Advanced Portfolio Construction Techniques › Risk Parity and Risk Budgeting

Risk parity represents a fundamentally different approach to portfolio construction. Rather than optimizing expected return per unit of risk, risk parity allocates capital so that each asset contributes equally to total portfolio risk. This approach gained prominence after the 2008 financial crisis, when traditional 60/40 stock-bond portfolios suffered large losses due to their concentration of risk in equities. Many investors holding 60% stocks and 40% bonds believed they were diversified, only to discover that over 90% of their portfolio risk came from the equity allocation.

To understand risk parity, we first need to define precisely what we mean by an asset's contribution to portfolio risk. Intuitively, we want to answer the question: if we slightly increase our position in Asset X, how much will our portfolio volatility change? This marginal contribution to risk, combined with the current position size, tells us how much each asset contributes to the total.

The total portfolio volatility can be decomposed into contributions from each asset:

$$\sigma_p = \sum_{i=1}^{n} w_i \frac{\partial \sigma_p}{\partial w_i} = \sum_{i=1}^{n} RC_i$$

where:

• $\sigma_p$: total portfolio volatility (standard deviation)
• $n$: number of assets
• $w_i$: weight of asset $i$
• $RC_i$: risk contribution of asset $i$

This decomposition follows from Euler's theorem for homogeneous functions: portfolio volatility is homogeneous of degree one in the weights, meaning if we scale all weights by a factor $k$, volatility also scales by $k$. Euler's theorem then guarantees that volatility can be exactly decomposed into the sum of each weight times its marginal contribution.

The marginal contribution to risk (MCR) for asset $i$ is derived from the portfolio volatility derivative using the chain rule:

$$\begin{aligned} \frac{\partial \sigma_p}{\partial \mathbf{w}} &= \frac{\partial}{\partial \mathbf{w}} (\mathbf{w}^T \boldsymbol{\Sigma} \mathbf{w})^{1/2} && \text{(substitute definition of } \sigma_p \text{)} \\ &= \frac{1}{2}(\mathbf{w}^T \boldsymbol{\Sigma} \mathbf{w})^{-1/2} \cdot (2\boldsymbol{\Sigma} \mathbf{w}) && \text{(chain rule)} \\ &= \frac{\boldsymbol{\Sigma} \mathbf{w}}{\sigma_p} && \text{(simplify)} \end{aligned}$$

where:

• $\sigma_p$: portfolio volatility
• $\mathbf{w}$: vector of portfolio weights
• $\boldsymbol{\Sigma}$: covariance matrix of asset returns

This formula has an intuitive interpretation. The term $\boldsymbol{\Sigma} \mathbf{w}$ is a vector whose $i$-th element represents the covariance between asset $i$ and the overall portfolio. An asset's marginal contribution to risk depends on its own volatility and on how it comoves with the portfolio as a whole. An asset with high own volatility but negative correlation with other holdings might reduce portfolio risk at the margin.

For a single asset $i$, the MCR is the $i$-th element:

$$MCR_i = \frac{\partial \sigma_p}{\partial w_i} = \frac{(\boldsymbol{\Sigma} \mathbf{w})_i}{\sigma_p}$$

where:

• $MCR_i$: marginal contribution to risk of asset $i$
• $\sigma_p$: portfolio volatility
• $\boldsymbol{\Sigma}$: covariance matrix of asset returns
• $\mathbf{w}$: vector of portfolio weights
• $(\boldsymbol{\Sigma} \mathbf{w})_i$: the $i$-th element of the vector product $\boldsymbol{\Sigma} \mathbf{w}$

The total risk contribution (RC) for asset $i$ combines the position size with the marginal contribution:

$$RC_i = w_i \cdot MCR_i = \frac{w_i (\boldsymbol{\Sigma} \mathbf{w})_i}{\sigma_p}$$

where:

• $RC_i$: total risk contribution of asset $i$
• $w_i$: weight of asset $i$
• $MCR_i$: marginal contribution to risk of asset $i$
• $\boldsymbol{\Sigma}$: covariance matrix of asset returns
• $\mathbf{w}$: vector of portfolio weights
• $(\boldsymbol{\Sigma} \mathbf{w})_i$: the $i$-th element of the vector product $\boldsymbol{\Sigma} \mathbf{w}$
• $\sigma_p$: portfolio volatility

By construction, the risk contributions sum to total portfolio volatility: $\sum_{i=1}^{n} RC_i = \sigma_p$. This additive decomposition allows us to express what fraction of portfolio risk comes from each asset, which allows meaningful comparisons across holdings.

A risk parity portfolio equalizes risk contributions across all assets. Rather than asking 'how should you allocate your capital?', risk parity asks 'how should you allocate your risk budget?' The answer is to find weights such that:

$$RC_i = RC_j \quad \forall i, j$$

where:

• $RC_i$: risk contribution of asset $i$
• $RC_j$: risk contribution of asset $j$

or equivalently, expressed in terms of weights and marginal contributions:

$$w_i (\boldsymbol{\Sigma} \mathbf{w})_i = w_j (\boldsymbol{\Sigma} \mathbf{w})_j$$

where:

• $w_i, w_j$: weights of assets $i$ and $j$
• $(\boldsymbol{\Sigma} \mathbf{w})_k$: the $k$-th element of the covariance-weight product (proportional to marginal risk)

This is a nonlinear system of equations because the marginal contributions themselves depend on the weights. There is no closed-form solution in general, so we must solve numerically. The optimization finds weights that minimize the dispersion of risk contributions, subject to the constraint that weights sum to one.

Each asset now contributes approximately 20% of total portfolio risk, as intended. The risk parity weights are inversely related to asset volatility: low-volatility assets receive higher capital allocations, while high-volatility assets receive lower allocations. Asset A, with the lowest volatility at 15%, receives the highest weight, while Asset C, with the highest volatility at 25%, receives the lowest weight. This inverse relationship ensures that despite unequal capital allocations, each asset contributes equally to portfolio risk.

Risk parity is a special case of a more general framework called risk budgeting, where we can specify arbitrary target risk contributions. Rather than requiring equal contributions from all assets, we might want specific allocations based on our views about asset classes or our risk tolerance. For example, we might want 50% of risk from equities and 50% from bonds, even if we hold only two equity assets and three bond assets. Or we might want to allocate more risk budget to assets we believe have higher Sharpe ratios.

The optimization successfully matches the target risk budget. Assets C and B, which we assigned higher risk budgets of 25%, receive larger risk allocations, while the capital weights adjust to account for both the target budget and the asset volatilities. Notice that even though Assets B and C have the same risk budget, their capital weights differ because they have different volatilities. The optimizer finds the unique set of weights that delivers the desired risk allocation.

Let us visualize how different portfolio construction methods allocate capital and risk, making the philosophical differences between approaches concrete and visible.