Risk contributions and risk parity

Definition Let $\Sigma$ be the symmetric positive semidefinite covariance matrix of the one-period returns $R = (R_1, \dots, R_n)$ of $n$ assets, let $w \in \mathbb{R}^n$ hold the positions as fractions of capital, and let the portfolio volatility be $\sigma_p(w) = \sqrt{w^{\top} \Sigma w}$. At every $w$ with $\sigma_p(w) > 0$ the function $\sigma_p$ is differentiable, and the marginal risk of asset $i$ is

The variance has gradient $2\Sigma w$ because $\Sigma$ is symmetric, and the chain rule for the square root divides it by $2\sigma_p$. Marginal risk is measured in volatility per unit of weight. It is the rate of change of $\sigma_p$ when $w_i$ alone changes and the other weights stay fixed, so the extra position is financed by cash with zero volatility, not by selling other assets. Since $(\Sigma w)_i = \operatorname{Cov}(R_i, R_p)$ for the portfolio return $R_p = w^{\top} R$, the marginal risk equals $\rho_{i,p}\, \sigma_i$, the asset’s volatility $\sigma_i$ times its correlation $\rho_{i,p}$ with the portfolio, whenever $\sigma_i > 0$. If $\sigma_p(w) = 0$ and $\Sigma \ne 0$, then $\Sigma w = 0$, and near $w$ the volatility equals $\sigma_p(w + h) = \sqrt{h^{\top} \Sigma h}$, which has a kink at $h = 0$, so there is no gradient.

Definition The component risk contribution of asset $i$, called its risk contribution for short, is

defined when $\sigma_p(w) > 0$ and measured in units of volatility. Its percentage contribution is

Marginal risk is a rate per unit of weight, and component risk is that rate times the weight already held. Since $w_i (\Sigma w)_i = \operatorname{Cov}(w_i R_i, R_p)$, a contribution is the covariance of the holding’s return with the portfolio’s return, divided by $\sigma_p$. It is negative exactly when $w_i$ and $(\Sigma w)_i$ have opposite signs, and then a small increase in the size of the holding lowers $\sigma_p$. Multiplying every weight by $c > 0$ multiplies every contribution by $c$ and leaves every percentage unchanged.

Definition Let $\mathcal{R}$ be a risk measure defined on an open cone $C \subseteq \mathbb{R}^n$ of positions, so that $\lambda w \in C$ whenever $w \in C$ and $\lambda > 0$, and suppose $\mathcal{R}$ is positively homogeneous of degree one, $\mathcal{R}(\lambda w) = \lambda \mathcal{R}(w)$ for every $\lambda > 0$. If $\mathcal{R}$ is differentiable at $w$, then

Differentiating both sides of $\mathcal{R}(\lambda w) = \lambda \mathcal{R}(w)$ with respect to $\lambda$ at $\lambda = 1$ gives $\nabla \mathcal{R}(w)^{\top} w$ on the left by the chain rule and $\mathcal{R}(w)$ on the right. Euler allocation assigns asset $i$ the contribution $w_i\, \partial \mathcal{R} / \partial w_i$. Volatility satisfies $\sigma_p(\lambda w) = \lambda \sigma_p(w)$ for $\lambda > 0$ and is differentiable on the open cone where $\sigma_p > 0$, so

The degree matters. Variance is homogeneous of degree two, and $w^{\top} \nabla (w^{\top} \Sigma w) = 2 w^{\top} \Sigma w$, so the same rule applied to variance gives twice the total unless the components are halved. At a point where $\mathcal{R}$ is not differentiable, suppose $\mathcal{R}$ is finite and convex on all of $\mathbb{R}^n$ as well as positively homogeneous. Then $\mathcal{R}(0) = 0$, and every subgradient $g$ at $w$ satisfies $g^{\top} w = \mathcal{R}(w)$, because the subgradient inequality at the points $0$ and $2w$ gives $g^{\top} w \ge \mathcal{R}(w)$ and $g^{\top} w \le \mathcal{R}(w)$. Each subgradient then gives an allocation that adds up, but different subgradients can give different allocations, so the allocation need not be unique.

Definition A risk budget is a vector $b = (b_1, \dots, b_n)$ with every $b_i > 0$ and $\sum_i b_i = 1$. A portfolio $w$ with $\sigma_p(w) > 0$ meets the budget when $RC_i(w) = b_i\, \sigma_p(w)$ for every $i$. Multiplying by $\sigma_p(w)$ gives the equivalent conditions

for $i = 1, \dots, n$. The budget fixes the proportions of risk, not of capital, and expected returns do not enter it. If $w$ meets a budget, so does $c w$ for every $c > 0$, so a budget determines a portfolio only up to its scale. Full investment, $\mathbf{1}^{\top} w = 1$, or a target volatility is a separate choice. Scaling the risky positions above full investment requires borrowing, which brings financing costs, margin calls and liquidity exposure that the covariance matrix does not describe.

Definition Let $\Sigma$ be symmetric positive definite and $b$ a risk budget with every $b_i > 0$ and $\sum_i b_i = 1$. On the open orthant $x > 0$ define

Its gradient has the coordinates $(\Sigma x)_i - b_i/x_i$, and its Hessian

is positive definite, so $F$ is strictly convex. The function tends to $+\infty$ when a coordinate tends to $0$ with the others bounded, through the logarithm, and when $\lVert x \rVert \to \infty$, because $\tfrac12 x^{\top} \Sigma x \ge \tfrac12 \lambda_{\min} \lVert x \rVert^2$ with $\lambda_{\min} > 0$ the smallest eigenvalue, which outgrows $\sum_i b_i \ln x_i \le \ln \lVert x \rVert$. So its sublevel sets are compact subsets of the open orthant, a minimizer $x^{*}$ exists, and strict convexity makes it unique. Setting the gradient to zero gives

for every $i$, and summing over $i$ gives $x^{*\top} \Sigma x^{*} = 1$. With $s = \mathbf{1}^{\top} x^{*}$, the weights $w^{*} = x^{*}/s$ are positive, add up to $1$ and satisfy $w_i^{*} (\Sigma w^{*})_i = b_i/s^2$ and $w^{*\top} \Sigma w^{*} = 1/s^2$, so $w^{*}$ meets the budget. Conversely, every positive $w$ that meets the budget, rescaled to $x = w/\sigma_p(w)$, satisfies the zero-gradient equations, so $w^{*}$ is the only fully invested long portfolio with this budget. Without positive definiteness the minimizer can fail to exist. Further weight constraints, such as caps or sector limits, change the problem and are not covered by this construction.

Definition Risk parity allocates so that every holding contributes the same amount to a chosen risk measure. With volatility as the risk measure and positive weights it is the equal risk contribution portfolio, which meets the risk budget $b_i = 1/n$ for every $i$ and exists and is unique among fully invested long portfolios when $\Sigma$ is positive definite. For two assets with positive weights and volatilities $\sigma_1, \sigma_2 > 0$, the cross term $w_1 w_2 \sigma_{12}$, with $\sigma_{12} = \rho \sigma_1 \sigma_2$ the covariance, appears in both $w_1 (\Sigma w)_1$ and $w_2 (\Sigma w)_2$, so equal contributions reduce to $w_1^2 \sigma_1^2 = w_2^2 \sigma_2^2$, that is, $w_1 \sigma_1 = w_2 \sigma_2$, and

the inverse-volatility weights, for every correlation $\rho$ with $-1 < \rho \le 1$. At $\rho = -1$ these weights have zero variance. With three or more assets, inverse-volatility weights give equal contributions when all pairwise correlations equal a common $\rho > -1/(n-1)$, since with $w_i = c/\sigma_i$ every product $w_i (\Sigma w)_i$ equals $c^2 \bigl(1 + (n - 1)\rho\bigr)$, but not for general correlations. Equal capital weights, inverse-volatility weights and equal risk contributions are different rules, and none of them uses expected returns.