Summary

Members	400
Annual payment	£36,000
Liability PV	£181,330,247
Bonds available	61
Bonds selected	46
Fit ratio	97.6%
Total notional	£234,498,517
PV mismatch	£+1,738,992
Transport cost	6,510,538

Selected Bonds (longest first)

#	Bond	Maturity	Notional (£)	MV (£)
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
9	1.625% 2054	28.6y	£1,943,388	£883,936
10	4.375% 2054	28.4y	£2,669,659	£2,309,147
11	3.75% 2053	27.6y	£1,569,657	£1,233,509
12	1.5% 2053	27.4y	£3,097,955	£1,376,197
13	3.75% 2052	26.4y	£6,244,314	£4,893,883
14	1.25% 2051	25.4y	£4,057,062	£1,747,993
15	0.625% 2050	24.6y	£6,046,105	£2,137,563
16	4.25% 2049	23.7y	£3,662,964	£3,199,881
17	1.75% 2049	22.9y	£9,526,799	£5,051,257
18	1.5% 2047	21.4y	£9,926,594	£5,112,617
19	4.25% 2046	20.7y	£2,421,753	£2,147,132
20	0.875% 2046	19.9y	£9,062,245	£4,122,100
21	3.5% 2045	18.9y	£7,526,387	£5,977,459
22	3.25% 2044	17.9y	£9,075,379	£6,996,724
23	4.5% 2042	16.7y	£9,965,015	£9,378,917
24	1.25% 2041	15.6y	£8,333,187	£4,911,098
25	4.25% 2040	14.7y	£7,161,477	£6,657,250
26	4.375% 2040	13.9y	£7,107,563	£6,693,925
27	4.25% 2039	13.5y	£1,248,089	£1,159,911
28	1.125% 2039	12.9y	£7,593,590	£4,822,464
29	3.75% 2038	11.9y	£5,427,051	£4,914,725
30	1.75% 2037	11.5y	£9,364,262	£6,818,098
31	4.25% 2036	10.0y	£11,646,554	£11,220,797
32	0.625% 2035	9.4y	£3,151,378	£2,192,523
33	4.5% 2034	8.5y	£6,580,188	£6,561,771
34	4.625% 2034	7.9y	£6,587,137	£6,664,178
35	0.875% 2033	7.4y	£625,554	£485,273
36	3.25% 2033	6.9y	£7,557,817	£7,078,411
37	4.25% 2032	6.2y	£809,338	£815,700
38	1% 2032	5.9y	£7,225,968	£6,008,843
39	0.25% 2031	5.4y	£2,814,174	£2,287,596
40	4.75% 2030	4.7y	£4,862,925	£5,060,887
41	4.375% 2030	4.0y	£8,584,976	£8,669,852
42	4.125% 2029	3.4y	£449,853	£454,001
43	0.5% 2029	2.9y	£7,108,661	£6,448,739
44	4.375% 2028	2.0y	£7,867,167	£7,937,065
45	0.125% 2028	1.9y	£90,020	£83,836
46	3.75% 2027	1.0y	£6,297,033	£6,294,142

Act 1 — Building the Liability

400 members, ages 55–65, each receiving £36,000/yr weighted by survival probability (Gompertz mortality model). Press ▶ to watch the liability staircase grow as members are layered in.

Act 2 — Available Gilt Instruments

61 gilts spanning the maturity spectrum. Click legend entries to toggle individual bond shapes.

Act 3 — Building the Hedge

Exposure-transport solver selects 46 bonds. Animation adds bonds longest-maturity-first, filling up the liability in -£/bp density space. Press ▶ to watch.

Exposure Transport Hedging

A shape-based alternative to NNLS

Russell Oxley
March 2026

Abstract

A nominal fixed-income liability has a continuous local forward-rate exposure profile over maturity. A single discounted cashflow generates a box. A coupon bond generates a staircase. A liability stream generates a larger staircase. Non-negative hedging can therefore be written as a shape-matching problem. Standard NNLS keeps this hedge object and minimises squared vertical error. The method proposed here keeps the same object and changes the metric. It treats exposure as mass spread across maturity and minimises the one-dimensional transport cost of moving hedge exposure into liability exposure, subject to exact exposure-area balance. The construction is close in spirit to cashflow transport, but it is applied directly to the rates target rather than to discounted payoff mass.

In a live 61-gilt comparison, exposure transport produces own-basis fit very close to NNLS, with a residual of 2.4% against 2.3%, while using 46 bonds rather than 43. In five HJM scenario families, it delivers lower tracking-error dispersion and less negative expected shortfall than NNLS in every family. The method preserves static hedge quality while improving robustness under stochastic forward-rate disturbances.

1. Exposure shapes

Let \(G_L(u)\) denote the liability's local instantaneous-forward exposure at maturity \(u\), and let \(G_j(u)\) denote the corresponding profile for bond \(j\).

\[ G(u)=p\,\mathbf 1_{\{u\le T\}} \]

For a discounted cashflow of present value \(p\) paid at time \(T\), the exposure profile is a box of height \(p\) from \(0\) to \(T\). A bond is a sum of boxes. A long-only bond portfolio is a non-negative linear combination of staircases. The liability is another staircase. The hedging problem is to choose non-negative weights \(x_j\) such that

\[ G_H(u;x)=\sum_{j=1}^N x_j\, G_j(u) \]

is close to \(G_L(u)\). The hedge object is already familiar from forward-segment hedging. Nothing new is being introduced at the level of basis or target. The change comes later, in the way distance is measured.

2. Exposure transport hedging

NNLS fits the exposure shapes with the usual vertical loss,

\[ \min_{x\ge 0}\int_0^U \bigl(G_H(u;x)-G_L(u)\bigr)^2\,du. \]

Exposure transport keeps the same hedge object and uses the maturity axis more directly. The cumulative exposure imbalance up to maturity \(u\) is

\[ H_x(u)\;=\;\int_0^u \bigl(G_H(s;x)-G_L(s)\bigr)\,ds. \]

If \(H_x(u)>0\), the hedge has too much exposure too early. If \(H_x(u)<0\), it is short early exposure. Exposure transport chooses the hedge that minimises the total imbalance carried across maturity:

\[ \min_{x\ge 0}\;\int_0^U |H_x(u)|\,du \qquad\text{subject to}\qquad \int_0^U G_H(u;x)\,du = \int_0^U G_L(u)\,du. \]

Proposition 1

Let \(G_L\) and \(G_H(\cdot;x)\) be non-negative integrable exposure profiles on \([0,U]\) with equal total area. Then the one-dimensional Wasserstein-1 distance between the exposure measures \(d\eta_L = G_L\,du\) and \(d\eta_H = G_H\,du\) is

\[ W_1(\eta_H,\eta_L)\;=\;\int_0^U |H_x(u)|\,du. \]

If both profiles are piecewise constant on an event grid \(0=e_0 < e_1 < \cdots < e_M = U\), then

\[ W_1(\eta_H,\eta_L) \;=\; \sum_{m=1}^{M} \int_0^{\Delta t_m} |h_m + r_m\, s|\,ds \]

where \(\Delta t_m = e_m - e_{m-1}\), \(r_m\) is the interval residual height, and \(h_m\) is the cumulative imbalance entering interval \(m\). The event-date formula follows because \(G_H - G_L\) is piecewise constant, so \(H_x\) is piecewise linear.

3. Relation to cashflow transport

Cashflow transport starts from discounted payoff mass at payment dates. Exposure transport starts from the forward-exposure profile itself. The geometry is similar. The target is different.

Cashflow transport tends naturally toward present-value balance. Exposure transport tends naturally toward exposure-area balance. For a rates hedge, that distinction is useful. The two transport ideas sit beside one another quite neatly. One is closer to dedication. The other is closer to rates hedging.

4. Relation to NNLS

NNLS and exposure transport work on the same hedge object. Both take the continuous local-forward exposure profile as given. The change is only in the norm.

NNLS penalises vertical differences point by point. Exposure transport penalises maturity displacement of exposure. The static fit should therefore be broadly similar, and in the live comparison it is. Own-basis residual moves from 2.3% under NNLS to 2.4% under exposure transport, while the number of active bonds rises from 43 to 46 in a 61-gilt universe.

5. Compute and implementation

The method looks continuous, but it is cheap to evaluate. Each exposure profile is piecewise constant, so \(H_x(u)\) is piecewise linear and the integral reduces to a finite sum. No full transport matrix is needed.

The optimisation is convex. Lot sizes, ticket costs and turnover penalties can be added without altering the geometry. Front-end weighting can be introduced by replacing \(du\) with \(d\phi(u)\).

6. HJM illustration

Five synthetic forward-rate families from an HJM engine: a smooth 3-factor family, a multiresolution 14-factor family, and three multiresolution families with increasing local residual components. Each family: 500 one-year scenarios, 15bp target mean absolute forward move.

Scenario family	Std TE (NNLS / CF / Exp)	Expected shortfall (NNLS / CF / Exp)	Wins (NNLS / CF / Exp)
Smooth 3-factor	7,010 / 16,037 / 237	−14,021 / −30,981 / −487	8 / 4 / 488
Multires 14-factor	7,681 / 16,792 / 6,157	−15,596 / −35,836 / −11,843	174 / 71 / 255
Multires 14f + local (low)	6,997 / 15,263 / 5,449	−14,418 / −31,905 / −10,489	169 / 66 / 265
Multires 14f + local (med)	6,160 / 13,256 / 4,604	−12,902 / −27,867 / −9,157	155 / 70 / 275
Multires 14f + local (high)	5,407 / 11,340 / 3,821	−11,680 / −23,247 / −7,839	146 / 79 / 275

Table 1. HJM tracking-error summary. Std TE and expected shortfall in pounds. Wins out of 500.

Exposure transport is very close to NNLS in static exposure fit, but in these five HJM families it has the lowest tracking-error dispersion and the least negative expected shortfall every time. Cashflow transport is weaker on these tests, which is unsurprising because it is aimed at a different target.

7. Discussion

Exposure transport gives a non-negative hedge that is easy to understand in the same visual language as cashflow transport while staying on the rates target.

A close first-order exposure match brings second-order and financing properties largely along with it. If \(G_H(u) \approx G_L(u)\) across maturity, then the convexity profiles are also closely matched, since the second-order sensitivity is determined by the shape of the first-order exposure. In the live comparison, funding RMSE is essentially identical between NNLS and exposure transport (both \(\approx\) £3.8m on a £180m liability), confirming that a tight exposure-shape match constrains cashflow timing well enough for the funding path to be well-behaved. These are not separate problems that need separate solutions: they are consequences of the same shape match.

The hedge is slightly denser than NNLS in the live comparison. Spread considerations sit outside the core metric. None of this weakens the central point: a rates hedge built from continuous forward-exposure shapes does not have to be fitted by squared vertical error.

8. Conclusion

Exposure transport keeps the standard hedge object and replaces the usual NNLS metric with a maturity-displacement metric. The resulting hedge matches total exposure area by construction and remains close to ordinary rates intuition. Static fit remains close to NNLS. Tracking under stochastic forward-rate disturbances improves materially. The method is easy to visualise and sits naturally beside cashflow transport.