How to use the risk parity calculator
- List the assets with their annual volatility (the annualized standard deviation of returns). The four defaults (stocks, bonds, gold and commodities) are example assumptions, not forecasts. The 60/40 comparison always uses the first two rows, so put stocks first and bonds second if you want the classic mix.
- Enter the correlations. Full matrix takes one row per asset in the same order as the list, with 1s on the diagonal; you can build one from your own return history with the correlation matrix calculator. One average correlation applies the same value to every pair.
- Set a target volatility if you want to see the leverage needed to run the portfolio at that risk level. Leave it empty to compare unlevered portfolios only.
The results show the volatility of the equal-risk portfolio and the leverage to reach your target, then the same figures for naive risk parity, equal weight and 60/40. The tables give every weight and every asset’s share of risk under each method.
The formulas
Naive risk parity sets each weight in proportion to the inverse of the asset’s volatility:
Equal risk contribution requires each asset’s share of portfolio variance to be the same:
There is no closed form for three or more assets with unequal correlations, so the calculator solves it numerically. It finds y > 0 with yi(Σy)i = 1/N by cyclical coordinate descent, updating one asset at a time with the positive root of a quadratic, and then rescales the weights to sum to one:
w = y ÷ Σi yiThis is the first-order condition of the strictly convex problem min ½yTΣy − (1/N) Σ ln yi, which has a unique solution whenever no long-only mix is riskless. Iteration stops when every risk share is within 10−12 of 1/N.
Leverage scales the whole portfolio to the target volatility:
Levered return = L × rp − (L − 1) × rborrowrp is the unlevered portfolio return and rborrow the financing rate on the borrowed (L − 1) of capital. With futures the financing cost is built into the futures price instead.
A worked example
The default portfolio holds stocks (16% volatility), bonds (6%), gold (15%) and commodities (18%), with the example correlation matrix.
Equal risk contribution puts 17.4% in stocks, 51.6% in bonds, 17.1% in gold and 13.9% in commodities. Each asset then carries exactly 25% of the risk, and the portfolio’s volatility is 6.69%.
Naive risk parity gives bonds less, 47.4%, and a volatility of 7.03%. Because commodities correlate with both stocks and gold, they end up carrying 28.7% of the naive portfolio’s risk, while bonds, the asset least correlated with the rest, carry only 19.7%.
To run the equal-risk portfolio at the 10% target, you would need 1.49× leverage: positions worth 1.49× your capital, with the excess borrowed or held through futures.
Naive versus true risk parity
Inverse-volatility weights equalize risk only when all assets are equally correlated with one another. It is a known result that equal correlations make the two methods identical, and with just two assets they always coincide, whatever the correlation. Rerunning the example with a single average correlation of 0.2 shows it: inverse volatility and ERC both give 47.4% in bonds and a volatility of 7.20%.
| Share of risk | Naive (inverse volatility) | Equal risk contribution |
|---|---|---|
| Stocks | 24.6% | 25.0% |
| Bonds | 19.7% | 25.0% |
| Gold | 27.0% | 25.0% |
| Commodities | 28.7% | 25.0% |
The rule is intuitive. An asset that correlates strongly with the rest of the portfolio contributes more risk per unit of volatility, so ERC trims it; an asset that diversifies the rest gets more weight than its volatility alone would suggest. With realistic correlation matrices the gap between the two methods is usually modest, a few points of weight, which is why inverse volatility is a popular and robust approximation. The gap grows when one asset is a strong diversifier or two assets are near-duplicates.
How it compares with equal weight and 60/40
| Portfolio | Volatility | Leverage for 10% | Largest risk share |
|---|---|---|---|
| Equal risk contribution | 6.69% | 1.49× | 25.0% each |
| Inverse volatility | 7.03% | 1.42× | 28.7% (commodities) |
| Equal weight | 9.19% | 1.09× | 40.9% (commodities) |
| 60/40 stocks and bonds | 9.90% | 1.01× | 94.1% (stocks) |
Equal capital weights do not mean equal risk: in the equal-weight portfolio, bonds hold a quarter of the money but only 3.9% of the risk. The 60/40 portfolio is, by risk, almost entirely a stock portfolio, with stocks supplying 94.1% of its volatility. Risk parity spreads the risk evenly but starts from a much lower volatility, which is why it needs leverage to match the others.
What changes the answer most
- The volatility of the calmest asset. If bonds were 8% volatile instead of 6%, their ERC weight would drop to 44.5% and the leverage for 10% would fall to 1.30×.
- The stock–bond correlation. With a correlation of 0.5 between stocks and bonds instead of zero, the equal-risk portfolio’s volatility rises to 7.48% and the bond weight moves to 48.6%. That regime, where inflation hurts both, removes much of the diversification the approach relies on.
- The asset list. Risk parity spreads risk across whatever you include. Adding a second, highly correlated equity asset raises the total equity risk, because each equity line still receives a full share.
Leverage, financing and All Weather
The best-known example of the idea is Bridgewater Associates’ All Weather strategy, launched in the 1990s, which balances risk across assets expected to do well in different economic environments (growth and inflation rising or falling). Many risk parity funds since then follow the same principle with their own asset lists and methods.
The leverage is not a detail. If the unlevered portfolio earns 5% and borrowing costs 4%, 1.5× leverage earns 1.5 × 5% − 0.5 × 4% = 5.5%, not 7.5%. When short-term rates approach the portfolio’s own return, leverage adds risk with little reward. Leverage also brings margin calls, the cost of rolling futures, and the danger of being forced to sell after a loss. Years when stocks and bonds fall together, such as 2022, are the hardest for the approach, because the large, levered bond position adds to losses instead of offsetting them.
What this calculator does not include
- Expected returns. Risk parity deliberately ignores them. Whether the portfolio is attractive depends on the returns of the assets, which the calculator does not forecast.
- Estimation error. Weights depend on volatilities and correlations estimated from the past. Volatility estimates are fairly stable; correlations are not.
- Tail risk. Equal contributions to volatility are not equal contributions to crash losses. Correlations tend to jump in a crisis, and volatility hides skew and fat tails.
- Costs. Financing, futures roll, trading and rebalancing costs are ignored, as is any constraint on leverage.
- Other risk budgets. The calculator gives every asset the same budget. Some investors prefer to budget risk by economic factor or to tilt the budgets toward assets they favor.