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Explain what convexity 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.f

The idea

Convexity measures how the relationship between price and yield curves. It is the second derivative of price with respect to yield, divided by price. From prices it is estimated using a decimal yield change Δy. Convexity = (Pup + Pdown − 2 × P0) ÷ (P0 × Δy²). Equivalently, convexity tells you how fast duration or DV01 changes as yield changes. With positive convexity a bond gains more when yields fall than it loses when they rise by the same amount, and the effect grows with the square of the move. The price estimate follows. Percent change = −Duration × Δy + 0.5 × Convexity × Δy². The convexity term is positive whichever way yields move. Callable bonds can have negative convexity. A frequent mistake is forgetting the 0.5 or squaring basis points instead of decimals.

Formulas

Convexity from up and down yield shifts

C = (Pup + Pdown − 2 × P0) ÷ (P0 × Δy²)

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

Square the decimal yield change, not basis points. A negative result, as with a callable bond, means the gain from a fall is smaller than the loss from a rise.

Percentage price change from duration and convexity

%ΔP = −D × Δy + 0.5 × C × Δy²

D
the effective duration
C
the convexity
Δy
the yield change, as a decimal

The convexity term is positive whichever way yields move for a positively convex bond. Forgetting the 0.5 is the usual error.

A practice question

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

Which of the following best defines convexity?

  1. The first derivative of price with respect to yield divided by price, with the sign reversed.Why not: Describes duration, which is the slope, not the curvature.
  2. The second derivative of price with respect to yield divided by price, a measure of the curvature of the price-yield relationship. - correct
  3. The dollar change in price for a one basis point change in yield.Why not: Describes DV01, a first-order measure in currency.
  4. The change in a bond's duration as time passes with yields unchanged.Why not: Describes a time effect on duration; convexity refers to how duration changes as yield changes.

Convexity is the second-order sensitivity, so it measures how the slope of the price-yield curve changes with yield. It is also the rate at which duration and DV01 change as yields change.

A harder one

A bond priced at 100.00 is repriced with yields 100bp lower and 100bp higher, giving 107.30 and 93.60. The convexity is closest to:

  1. 0.9Why not: Divides by the yield change instead of its square.
  2. 45Why not: Averages the two shifted prices instead of adding them, halving the numerator.
  3. 90 - correct
  4. 9,000Why not: Divides by the squared yield change in decimals but omits the price P0 from the denominator.

Convexity = (107.30 + 93.60 - 2 x 100.00) / (100.00 x 0.01²) = 0.90 / 0.01 = 90.

The hardest tier

A callable bond priced at 100.00 is repriced with yields 100bp lower and 100bp higher, giving 102.40 and 97.10. Which pair gives the convexity and its interpretation?

  1. +50: the price gain from a 100bp fall is larger than the price loss from a 100bp rise.Why not: Drops the negative sign of the numerator (102.40 + 97.10 - 200 = -0.50).
  2. -50: the price gain from a 100bp fall is smaller than the price loss from a 100bp rise. - correct
  3. -0.5: the price gain from a 100bp fall is smaller than the price loss from a 100bp rise.Why not: Divides by the yield change in decimals but not by its square, so the magnitude is wrong.
  4. -5,000: the price gain from a 100bp fall is smaller than the price loss from a 100bp rise.Why not: Omits P0 from the denominator and divides by the squared yield change only.

Convexity = (102.40 + 97.10 - 200.00) / (100 x 0.0001) = -0.50 / 0.01 = -50. The gain is 2.40 and the loss is 2.90, so convexity is negative, as the call caps the upside.

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