Lesson 4 of 6 · All lessons in this module
  1. Convexity: measuring the curve
  2. What drives convexity, and why investors value it
  3. Estimating price changes with duration and convexityVideo lesson
  4. Integrated cases: a negative-yield zero priced by hand
  5. Money convexity: the adjustment in currency
  6. Portfolio duration and convexityVideo lesson

Lesson 4 of 6 · 15 min

Integrated cases: a negative-yield zero priced by hand

With a negative yield and a settlement date inside the year, a spreadsheet price function is no help: you price the zero by hand, shift the yield by 1 bp to estimate duration and convexity, and then test the duration-plus-convexity estimate against a full repricing.

In short

  • The full price of a zero is 100/(1+r)N×(1+r)t/T100/(1+r)^{N}\times(1+r)^{t/T}: discount over N whole years, then carry forward t/T of a year. With r<0r<0 that is still the same formula.
  • Estimate duration and convexity by repricing at r±1r\pm1 bp, keeping the same full-price base in every denominator.
  • For a zero, ModDur is (time to maturity) ÷ (1 + r). With a negative yield 1+r<11+r<1, so ModDur is larger than the time to maturity.
  • Convexity adjustment = ½ × Convexity × (Δy)². It is positive for both directions of yield change, and it matters more for big moves.
  • Compare the estimate with the exact repricing: duration alone misses by the convexity term; adding it should land within a few thousandths of a percent for a 60 bp move.

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