Marginal Contribution to Risk

Suppose your portfolio's volatility is 11.5% and you're about to nudge a bit more capital into one holding. Does total risk go up a lot, a little, or — if the holding is a hedge — actually down? The marginal contribution to risk (MCR) answers exactly that. It's the rate at which total portfolio volatility changes as you add more of one position: the slope of portfolio risk with respect to that weight.

$$\text{MCR}_i = \frac{\partial \sigma_p}{\partial w_i} = \frac{(\Sigma w)_i}{\sigma_p}.$$

$\sigma_p = \sqrt{w^\top \Sigma w}$ is the portfolio volatility, $w$ the weights, and $\Sigma$ the covariance matrix. The expression $(\Sigma w)_i$ is the covariance between asset $i$ and the whole portfolio, and dividing by $\sigma_p$ turns it into a marginal volatility impact. There's a much friendlier way to read the same quantity:

$$\text{MCR}_i = \rho_{i,p}\,\sigma_i .$$

That is, an asset's marginal contribution to risk is its own volatility $\sigma_i$ times its correlation $\rho_{i,p}$ with the total portfolio. This is the intuition worth keeping: what makes a position risky to your book isn't just how volatile it is on its own — it's how volatile it is and how much it moves with everything else you hold. A wild asset that zigs when your portfolio zags can have a small, even negative, marginal risk.

Worked example

Take a 60/40 book: equities (weight 0.6, vol 18%) and bonds (weight 0.4, vol 6%), correlation 0.2. The portfolio volatility works out to $\sigma_p = 11.5\%$ (the same computation as in Risk Budgeting). Now compute each asset's marginal and total contributions.

Asset	Weight	Vol	Corr. w/ portfolio	MCR $=\rho_{i,p}\sigma_i$	RC $=w_i\,\text{MCR}_i$	% of risk
Equities	0.60	18%	0.98	17.6%	10.6%	92%
Bonds	0.40	6%	0.40	2.4%	0.95%	8%
Total					11.5%	100%

Read the equity row. Its correlation with the whole portfolio is 0.98 — the book is mostly equities, so equities move almost in lockstep with it — and $0.98 \times 18\% = 17.6\%$ is its marginal contribution. That means adding a small slug $\mathrm{d}w$ of equities raises portfolio volatility by about $17.6\% \times \mathrm{d}w$. Bonds, only 40% correlated with the book, contribute a marginal $2.4\%$ despite being real, volatile assets. Multiply each MCR by its weight and the two risk contributions ($10.6\%$ and $0.95\%$) sum to exactly the total $11.5\%$ — the clean decomposition that makes Risk Budgeting and Risk Parity possible.

Where it misleads

• It's a local slope, valid only for small moves. MCR is a first derivative computed at the current weights. Double a position and $\sigma_p$, the correlations, and hence the MCRs all shift — the number you started with no longer applies. Use it for marginal, incremental decisions, not wholesale reallocations.
• It inherits every flaw of $\Sigma$. Both forms lean entirely on the covariance matrix. A stale or noisy $\Sigma$ gives confidently wrong marginal risks (shrinkage helps — see Ledoit-Wolf Covariance Shrinkage).
• Signs matter. A position that's negatively correlated with the book has a negative MCR — adding it lowers total risk. That's the mathematical fingerprint of a hedge, and it's easy to misread if you assume all contributions are positive.