Asset-Liability Management

Ziemba & Mulvey (1998) · Multi-stage Stochastic Programming

Family 06ALMMulti-stage SP

Pension funds, insurers, and banks must match a stream of long-horizon liabilities (retirement payouts, insurance claims, depositor withdrawals) with a portfolio whose returns evolve stochastically over the same decades. The canonical formulation is a multi-stage stochastic programme (Ziemba & Mulvey 1998) over a scenario tree: at each node, decide allocation and contributions conditional on the realised history up to that node; across nodes, enforce non-anticipativity.

Educational Purpose

Educational demonstration of multi-stage stochastic programming applied to ALM. Not pension, insurance, or actuarial advice. Real ALM uses $10^4$-$10^6$ scenarios, calibrated econometric models, and regulatory-specific liability valuations.

The model

Scenario tree, funding ratio, multi-stage objective

OR family: Asset-liability management Solver class: Multi-stage LP/QP Measure: Physical $\mathbb{P}$ Realism: ★☆☆ Simulator

Setting

Time horizon $t = 0, 1, \ldots, T$. Asset universe $\{1, \ldots, n\}$ with stochastic returns $r_{i,t}(\omega)$ across scenarios $\omega$. Deterministic or stochastic liability stream $L_t(\omega)$. Funding ratio $F_t = A_t / \text{PV}(L_{\ge t})$. The plan sponsor wants to maximise expected surplus or probability that $F_t \ge 1$ at every stage, subject to regulatory floors.

Multi-stage stochastic programme

Canonical ALM formulation (Ziemba & Mulvey 1998) $$ \max_{\,\{w_{i,t,\omega}, c_{t,\omega}\}} \; \mathbb{E}^{\mathbb{P}}\!\left[\, U\!\bigl(A_T - L_T\bigr) \,\right] $$ $$ \text{s.t.} \quad A_{t+1,\omega} = \sum_i w_{i,t,\omega} (1 + r_{i,t,\omega}) - L_{t,\omega} + c_{t,\omega}, $$ $$ \qquad\quad \sum_i w_{i,t,\omega} = A_{t,\omega}, \quad w_{i,t,\omega} \ge 0 $$ $$ \qquad\quad w_{i,t,\omega} = w_{i,t,\omega'} \text{ if } \omega, \omega' \text{ agree up to } t \text{ (non-anticipativity)} $$

$U$ is a concave utility (or a CVaR-style tail penalty). $c_{t,\omega}$ are plan-sponsor contributions. Non-anticipativity couples scenarios sharing history. For scenario trees with $|N|$ nodes the programme has $\sim n |N|$ decisions.

Funding-ratio dynamics

Funding ratio evolution $$ F_{t+1}(\omega) \;=\; F_t(\omega) \cdot \frac{\sum_i w_{i,t,\omega}(1+r_{i,t,\omega}) / A_{t,\omega} \cdot A_{t,\omega} - L_{t,\omega} + c_{t,\omega}}{\text{PV}_{t+1}(L_{\ge t+1,\omega})} $$

Under-funded plans ($F_t < 1$) are in underfunding; ALM aims to keep a target funding ratio with high probability.

Solution approaches

For moderate trees the deterministic equivalent LP/QP is solved directly. For realistic $10^4$-$10^6$ scenarios, Benders decomposition, progressive hedging (Rockafellar-Wets), or stochastic dual dynamic programming (Pereira-Pinto 1991) are used. Nested scenario generation (moment-matching, Importance sampling) reduces tree size while preserving decision-relevant moments.

Interactive ALM simulator

Fund evolution under a fixed allocation policy

Pension funding-ratio simulator

Stocks share

60%

Bond return $\mu_B$

2.5%

Equity return $\mu_E$

7.0%

Equity vol $\sigma_E$

17%

Horizon (years)

Liabilities growth

3.0%

Initial funding ratio

1.00

Scenarios

200

Seed

Mean $F_T$

—

5th pctile $F_T$

—

95th pctile $F_T$

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$\mathbb{P}(F_T < 1)$

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Funding-ratio fan chart: median (gold) with 5-95 percentile band (light). Dashed red line = fully funded ($F = 1$). Higher stock shares lift the median but widen the underfunding tail.

Assumptions & extensions

Assumptions of this demo

Fixed allocation policy across time (the interactive demo); real ALM optimises the allocation path conditional on the scenario tree.
Two asset classes (stocks + bonds); real plans have 8-20 building blocks.
Deterministic liability growth; real liabilities depend on mortality, wage growth, discount-rate dynamics.
Lognormal equity returns; fat tails, regime shifts, and equity-bond correlation dynamics are all ignored.
No contributions, no regulatory floor; real plans have sponsor contribution rules, smoothing corridors, and solvency floors (Solvency II for insurers, PBGC for US pensions).

Extensions in the literature

Russell-Yasuda Kasai (Cariño et al.\ 1994): multi-period, multi-currency insurance ALM at Japan’s Yasuda Kasai.
InnoALM (Geyer & Ziemba 2008): Austrian pension ALM using stochastic programming.
Liability-Driven Investment (LDI): cash-flow-matching under yield-curve dynamics; overlaps with fixed-income and rate-model pages.
CVaR-based ALM: replace expected-utility objective with tail-risk objective (Rockafellar-Uryasev applied here).

Key references

Ziemba, W. T. & Mulvey, J. M. (eds.) (1998).

Worldwide Asset and Liability Modeling.

Cambridge University Press. ISBN 978-0-521-57108-3.

Cariño, D. R. et al. (1994).

The Russell-Yasuda Kasai Model: An Asset/Liability Model for a Japanese Insurance Company Using Multistage Stochastic Programming.

Interfaces, 24(1), 29–49. doi:10.1287/inte.24.1.29

Geyer, A. & Ziemba, W. T. (2008).

The Innovest Austrian Pension Fund Financial Planning Model: InnoALM.

Operations Research, 56(4), 797–810. doi:10.1287/opre.1080.0535

Consigli, G. & Dempster, M. A. H. (1998).

Dynamic Stochastic Programming for Asset-Liability Management.

Annals of Operations Research, 81, 131–162. doi:10.1023/A:1018992620324

Pereira, M. V. F. & Pinto, L. M. V. G. (1991).

Multi-stage Stochastic Optimization Applied to Energy Planning.

Mathematical Programming, 52, 359–375. doi:10.1007/BF01582895 (SDDP.)

Birge, J. R. & Louveaux, F. (2011).

Introduction to Stochastic Programming, 2nd ed.

Springer. ISBN 978-1-4614-0236-7.

Rockafellar, R. T. & Wets, R. J.-B. (1991).

Scenarios and Policy Aggregation in Optimization Under Uncertainty.

Mathematics of Operations Research, 16(1), 119–147. doi:10.1287/moor.16.1.119