Risk Parity

A 60/40 stock-bond portfolio looks balanced by capital but is wildly imbalanced by risk: because equities are three to four times as volatile as bonds, roughly 90% of the portfolio's variance comes from the equity sleeve. Risk parity flips the design principle, allocate so that each asset contributes the same amount of risk, then use leverage to reach the desired total risk. It sidesteps the estimation of expected returns entirely (Pitfalls of Mean-Variance Optimization), relying only on the more-estimable covariance matrix, which is why it became the flagship of Bridgewater's All Weather and a standard institutional allocation.

Risk contributions via Euler's theorem

Portfolio volatility is $\sigma(w) = \sqrt{w^\top\Sigma w}$. It is homogeneous of degree 1 in $w$, so by Euler's theorem it decomposes exactly into additive per-asset contributions:

$\sigma(w) = \sum_{i=1}^N w_i\frac{\partial \sigma}{\partial w_i}, \qquad \frac{\partial\sigma}{\partial w_i} = \frac{(\Sigma w)_i}{\sqrt{w^\top\Sigma w}}.$

The marginal risk contribution of asset $i$ is $\partial\sigma/\partial w_i$ (how much total vol rises per unit more of $i$), and its total risk contribution is

$\text{RC}_i = w_i\frac{\partial\sigma}{\partial w_i} = \frac{w_i(\Sigma w)_i}{\sqrt{w^\top\Sigma w}}.$

These sum to $\sigma(w)$ exactly, a clean, complete budgeting of risk across assets. The percentage risk contribution is $\text{RC}_i/\sigma(w)$.

The equal-risk-contribution condition

The equal risk contribution (ERC) portfolio requires $\text{RC}_i = \text{RC}_j$ for all $i,j$, i.e.

$\boxed{\;w_i(\Sigma w)_i = w_j(\Sigma w)_j = \frac{\sigma(w)^2}{N}\quad\text{for all }i,j.\;}$

This is a system with no general closed form, it is solved numerically (e.g. by minimizing $\sum_{i,j}(\text{RC}_i - \text{RC}_j)^2$, a convex problem, or by cyclical coordinate descent). Two special cases build intuition:

• Uncorrelated assets. If $\Sigma$ is diagonal, $(\Sigma w)_i = \sigma_i^2 w_i$, so $\text{RC}_i \propto w_i^2\sigma_i^2$, and equalizing gives $w_i \propto 1/\sigma_i$, the inverse-volatility portfolio. This is the risk-parity heuristic most people know, but it is only correct when correlations are zero.
• Equal correlations / equal vols. ERC reduces to equal weight, $w_i = 1/N$.

For general $\Sigma$ the inverse-vol rule is wrong because it ignores correlations; the ERC solution accounts for the fact that a highly-correlated asset "double counts" risk.

The role of leverage

Because ERC balances risk without regard to return, its unlevered expected return is low, it holds a lot of low-vol bonds. Risk parity therefore applies leverage to scale the whole portfolio up to a target volatility (see Vol Targeting): if the unlevered ERC portfolio has 6% vol and you want 12%, you lever 2×. The economic thesis is that leverage-constrained investors overpay for high-beta assets (the same low-beta anomaly that flattens the The Capital Asset Pricing Model (CAPM) SML), so a levered low-risk portfolio harvests a better risk-adjusted return than an unlevered high-risk one. This is the crux, and the vulnerability: risk parity needs leverage and cheap financing to work.

Comparison to 60/40

In 60/40, the equity contribution to variance is $\approx w_e^2\sigma_e^2/\sigma_p^2$. With $w_e = 0.6$, $\sigma_e = 15\%$, $w_b = 0.4$, $\sigma_b = 5\%$, and low correlation, equity variance is $0.6^2\times 0.15^2 = 0.0081$ versus bond $0.4^2\times 0.05^2 = 0.0004$, so equities supply about $0.0081/0.0085 \approx 95\%$ of the risk. Risk parity re-weights to, say, $\sim 25\%$ equities / $75\%$ bonds so each contributes half the risk, then levers the whole thing back up to the 60/40 volatility. The claim is a more diversified portfolio at the same risk level, better Sharpe, provided bonds keep diversifying equities.

Worked example

Two assets: $\sigma_1 = 20\%$ (equities), $\sigma_2 = 8\%$ (bonds), correlation $\rho = 0$. With zero correlation, inverse-vol weighting is exact: $w_1 = (1/0.20)/(1/0.20 + 1/0.08) = 5/(5+12.5) = 0.286$, $w_2 = 0.714$. Check the risk contributions: $\text{RC}_1 \propto w_1^2\sigma_1^2 = 0.286^2\times 0.04 = 0.00327$ and $\text{RC}_2 \propto 0.714^2\times 0.0064 = 0.00326$, equal, as required. The portfolio vol is $\sqrt{0.00327+0.00326} = 8.08\%$; to run at 12% target vol you lever $\approx 1.49\times$. Now introduce $\rho = 0.4$: inverse-vol is no longer ERC, and the true solution shifts weight off the more-correlated (equity) asset because its risk is partly redundant.

Failure modes

• Leverage risk. Risk parity's Achilles heel is its dependence on leverage and financing. In a joint stock-bond selloff (e.g. 2022, 2013 "taper tantrum") both legs fall, correlations turn positive, and levered risk parity suffers outsized drawdowns, de-levering into the fall amplifies losses.
• Correlation instability. ERC depends entirely on $\Sigma$; when correlations regime-shift the risk budget is stale (Covariance Matrix Estimation, Regime Detection).
• No return input. By ignoring $\mu$, risk parity implicitly assumes all assets have equal Sharpe, a strong, unhedged assumption. It maximizes diversification, not return.
• Bond-yield regime. The historical success leaned on a 40-year bond bull market; near the zero lower bound, levered bonds offer little cushion.

In interviews

Lead with the framing: 60/40 is balanced by capital but ~90% of its risk is equities; risk parity balances risk contributions instead. Derive the risk contribution $\text{RC}_i = w_i(\Sigma w)_i/\sigma(w)$ from Euler's theorem and state the ERC condition $w_i(\Sigma w)_i$ equal across assets. Know the two special cases, inverse-volatility when uncorrelated, equal-weight when correlations and vols are equal, and be ready to explain that inverse-vol is only exact for diagonal $\Sigma$. Cover leverage (needed to reach target return/vol) and the main risk (levered exposure when stock-bond correlation flips positive). See Hierarchical Risk Parity (López de Prado) and Vol Targeting.