FRM Exam Part II · VaR Mapping

Mapping Fixed-Income Portfolios for VaR: Principal, Duration and Cash-Flow

Q: What is the difference between principal, duration and cash-flow mapping?

Principal mapping places the bond at its maturity vertex using face value. Duration mapping places the bond's present value at the vertex matching its duration. Cash-flow mapping splits each cash flow's present value between two adjacent vertices, so it is the most accurate.

Q: Why must cash-flow mapping preserve both value and risk?

Preserving value keeps the market value of the position unchanged. Preserving risk keeps the variance the same, so VaR is not distorted by the mapping. Together they make the vertex positions a faithful stand-in for the original cash flow.

Q: How do I find the weight α in cash-flow mapping?

Interpolate the price volatility between the two vertex price volatilities. Then solve the variance equation using the vertex volatilities and correlation for α on the shorter vertex. Choose the root between 0 and 1.

Q: Does principal mapping overstate or understate VaR?

For a coupon bond it typically overstates VaR when vertex volatility increases with maturity. It treats all value as arriving at maturity, which carries more rate risk than the true earlier-weighted cash flows. This is not universal, because it depends on the shape of the volatility term structure.

Updated 11 October 2026 · Fact-checked

Mapping a bond portfolio means replacing each bond with positions in a few standard vertices (maturities) whose risk you measure. Principal mapping uses the maturity vertex only. Duration mapping uses the vertex matching duration. Cash-flow mapping splits every cash flow between two adjacent vertices, preserving present value and risk.

Understand Mapping Fixed-Income Portfolios

A bond portfolio has hundreds of cash flows at many dates. You cannot estimate volatilities and correlations for every date. So risk systems pick a small set of vertices, such as 1, 3, 5 and 10 years, and measure risk only for zero-coupon rates at those points. Mapping moves each bond's exposure onto those vertices.

There are three levels of detail. Principal mapping treats the bond as one zero-coupon payment of its face value at maturity, so it maps the bond to the vertex nearest its maturity. It ignores coupons. For a coupon bond it typically overstates risk when vertex volatility rises with maturity, because coupons arrive earlier. Duration mapping treats the bond as a zero with maturity equal to its duration, and puts the bond's present value at the vertex matching that duration. It is better, but still one point for a whole bond.

Cash-flow mapping is the most accurate. You discount each cash flow to present value using the zero rate for its date. If the date falls between two vertices, you split its present value between the two neighbouring vertices. The split is chosen so that the total value and the total risk (variance) are unchanged.

The risk-preserving split works like this. You interpolate the price volatility between the two vertex price volatilities. Then you solve for the weight α on the shorter vertex so that the variance of the two-vertex portfolio equals the variance of the original cash flow. This needs the correlation between the two vertices. The result is a pair of vertex positions that you feed into the usual VaR formula.

Key formulas to remember

Present value of a cash flow: PV = CF ÷ (1 + y)^t; Use the zero (spot) rate y for that date t, with the compounding convention in the question.
Value-preserving split: PV = V_a + V_b, with V_a = α × PV and V_b = (1 − α) × PV; V_a sits at the shorter vertex and V_b at the longer vertex.
Interpolated volatility: σ_cf = σ_a + (t − t_a) ÷ (t_b − t_a) × (σ_b − σ_a); Linear interpolation of vertex price volatilities, where t_a < t < t_b.
Risk-preserving condition: σ_cf² = α² σ_a² + (1 − α)² σ_b² + 2 α (1 − α) ρ σ_a σ_b; Solve this quadratic for α. Take the root between 0 and 1.
Duration mapping vertex: Map the bond's PV to the vertex at its (Macaulay) duration; If duration lies between vertices, split between the two, or use the nearest one, as the question states.
Portfolio VaR from vertex positions: VaR = z × √(x′ Σ x) , with x = vector of vertex PVs; Σ is the covariance matrix of vertex price changes. Individual vertex VaR = z × σ × PV.

How to solve Mapping Fixed-Income Portfolios questions

Use this order for any bond mapping question. It works for single bonds and portfolios.

1List the cash flows and their dates. Note the vertices given and the VaR confidence level and horizon.
2Identify the method asked: principal, duration or cash-flow mapping.
3Compute present values with the zero rate for each date. For principal mapping use only the face value. For duration mapping use the bond's total PV.
4Decide the vertex or vertices. Principal: maturity vertex. Duration: vertex at duration. Cash-flow: the two vertices that bracket each date.
5For cash-flow mapping, interpolate volatility, then solve the variance-matching equation for α. Check that 0 ≤ α ≤ 1.
6Allocate PV: α to the shorter vertex, (1 − α) to the longer. Add up positions at each vertex across all cash flows.
7Compute VaR with the vertex volatilities and correlations: z × √(x′Σx). Check that total mapped PV equals the bond's PV.
8Interpret: principal mapping typically overstates VaR for coupon bonds when vertex volatility rises with maturity; cash-flow mapping is the most accurate.

Quickest way: Shortcut for exam time pressure

When to use it: Use when the question gives you α, or when only the method choice or direction of error is tested.

If the question asks which method is conservative, answer principal mapping for a coupon bond. It uses the maturity, the longest point, which typically has the highest volatility when volatility rises with maturity.
For duration mapping, put the whole PV at the duration point and compute VaR = z × σ × PV directly, using the interpolated volatility if the duration is not on a vertex.
For cash-flow mapping with α given, split PV and check the totals match before doing anything else.
If the cash flow lies exactly on a vertex, no split is needed: α = 1 or 0.
With perfect correlation (ρ = 1), σ_cf = α σ_a + (1 − α) σ_b, so α is linear and easy to solve.

Common mistakes in Mapping Fixed-Income Portfolios

Mapping a coupon bond to its maturity vertex and calling it exact.
Principal mapping is simple, so students apply it by default.
Fix: Remember it ignores coupons. It typically overstates the bond's risk when vertex volatility rises with maturity, because the true average timing is shorter than maturity.
Using the bond's maturity instead of its duration in duration mapping.
Both are called 'time' to the bond.
Fix: Duration is the weighted average time of cash flows. Use it, not maturity, to pick the vertex.
Splitting cash flows by time distance only and not by variance.
Linear weights feel natural.
Fix: The weight α must preserve risk. Solve the variance equation, not a simple time-proportion rule.
Allocating face value or coupon amounts rather than present values.
Students forget that VaR needs current market value exposure.
Fix: Always discount first. Mapped amounts are PVs, and they must sum to the bond's PV.
Picking the wrong quadratic root for α.
The equation gives two roots.
Fix: Keep the root between 0 and 1. A weight outside that range does not make sense for a split.
Adding vertex VaRs together as the portfolio VaR.
It skips the correlation step.
Fix: Combine vertex positions using the covariance matrix. Simple addition holds only when ρ = 1.

Worked examples

Example 1

A bank holds a cash flow with present value $1,000,000 due in 4 years. Vertices are at 3 years and 5 years, with price volatilities of 1.0% and 1.6% (monthly). Assume correlation between the vertices is 1. Find the cash-flow map.

Show the solution

Interpolate volatility at 4 years: 1.0% + (4 − 3) ÷ (5 − 3) × (1.6% − 1.0%) = 1.0% + 0.5 × 0.6% = 1.3%.
With ρ = 1, the risk condition is σ_cf = α σ_a + (1 − α) σ_b.
So 1.3 = α × 1.0 + (1 − α) × 1.6 = 1.6 − 0.6α.
Solve: 0.6α = 0.3, so α = 0.5.
Allocate: 3-year vertex = 0.5 × $1,000,000 = $500,000; 5-year vertex = $500,000.
Check: $500,000 + $500,000 = $1,000,000, so value is preserved.

Answer: Map $500,000 to the 3-year vertex and $500,000 to the 5-year vertex (α = 0.5).

Example 2

A bond maturing in 2 years has a present value of $2,000,000 and a duration of 1.8 years. The 1-year vertex has monthly price volatility of 0.8% and the 2-year vertex has 1.2%. Use duration mapping at the 1.8-year point, with price volatility interpolated linearly between the vertices, and a 95% confidence level (z = 1.645). Find the monthly VaR. Then compare it with principal mapping to the 2-year vertex.

Show the solution

Duration mapping puts the full PV at the 1.8-year point, so position = $2,000,000.
Interpolate volatility at 1.8 years: 0.8% + (1.8 − 1) ÷ (2 − 1) × (1.2% − 0.8%) = 0.8% + 0.8 × 0.4% = 1.12%.
Duration-mapped VaR = z × σ × PV = 1.645 × 0.0112 × 2,000,000.
1.645 × 0.0112 = 0.018424.
0.018424 × 2,000,000 = 36,848.
Principal mapping uses the maturity (2 years) and its volatility of 1.2%: 1.645 × 0.012 × 2,000,000 = 0.01974 × 2,000,000 = 39,480.
Difference: 39,480 − 36,848 = 2,632.

Answer: Duration-mapped monthly VaR = $36,848. Principal mapping to the 2-year vertex gives $39,480, which is $2,632 higher. Here volatility rises with maturity, so principal mapping overstates the risk.

Exam tips

Know the ranking: principal mapping is least accurate, cash-flow mapping is most accurate, and duration mapping sits between.
Check that mapped PVs add up to the bond PV. It catches most arithmetic slips quickly.
Questions often give ρ = 1 to simplify α. Look for it before building the quadratic.
Read whether the volatility given is for price or yield. Mapping uses price volatility.
Be ready to explain why value and risk are both preserved, not just value.

Practice questions from VaR Mapping

Mapping Fixed-Income Portfolios: frequently asked questions

What is the difference between principal, duration and cash-flow mapping?

Principal mapping places the bond at its maturity vertex using face value. Duration mapping places the bond's present value at the vertex matching its duration. Cash-flow mapping splits each cash flow's present value between two adjacent vertices, so it is the most accurate.

Why must cash-flow mapping preserve both value and risk?

Preserving value keeps the market value of the position unchanged. Preserving risk keeps the variance the same, so VaR is not distorted by the mapping. Together they make the vertex positions a faithful stand-in for the original cash flow.

How do I find the weight α in cash-flow mapping?

Interpolate the price volatility between the two vertex price volatilities. Then solve the variance equation using the vertex volatilities and correlation for α on the shorter vertex. Choose the root between 0 and 1.

Does principal mapping overstate or understate VaR?

For a coupon bond it typically overstates VaR when vertex volatility increases with maturity. It treats all value as arriving at maturity, which carries more rate risk than the true earlier-weighted cash flows. This is not universal, because it depends on the shape of the volatility term structure.