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For a set of key rates, work out a portfolio's KR01s.

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

The idea

To find a portfolio's KR01s, reprice every cash flow after a one basis point fall in each key rate, with the shift tapering linearly to zero at the adjacent key maturities. For a cash flow with present value PV at maturity t, a full one basis point fall in its own rate adds about PV × t × 0.0001, with continuous compounding. If t lies between two key maturities, split that amount between them by distance. The nearer key rate receives the larger weight, and the weights sum to one. A cash flow at a key maturity belongs entirely to that key rate. Add the contributions across all cash flows, and across positions, keeping the sign of shorts. A common error is giving the whole amount to the nearest key rate or reversing the weights. As a check, the KR01s should sum to the portfolio DV01.

Formulas

Contribution of one cash flow to a KR01

KR01 = PV × t × w × 0.0001

PV
the present value of the cash flow
t
its maturity in years
w
its weight at that key rate, from linear interpolation between neighbouring key maturities

Holds with continuously compounded rates. The weights for one flow sum to one, the nearer key rate taking more; reversing them, or dropping t, is the usual error.

Weight of a cash flow at the next key rate above it

w = (t − tL) ÷ (tU − tL)

t
the cash flow's maturity
tL
the key maturity just below it
tU
the key maturity just above it

The key rate below takes 1 − w. A 7-year flow between key rates at 5 and 10 years gives 0.4 to the 10-year key rate, not 0.6.

A practice question

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

Spot rates are continuously compounded. A portfolio consists only of a zero coupon bond with a present value of USD 2,000,000 that matures in exactly 5 years, and 5 years is one of the key rates. Using the approximation of present value x maturity x 0.0001, the KR01 at the 5-year key rate is closest to:

  1. USD 200Why not: Leaves out the maturity multiplier: computes PV x 0.0001, which is the effect on a bond with one year to maturity.
  2. USD 1,000 - correct
  3. USD 10,000Why not: Scales by 0.001 (ten basis points) instead of 0.0001: 2,000,000 x 5 x 0.001.
  4. USD 0Why not: Assumes a zero coupon bond has no KR01 because it pays no coupons; it is fully exposed to its own maturity rate.

KR01 = 2,000,000 x 5 x 0.0001 = USD 1,000. The bond matures at the key rate, so the whole sensitivity is at the 5-year key rate and other key rates have zero KR01.

A harder one

Key rates are 2, 5 and 10 years; each key rate shift is one basis point at its own maturity and falls linearly to zero at the adjacent key maturities. Spot rates are continuously compounded. A portfolio is a single 7-year zero coupon bond with a present value of USD 4,000,000. Using KR01 contributions of PV x maturity x weight x 0.0001, the KR01 at the 10-year key rate is closest to:

  1. USD 1,680Why not: Reverses the weights, giving the 10-year rate 0.6 instead of 0.4 (for 7 years the 10-year weight is 2/5).
  2. USD 1,400Why not: Splits the risk equally between the 5-year and 10-year key rates.
  3. USD 1,120 - correct
  4. USD 2,800Why not: Gives the entire DV01 (4,000,000 x 7 x 0.0001) to a single key rate.

DV01 = 4,000,000 x 7 x 0.0001 = 2,800. The 7-year point is (7 - 5)/(10 - 5) = 0.4 of the way from 5 to 10 years, so the 10-year key rate gets 0.4 x 2,800 = 1,120 and the 5-year key rate gets 0.6 x 2,800 = 1,680.

The hardest tier

Key rates are 2, 5 and 10 years with tent shaped shifts and continuous compounding. A portfolio is long a 3-year zero coupon bond with a present value of USD 1,000,000 and short an 8-year zero coupon bond with a present value of USD 600,000. Using PV x maturity x weight x 0.0001 for each position, the portfolio's net KR01 at the 5-year key rate is closest to:

  1. USD 292Why not: Ignores that the 8-year bond is short, adding its 5-year KR01 of 192 to the long bond's 100.
  2. -USD 92 - correct
  3. -USD 188Why not: Weights the 8-year bond 0.6 instead of 0.4 at the 5-year key rate: 100 - 288.
  4. -USD 192Why not: Counts only the short 8-year bond and misses the 3-year bond's 1/3 allocation to the 5-year key rate.

3-year bond: DV01 = 1,000,000 x 3 x 0.0001 = 300, of which 1/3 = 100 sits at 5 years. 8-year bond: DV01 = 600,000 x 8 x 0.0001 = 480, of which 0.4 = 192 sits at 5 years, and it is short. Net = 100 - 192 = -92.

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