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Convert a KR01 or a forward bucket 01 into the duration measure that corresponds to it.

FRM® Part I · Valuation and Risk Models · Modeling Non-Parallel Term Structure Shifts and Hedging · FRM1.VR.13.g

The idea

A KR01 or forward bucket 01 is a value change per basis point, so dividing by value and scaling to a one unit change in rate gives the matching duration. Key rate duration = KR01 × 10,000 ÷ portfolio value. Here KR01 is the gain from a one basis point fall, and multiplying by 10,000 is the same as dividing by 0.0001. The forward bucket duration is found the same way from the forward bucket 01. A duration of 2 means the value rises about 0.02 percent when that rate falls one basis point. Key rate durations add up to the effective duration, and so do forward bucket durations, when the key rates, their interpolation and the size of the bump are defined consistently with the parallel shift used for the effective duration. Divide by the position's market value, not face value, and for a net position with longs and shorts use the net value. Dollar duration = KR01 × 10,000. It is the same quantity before dividing by value.

Formulas

Key rate duration from a KR01

KRD = KR01 × 10,000 ÷ V

KR01
the gain from a one basis point fall in the key rate
V
the market value of the position, net of shorts

The same formula turns a forward bucket 01 into a forward bucket duration. Dividing by face value, or leaving out the 10,000, is the usual mistake.

A practice question

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

A portfolio has a market value of USD 2,000,000 and a KR01 at the 5-year key rate of USD 450 (gain from a one basis point fall). The 5-year key rate duration is closest to:

  1. 0.000225Why not: Divides KR01 by value without scaling for the basis point (omits the 10,000 factor).
  2. 4,444Why not: Inverts the ratio, computing value / KR01 instead of KR01 / value.
  3. 2.25 - correct
  4. 22.5Why not: Divides KR01 by value x 0.00001 instead of value x 0.0001, mis-scaling the basis point.

Key rate duration = KR01 / (value x 0.0001) = 450 / (2,000,000 x 0.0001) = 450 / 200 = 2.25. It says the value rises about 2.25% per 1 percentage point fall in that key rate.

A harder one

A portfolio has a market value of USD 50 million. Its KR01s (gain from a one basis point fall) are USD 1,500 at the 2-year key rate, USD 9,500 at the 5-year and USD 14,200 at the 10-year, with no exposure at other key rates. What is the portfolio's effective duration for a parallel shift?

  1. 2.84Why not: Uses only the 10-year key rate duration, 14,200 / 5,000.
  2. 5.04 - correct
  3. 1.68Why not: Averages the three key rate durations (0.30, 1.90 and 2.84) instead of adding them.
  4. 0.000504Why not: Divides total KR01 by market value without the 10,000 scaling factor.

Key rate durations: 1,500 / 5,000 = 0.30; 9,500 / 5,000 = 1.90; 14,200 / 5,000 = 2.84, where 5,000 = 50m x 0.0001. They add to the effective duration: 0.30 + 1.90 + 2.84 = 5.04, equivalent to total KR01 25,200 / 5,000.

The hardest tier

A portfolio is long USD 80 million market value of Bond L with a 5-year key rate duration of 3.0, and short USD 50 million market value of Bond S with a 5-year key rate duration of 4.2. What is the 5-year key rate duration of the net position, measured on the net market value?

  1. 0.23Why not: Divides the net KR01 of 3,000 by the gross market value of 130 million instead of the net 30 million.
  2. 1.00 - correct
  3. -1.20Why not: Subtracts the durations (3.0 - 4.2) without weighting by market values.
  4. 0.38Why not: Divides the net KR01 by the long market value of 80 million rather than the net value of 30 million.

KR01 of L = 80m x 3.0 x 0.0001 = 24,000. KR01 of S = -50m x 4.2 x 0.0001 = -21,000. Net KR01 = 3,000. Net market value = 30m. Net duration = 3,000 / (30m x 0.0001) = 1.00, so the net position has small exposure per unit of capital tied up.

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