Hedging liabilities from their cashflows

Matching present value and duration leaves a choice about when the assets supporting a pension scheme will pay. Receipts that arrive early must be invested; a payment falling due before its supporting receipt requires cash or financing. The rates applied across those gaps affect the cost of meeting the pension promise.

For fixed payments valued on a common discount curve, the residual forward-rate exposure is the negative of the running discounted cash balance when initial present values match. The exposure residual therefore records the financing and reinvestment position left by the hedge. A model of relative-rate changes then determines how to rank imperfect hedges. Exposure transport ranks them by the amount of exposure and the maturity distance to its offset. Under a bound on the maturity slope of forward-rate changes, it minimises the worst first-order change in assets less liabilities.

This paper derives the result, gives an exact numerical formulation and tests four objectives on a fully specified pension and gilt example. The comparisons use the same initial capital, total rate sensitivity and permitted assets. Transport has the smallest bounded-slope loss. Accumulated-residual least squares has lower dispersion in the two co-moving scenario families tested; pointwise least squares has lower dispersion under independent local shocks. The financing identity is common to these methods. Their different objectives express different assumptions about the risk left between payment dates.

1. The cash balance is the rates position

Put the asset receipts and liability payments on one schedule. Let $p_i=C_iD(t_i)$ be the present value of a payment $C_i$ at time $t_i$, using the discount factor

Every payment after date $u$ is discounted through the instantaneous forward rate at $u$. A payment before it is not. The positive magnitude of forward-rate exposure is therefore

One payment produces a rectangle, extending from valuation to payment. A coupon bond is a descending staircase. A pension payment stream is another staircase. Their difference locates the residual position across the curve.

Write $n_i$ for asset receipts less liability payments, and define

The payments already made and the payments remaining add to the initial net present value. Hence

With equal initial present values, $r(u)=-c(u)$. A positive exposure residual corresponds to a discounted financing requirement: more liability cash has gone out than asset cash has arrived. A negative residual corresponds to cash awaiting a later use.

This has a direct cash-account interpretation. Start with no separate cash balance and invest or borrow at the deterministic rates implied by the common curve. If $B(u)$ is the nominal cash-account balance after that date’s payments, then $D(u)B(u)=c(u)=-r(u)$. A different initial cash allocation, borrowing spread or collateral call enters that account explicitly. For unequal present values, the general identity above retains the initial difference.

The area under $G$, multiplied by $10^{-4}$, is parallel-forward PV01: the magnitude of the first-order value change for a one basis point parallel rise in forwards. One basis point is 0.01 percentage points. Equal present values and equal PV01 are two separate constraints:

The first matches the initial investment under the stated valuation basis. The second removes the common first-order response to a parallel move. Neither fixes the intervening cash balance.

2. Why the distance to an offset enters the objective

A cash surplus and a financing requirement respond in opposite directions to a common movement in their forward rates. The offset is dependable to the extent that the two rates move together. Matching exposure close by reduces reliance on rates remaining aligned across a longer part of the curve.

Let the accumulated exposure residual be

Matched PV01 gives $H(T)=0$. For a small, absolutely continuous forward-rate change $g(u)$, integration by parts gives the first-order surplus change:

An explicit uncertainty set now produces an explicit ranking. Suppose the maturity slope of the rate change satisfies $|g'(u)|\leq\kappa$. Then

The upper bound follows by taking absolute values. It is attained by choosing $g'(u)=\kappa\,\operatorname{sign}H(u)$, or its negative. Thus a portfolio that minimises $J$ minimises the worst first-order surplus change over this set. The common level of $g$ cancels because PV01 is matched.

For nonnegative exposure profiles of equal total area, $J$ is also the least maturity-distance cost of pairing asset exposure with liability exposure. At each cut in the maturity line, $H(u)$ is the net exposure that has to cross it. Integrating the absolute amount crossing each cut gives the one-dimensional transport cost. Dividing by the common exposure area gives a Wasserstein-1 distance in years. The cumulative and dual representations are standard optimal-transport results. Peyré and Cuturi

The assumption concerns changes in rates. It does not follow merely from fitting a smooth initial discount curve. The initial curve values the payments; the uncertainty set specifies how their relative rates may move.

3. A nearby bridge and a distant bridge

Consider a liability with present value £100m, payable in year ten. Value all payments on a flat 4% continuously compounded curve. Each hedge invests £50m today in each of two hypothetical zero-coupon bonds. The nearby bonds mature in years nine and eleven; the distant bonds mature in years five and fifteen. The investment amounts are present values; each asset’s face amount is its present value divided by the discount factor to payment.

Both hedges have £100m of initial value and £100,000 per bp of parallel PV01. Consider the steepening $g(u)=10^{-4}u$: forward rates rise by one additional basis point for each year of maturity.

The distant hedge has twenty-five times the transport cost and first-order steepening loss. For a single liability at $h$, positive asset cashflows, matched value $P$ and matched money duration,

This is a connection to the payment-date dispersion used in classical immunisation. It follows because $\sum_i p_i\min(u,t_i)-P\min(u,h)\leq0$, by concavity of the minimum function. Integrating its negative gives the displayed expression. Fong and Vasicek

The two hedges nevertheless reach the same £50m discounted financing deficit. Proximity shortens the interval over which relative rates matter; it does not eliminate the cash required during that interval. A financing limit can therefore change which otherwise attractive hedge is feasible.

4. Four ways to rank the same residual

The comparisons below hold the assets, valuation basis and portfolio constraints fixed. Only the loss function changes.

Under equal present values, the cumulative difference between discounted payments is $-r$. Cashflow transport therefore integrates $|r|$. Exposure transport accumulates $r$ once more before taking the absolute value. The two transport criteria price different displacements.

This distinction has established predecessors. M-Absolute measures the absolute distance of discounted asset payments from an investment horizon. Kurochkin and Rodina formulate transport of discounted payment streams for multiple liabilities. Kondratiuk-Janyska and Kałuszka’s equations (21)–(22) use the squared accumulated residual, after translating their horizon-valued cumulative cashflows into the notation here. The present construction uses an absolute penalty and the corresponding bounded-slope interpretation. Nawalkha and Chambers, Kurochkin and Rodina, Kondratiuk-Janyska and Kałuszka

Key-rate and factor reports can be obtained by applying their specified shocks to $r$. Constructing a portfolio to match those reports requires choosing the nodes, factor shapes and interpolation. The full exposure profile keeps the dated position visible before those choices are made.

5. A pension liability and a gilt portfolio

The larger example uses 400 pensioners and 61 conventional-gilt coupon and redemption schedules. All inputs are supplied with the paper. The curve and mortality law are illustrative model assumptions, stated here so that every price, payment and portfolio weight can be regenerated.

For a member aged $a$, conditional survival to $t$ years is

The expected pension payment in year $t$ is $36{,}000\sum_a S_a(t)$. Discounting these payments gives initial value £173,640,556, duration 9.5778 years, and parallel PV01 @@PV01@@ per bp.

Each instrument’s coupon dates, redemption date and cash amounts per £100 nominal are listed in gilt-schedules.json. Times are measured in years from the model’s valuation point. The amounts at each semiannual date are half the annual coupon; the final date also returns £100 principal. A portfolio weight denotes the fraction of the common initial budget invested in that bond. Its nominal holding is the invested amount divided by its price per pound of nominal.

The four objectives give the following results. The transport column is $J/P$, in years squared. The fit percentage is $100[1-\int|r|/\int G_L]$; it is a description of the pointwise gap, not the objective used by the transport optimiser.

The exposure-transport solution uses 46 gilts and gives a 97.42% pointwise fit. Its absolute residual is £4,292 per bp. Positive and negative residuals cancel for a parallel move to first order, because every method is constrained to match total PV01. Both liability and hedge start at the same present value.

6. Repricing under specified curve changes

The scenario experiment measures an instantaneous change in the value of the fixed payment streams. Divide the sixty-year maturity range into quarter-year intervals. The forward-rate shock is constant within each interval. For a payment at $t$, calculate the integral $F(t)=\int_0^t g(u)du$ exactly by intersecting its life with those intervals. Its stressed present value is $p_i e^{-F(t_i)}$.

Each family contains 5,000 scenarios. Within a family every portfolio faces exactly the same shocks. The random generator is NumPy PCG64 with seed 20260915. Shocks are normalised so that their expected squared value, averaged over the maturity intervals, has square root 15bp. This normalisation is fixed from the model covariance, rather than from each realised path.

For each scenario, the reported surplus is stressed assets less stressed liabilities. Initial surplus is zero. Standard deviation uses the sample estimate with denominator 4,999. The downside measure is the mean surplus in the worst 250 scenarios, the lowest 5% of that family. The downloadable outputs also report means, root mean squared errors and first-order approximations.

Accumulated least squares has the lowest surplus dispersion in the smooth and correlated families. Exposure transport is close to it in the correlated family, while pointwise least squares has the lowest dispersion under local independent shocks. Cashflow transport produces a small smooth-factor error but leaves more exposure to the correlated family.

For independent interval shocks of common variance $\sigma^2$, first-order surplus variance is $\sigma^2\sum_k(\int_{I_k}r)^2$. This is proportional to $\int r^2$ when the intervals have equal widths and the residual is constant within each. In this large example some payments fall inside the quarter-year shock intervals, so the pointwise objective and scenario variance need not have exactly the same minimiser. The small example below aligns the intervals and demonstrates the exact result.

These rankings depend on how rates move together. A norm that permits adjacent residuals to offset can perform well when those rates co-move. The distinction between an absolute and a squared accumulated residual then determines the trade-off between the largest possible loss and the distribution of losses under a particular model. A covariance model supplies a different decision rule from a bound on maturity slopes.

7. A case where the ranking can be checked by hand

A smaller example isolates that choice. At a zero initial discount rate, a liability pays £50m in year two and £50m in year four. The available zero-coupon assets pay in years one, three and five. Equal initial value and money duration force their budget weights to be

Per pound of liability value, the exposure residual on the five unit intervals is $(0,-x,\frac12-x,x-\frac12,x)$. All quantities can therefore be integrated directly.

Exposure transport chooses $x=(1-1/\sqrt2)/2$; pointwise least squares chooses $x=1/4$; accumulated least squares chooses $x=5/32$. With independent equal-variance shocks on the unit intervals, surplus variance is proportional to $\int r^2$, so the pointwise solution is optimal for that model.

For a continuous Brownian shock across maturity, $g(u)=\sigma W(u)$, the matched-duration condition gives $-\int rW=\int H\,dW$. Itô isometry then gives variance $\sigma^2\int H^2$, making accumulated least squares optimal. Adjacent rates co-move in this model. Proximity alone therefore does not select the absolute penalty.

Transport instead minimises the maximum first-order loss for bounded maturity slopes. In the small example it sacrifices about 1.4% in Brownian-model variance relative to accumulated least squares, while achieving the smaller robust bound. This is an explicit choice about the loss criterion.

8. Financing limits enter the portfolio calculation

The exposure residual also records the financing position. Under the common deterministic curve, a positive $r(u)$ requires nominal funding $r(u)/D(u)$. The unconstrained transport portfolio reaches £5.82m of discounted financing need in the large example. Different objectives produce different dated balances even though they start with equal capital.

Impose a £3m ceiling on the discounted financing requirement at every event interval. In these units the constraint is $r(u)\leq£3\text{m}$; its nominal equivalent rises as $£3\text{m}/D(u)$. The cap is part of the same optimisation. The new solution respects that ceiling and has J/P = 0.054783 years squared, compared with 0.037114 without the ceiling: a 47.6% increase in the bounded-slope risk measure.

The limit changes the feasible holdings. It does not change the survival model or the liability estimate. An actual nominal credit line, collateral schedule or borrowing spread would instead enter through its own dated cash-account specification. Fixed cashflow and financing assumptions make those balances affine in holdings, so linear limits can be added to the portfolio problem.

A constraint multiplier measures the local improvement in the objective from relaxing its bound. When the objective is $\kappa J$, the multiplier has the corresponding worst-case first-order surplus units. A finite relaxation can change the binding set and is evaluated by solving the revised problem. Discrete lots and fixed ticket costs require an integer formulation; continuous linear constraints retain convexity.

9. Allowing the maturity distance to vary

Calendar distance treats a year of separation in the same way at the front and the long end. An increasing maturity clock $z(u)$ makes that assumption explicit and adjustable. Under $|g'(u)|\leq\kappa z'(u)$, the same derivation gives

The clock changes the distance assigned to offsets. It leaves the payment records, exposure measure and PV01 constraint intact.

A logarithmic clock, $z(u)=\psi\log(1+u/\psi)$, has local weight $\psi/(\psi+u)$. Le Coz and Bouchaud examine a compressed maturity coordinate in a model of Eurodollar and SOFR forward-rate correlations. Their work motivates measuring how the maturity axis should be weighted. Applying it here requires a chosen horizon, raw-rate volatilities and a sterling calibration extending through the maturities of the hedge. The numerical examples in this paper use calendar time throughout. Le Coz and Bouchaud

A distance specification and a resource limit answer different questions. The first expresses how relative rates may move. The second states what the institution can hold or finance. Keeping them separate makes both the portfolio and the marginal cost of its constraints interpretable.

Appendix A. Exact construction and verification

Event grid. Combine every asset payment date and liability payment date with zero and the terminal horizon. Between successive dates the exposure residual is constant and $H$ is linear. The grid retains both sides of each cashflow jump when plotting the exposure.

Exact integral. On an interval of width $\Delta$, let the endpoints of $H$ be $a,b$. Its contribution to $J$ is

A trapezoid on the absolute endpoints is exact in the first case. In the second it misses the crossing. For $a=1,b=-1,\Delta=1$, the exact integral is one-half, while that trapezoid gives one.

Convex representation. The exact integral has a second-order cone formulation. Introduce $z,q$ for each interval and minimise the sum of $\Delta(z+q)/4$, subject to

The cone imposes $qz\geq(a+b)^2$, with nonnegative $q,z$. Minimising over them chooses $z=\max(|a-b|,|a+b|)$, yielding the exact piecewise formula above. Zero crossings are handled by the optimisation itself.

Accumulated least squares is also integrated exactly: each interval contributes $\Delta(a^2+ab+b^2)/3$. Pointwise least squares and cashflow transport use the constant residual and the exact interval widths. The implementation solves the conic problem with Clarabel, the quadratic objectives with Clarabel, and the payment-transport linear programme with HiGHS.

Checks. The replication verifies the cashflow-to-exposure identity, equal initial value, equal PV01, exact interval integrals, the analytic small-example optima and independent direct repricing. It evaluates the transport objective from the returned weights and compares it with a global lower bound. For the large unconstrained example, the gap between the feasible portfolio and the independently checked lower bound is 1.13e-7% of the objective. The package includes the endpoint support coefficients and feasible linear-programme dual variables needed to check the bound.

Finite rate changes. The main robust result is first order. For equal initial PV and matched PV01, fixed payments and a smooth finite shock, define $F(t)=\int_0^t g$. Twice integrating the payment measure by parts gives the exact identity

The corresponding absolute-value bound is $J\sup_t|(g'-g^2)e^{-F}|$. In particular, a parallel shock has a second-order residual. Parallel convexity mismatch is $-2\int H$, whose magnitude is bounded by $2J$. The scenario tables use full exponential repricing rather than substituting the first-order bound for realised outcomes.

Appendix B. Reproduce the results

The replication package contains the input specification, all 61 payment schedules, the calculation and verification scripts, pinned Python dependencies, the generated portfolio weights, every scenario outcome, and the figures. The same generated exposure arrays supply the interactive gilt example.

Run the following from the extracted package directory:

python -m pip install -r requirements.txt
python reproduce.py --output results
python verify.py --results results

config.json specifies the economic assumptions, generator seed and tolerances. gilt-schedules.json supplies payment dates and amounts. all-portfolios.csv gives each method’s weights and nominal holdings. scenario-outcomes.csv contains the 60,000 paired method/scenario observations, and results.json records the summaries, software versions and input hashes. Monetary figures in the text are rounded from those outputs.

References

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2. Nawalkha, S. K. and Chambers, D. R. (1996). An Improved Immunization Strategy: M-Absolute. Financial Analysts Journal, 52(5), 69–76. doi:10.2469/faj.v52.n5.2026.
3. Kondratiuk-Janyska, A. and Kałuszka, M. (2006). Assets/liabilities portfolio immunization as an optimization problem. Control and Cybernetics, 35(2), 335–349. Original paper.
4. Kurochkin, S. V. and Rodina, V. A. (2023). Optimal solution for immunizing arbitrarily scheduled multiple liabilities. Economics and Mathematical Methods, 59(2), 87–99. doi:10.31857/S042473880025861-6.
5. Peyré, G. and Cuturi, M. (2019). Computational Optimal Transport. Foundations and Trends in Machine Learning, 11(5–6), 355–607. Author edition.
6. Le Coz, V. and Bouchaud, J.-P. (2024). Revisiting Elastic String Models of Forward Interest Rates. Quantitative Finance, 24(11), 1561–1578. arXiv:2403.18126v2.