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Evaluate where duration falls short and how convexity covers some of those shortcomings.

FRM® Part I · Financial Markets and Products · Properties of Interest Rates · FRM1.FM.16.f

The idea

Duration is the slope of the price-yield curve, so it is a straight-line approximation. It works only for small, parallel yield changes. It breaks down for large moves because the actual curve bends. It also ignores changes in the shape of the curve, so portfolios with equal duration can behave differently when the curve twists. Convexity measures that curvature. It is the second derivative of price with respect to yield, divided by price. For a plain bond convexity is positive, so the actual price is above the duration estimate whether yields rise or fall. Of two bonds with equal duration, the more convex one benefits more from large moves. For a zero-coupon bond with annual compounding, Convexity = T(T + 1) ÷ (1 + y)². Bonds with embedded options, such as callable bonds and mortgage securities, can have negative convexity, and the exam tests this contrast.

Formulas

Convexity of a zero-coupon bond

C = T × (T + 1) ÷ (1 + y)²

T
the bond's maturity, in years
y
the annually compounded yield

Annual compounding only. Plain bonds have positive convexity; callable bonds and mortgage securities can have negative convexity.

A practice question

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

Which statement describes the role of convexity in bond price analysis?

  1. Convexity is the percentage price change for a one percentage point change in yieldWhy not: This is the definition of duration, the first-order sensitivity, not convexity.
  2. Convexity is the average time to a bond's cash flows, weighted by present valueWhy not: This is Macaulay duration. Convexity concerns the second derivative of price with respect to yield.
  3. Convexity measures the bond's sensitivity to changes in credit spreads rather than risk-free yieldsWhy not: Convexity is a property of the price-yield curve; it is not about credit spreads.
  4. Convexity measures the curvature of the price-yield relationship and corrects the straight-line duration estimate for the second-order effect - correct

Duration is the slope of the price-yield curve, a straight-line approximation. Convexity, the second derivative divided by price, captures how that slope changes and therefore how bad the linear approximation becomes for larger yield moves.

A harder one

Two portfolios have the same modified duration of 5. After the yield curve steepens, with short rates falling and long rates rising, portfolio A gains while portfolio B loses. Which limitation of duration does this illustrate?

  1. Duration cannot be calculated for portfolios that contain more than one bondWhy not: Duration is additive across bonds as a value-weighted average, so portfolios are not a problem.
  2. Duration overstates the effect of small yield changes because it is a second-order measureWhy not: Duration is a first-order measure; its error appears for large yield changes, and the case here is about curve shape.
  3. Duration measures sensitivity to a parallel shift in the yield curve and does not capture changes in its shape - correct
  4. Duration ignores the credit quality of the issuersWhy not: Credit quality is not the issue in this scenario; the portfolios differ in how their cash flows are spread along the curve.

A single duration number assumes every cash flow's yield moves by the same amount. Portfolios with equal duration can hold different mixes of short and long maturities, so a twist in the curve affects them differently. This is a key limitation that key-rate durations address.

The hardest tier

A barbell portfolio of 2-year and 30-year bonds and a bullet portfolio of 10-year bonds have the same price, yield and modified duration. The barbell has higher convexity. If the yield curve then shifts in a large parallel move, which statement is most accurate?

  1. The barbell outperforms the bullet whether yields rise or fall, because the extra convexity adds a positive second-order term in both cases - correct
  2. The barbell outperforms only if yields fall, because convexity matters only for price gainsWhy not: Convexity adds one half x convexity x (yield change)², which is positive for either sign of the change.
  3. The bullet outperforms in both cases, because its lower convexity makes its price less volatileWhy not: Lower convexity means a smaller second-order gain; it does not reduce the first-order exposure, which is the same for both.
  4. The two perform identically, because equal modified duration fully determines the price changeWhy not: Equal duration matches only the first-order change; the convexity term differs and grows with the square of the move.

With equal duration, the first-order price change is identical. The second-order term, 0.5 x convexity x (change in yield)², is positive and larger for the barbell whatever the direction. In practice, higher convexity usually costs something (a lower yield), so the advantage is not free.

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