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Risk Parity and Stress Testing

Mean-variance optimisation needs expected returns, and expected returns are the inputs nobody can estimate well. Risk parity is the best-known method that avoids them: it sizes positions so each contributes the same amount of risk. Interviewers ask how to compute a risk contribution, why a 60/40 portfolio is not diversified, and what risk parity quietly assumes. They follow with stress testing, because a covariance matrix describes normal days and portfolios fail on the others.

Risk contributions

A portfolio's variance splits exactly into one contribution per position:

Risk contribution
σp2=w⊤Σ w=∑iRCi,RCi=wi (Σw)i\sigma_p^2 = w^\top \Sigma\, w = \sum_{i} RC_i, \qquad RC_i = w_i \,(\Sigma w)_i

Each position's weight times its covariance with the whole portfolio. The contributions add up to the portfolio variance, so they answer where the risk comes from.

(Σw)i(\Sigma w)_i is the covariance of asset ii with the portfolio. A position contributes a lot of risk when it is large, volatile, or highly correlated with everything else, and a position that hedges the rest can contribute negative risk.

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