FRM Exam Part II · Portfolio Construction

Risk Budgeting and Risk Parity for FRM Part II

Updated 11 October 2026 · Fact-checked

Risk budgeting allocates a portfolio's total risk, not just capital, across assets or managers. Marginal contribution to risk (MCR) = (Σw)ᵢ ÷ σp. Component contribution = wᵢ × MCR, and components sum to σp. Risk parity sets all component contributions equal. To solve: compute σp, MCR, components, then percentages.

Understand Risk Budgeting and Risk Parity

Capital allocation says how much money goes to each asset. Risk budgeting asks a different question: how much of the portfolio's total risk comes from each asset or manager? A 60/40 equity-bond portfolio looks balanced in capital, but equities often supply well over 80% of the volatility.

To split risk, you need a measure that adds up. Volatility does not add up across assets, because correlation matters. The fix is Euler decomposition. Portfolio volatility is homogeneous of degree one in the weights, so it can be written as the sum of each weight times its marginal contribution to risk (MCR). MCR is the change in portfolio volatility from a small increase in that asset's weight. Each weight times its MCR is the component contribution to risk, and the components sum exactly to portfolio risk.

The same logic applies to VaR. For a normal (parametric) portfolio, component VaR = wᵢ × marginal VaR, and component VaRs sum to total VaR. This differs from incremental VaR, which is the actual change in VaR when you add or remove a whole position. Incremental VaR is only approximately equal to component VaR, and only for small positions.

A risk budget is a target for each component's share of total risk. Risk parity is the special case where every asset contributes an equal share, so wᵢ × MCRᵢ is the same for all i. It is not the same as equal weight. Equal weight gives equal capital. Risk parity gives equal risk, so low-volatility assets such as bonds get larger weights. With zero correlations, risk parity weights are proportional to 1/σᵢ. Risk parity portfolios often need leverage to reach a target return, which adds funding and liquidity risk.

Key formulas to remember

Portfolio volatility: σp = √(wᵀ Σ w); w is the weight vector and Σ is the covariance matrix. For two assets: σp² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂.
Marginal contribution to risk (MCR): MCRᵢ = ∂σp/∂wᵢ = (Σw)ᵢ ÷ σp = Cov(Rᵢ, Rp) ÷ σp; Equivalent form: βᵢ × σp, where βᵢ is asset i's beta to the portfolio.
Component contribution to risk (CCR): CCRᵢ = wᵢ × MCRᵢ, and Σ CCRᵢ = σp; Percentage contribution = CCRᵢ ÷ σp = wᵢ × βᵢ. Percentages sum to 100%.
Marginal VaR (normal): Marginal VaRᵢ = z × MCRᵢ; z is the confidence-level quantile, e.g. 1.645 at 95% and 2.326 at 99%. Use the same time horizon as σ.
Component VaR: Component VaRᵢ = wᵢ × Marginal VaRᵢ (if weights, in currency: positionᵢ × marginal VaR per unit); Component VaRs sum to total portfolio VaR. Valid under the normal, linear model.
Risk parity condition: wᵢ × (Σw)ᵢ = wⱼ × (Σw)ⱼ for all i, j; Equal risk contribution. With zero correlations, wᵢ ∝ 1/σᵢ.
Risk budget check: Optimal when (excess returnᵢ ÷ MCRᵢ) is equal across assets; Return per unit of marginal risk should be equal. Otherwise shift risk toward the higher ratio.

How to solve Risk Budgeting and Risk Parity questions

Use this order for any risk budgeting or risk parity question. Stay in one unit (percent or currency) and one horizon throughout.

1Write down the weights (or position sizes), volatilities and correlations or covariances.
2Compute portfolio variance wᵀΣw, then take the square root to get σp.
3Compute each asset's covariance with the portfolio, (Σw)ᵢ. This is Σⱼ wⱼ Cov(i, j).
4Divide by σp to get MCRᵢ. If the question asks for VaR, multiply by z to get marginal VaR.
5Multiply by the weight to get the component contribution. Check that components sum to σp (or total VaR).
6Divide each component by the total to get percentage risk contributions.
7Interpret: compare risk share with capital share, or compare with the risk budget or equal-risk target. For risk parity, adjust weights toward low-risk-share assets.
8Check any return-to-marginal-risk condition if the question asks whether allocation is optimal.

Quickest way: Beta shortcut for percentage risk contribution

When to use it: Use when the question gives betas to the portfolio or when you only need risk shares, not the absolute level.

Percentage risk share of asset i = wᵢ × Cov(i, p) ÷ σp².
Compute Cov(i, p) = Σⱼ wⱼ Cov(i, j) for each asset.
Multiply by wᵢ and divide by σp². The shares must sum to 100%, which is your built-in check.
For two uncorrelated assets, share of i = wᵢ²σᵢ² ÷ σp².
For risk parity with zero correlation, set wᵢ = (1/σᵢ) ÷ Σ(1/σⱼ). Skip the optimisation.

Common mistakes in Risk Budgeting and Risk Parity

Treating equal weight as equal risk, or equating risk parity with equal weight.
Candidates focus on capital shares and forget volatility and correlation differ across assets.
Fix: Risk parity equalises wᵢ × MCRᵢ. Equal weight only equals risk parity if all assets have the same volatility and the same pairwise correlations.
Using wᵢ ÷ σᵢ-style weights (1/σ) when correlations are not zero.
The inverse-volatility shortcut is memorised without its condition.
Fix: State that inverse volatility is exact only with zero correlations (or equal correlations across all pairs). Otherwise solve using the covariance matrix.
Adding standalone volatilities or VaRs to get portfolio risk.
Candidates forget diversification benefit.
Fix: Add component contributions, not standalone risks. Standalone VaRs sum to more than portfolio VaR unless correlation is 1.
Confusing component VaR with incremental VaR.
Both measure an asset's role in portfolio VaR and the names sound alike.
Fix: Component VaR is a marginal-based decomposition that sums to total VaR. Incremental VaR is the actual change from adding or removing a position and does not sum to total.
Forgetting to multiply by the weight when moving from MCR to component risk.
MCR looks like a final answer.
Fix: MCR is per unit of weight. Always compute CCR = w × MCR and check the sum equals σp.
Ignoring leverage when judging risk parity.
Weights are viewed as summing to 100%.
Fix: Low-volatility assets get big weights, so reaching a return target may need leverage. Mention funding, margin and liquidity risk in interpretation.

Worked examples

Example 1

A portfolio holds 60% in equities (σ = 20%) and 40% in bonds (σ = 5%). The correlation is 0.2. Find the percentage of portfolio risk from equities.

Show the solution

Covariances: Cov(E,E) = 0.04; Cov(B,B) = 0.0025; Cov(E,B) = 0.2 × 0.20 × 0.05 = 0.002.
Variance: 0.6² × 0.04 + 0.4² × 0.0025 + 2 × 0.6 × 0.4 × 0.002 = 0.0144 + 0.0004 + 0.00096 = 0.01576.
σp = √0.01576 = 0.12554, about 12.55%.
Cov(E, p) = 0.6 × 0.04 + 0.4 × 0.002 = 0.024 + 0.0008 = 0.0248.
MCR_E = 0.0248 ÷ 0.12554 = 0.1975. CCR_E = 0.6 × 0.1975 = 0.1185, i.e. 11.85%.
Share = 0.1185 ÷ 0.12554 = 94.4%. Check via variance: 0.6 × 0.0248 = 0.01488; 0.01488 ÷ 0.01576 = 94.4%.

Answer: Equities contribute about 94.4% of portfolio risk (component 11.85% of the 12.55% volatility), despite holding 60% of capital.

Example 2

A $200 million portfolio has a 1-day 99% normal VaR of $4.65 million. Position A is $120 million with marginal VaR of 0.0300 per dollar invested, and position B is $80 million. Find component VaR of A and of B, and the percentage for each.

Show the solution

Component VaR of A = 120 × 0.0300 = $3.60 million.
Components must sum to total VaR of $4.65 million, so component VaR of B = 4.65 − 3.60 = $1.05 million.
Implied marginal VaR of B = 1.05 ÷ 80 = 0.013125 per dollar.
Percentage of A = 3.60 ÷ 4.65 = 77.4%. Percentage of B = 1.05 ÷ 4.65 = 22.6%.
Compare with capital shares: A is 60% of capital but 77.4% of VaR.

Answer: Component VaR is $3.60 million for A (77.4%) and $1.05 million for B (22.6%). A is over-represented in risk relative to its capital share.

Exam tips

Always check that your components sum to total risk. It is the fastest way to catch arithmetic slips and eliminate wrong options.
Read whether the question wants risk in volatility terms, VaR terms or percentage terms. Wrong units are a common trap.
Questions often ask for interpretation: which asset to cut, or whether the allocation matches the budget. Compare risk share to capital share and to the stated target.
Know the zero-correlation shortcut for risk parity (weights ∝ 1/σ) and when it does not apply.
Be ready to separate component, marginal and incremental VaR definitions in conceptual questions.

Practice questions from Portfolio Construction

Risk Budgeting and Risk Parity: frequently asked questions

What is the difference between risk parity and equal weight?

Equal weight gives each asset the same capital share. Risk parity gives each asset the same contribution to portfolio risk. Because bonds are less volatile than equities, risk parity usually holds much more in bonds and may use leverage.

How do you calculate component VaR?

Compute marginal VaR for each position (z × the asset's covariance with the portfolio ÷ σp), then multiply by the position size. Component VaRs add up to total portfolio VaR under the normal linear model.

What is the marginal contribution to risk formula?

MCRᵢ = (Σw)ᵢ ÷ σp, which is the covariance of asset i with the portfolio divided by portfolio volatility. It equals the asset's beta to the portfolio times σp.

What is risk budgeting in practice?

It sets limits on how much of total risk each asset class, strategy or manager may use, then monitors actual risk contributions against those limits. Optimal budgets tilt toward assets with higher excess return per unit of marginal risk.