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 general formulation as background, and states the shipped configuration beside it.

The budgets are always equal. No request field supplies them, and the estimator is constructed without a budget argument, so every run of this method uses . Where the text below writes "user-supplied budgets", read it as the general theory rather than as a setting you can enter. On a run that carries constraints, the result records the requested share , the achieved contribution share per asset, the largest deviation between the two, and the assets held at a bound. A binding weight bound prevents exactly equal contributions, so this record shows how far the run is from the budget.

FolioLab's production path uses CVaR as the risk measure by default (the API adapter passes risk_measure="CVAR" to skfolio's RiskBudgeting estimator, with skfolio's default CVaR confidence level of 0.95 and the daily returns of the run window as scenarios). 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 measure is a choice. The request field risk_budgeting_measure, the optimize form and the MCP submit tools accept eight measures. The result records the one it used.

  • cvar: CVaR at 95%, the default
  • variance: equal variance contributions, which is risk parity
  • semi_variance: variance of the returns below the mean
  • mean_absolute_deviation: mean absolute deviation
  • cdar: CDaR at 95%
  • evar: EVaR at 95%
  • edar: EDaR at 95%
  • ulcer_index: the ulcer index

Two measures skfolio accepts are left out. With worst realization the problem does not solve, and with the Gini mean difference a solve took 107 seconds on a 10-stock, 12-year book. Each measure gives a different portfolio. None is validated to perform better than another, and none is validated out of sample.

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. skfolio solves the same idea as one convex program for any supported risk measure: it minimises subject to and , with the total weight left free, and then rescales the weights to sum to one. For CVaR, uses the Rockafellar-Uryasev (2002) representation.

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 minimum-risk, capital-diversification and risk-diversification strategies on several equity data sets. Their findings are specific to those samples and risk measures, and they are not a forecast for any universe run here.

Advantages & Limitations

Advantages

  • Tail-risk aware: CVaR contributions reflect actual loss-quantile co-movement.
  • Generalises ERC: Same framework, swap the risk measure.
  • Diversified by construction: Each name targets an equal share of the tail loss, so no single name is meant to dominate it.
  • Interpretable budgets: Each reads as "X% of total tail loss". They are equal on every run here, so each name targets of the tail loss.

Limitations

  • Numerical solve: No closed form; the solve depends on the conditioning of the joint loss distribution.
  • No exclusions: The program takes logarithms of the weights, so a ceiling of zero is outside its domain. Per-name bounds and group sum limits enter the program, but a run that excludes a name does not run this method. Remove the name from the universe instead.
  • 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.

Not investment advice. Past performance is not indicative of future results.