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Break a bond position or portfolio's profit and loss (P&L) down into separate drivers, namely carry roll-down, rate change and spread change effects, and explain each.

FRM® Part I · Valuation and Risk Models · Bond Yields and Return Calculations · FRM1.VR.11.h

The idea

A bond position's profit and loss over a horizon is split into pieces so each source of risk is visible. Carry roll-down is the return earned if nothing in the market changes. It is coupon income, less the cost of financing the position, plus the price change that comes simply from time passing as the bond rolls down the unchanged curve to a shorter maturity. The rate change effect is the P&L from movement in the benchmark curve, and it is approximately the following. Rate effect = −DV01 × change in yield in basis points. The spread change effect is the P&L from movement in the bond's spread over that curve, with a similar sensitivity. Total P&L = carry roll-down + rate change effect + spread change effect. A hedge with the benchmark removes the rate effect and leaves the spread. Typical errors: sign mistakes (rising yields lose money) and counting the roll-down price change twice.

Formulas

Bond P&L attribution

P&L = CRD + RE + SE

CRD
carry roll-down: coupon less financing plus the price change from rolling down an unchanged curve
RE
the rate change effect from benchmark curve moves
SE
the spread change effect from moves in the bond's spread

Count the roll-down price change once, inside CRD. A benchmark hedge removes RE and leaves SE.

Rate change effect on a bond position

RE = −DV01 × Δy

DV01
the position's price change per basis point, quoted positive
Δy
the change in benchmark yield, in basis points

The spread effect has the same form with the spread change. Rising yields lose money, hence the minus sign.

A practice question

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

A bond position's profit and loss is separated into drivers for attribution. Which set of drivers is used?

  1. Delta, gamma and vega effects of the bond's embedded options.Why not: These are option Greeks, not the drivers of a plain bond's return over time.
  2. Duration, convexity and a residual from higher-order terms.Why not: These are terms in a price approximation, not the economic sources of the P&L.
  3. Coupon income, price appreciation and dividend yield.Why not: Dividend yield belongs to equities, and the split is not a risk-factor attribution.
  4. Carry roll-down, the change in the benchmark curve, and the change in the bond's spread. - correct

The standard attribution separates P&L into carry roll-down (what accrues if nothing changes), the rate change effect (benchmark curve moves) and the spread change effect (movement of the bond's spread).

A harder one

A bond position has carry roll-down of USD 40,000 over a month. Its DV01 to the benchmark curve is USD 7,500 per bp, and its spread DV01 is also USD 7,500 per bp. Over the month the benchmark yield rises by 6 bp and the bond's spread widens by 3 bp. What is the total P&L, closest to?

  1. -USD 5,000Why not: Leaves out the spread change effect of -22,500.
  2. -USD 27,500 - correct
  3. -USD 107,500Why not: Subtracts the carry roll-down instead of adding it.
  4. -USD 67,500Why not: Leaves out the positive carry roll-down.

Rate effect = -7,500 x 6 = -45,000. Spread effect = -7,500 x 3 = -22,500. Total = 40,000 - 45,000 - 22,500 = -27,500.

The hardest tier

A position was bought at a price of 100.50. Over the horizon it received a coupon of 4.00 and paid 1.20 to finance the position. The bond's price at the horizon would have been 100.10 had the benchmark curve and the spread stayed unchanged; the actual price was 99.30. Of the difference between the actual and unchanged-curve prices, benchmark curve moves explain -1.10. Which set of effects (carry roll-down, rate change, spread change), in price points, is correct?

  1. +3.60, -1.10, +0.30Why not: Leaves out the 1.20 financing cost from carry roll-down.
  2. +2.80, -1.10, +0.30Why not: Counts coupon less financing only and omits the -0.40 roll-down price change from the same-curve repricing.
  3. +2.40, -1.10, -0.10Why not: Computes the spread effect from the change versus the purchase price, which double counts the roll-down move.
  4. +2.40, -1.10, +0.30 - correct

Carry roll-down = coupon 4.00 - financing 1.20 + roll-down (100.10 - 100.50 = -0.40) = +2.40. Actual price minus unchanged-curve price = 99.30 - 100.10 = -0.80 = rate (-1.10) + spread (+0.30). Total = +2.40 - 0.80 = +1.60.

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