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Key Takeaways
- Equal risk contribution balances each holding’s contribution to portfolio volatility. The weights use the complete covariance structure.
- Each holding’s risk contribution is its weight times its marginal risk, and the contributions add up exactly to portfolio volatility.
- At 16 percent equity volatility, 6 percent bond volatility, and 0.20 correlation, a 60/40 portfolio draws 90.3 percent of its volatility from equities, the imbalance risk parity is designed to remove.
- Set the equal-risk allocation first, then size total exposure to your volatility target; leverage is one way to get there.
- Equal risk sets how risk is shared. Judge each holding’s expected return separately before you fund it.
- For a bond allocation, connect yield changes with duration, portfolio weight and financing. Recalculate the combined stock-bond scenario.
What is risk parity?
Risk parity is a portfolio allocation method that sets each holding’s contribution to a chosen risk measure equal. For volatility-based equal risk contribution, each of n assets contributes one nth of total portfolio volatility. In a two-asset stock and bond mix, that puts the larger capital weight on the lower-volatility bonds. Total exposure is a separate decision, sometimes made with leverage to reach a volatility target.
The allocation decides how risk is shared; a separate exposure decision sets how much of it you carry.
How are risk contributions calculated?
For weights w and covariance matrix Sigma, portfolio volatility is √(wᵀSigma w). The marginal contribution of asset i is (Sigma w)i divided by portfolio volatility. Multiply it by wi to obtain the component contribution.
These component contributions add to portfolio volatility. Divide each by that total to express the risk budget as percentages.
Marginal contribution is how much portfolio volatility rises per unit of extra weight. Component contribution multiplies it by the weight you actually hold, and only the component figures add up to the total.
Formula
σ_p = √(w^T Σ w); MRC_i = (Σ w)_i / σ_p; RC_i = w_i × MRC_i; risk parity solves RC_i = σ_p / n for every asset; for two assets this reduces to w_i = (1 / σ_i) / sum of (1 / σ_j)- w is the vector of portfolio weights and Sigma is the covariance matrix of asset returns, so σ_p is portfolio volatility.
- MRC_i is the marginal risk contribution, the partial derivative of portfolio volatility with respect to weight i.
- RC_i is the risk contribution of asset i. Because volatility is homogeneous of degree one in the weights, Euler decomposition makes the RC_i sum exactly to σ_p, so percent contributions RC_i / σ_p sum to 100 percent.
- n is the number of assets. Equal risk contribution means every RC_i equals σ_p / n.
- For two positive-volatility assets with portfolio volatility above zero, positive equal-risk weights are proportional to inverse volatility. For more assets, inverse-volatility weights give equal contributions when the correlation matrix has equal row sums; identical pairwise correlations are one sufficient case.
- A positive multiplier k scales all risky weights, volatility and contributions by k when the financing or cash return adds no volatility to this model.
Risk parity example: equal risk vs 60/
A two-asset sleeve holds an equity index at 16 percent annualized volatility and a Treasury position at 6 percent annualized volatility, with a return correlation of 0.20. The comparison is against a conventional 60 percent equity, 40 percent Treasury mix.
- Risk parity weights (two assets): (1/16) / ((1/16) + (1/6)) = 27.3 percent equities and 72.7 percent Treasuries
- Unlevered portfolio volatility: each leg carries weight × volatility of 4.364 points, so √(4.364² + 4.364² + 2 × 0.20 × 4.364 × 4.364) = 6.76 percent
- Risk contributions: equities 3.38 points and Treasuries 3.38 points, which sum to the 6.76 percent total and split it 50/50
- Same math on the 60/40 mix: volatility √(107.136) = 10.35 percent, with an equity risk contribution of 9.35 points
- Equity share of 60/40 risk: 9.35 / 10.35 = 90.3 percent, against a 60 percent capital weight
- Match 10.35% volatility with a multiplier of 1.531: hold 41.76% equities and 111.36% Treasuries.
- Total risky exposure is 153.1%. A cash-security implementation borrows 53.1% of starting capital; a futures implementation uses its own notional, collateral and margin structure.
The 60/40 mix runs 10.35% volatility with 90.32% of it coming from equities. Equal-risk weights of 27.27% equities and 72.73% Treasuries cut volatility to 6.76% and split it 50/50, and scaling by 1.531 brings volatility back to 10.35% with the risk still split evenly.
Use 16% equity volatility, 6% Treasury volatility and 0.20 correlation. A 60/40 allocation has 10.35% volatility. Equity contributes 9.35 percentage points and Treasuries 1.00, so equities carry 90.3% of the risk on 60% of the capital.
Two-asset equal-risk weights are proportional to inverse volatility. The solution is 27.27% equities and 72.73% Treasuries. Total volatility is 6.76%, split equally into two 3.38-point contributions.
Risk parity leverage and the volatility target
To match the 60/40 portfolio’s 10.35% volatility, multiply both equal-risk weights by 1.531. The exposures become 41.76% equities and 111.36% Treasuries, 153.1% of capital in total.
A cash-security portfolio borrows 53.1% of starting capital. Futures carry the same exposure through contract notional and collateral, with their own margin and financing terms. The unlevered version is a complete risk-parity portfolio too, at 6.76% volatility.
Risk parity vs inverse-volatility weighting
With two positive-volatility assets and nonzero portfolio volatility, inverse-volatility weights give equal contributions. With more assets, the shortcut requires equal row sums in the correlation matrix. Equal pairwise correlations are one sufficient arrangement.
For a general portfolio, solve the risk budgets using the full covariance matrix. Set different contribution targets when the portfolio calls for unequal budgets.
Equal risk contributions vs diversification
Five equally weighted assets with 40% volatility and 0.80 pairwise correlation each contribute 20% of total risk. Portfolio volatility is 36.66%. At zero correlation, it is 17.89%.
The diversification ratios, weighted asset volatility divided by portfolio volatility, are 1.09 and 2.24. Squared, they give the equivalent number of independent exposures in this equal-volatility, equal-weight example: 1.19 at 0.80 correlation and 5 at zero. Five holdings at 0.80 correlation behave like barely more than one.
Compare total volatility, component contributions and concentration together.
Limitations of risk parity
- Rebalance when the inputs move. The weights come from volatilities and correlations measured over a look-back window, so a jump in bond volatility or correlation shifts the equal-risk weights and the realized risk split.
- Stress the stock-bond correlation. It has changed sign over time, and when it turns positive the bond leg falls with equities instead of cushioning them; with more than two assets the weights move too.
- Size for the tail separately. Equal volatility contributions describe how day-to-day swings are shared, so check expected returns and crash losses before you fund the mix.
- Price the leverage. Matching 60/40 volatility takes 153.1% exposure, so the 53.1% borrowed adds financing, margin and collateral terms, and the 111.36% Treasury weight ties half the portfolio’s risk to duration.
- Check correlation as well as the risk split. The five assets at 0.80 correlation each contribute 20% of risk, and their 1.09 diversification ratio means they trade like barely more than one position.
Bottom line
Risk parity spreads portfolio risk evenly across holdings instead of spreading capital evenly, which removes the equity-heavy risk split of a mix like 60/40. Use it with current covariance estimates, set total exposure as a separate decision, and test the combined portfolio under rate, correlation and financing scenarios.
Clear answers
Frequently Asked Questions
Sources and Further Reading
- Asness, Frazzini and Pedersen, Leverage Aversion and Risk Parity, Financial Analysts Journal 2012
- Thierry Roncalli, Introduction to Risk Parity and Budgeting (arXiv:1403.1889)
- Campbell, Pflueger and Viceira, Macroeconomic Drivers of Bond and Equity Risks (NBER Working Paper 20070)
- Board of Governors of the Federal Reserve System, Financial Stability Report, November 2024
How entries are sourced, calculated and updated: editorial methodology.