Risk budgeting: contribution to risk versus capital weights

What you will learn

• Why a 60/40 portfolio is often a 90/10 portfolio when measured in risk
• How to calculate marginal and total risk contributions for three assets
• What the risk-parity idea does to weights, volatility and expected return
• Practical rules of thumb for setting and reviewing a risk budget in euros and percent

Explanation

A capital weight is the share of money placed in an asset. A risk contribution is the share of total portfolio volatility that the asset is responsible for. The two are usually very different, because volatile assets dominate the risk of a mix even when they hold a modest share of the capital. Risk budgeting means deciding the risk contributions first and deriving the capital weights from them.

The two-asset volatility formula and correlation were covered in the course ETFs, Funds & Long-Term Investing (lesson on diversification). Here we go one step further. For any number of assets, portfolio variance is the sum over all pairs of weight × weight × covariance, where covariance = correlation × volatility of A × volatility of B. Portfolio volatility is the square root of the variance.

The marginal contribution to risk (MCR) of asset i is the change in portfolio volatility when its weight rises by a tiny amount: MCR(i) = (covariance of asset i with the whole portfolio) ÷ portfolio volatility. The total risk contribution is RC(i) = weight(i) × MCR(i). A useful property: all RC(i) add up exactly to the portfolio volatility, so each one can be expressed as a percentage of total risk.

Risk parity is the special case where every asset is given the same risk contribution. In practice this moves capital out of equities and into bonds. Because a bond-heavy mix has a low expected return, professional risk-parity funds usually add leverage to lift the volatility back to a target; leverage is treated in a later lesson of this course and can lose money quickly. For a private investor the unlevered version is still a useful benchmark: it shows how far your actual portfolio sits from an equal-risk mix.

Worked example

Hypothetical example: a €100,000 portfolio with 60% global equities, 30% euro government bonds and 10% gold. Assumed annual volatilities: equities 16%, bonds 6%, gold 15%. Assumed correlations: equities–bonds 0.0, equities–gold 0.1, bonds–gold 0.2. Fees, spreads and taxes are excluded; all numbers are assumptions, not forecasts.

Step 1 – covariances. Equities–gold: 0.1 × 0.16 × 0.15 = 0.0024. Bonds–gold: 0.2 × 0.06 × 0.15 = 0.0018. Equities–bonds: 0.

Step 2 – variance. 0.6² × 0.16² + 0.3² × 0.06² + 0.1² × 0.15² + 2 × 0.6 × 0.1 × 0.0024 + 2 × 0.3 × 0.1 × 0.0018 = 0.009216 + 0.000324 + 0.000225 + 0.000288 + 0.000108 = 0.010161. Volatility = √0.010161 ≈ 10.08%.

Step 3 – marginal contributions. Covariance of equities with the portfolio = 0.6 × 0.0256 + 0.3 × 0 + 0.1 × 0.0024 = 0.0156; MCR = 0.0156 ÷ 0.1008 ≈ 0.1548. Bonds: 0.3 × 0.0036 + 0.1 × 0.0018 = 0.00126; MCR ≈ 0.0125. Gold: 0.6 × 0.0024 + 0.3 × 0.0018 + 0.1 × 0.0225 = 0.00423; MCR ≈ 0.0420.

In euros: the portfolio's one-standard-deviation yearly move is about €10,080, and roughly €9,290 of it comes from the €60,000 in equities. The 30% bond block barely moves the needle.

Step 4 – risk parity. A quick first guess is inverse-volatility weighting: weights proportional to 1/0.16, 1/0.06 and 1/0.15, which gives about 21% / 56% / 23%. Solving exactly for equal risk contributions (an iterative calculation) gives about 22% equities, 56% bonds and 21% gold, with a portfolio volatility of about 6.4%. Each asset then contributes one third of that 6.4%.

Common mistake

Practice activity

On paper, take a two-asset portfolio of 70% equities (volatility 18%) and 30% bonds (volatility 5%) with correlation 0.0. Calculate (a) the portfolio volatility, (b) the risk contribution of each asset as a percentage of the total, and (c) the equity weight at which both assets contribute equal risk.

Key takeaways

• Capital weights tell you where the money is; risk contributions tell you where the losses will come from.
• Risk contributions add up to total volatility, so they can be expressed as a percentage budget.
• A classic 60/30/10 mix is roughly 90% equity risk; unlevered risk parity cuts volatility sharply but also expected return.
• Write your target risk shares into your written plan and recheck them once a year with updated volatilities and correlations.

Next step

The risk numbers above depend on correlations, and correlations are not stable: the next lesson looks at correlation regimes and what happens when the stock–bond relationship flips.