described honestly, is far more concentrated in risk terms than its weights suggest.

This is not a criticism of the allocation. It's a description that the weights alone conceal — and one that anyone reporting on a portfolio ought to be able to produce.

Why the effect is so strong

Because risk scales with volatility, and volatility differences compound through the calculation. Equities here are roughly 2.7 times as volatile as bonds. Squared and weighted, that ratio dominates everything else in the variance. Low correlation helps the total, but it cannot move the split much when one asset is so much more volatile than the other.

The general lesson: whenever one holding is much more volatile than the others and carries real weight, its share of the risk runs far ahead of its share of the money. The weight still matters, though. Trim the equity sleeve here from 60% to 20% and its risk share falls from 92% to about 32%; at 5% it produces barely 3% of the risk, less than its share of the capital. Volatility tilts the split; it does not decide it on its own.

Marginal contribution

A related question: if I add one more dollar to this holding, how much does portfolio risk change? That's the marginal contribution to risk, and it's what tells you which position is expensive at the margin. In our example, adding to equities raises portfolio volatility far more than the same dollar added to bonds — which is the same fact, viewed from the edge rather than the total.

Try it now

1. Both sleeves are below. Measure a dozen ordinary sessions on each to estimate their daily volatilities, annualise by × √252, and estimate the correlation between them from how often they lean the same way.

2. Build a 60/40 portfolio and compute σₚ and each asset's risk contribution using the formulas above. Verify the two contributions sum to σₚ.
3. Re-run it with 30/70 weights instead. At what weighting do the two contributions get close to equal? Note the answer as arithmetic — the course makes no claim that any split is appropriate for anyone.