Contribution to risk versus capital
Half the capital is not half the risk. Which position really drives the book's volatility, and what an equal split of risk looks like.
- Split portfolio volatility into each position's contribution
- Compare each position's share of the risk with its share of the capital
- Find the risk-parity weights for two assets
- Compare the volatility of the current book with the risk-parity book
The intuition
Two flatmates split the rent 50/50. One of them plays the drums at midnight. The rent is shared equally; the neighbors' complaints are not. Who pays is one question; who causes the trouble is another.
A portfolio's capital split is how the money is divided. Its risk split is how much of the portfolio's volatility each position causes, counting how it moves with the rest. The contributions add up exactly to the portfolio's volatility. Risk parity sizes positions so the contributions are equal.
Contribution of A = wA × cov(A, portfolio) ÷ σp, where cov(A, portfolio) = wA σA² + wB ρ σA σB. A's and B's contributions add to σp. Risk share = contribution ÷ σp. Two-asset risk parity: wA σA = wB σB, so wA = σB ÷ (σA + σB).
Why it works
- The conventions here: two positions whose weights add to one, annual volatilities, one correlation. A position's contribution is its weight × its covariance with the whole portfolio ÷ the portfolio's volatility.
- Contributions add up; the positions' own volatilities do not. That is why risk reports list contributions: you can see who is driving the total.
- The volatile position usually takes far more than its share of the capital, because its variance grows with the square of its volatility.
- A contribution can be negative. A position that moves against the rest of the book lowers portfolio volatility; that is the mark of a real diversifier.
- For two assets, parity ignores correlation. The cross term is shared equally, so only the volatilities set the weights: the quieter asset gets proportionally more capital.
- A risk-parity book is more diversified, not automatically better. It holds more of the quiet asset, so its return is usually lower unless that asset is levered up, which is what risk-parity funds do.
| Variance: 50%² × 40%² + 50%² × 10%² + 2 × 50% × 50% × 0.2 × 40% × 10% | 465 %² |
| Portfolio volatility: √465 | 21.6% |
| cov(A, portfolio): 50% × 40%² + 50% × 0.2 × 40% × 10% | 840 %² |
| A's contribution: 50% × 840 ÷ 21.6 | 19.5 points, 90% of the risk |
| B's contribution: 21.6 − 19.5 | 2.1 points, 10% of the risk |
| Risk parity: weight in A = 10 ÷ (40 + 10) | 20% in A, 80% in B, at 12.4% volatility |
Equal capital, but A drives nine-tenths of the risk. At the parity weights each contributes exactly half.
The formulas
The portfolio's variance.
How A moves with the whole book, itself included.
A's slice of the volatility; A's and B's add to σp.
Set it against A's share of the capital.
Weight × volatility the same for both.
Worked example
The portfolio's volatility first, then A's covariance with the portfolio, then A's contribution. B's contribution is what is left.
See it move
Same two positions: A is the first company in the worked example, B the second. Change the weight in A, both volatilities and the correlation.
- Raise B's volatility. The risk-parity weight in A rises.
- Raise A's volatility. The risk-parity weight in A falls.
- Change the correlation. The risk-parity weights do not move: for two assets only the volatilities set them.
- With a correlation of zero or above, raise the weight in A. Its share of the risk rises.
Run it backwards
Same two positions, reversed: what weights make them contribute equally to the portfolio's risk?
For two assets, equal contributions means weight × volatility is the same for both. Solve for the weight in A: B's volatility ÷ the sum of the two volatilities.
The correlation drops out because the cross term is shared equally between the two. With three or more assets it does not, and the weights have to be solved numerically.
Traps
Say it in the interview
“A book is split 50/50 between two positions. Is its risk split 50/50?”
Check yourself
4 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.
Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.
- Contribution = weight × cov(asset, portfolio) ÷ σp.
- Contributions add up to σp; own volatilities do not.
- Two-asset risk parity: wA σA = wB σB.
- A share of the capital is not a share of the risk.