How does cash flow mapping work for VaR calculations on fixed income positions?

Cash flow mapping is the process of converting complex positions into exposures at standard maturity points (vertices) so that VaR can be computed using a manageable covariance matrix. Without mapping, you'd need volatilities and correlations for every possible maturity — which is impractical.

The Process

Step 1: Identify cash flows

For a coupon bond, list every cash flow with its timing.

Step 2: Map to vertices

Standard vertices might be: 1M, 3M, 6M, 1Y, 2Y, 5Y, 10Y, 30Y. Each cash flow gets split between the two nearest vertices.

Step 3: Splitting rules

The split preserves three properties:

1. Market value — the PV of the mapped cash flows equals the PV of the original
2. Duration — the duration contribution is preserved
3. Variance — the variance contribution is preserved (uses the correlation between the two vertices)

Worked Example

Falconridge Securities holds a bond with a $500,000 cash flow due in 3.5 years. The nearby vertices are 2Y and 5Y.

Market data:

• 2Y zero rate: 4.2%, volatility: 0.65%
• 5Y zero rate: 4.8%, volatility: 0.90%
• Correlation between 2Y and 5Y: 0.92
• 3.5Y zero rate (interpolated): 4.5%

PV of the cash flow:

PV = 500,000 / (1.045)^3.5 = 500,000 / 1.1647 = $429,322

Duration-preserving split (linear interpolation in time):

• Weight on 2Y: (5 - 3.5) / (5 - 2) = 1.5 / 3 = 0.50
• Weight on 5Y: (3.5 - 2) / (5 - 2) = 1.5 / 3 = 0.50

Mapped amounts:

• 2Y vertex: 0.50 x $429,322 = $214,661
• 5Y vertex: 0.50 x $429,322 = $214,661

Step 4: Compute mapped VaR

Now each vertex has a known volatility and the correlation matrix is defined only at the vertices:

sigma_mapped^2 = w_2^2 x sigma_2^2 + w_5^2 x sigma_5^2 + 2 x w_2 x w_5 x rho_{2,5} x sigma_2 x sigma_5

This is a standard 2-asset portfolio variance calculation, which then feeds into the parametric VaR formula.

Key Exam Points

• Mapping reduces the dimensionality of the VaR problem
• More vertices = better accuracy but larger covariance matrix
• The RiskMetrics approach popularized this technique
• Cash flow mapping works for bonds, swaps, FRAs, and any instrument with deterministic cash flows

Explore more VaR methodology in our FRM course.