14.4 Risk parity: Euler contributions and equal risk contribution

$\sigma(cw) = |c|\sigma(w)$, so for $c > 0$, $\sigma(cw) = c\sigma(w)$. Differentiate both sides in $c$ and set $c = 1$: $\sum_i w_i\,\partial_i\sigma(w) = \sigma(w)$ (Euler's theorem for homogeneous functions). Directly, $\nabla\sigma = \Sigma w/\sigma$, so $\sum_i w_i(\Sigma w)_i/\sigma = w^\top\Sigma w/\sigma = \sigma$.∎

$f$ is the sum of a strictly convex quadratic and the convex function $-\sum b_i\ln y_i$, so it is strictly convex. It tends to $+\infty$ as any $y_i \to 0^+$ (the log term) and as $\|y\| \to \infty$ (the quadratic dominates the logarithm), so it attains a minimum on the open orthant, which is unique by strict convexity. The first-order condition $(\Sigma y)_i - b_i/y_i = 0$ gives $y_i(\Sigma y)_i = b_i$. Then $RC_i(y) = b_i/\sigma(y)$ and $\sum_i RC_i = 1/\sigma(y) = \sigma(y)$, so $\sigma(y)^2 = 1$ and $RC_i(y)/\sigma(y) = b_i$. Risk shares are invariant to scaling $y$, so $w = y/\mathbf 1^\top y$ has the same shares. Conversely, any long-only $w$ with shares $b$ satisfies $w_i(\Sigma w)_i = b_i\sigma(w)^2$; then $y = w/\sigma(w)$ satisfies the FOC, so $w$ is the normalised unique minimiser.∎

With $w_i = c/\sigma_i$, $(\Sigma w)_i = \sum_j \rho_{ij}\sigma_i\sigma_j c/\sigma_j = c\sigma_i\sum_j\rho_{ij}$, so $w_i(\Sigma w)_i = c^2\sum_j\rho_{ij}$. These are equal across $i$ iff the row sums are equal. For two assets both row sums are $1 + \rho$.∎