Risk Budgeting

Capital weights lie about risk. A classic 60/40 stock-bond portfolio looks balanced — 60% of your money in stocks, 40% in bonds — but because stocks are three or four times as volatile as bonds, something like 90% of the portfolio's actual risk comes from the equity sleeve. Risk budgeting is the discipline of allocating by that second number instead of the first. You decide up front how much of your total risk each position or strategy is allowed to contribute, then choose weights to hit those targets.

The machinery comes from splitting total portfolio volatility into per-asset pieces that add up exactly. Portfolio volatility is $\sigma(w) = \sqrt{w^\top \Sigma w}$, where $w$ is the vector of weights and $\Sigma$ is the covariance matrix. It decomposes cleanly (this is Euler's theorem) into risk contributions:

$$\text{RC}_i = w_i\,\frac{(\Sigma w)_i}{\sigma(w)}, \qquad \sum_i \text{RC}_i = \sigma(w).$$

$\text{RC}_i$ is the amount of total volatility that position $i$ is responsible for. The term $(\Sigma w)_i/\sigma(w)$ is its marginal contribution to risk — how much total vol rises if you add a bit more of $i$ — and multiplying by the weight $w_i$ gives its full share. The beautiful part: these shares sum to exactly the total risk, so "position $i$'s percent of risk budget" is just $\text{RC}_i / \sigma(w)$, and they add to 100%.

Worked example

Take a 60/40 portfolio: equities at weight $0.6$ with volatility 18%, bonds at $0.4$ with volatility 6%, correlation $0.2$.

• Portfolio variance: $0.6^2(0.18^2) + 0.4^2(0.06^2) + 2(0.6)(0.4)(0.2)(0.18)(0.06) = 0.01166 + 0.00058 + 0.00104 = 0.01328$, so $\sigma_p = \sqrt{0.01328} = 11.5\%$.
• Equity's risk contribution: its row of $\Sigma w$ is $0.6(0.0324) + 0.4(0.00216) = 0.02030$, so $\text{RC}_\text{eq} = 0.6 \times 0.02030 / 0.115 = 0.1057$. As a share of total: $0.1057/0.115 = 92\%$.
• Bond's risk contribution: its $\Sigma w$ row is $0.6(0.00216) + 0.4(0.0036) = 0.00274$, so $\text{RC}_\text{bd} = 0.4 \times 0.00274/0.115 = 0.0095$, a share of $8\%$.

So a portfolio that is 60/40 by capital is 92/8 by risk. If your risk budget instead calls for a 50/50 split of risk, you'd have to slash the equity weight dramatically and lever the bond sleeve up — which is precisely the recipe Risk Parity follows, then scales the whole thing to a target volatility (Vol Targeting).

Where it misleads

• The budget is only as stable as the covariance matrix. Risk contributions depend entirely on $\Sigma$. When correlations regime-shift — bonds stop hedging stocks, say — yesterday's carefully set budget is silently wrong today (Regime Detection).
• Contributions can go negative. A genuine hedge has a negative risk contribution: it reduces total risk. That's correct and useful, but it means a naive "everyone gets a positive slice" mental model breaks for hedged books.
• No return input. Budgeting risk says nothing about reward. Handing equal risk to a great strategy and a mediocre one wastes budget; risk budgeting maximises diversification, not expected return, unless you feed in views.