What is risk contribution?

Risk contribution measures how much of a portfolio's total risk comes from each holding. It is the counterpart of contribution to return. The key insight is that a holding's share of the risk can be very different from its share of the value: a volatile position that moves with the rest of the portfolio can dominate the risk even at a modest weight.

The formula

The contribution of holding $i$ to the portfolio's volatility $\sigma_p$ is its weight multiplied by its marginal contribution — how much the portfolio's volatility changes when that holding's weight increases slightly:

\text{RC}_i = w_i \, \frac{\partial \sigma_p}{\partial w_i} = w_i \, \frac{\operatorname{Cov}(r_i, r_p)}{\sigma_p}

where $r_i$ and $r_p$ are the returns of the holding and of the portfolio. The contributions of all holdings add up exactly to the portfolio's volatility:

\sum_i \text{RC}_i = \sigma_p

so each one can also be expressed as a percentage share of the total risk.

A simple illustration

A portfolio holds 80% in a bond fund with 5% volatility and 20% in a stock fund with 20% volatility. For simplicity, assume the two are uncorrelated. The portfolio's volatility is:

\sigma_p = \sqrt{0.8^2 \times 0.05^2 + 0.2^2 \times 0.20^2} \approx 5.7\%

Both terms under the root are equal — 0.0016 each — so each holding carries exactly half of the risk:

Holding	Share of value	Share of risk
Bond fund	80%	50%
Stock fund	20%	50%

The stock fund makes up a fifth of the value and half of the risk. In a typical balanced portfolio, where stocks are much more volatile than bonds and the two are often positively correlated, the imbalance is even larger: a portfolio split 60/40 by value commonly carries around 80–90% of its risk in the stocks.

Why it matters

Diversification by value can be an illusion. A portfolio spread evenly over ten positions can still have most of its risk in two or three of them.
Reducing risk efficiently. Trimming the holding with the largest risk contribution lowers the portfolio's volatility the most per euro moved.
Risk parity. Some strategies build portfolios so that every asset class contributes the same share of risk rather than the same share of money.

Limits

It rests on estimates. Volatilities and correlations change, especially in crises, and risk contributions shift with them.
Volatility is not the only risk. A holding with a small risk contribution can still carry a risk that volatility does not capture, such as a default.

A worked example

Worked through on a sample portfolio. The figures below are that portfolio’s, not yours.

Which holdings contribute most to portfolio volatility?

Top five holdings by contribution to portfolio volatility:

Asset	Weight (%)	Volatility p.a. (%)	Vol. contrib. (%)
Amazon.com Inc.	6,39%	27,76%	12,23%
Vanguard FTSE All-World UCITS ETF	13,96%	9,43%	10,92%
iShares Core S&P 500 UCITS ETF USD (Acc)	12,82%	9,40%	9,74%
Bitcoin	4,80%	36,07%	9,43%
Tesla Inc.	4,52%	35,08%	8,71%

Together the top three holdings account for 32,89% of the portfolio's volatility.

See these numbers for your own portfolio

Floreo works out every figure on this page from your own holdings — returns, risk, allocation, currencies — and the assistant explains them the way this page does. Import from your broker, or try it on the sample portfolio first.

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