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Risk Contribution

Definitions

Marginal risk contribution: MCR_i = (Sigma w)_i / sigma_p

Component risk contribution: CR_i = w_i * MCR_i

Identity

sum_i CR_i = sigma_p

Why PMs care

Capital weights and risk weights can be radically different.

PM question

Which positions are actually driving portfolio volatility?

Common mistakes

  • Reading CR_i as "the volatility this position would have on its own." It is the position's share of portfolio volatility given everything else held — remove the position and the remaining contributions all change.
  • Expecting the components to sum to portfolio variance. The identity is sum_i CR_i = sigma_p, the volatility, not sigma_p^2.
  • Assuming a contribution cannot be negative. A position that hedges the rest of the book has a negative marginal contribution, and that is the correct answer, not a sign error.
  • Confusing this with contribution to tracking error. Same algebra, different input: MCTE uses active weights against a benchmark. See MCTE and group risk decomposition.