5.4.4 Risk Budgeting

Basic Prerequisites: 03 · Equal Risk Contribution and Convex optimization.

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1. Intuition & Practical Objective

ERC equalizes - but an allocator rarely wants equal slices of risk. A pension fund may want to give equities 40% of the risk budget, bonds 30%, commodities 20%, credit 10%. Risk budgeting generalizes ERC in one move: pick a vector of risk budgets $b=(b_1,\dots ,b_N)$ with $\sum b_i=1$, and find the weights $w$ whose contributions match them:

$$R{C}_i(w)=b_i\sigma (w)\text{for all }i.$$

ERC is simply the special case $b_i=1/N$. The objective of this page: (1) state the risk-budgeting program, (2) show it is solved by the same convex machinery as ERC (just with a weighted log-barrier), (3) verify arbitrary budgets land exactly, and (4) explain the discipline’s second name - risk budgets as a pre-commitment device, made credible by Qian (2006)‘s proof that budgets do add up and track actual losses.

Why is this a management tool and not just an optimization trick? Because a budget is a contract with yourself about where the risk will be, before the losses arrive. ERC is the “fair” default; risk budgeting is the knob that says “I am willing to be deliberately concentrated in strategy A, up to 40% of my risk, and no more.” It is how multi-asset and portable-alpha shops declare their exposures, and how serious funds run ex-ante risk limits (Litterman 1996’s “hot spots”).

The one-sentence essence. “Risk budgeting replaces ‘equalize the risk’ with ‘target a chosen risk decomposition’: solve for the weights whose contributions hit $b_i$, and ERC is just the equal-budget instance of the same program.”

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2. Mathematical Ground Truth & Derivations

The program. Find $w>0$, ${\sum }_i{w}_i=1$, such that

$$R{C}_i(w)\equiv \frac{w_i(\Sigma w{)}_i}{\sigma (w)}=b_i\sigma (w),\sum _{i=1}^N{b}_i=1.$$

Divide the $i$-th equation by $\sigma (w)$ and use $w_i(\Sigma w{)}_i={\text{RC}}_i\sigma (w)$:

$$R{C}_i(w)=b_i\sigma (w)⟺\frac{w_i(\Sigma w{)}_i}{\sigma (w{)}^2}=b_i⟺w_i(\Sigma w{)}_i\propto b_i.$$

So the condition is weight-times-portfolio-covariance proportional to the budget - the natural extension of the ERC condition ($b_i\propto 1$).

Convex embedding (extension of page 03). Weighted log-barrier with the budget active:

$$\underset{w>0}{\min }\frac{1}{2}{w}^{\top }\Sigma w\text{s.t.}{\sum }_i{b}_i\ln w_i\ge c.$$

KKT stationarity: $(\Sigma w{)}_i=\lambda b_i/w_i\Rightarrow w_i(\Sigma w{)}_i=\lambda b_i$ - proportional to $b_i$ as required. In practice one minimizes the unconstrained barrier $f(w)=\frac{1}{2}{w}^{\top }\Sigma w-{\sum }_i{b}_i\ln w_i$ at a fixed barrier weight and renormalizes the optimum to $\sum w=1$; the contribution ratios are scale-invariant so normalization does not disturb the budgets.

Feasibility & uniqueness. For any positive budgets $b>0$ the program is strictly convex in the interior (quadratic term PD + concave $\ln$) so a unique long-only solution exists (provided, as always, $\Sigma$ is positive-definite). If any $b_i=0$, that weight is driven to zero and the asset leaves the portfolio - which is itself a meaningful decision.

Special cases worth knowing.

• $b_i=1/N$ (equal): the ERC portfolio of page 03.
• $b_i\propto 1/{\sigma }_i$ or other “vol-matched” budgets: a risk-parity variant.
• Budget set equal to asset’s standalone Sharpe or to expected contribution to return: connects risk budgeting back to mean-variance thinking, where Qian (2006) shows budgets become expected return-budgets precisely at the mean-variance optimum.

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3. Computational Implementation - arbitrary budgets, exactly hit

Same cyclical coordinate-descent solver as page 03, but with a weighted barrier - budget $b_i$ enters the coefficient of the $\ln w_i$ term. Solve three different budget vectors on the Maillard universe and check the realized contribution shares hit the targets.

The realized percentage risk contributions match the requested budgets to the first decimal - the discipline works: the portfolio manager who commits to “equities may not exceed 40% of my risk” gets, out of the box, a portfolio that obeys exactly that rule. Note how the tilted 40/30/20/10 portfolio weights (49.3/19.0/19.4/12.3%) are not proportional to the budgets - weights and risk shares are different currencies, and risk budgeting is explicit that it prices risk, not capital.

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4. Failure Modes & First-Principles Breakdowns

1. Budgets live in risk-space, not capital-space. A 40% risk budget is not a 40% weight. Confusing the two is the most common implementation error; the code’s gap between budgets and weights is the reminder.
2. Zero budget = zero position. Setting $b_i=0$ exits the asset - and re-estimating $\Sigma$ then freezes that exit decision, which can be wrong if a low budget was set on stale correlation assumptions.
3. The whole budget sheet is a function of $\Sigma$. Budgets are honored only to the extent the covariance used to solve them is the covariance that realizes. When correlations move, a portfolio that was “40% equity risk” silently becomes 55% on the first crash day (05 · Failure Modes).
4. Institutional anchoring. “Risk-budget additivity” (Qian 2006) is precise only for volatility/VaR-type (linear-homogeneous) measures; sloppy shops budget on stand-alone ${\sigma }_i$ totals that do not add up and misreport the true decomposition.

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5. References

• Qian, Edward (2006): On the Financial Interpretation of Risk Contribution - “risk budgets do add up”: budgets = expected contributions to loss, and become expected-return budgets at the mean-variance optimum.
• Litterman, Robert: Hot Spots and Hedges, Journal of Portfolio Management (1996)
• Roncalli, Thierry: Introduction to Risk Parity and Budgeting, CRC (2013)
• Maillard, Roncalli & Teïletche (2010)

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6. Connected Graph Bridges

• Back: 03 · Equal Risk Contribution · Index Hub
• Forward: 05 · Failure Modes · 06 · Advanced Extensions
• Base: Convex Optimization · CVaR risk limits)