How does risk budgeting work using marginal VaR and component VaR?

Risk budgeting decomposes total portfolio risk into contributions from each position, enabling managers to allocate risk capital efficiently. The key tools are marginal VaR, component VaR, and incremental VaR.

Definitions:

Critical Property:

Sum of all Component VaRs = Total Portfolio VaR

This is Euler's theorem in action — the components perfectly decompose total risk.

Marginal VaR Formula:

MVaR_i = z_alpha x (Sigma x w)_i / sigma_p

This is the contribution of asset i to portfolio volatility, scaled by the confidence z-score.

Worked Example — Silveridge Asset Management:

Silveridge has a $50M portfolio:

• US Equities: 50% ($25M), vol = 18%
• Intl Equities: 30% ($15M), vol = 22%
• Fixed Income: 20% ($10M), vol = 5%

Correlations: US/Intl = 0.65, US/FI = -0.15, Intl/FI = 0.05

Step 1: Portfolio Volatility

After the matrix calculation: sigma_p = 12.84%

Step 2: Marginal VaR (per unit)

MVaR_US = z x [w_US x sigma_US^2 + w_Intl x rho_{US,Intl} x sigma_US x sigma_Intl + w_FI x rho_{US,FI} x sigma_US x sigma_FI] / sigma_p

For 99% confidence (z = 2.326):

• MVaR_US = 2.326 x 0.1578 = 0.367
• MVaR_Intl = 2.326 x 0.1712 = 0.398
• MVaR_FI = 2.326 x (-0.0006) = -0.001

Step 3: Component VaR ($)

• CVaR_US = 0.50 x 0.367 x $50M = $9.175M
• CVaR_Intl = 0.30 x 0.398 x $50M = $5.970M
• CVaR_FI = 0.20 x (-0.001) x $50M = -$0.010M

Total VaR = $9.175M + $5.970M - $0.010M = $15.135M

Insight: Fixed income has negative component VaR — it's actually reducing total portfolio risk. This means adding more fixed income would decrease VaR.

Risk Budgeting in Practice:

Silveridge allocates risk budgets to each desk:

• US Equity desk: budget = $10M VaR
• Intl Equity desk: budget = $6M VaR
• Fixed Income desk: budget = $1M VaR (hedging benefit counted separately)

If the US desk's component VaR exceeds $10M, they must reduce positions. If below, they have capacity to take more risk.

Optimal Portfolio: In a risk-budgeting framework, the optimal portfolio equalizes marginal VaR per unit of expected return across all positions. If MVaR_i / E(R_i) is higher for one asset, you're taking too much risk per unit of return there.

For more risk management frameworks, explore our FRM Part II course.