Immunisation playground
Redington's three conditions, built rather than memorised: two bonds solved to match a liability's present value and duration, and the surplus curve that smiles upward from zero — while the PV-only portfolio beside it shows exactly what duration matching is for.
Surplus PV(assets) − PV(liability) as the whole yield curve shifts
The solved portfolio
Bond at 4y pays 426
Bond at 18y pays 633
PV each side 614
Duration each side 10.0y
✓ PV of assets = PV of liabilities (614)
✓ Discounted mean terms equal (10.0 years)
✓ Asset spread 148 > liability spread 100 — barbells beat bullets on convexity
The amounts are solved, not chosen: with PV and duration pinned, the split between the bonds is w₁ = (t₂ − t_L)/(t₂ − t₁) — the liability’s time sits at the weighted middle of the asset times. That bracket t₁ < t_L < t₂ automatically delivers Redington’s third condition, so the purple surplus curves UP from zero both ways: small shifts can only help. Push the bonds far apart and the smile deepens (more convexity); pull them toward t_L and it flattens toward the knife-edge. The grey line is what PV-matching alone buys: nothing, the moment yields move the wrong way. The small print the exam wants: this holds for SMALL, PARALLEL shifts, and the portfolio must be rebalanced as time and rates move.
Why the smile happens
Matching present values makes the surplus zero at today’s rate; matching durations makes its FIRST derivative zero there too, so small shifts have no first-order effect. What remains is the second-order term, and its sign is the spread condition: assets spread more widely in time than the liability (a barbell around a bullet), so the asset side has more convexity and the surplus bends upward both ways. That is the whole theorem in one curve. The catches are the same ones the examiner wants listed: it protects against small parallel shifts only, profits at the expense of nobody cannot persist in equilibrium, real liability spreads can defeat the construction, and the match decays as time passes — immunisation is a rebalancing discipline, not a one-off trade.
The ingredients. Duration and convexity for a single bond live in the bond calculator, and the discounting behind every PV here is the annuity calculator’s bread and butter. Memori is a flashcard app built by actuarial students, with ready-made CM1 and SP5 sets in the shop. Join the beta.
For education only.