Three assets have volatilities of 8%, 16% and 24% and are mutually uncorrelated. What weight does the equal risk contribution portfolio put on the 24% asset?

Three assets have volatilities of 8%, 16% and 24% and are mutually uncorrelated. What weight does the equal risk contribution portfolio put on the 24% asset?

Approach: With zero correlation each asset's risk contribution is its own weighted variance, so set those equal and normalise.

2/11. With zero correlation asset i contributes w_i^2*sigma_i^2 of variance, so equal risk contribution requires w_i proportional to 1/sigma_i. The inverse volatilities 1/8, 1/16 and 1/24 are 6, 3 and 2 over a common denominator of 48, so the risk parity weights are 6/11, 3/11 and 2/11 and the 24% asset takes 2/11 = 18.2%. Each name then contributes (2/11*0.24)^2 = 0.00190 of variance, one third of the total 0.00571, so the portfolio volatility is 7.56%. Once correlations are non-zero the inverse volatility rule stops being exact and the weights solve w_i*(Sigma*w)_i equal across i, which needs a numerical solver.

Follow-up: How do the weights change if the 8% and 16% assets have correlation 0.8 and the third is independent?

Key concepts: risk parity, risk contribution, inverse volatility, portfolio volatility.