Risk parity: equal risk contributions instead of equal weights

Advanced9 min read · 4 sources

Risk parity means that each asset should contribute the same amount to the portfolio's total risk, rather than receive the same amount of capital. In a classic 60/40 portfolio of equities and bonds, the risk comes almost entirely from the equities. Risk parity therefore gives the low-volatility bonds much more weight and usually needs leverage to achieve an equity-like return.

The idea: spreading risk, not capital

If you split your money 60 to 40 between equities and bonds, you have diversified your capital but hardly your risk. Equities fluctuate about three to four times as much as high-quality government bonds. As a result, they almost single-handedly determine how much the portfolio fluctuates and how deeply it falls in crises. Risk parity turns the question around: which weights ensure that each asset carries the same share of the risk?

The idea has two roots. Practitioners wanted portfolios that do not depend on a single asset class. And estimates of expected returns are so uncertain that many investors prefer to use only volatilities and correlations. These can be measured much more accurately. Risk parity does without return estimates, much like the minimum variance portfolio and equal weighting.

Formula: the risk contribution of each asset

The basis is the decomposition of portfolio volatility into the contributions of the individual assets. An asset's share of risk is its weight times its covariance with the whole portfolio, divided by the portfolio's variance. The shares always add up to exactly 100%, as the article on risk contribution shows.

For two assets, there is a simple solution: the weights are inversely proportional to the volatilities, regardless of the correlation. With more assets, this only holds if all correlations are equal. Otherwise, the weights have to be found numerically; for portfolios without short sales, the problem has a unique solution.

Example with two assets: 60/40 versus risk parity

This brings us to the heart of the strategy. With an expected return of 7% for equities, 2% for bonds and a money market rate of 1%, 60/40 delivers around 5.00% at 10% volatility, i.e. a Sharpe ratio of 0.40. Risk parity comes to 3.19% at 5.65% volatility, a Sharpe ratio of 0.39. To reach the same volatility as 60/40, you would have to lever the risk parity portfolio by a factor of 1.77. Financed at the money market rate, that would give 4.88%, slightly less than 60/40.

That is the honest short version: risk parity is not a trick that creates return out of nothing. It pays off if low-volatility assets have a higher Sharpe ratio than high-volatility ones, and if leverage is cheap. That is exactly what its proponents claim (see the research section).

Example with three assets: correlations count

With gold as a third asset (σ = 15%) and correlations of 0.1 between equities and bonds, 0.1 between equities and gold and 0.3 between bonds and gold, the simple formula no longer suffices. The table compares four approaches:

Inverse volatility is a good approximation but does not distribute risk exactly equally: bonds and gold are more strongly correlated with each other, so both carry slightly more risk. The minimum variance portfolio would have the lowest volatility at 4.89%, but puts 93% in bonds. At 6.14%, risk parity lies between minimum variance and equal weighting. Maillard et al. (2010) proved exactly this ordering in general.

Advanced: risk budgets, leverage and solution methods

Risk budgets instead of equal shares

Risk parity is the special case of a risk budget: instead of 1/n, each asset can be given its own target share b_i, such as 50% equities, 30% bonds and 20% commodities. The condition is then RiskShare_i = b_i. The solution can be written as the minimization of wᵀΣw/2 − Σ b_i ln w_i, which can then be normalized to sum to 1; uniqueness for positive weights follows from this. Roncalli (2013) describes the theory and algorithms in detail.

Factors instead of asset classes

Equal risk contributions from asset classes do not yet mean equal risk contributions from risk drivers. Corporate bonds, for example, carry interest rate and credit risk, and equities carry growth and credit risk. Advanced variants therefore distribute risk across factors (growth, inflation, interest rates) or across uncorrelated principal components. One approach without inverting the covariance matrix is hierarchical risk parity.

Related approaches

The maximum diversification portfolio of Choueifaty and Coignard (2008) maximizes the ratio of weighted individual volatility to portfolio volatility. It coincides with risk parity when all correlations are equal, but otherwise gives more weight to assets with low correlation. Neither needs a return estimate, but both react to errors in the covariance matrix. Covariance estimation is therefore important.

What the research shows

Maillard et al. (2010) derived the properties of portfolios with equal risk contributions: their volatility lies between that of the minimum variance portfolio and that of equal weighting, and they are more robust to estimation errors than mean-variance portfolios. The authors see risk parity as a compromise between absolute risk, risk budgeting and diversification.

The most important explanation of why risk parity can work even after costs comes from Asness et al. (2012): many investors are not allowed or not willing to borrow. Those who want more return therefore buy risky assets instead of safe ones with leverage. This drives up the prices of risky assets and makes safe assets relatively cheap. Risk parity exploits this by overweighting and levering safe assets. The same logic lies behind the low-volatility anomaly in equities.

Many risk parity funds achieved their good results at a time when falling interest rates favored bonds for decades. When interest rates rise and equities come under pressure at the same time, as in 2022, a levered bond portfolio suffers twice over. Historical results therefore depend heavily on the interest rate phase.

Limits and common mistakes

• Leverage costs money: levered bonds are financed through futures or borrowing. If financing costs rise, the advantage shrinks. Without leverage, risk parity is a defensive portfolio with a rather low return.
• Correlations are not stable: in crises, many correlations rise. Equities and bonds moved in opposite directions over long phases, but in the same direction during inflationary phases. A portfolio based on historical correlations can then fluctuate considerably more than planned.
• Volatility is not everything: equal volatility contributions do not mean equal loss contributions. Assets with fat tails or rare large losses (credit, option strategies) look more harmless in terms of volatility than they are.
• The estimation window affects the result: volatilities from calm years make an asset appear safe. Short windows react quickly but often lead to reallocation.
• No return estimate does not mean no assumption. Risk parity implicitly assumes that all assets have similar Sharpe ratios. If this is not the case, the portfolio is not efficient.

How Portfolio Lab calculates it

In Optimization, the objective is called “Risk parity (equal risk contributions)”. Portfolio Lab estimates the covariance matrix from monthly returns (by default with Ledoit-Wolf shrinkage, alternatively from the sample) and numerically searches for the weights at which the squared deviations of the shares of risk from 1/n are minimal. The starting point is inverse volatility. The weights add up to 100% and are not negative; the optimization does not calculate leverage. If you set minimum or maximum weights, the shares of risk may no longer be exactly equal. Custom risk budgets (such as 50/30/20) are not yet available.

The results table shows the weight and risk contribution of each asset. With the “Rolling” mode, the portfolio is recalculated regularly using only the data available up to each point in time. This is an honest out-of-sample test. In the Backtest, you can also see the risk contributions for each of your own portfolios. All details are in the optimization methodology.

This article explains a method; it is not investment or tax advice. Worked examples are simplified and rounded. All terms in the glossary.