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11.8 Summary

Diversification is a crucial principle in portfolio design, exemplified by the well-known phrase, “don’t put all your eggs in one basket.” Some key points are:

• The widely-used \(1/N\) portfolio (or equally-weighted portfolio) effectively diversifies capital allocation. However, a more advanced strategy is to diversify risk allocation, as implemented by risk parity portfolios.

• In risk parity portfolios, the risk measure of interest (e.g., volatility) is expressed as the sum of individual risk contributions from each asset. This provides a more refined control of the risk compared to simply using a single risk value for the overall portfolio.

• Risk parity formulations can be classified into three levels of complexity:

  • naive diagonal formulation: the covariance matrix is assumed diagonal and the solution simplifies to the inverse-volatility portfolio (which ignores the assets’ correlations);

  • vanilla convex formulations: simple long-only portfolios are considered and the problems can be rewritten in convex form with efficient algorithms; and

  • general nonconvex formulations: admit any realistic constraint and extended objective functions at the expense of becoming nonconvex problems that require a more careful resolution (but still efficient iterative algorithms can be derived).