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Explain what effective duration is, interpret it, and calculate it for a fixed-income security from a given change in rates and the price change that results.

FRM® Part I · Valuation and Risk Models · Applying Duration, Convexity, and DV01 · FRM1.VR.12.d

The idea

Effective duration is the percentage change in a security's price per unit change in yield, so a rise of 1 percentage point in yield moves the price by about the duration in percent, and it is measured by repricing. Shift the yield down and up by the same amount and take the two prices. Duration = (Pdown − Pup) ÷ (2 × P0 × Δy). Here Δy is the yield change as a decimal, such as 0.0025 for 25bp, and P0 is the starting price. A duration of 6.5 means the price falls about 6.5 percent for a rise of 1 percentage point in yield. Full repricing makes it usable for bonds with embedded options. With only one shift, Duration = (price change ÷ P0) ÷ Δy, signed to be positive. Common errors are forgetting the factor of 2 and using basis points instead of decimals.

Formulas

Effective duration from up and down yield shifts

D = (Pdown − Pup) ÷ (2 × P0 × Δy)

Pdown
the price after yields shift down
Pup
the price after yields shift up by the same amount
P0
the starting price
Δy
the size of each shift, as a decimal

Use a decimal yield change, 0.0025 for 25bp. Forgetting the 2 doubles the answer.

Effective duration from a single yield shift

D = −(ΔP ÷ P0) ÷ Δy

ΔP
the price change
P0
the starting price
Δy
the yield change, as a decimal

Signed to come out positive. Less accurate than the two-sided version because it ignores curvature.

A practice question

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

A bond has an effective duration of 6.5. If its yield rises by 1 percentage point, the bond's price will fall by approximately:

  1. 0.65 percentWhy not: Treats the 1 percentage point change as 10bp when scaling duration.
  2. 6.5 percent - correct
  3. 65 percentWhy not: Applies duration to a 1 percentage point move as if it were 10 percentage points.
  4. 0.065 percentWhy not: Treats the 1 percentage point change as 1bp when scaling duration.

Effective duration is the percentage price change per unit (1.00 or 100 percent) change in yield, so a 1 percentage point rise gives about 6.5 x 1% = 6.5%.

A harder one

A bond has a price of 99.50. Repricing with yields 50bp lower gives 103.35, and with yields 50bp higher gives 95.90. The effective duration is closest to:

  1. 14.97Why not: Omits the factor of 2 in the denominator, treating the price range as a one-sided change.
  2. 7.74Why not: Uses only the yield-down repricing, which overstates duration because of positive convexity.
  3. 7.49 - correct
  4. 0.0749Why not: Uses the 0.5 percent yield change as 0.5 instead of 0.005, so the result is off by the unit conversion.

Duration = (103.35 - 95.90) / (2 x 99.50 x 0.005) = 7.45 / 0.995 = 7.49. Both shifts are used so that curvature largely cancels.

The hardest tier

A callable bond is priced at 100.00. A 100bp fall in yields takes the price to 103.10, and a 100bp rise takes it to 93.60. The effective duration is closest to:

  1. 9.50Why not: Omits the factor of 2 in the denominator.
  2. 6.40Why not: Uses only the yield-up repricing, ignoring the capped upside from the call.
  3. 3.10Why not: Uses only the yield-down repricing, which understates duration because the call limits the price gain.
  4. 4.75 - correct

Duration = (103.10 - 93.60) / (2 x 100 x 0.01) = 9.50 / 2.00 = 4.75. The two-sided estimate averages the capped upside and the larger downside.

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