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Tell apart the trio of techniques available for mapping bond portfolios.

FRM® Part II · Market Risk Measurement and Management · VaR Mapping · FRM2.MR.5.c

The idea

There are three standard ways to map a bond to zero-coupon risk factors, and they differ in how finely they treat its cash flows. Principal mapping places the bond's whole market value at the vertex matching its final maturity. It is the simplest, but it overstates the risk of a coupon bond because coupons make its effective maturity shorter. Duration mapping places the whole market value at the vertex equal to the bond's duration, interpolating between vertices if needed. It captures the coupon effect in one number but treats the bond as sensitive only to a parallel shift. Cash-flow mapping discounts every coupon and principal payment at its own spot rate and allocates each present value to the neighbouring vertices, preserving value and risk. It is the most accurate and needs the most data. Exam point: accuracy ranks cash-flow mapping first, then duration, then principal.

Formulas

Macaulay duration used for duration mapping

D = Σ(t × PVₜ) ÷ P

t
the time of each payment, in years
PVₜ
the present value of the payment at time t
P
the bond price, the sum of the PVs

The vertex is set by Macaulay duration in years, not modified duration. It is shorter than maturity for a coupon bond.

A practice question

From the question bank, with the reason each wrong option is wrong.

Under principal mapping, a coupon bond is represented by:

  1. its full market value placed at the vertex equal to the bond's duration.Why not: Describes duration mapping, not principal mapping.
  2. each coupon and principal payment allocated to the vertices nearest its own payment date.Why not: Describes cash-flow mapping, which treats every payment separately.
  3. its full market value placed at the vertex closest to the first coupon date.Why not: The first coupon date has no role; principal mapping is anchored on the final maturity.
  4. its full market value placed at the vertex corresponding to its final maturity. - correct

Principal mapping is the crudest of the three techniques: the whole bond is treated as a zero-coupon bond maturing when the bond does. For a coupon bond this overstates risk, since duration is shorter than maturity.

A harder one

A 3-year bond pays an 8 percent annual coupon and yields 6 percent with annual compounding. Under duration mapping, the bond is mapped to the vertex whose maturity is closest to:

  1. 2.79 years - correct
  2. 3.00 yearsWhy not: Uses the bond's maturity, which is principal mapping rather than duration mapping.
  3. 2.63 yearsWhy not: Uses modified duration instead of Macaulay duration; the vertex is set by Macaulay duration in years.
  4. 2.00 yearsWhy not: Takes the simple average of the payment dates (1, 2 and 3) instead of weighting by present value.

Price = 105.35. Macaulay duration = (1 x 8/1.06 + 2 x 8/1.06² + 3 x 108/1.06³)/105.35 = 2.79 years. Modified duration (2.63) is a sensitivity measure, not a maturity, so it is not the vertex.

The hardest tier

A 5-year bond with a 6 percent annual coupon is priced at par, so its yield is 6 percent. Assume the 95 percent VaR of a zero-coupon position is 0.35 percent of value for each year of maturity. For a USD 10 million holding, principal mapping gives a VaR higher than duration mapping (using Macaulay duration) by an amount closest to:

  1. USD 18,700 - correct
  2. USD 27,600Why not: Uses modified duration (4.21) instead of Macaulay duration (4.47) as the mapping horizon.
  3. USD 70,000Why not: Takes duration as the simple average payment date of 3 years instead of the present-value-weighted 4.47.
  4. USD 156,300Why not: Reports the duration-mapped VaR itself rather than the difference from principal mapping.

Macaulay duration of the par 6 percent 5-year bond is 4.465 years. Principal-mapped VaR = 10m x 0.35% x 5 = 175,000; duration-mapped VaR = 10m x 0.35% x 4.465 = 156,300. The difference is 18,700.

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