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Given the move in rates and the price change that results, define a fixed-income security's DV01 and compute it.

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

The idea

DV01 is the dollar value of one basis point: the change in a security's price when its yield moves by 0.01 percent. It is quoted as a positive number for ordinary bonds. To compute it from a scenario, divide the absolute price change by the yield change measured in basis points. DV01 = price change ÷ yield change in bp. A move of 12bp that lowers a position by USD 4,740 gives a DV01 of USD 395. Check the units. A yield change quoted in percent must be multiplied by 100 to get basis points, and a price quoted per 100 of face must be scaled by the face amount to get the position DV01. Averaging an up shift and a down shift is more accurate, because the price-yield curve bends. DV01 is a dollar figure, so it depends on position size.

Formulas

DV01 from a single rate scenario

DV01 = |ΔP| ÷ Δy

ΔP
the price change of the position or per 100 of face
Δy
the yield change in basis points

Convert a yield change in percent to basis points first. Scale a per 100 of face figure by the face amount for the position.

DV01 from symmetric up and down shifts

DV01 = (Pdown − Pup) ÷ (2 × Δy)

Pdown
the price after yields shift down
Pup
the price after yields shift up by the same amount
Δy
the size of each shift in basis points

More accurate than one shift because it averages out most of the curvature. Forgetting the 2 doubles the answer.

A practice question

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

A bond's price per 100 of face value falls from 102.40 to 101.95 when its yield rises by 5bp. The DV01 per 100 of face value is:

  1. 0.090 - correct
  2. 0.450Why not: Uses the whole price change and does not divide by the 5bp yield change.
  3. 2.250Why not: Multiplies the price change by the yield change in bp instead of dividing.
  4. 0.0045Why not: Divides the 0.45 price change by 100 instead of by the 5bp yield change.

DV01 = price change / yield change in bp = (102.40 - 101.95) / 5 = 0.45 / 5 = 0.090 per 100 of face.

A harder one

A holder of USD 50 million face of a bond sees its price per 100 of face rise from 98.50 to 98.94 when the yield falls by 10bp. The DV01 of the position is closest to:

  1. USD 220,000Why not: Uses the whole 10bp price change (0.44 per 100) instead of the per-basis-point change of 0.044.
  2. USD 22,000 - correct
  3. USD 2,200,000Why not: Applies the 0.044 per bp to the face amount in millions without dividing the face by 100.
  4. USD 2,200Why not: Uses 5 million face instead of 50 million when scaling the per-100 DV01.

DV01 per 100 = 0.44 / 10 = 0.044. Position DV01 = 0.044 x 50,000,000 / 100 = USD 22,000 per basis point.

The hardest tier

A dealer is short USD 20 million face of a bond. When yields rise 25bp the price per 100 of face falls from 96.00 to 94.85. The change in value of the dealer's position for a 1bp rise in yield is closest to:

  1. A loss of USD 9,200Why not: Sign error: a short position gains when yields rise and the price falls.
  2. A gain of USD 230,000Why not: Uses the whole 25bp price change (1.15 per 100) instead of dividing by 25 to get a per-basis-point figure.
  3. A gain of USD 9,200 - correct
  4. A gain of USD 920,000Why not: Divides by 25bp but does not divide by 100 when scaling the per-100 figure to the 20 million face amount.

Price change per bp = 1.15 / 25 = 0.046 per 100. On 20,000,000 face this is 0.046 x 200,000 = USD 9,200. The short position gains that amount for each 1bp rise.

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