Folio Lab/Documentation

Risk Budgeting (CVaR)

Allocate capital so that each asset contributes a pre-specified share of total tail risk. Generalises the Equal Risk Contribution portfolio of Maillard, Roncalli and Teiletche (2010): with equal budgets and CVaR replaced by variance, this estimator reduces to classical risk parity.

Overview

Risk Budgeting decomposes a chosen risk measure additively across assets and forces each asset to bear a pre-specified fraction of the total. The classical instance (Equal Risk Contribution, ERC) uses portfolio variance and equal budgets for all assets (Maillard et al., 2010; Qian, 2005). This document treats the more general formulation: CVaR as the risk measure and arbitrary user-supplied budgets.

FolioLab's production path uses CVaR as the risk measure (the API adapter passes risk_measure="CVAR" to skfolio's RiskBudgetingestimator). With equal budgets it gives an "Equal CVaR Contribution" portfolio: each asset contributes the same absolute amount to the portfolio's tail loss. Bruder and Roncalli (2012) and Roncalli (2013) develop the general theory of risk budgeting under arbitrary coherent risk measures, including CVaR.

The method is attractive when the analyst's preference is genuinely on the tail: variance-based ERC equalises overall variance contributions but can still leave the portfolio concentrated in the names that drive the worst quantile losses. CVaR Risk Budgeting addresses that directly.

Mathematical Formulation

Notation

  • — portfolio risk measure (positively homogeneous, e.g. variance, CVaR, EVaR)
  • — risk contribution of asset
  • — risk budget for asset , with

Euler decomposition

Positive homogeneity ( for ) means the total risk decomposes additively into per-asset contributions via Euler's theorem. This is what makes the budgeting problem well-defined.

Risk-budgeting problem

For variance the problem has a convex Lagrangian formulation (Spinu, 2013; Bruder & Roncalli, 2012) that admits a fast iterative solver. For CVaR the contributions are computed via the Rockafellar-Uryasev (2002) representation and the resulting fixed-point problem is solved by skfolio's built-in iteration.

CVaR risk contributions

The contribution of asset to portfolio CVaR is its weight times the conditional expectation of its loss in the tail-event regime. This follows directly from differentiating the Rockafellar-Uryasev convex CVaR objective with respect to .

Special cases and connections

Setting all budgets equal () and the risk measure to variance recovers the Maillard-Roncalli-Teiletche (2010) Equal Risk Contribution portfolio. Setting all budgets equal and the risk measure to CVaR gives the Equal CVaR Contribution portfolio. Setting all budgets equal and ignoring correlations (i.e. variance with diagonal covariance) recovers Inverse Volatility.

Cesarone and Colucci (2018) compare risk-budgeting solutions across risk measures on equity universes; their results suggest CVaR budgeting tends to improve drawdown control with little cost to long-run Sharpe in regimes where tail co-movement diverges from variance co-movement.

Advantages & Limitations

Advantages

  • Tail-risk aware: CVaR contributions reflect actual loss-quantile co-movement.
  • Generalises ERC: Same framework, swap the risk measure.
  • Fully diversified by construction: No single name dominates the tail.
  • Interpretable budgets: Each reads as "X% of total tail loss".

Limitations

  • Iterative solve: No closed form; convergence depends on the conditioning of the joint loss distribution.
  • Sample CVaR is noisy: Tail estimates need long histories or shrinkage.
  • No expected-return tilt: Cannot favour high-conviction names.
  • Dependence on confidence level: CVaR and CVaR can give different portfolios.

References

  • Maillard, S., Roncalli, T., & Teiletche, J. (2010). "The Properties of Equally Weighted Risk Contribution Portfolios." The Journal of Portfolio Management, 36(4), 60-70.
  • Qian, E. (2005). "Risk Parity Portfolios: Efficient Portfolios Through True Diversification." PanAgora Asset Management White Paper.
  • Bruder, B., & Roncalli, T. (2012). "Managing Risk Exposures Using the Risk Budgeting Approach." SSRN Working Paper, available at doi:10.2139/ssrn.2009778.
  • Roncalli, T. (2013). Introduction to Risk Parity and Budgeting. Chapman & Hall / CRC Financial Mathematics Series.
  • Spinu, F. (2013). "An Algorithm for Computing Risk Parity Weights." SSRN Working Paper.
  • Cesarone, F., & Colucci, S. (2018). "Minimum Risk Versus Capital and Risk Diversification Strategies for Portfolio Construction." Journal of the Operational Research Society, 69(2), 183-200.
  • Rockafellar, R. T., & Uryasev, S. (2002). "Conditional Value-at-Risk for General Loss Distributions." Journal of Banking & Finance, 26(7), 1443-1471.
  • Palomar, D. P. (2025). Portfolio Optimization: Theory and Application. Cambridge University Press, Chapter 11 (Risk Parity Portfolios).
  • skfolio documentation — skfolio.optimization.RiskBudgeting.