01 / Project
Member cashflows
Project each pension payment and weight it by the probability that the member is alive when it falls due.
PV = Σ Ct · St · Dt
Liability hedging research
From pension promises to a gilt hedge
Residual exposure is 2.4%: £4.4k of absolute residual PV01 remains unmatched on a £181.3m liability. Build the projected member cashflows, translate them into forward-rate exposure and inspect the 97.6% fit across the curve.
01 / Interactive model
Build the hedge
Move from projected member payments to the final 46-line hedge. Each control updates the calculated exposure or portfolio state.
Stage 01
Build the liability member by member
The worked scheme contains 400 members aged 55–65, all in payment from the valuation date. Each receives a level nominal pension of £36,000 a year, weighted by survival under the model. No indexation is modelled; index-linked liabilities are a natural extension.
Members included400 / 400
March 2026 worked example400 members
Members400
Liability PV£181.3m
Absolute residual PV01£4.4k
Exposure fit97.6%
2.4% residual. The discrete, long-only gilt universe ends in 2073, so finite gilt shapes cannot reproduce every maturity point. The optimiser minimises cumulative transport rather than pointwise mismatch.
Liability assumptions. All 400 members are in payment from the valuation date and receive level pensions with no indexation. An index-linked liability and gilt hedge is a natural extension.
View the 46-line hedge
| Line | Bond | Maturity | Notional | Market value |
|---|---|---|---|---|
| 1 | 1.625% 2071 | 45.6y | £4,026 | £1,482 |
| 2 | 3.5% 2068 | 42.4y | £279,231 | £194,326 |
| 3 | 2.5% 2065 | 39.4y | £707,808 | £379,551 |
| 4 | 4% 2063 | 37.6y | £1,187,258 | £957,190 |
| 5 | 0.5% 2061 | 35.6y | £1,829,796 | £433,939 |
| 6 | 4% 2060 | 33.9y | £4,220,399 | £3,407,358 |
| 7 | 1.75% 2057 | 31.4y | £6,122,453 | £2,772,537 |
| 8 | 4.25% 2055 | 29.7y | £4,858,307 | £4,113,456 |
Construction order runs from the longest maturity to the shortest. Lines 1 (£4,026 notional) and 45 (£90,020) are retained to show the optimiser’s mathematical solution; a tradable implementation would apply desk-specific minimum notionals and re-optimise.
02 / Method
The hedge begins with the promise
Cashflows, discounting and survival determine the liability. The resulting forward-rate exposure is then matched against the instruments that can actually be held.
02 / Translate
Forward exposure
Reprice the liability against small movements in the forward curve. This reveals where rate risk sits through time.
E(u) = −∂PV / ∂f(u)
03 / Construct
Gilt portfolio
Select and size long-only gilt positions to minimise the cumulative transport mismatch across maturity, subject to equal total exposure area.
min ∫ |Hx(u)| duHx(u) = ∫0u [GH(s) − GL(s)] ds
03 / Hedging policy
From a static hedge to a hedging policy
A state-space policy keeps actuarial, market and portfolio uncertainty separate. Rebalancing responds to the current position, market state and expected value of trading.
Actuarial state
Cashflows, options and demographic uncertainty
Market translation
Curves, inflation and model residuals
Benchmark
The clean liability-matching portfolio
Policy
Rebalance only when the expected gain justifies action
Actual portfolio
Liquidity, costs, inventory and governance
Trading costs and liquidity belong in the implementation layer. They do not rewrite the liability estimate.
04 / Worked result
A measurable match across the curve
The worked example constructs a 46-gilt hedge for a £181.3m liability from a 61-gilt universe. It leaves £4.4k of absolute residual PV01 unmatched: 2.4% residual exposure and a 97.6% fit.
Pension members400
Liability PV£181.3m
Gilts selected46 / 61
Exposure fit97.6%