How do you calculate parametric VaR for a multi-asset portfolio using the correlation matrix?

Portfolio parametric VaR is one of the most testable topics in FRM Part II Market Risk. The key insight is that diversification reduces VaR when correlations are less than 1.

Single-Asset VaR:

VaR = z_alpha x sigma x Portfolio_Value

Portfolio VaR (Matrix Approach):

VaR_p = z_alpha x sqrt(w' x Sigma x w) x Portfolio_Value

Where:

• w = vector of portfolio weights
• Sigma = covariance matrix
• z_alpha = confidence level z-score (2.326 for 99%)

Worked Example — Beacon Capital:

Beacon holds a $10M portfolio:

• Asset A (Equities): 60% weight, daily vol = 1.5%
• Asset B (Bonds): 25% weight, daily vol = 0.6%
• Asset C (Commodities): 15% weight, daily vol = 2.0%

Correlation matrix:

Step 1: Individual VaRs (undiversified)

• VaR_A = 2.326 x 0.015 x $6M = $209,340
• VaR_B = 2.326 x 0.006 x $2.5M = $34,890
• VaR_C = 2.326 x 0.020 x $1.5M = $69,780
• Undiversified VaR = $314,010 (simple sum)

Step 2: Build the Covariance Matrix

Cov(i,j) = rho(i,j) x sigma_i x sigma_j

Step 3: Portfolio Variance

sigma_p^2 = w' x Sigma x w

= 0.60^2 x 0.000225 + 0.25^2 x 0.000036 + 0.15^2 x 0.000400

+ 2 x 0.60 x 0.25 x (-0.000018)

+ 2 x 0.60 x 0.15 x 0.000105

+ 2 x 0.25 x 0.15 x 0.000012

= 0.0000810 + 0.0000023 + 0.0000090

+ (-0.0000054) + 0.0000189 + 0.0000009

= 0.0001067

sigma_p = sqrt(0.0001067) = 1.033%

Step 4: Diversified VaR

VaR_p = 2.326 x 0.01033 x $10M = $240,278

Diversification Benefit:

$314,010 - $240,278 = $73,732 (23.5% reduction)

The negative correlation between equities and bonds (-0.20) is the primary driver of this benefit.

Key Exam Points:

• Undiversified VaR is always >= Diversified VaR
• They're equal only when all correlations = 1
• Negative correlations provide the most diversification benefit
• The square root in the formula is why VaR doesn't scale linearly with portfolio size

Practice more VaR calculations in our FRM Part II question bank.