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Mathematical Programming Models for Asset and Liability Management Katharina Schwaiger, Cormac Lucas and Gautam Mitra, C

Mathematical Programming Models for Asset and Liability Management Katharina Schwaiger, Cormac Lucas and Gautam Mitra, CARISMA, Brunel University West London. 22 nd European Conference on Operational Research Prague, July 8-11, 2007 Financial Optimisation I, Monday 9 th July, 8:00-9.30am.

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Mathematical Programming Models for Asset and Liability Management Katharina Schwaiger, Cormac Lucas and Gautam Mitra, C

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  1. Mathematical Programming Models for Asset and Liability ManagementKatharina Schwaiger, Cormac Lucas and Gautam Mitra,CARISMA, Brunel University West London 22nd European Conference on Operational Research Prague, July 8-11, 2007 Financial Optimisation I, Monday 9th July, 8:00-9.30am

  2. Outline Problem Formulation Scenario Models for Assets and Liabilities Mathematical Programming Models and Results: Linear Programming Model Stochastic Programming Model Chance-Constrained Programming Model Integrated Chance-Constrained Programming Model Discussion and Future Work

  3. Problem Formulation • Pension funds wish to make integrated financial decisions to match and outperform liabilities • Last decade experienced low yields and a fall in the equity market • Risk-Return approach does not fully take into account regulations (UK case) use of Asset Liability Management Models

  4. Pension Fund Cash Flows Sponsoring Company Figure 1: Pension Fund Cash Flows • Investment: portfolio of fixed income and cash

  5. Mathematical Models • Different ALM models: • Ex ante decision by Linear Programming (LP) • Ex ante decision by Stochastic Programming (SP) • Ex ante decision by Chance-Constrained Programming • All models are multi-objective: (i) minimise deviations (PV01 or NPV) between assets and liabilities and (ii) reduce initial cash required

  6. Asset/Liability under uncertainty • Future asset returns and liabilities are random • Generated scenarios reflect uncertainty • Discount factor (interest rates) for bonds and liabilities is random • Pension fund population (affected by mortality) and defined benefit payments (affected by final salaries) are random

  7. Scenario Generation • LIBOR scenarios are generated using the Cox, Ingersoll, and Ross Model (1985) • Salary curves are a function of productivity (P), merit and inflation (I) rates • Inflation rate scenarios are generated using ARIMA models

  8. Linear Programming Model • Deterministic with decision variables being: • Amount of bonds sold • Amount of bonds bought • Amount of bonds held • PV01 over and under deviations • Initial cash injected • Amount lent • Amount borrowed • Multi-Objective: • Minimise total PV01 deviations between assets and liabilities • Minimise initial injected cash

  9. Linear Programming Model • Subject to: • Cash-flow accounting equation • Inventory balance • Cash-flow matching equation • PV01 matching • Holding limits

  10. Linear Programming Model PV01 Deviation-Budget Trade Off

  11. Stochastic Programming Model • Two-stage stochastic programming model with amount of bonds held , sold and bought and the initial cash being first stage decision variables • Amount borrowed , lent and deviation of asset and liability present values ( , ) are the non-implementable stochastic decision variables • Multi-objective: • Minimise total present value deviations between assets and liabilities • Minimise initial cash required

  12. SP Model Constraints • Cash-Flow Accounting Equation: • Inventory Balance Equation: • Present Value Matching of Assets and Liabilities:

  13. SP Constraints cont. • Matching Equations: • Non-Anticipativity:

  14. Stochastic Programming Model Deviation-Budget Trade-off

  15. Chance-Constrained Programming Model • Introduce a reliability level , where , which is the probability of satisfying a constraint and is the level of meeting the liabilities, i.e. it should be greater than 1 in our case • Scenarios are equally weighted, hence • The corresponding chance constraints are:

  16. CCP Model Cash versus beta

  17. CCP Model SP versus CCP frontier

  18. Integrated Chance Constraints • Introduced by Klein Haneveld [1986] • Not only the probability of underfunding is important, but also the amount of underfunding (conceptually close to conditional surplus-at-risk CSaR) is important Where is the shortfall parameter

  19. Discussion and Future Work Generated Model Statistics:

  20. Discussion and Future Work • Ex post Simulations: • Stress testing • In Sample testing • Backtesting

  21. References • J.C. Cox, J.E. Ingersoll Jr, and S.A. Ross. A Theory of the Term Structure of Interest Rates, Econometrica, 1985. • R. Fourer, D.M. Gay and B.W. Kernighan. AMPL: A Modeling Language for Mathematical Programming. Thomson/Brooks/Cole, 2003. • W.K.K. Haneveld. Duality in stochastic linear and dynamic programming. Volume 274 of Lecture Notes in Economics and Mathematical Systems. Springer Verlag, Berlin, 1986. • W.K.K. Haneveld and M.H. van der Vlerk. An ALM Model for Pension Funds using Integrated Chance Constraints. University of Gröningen, 2005. • K. Schwaiger, C. Lucas and G. Mitra. Models and Solution Methods for Liability Determined Investment. Working paper, CARISMA Brunel University, 2007. • H.E. Winklevoss. Pension Mathematics with Numerical Illustrations. University of Pennsylvania Press, 1993.

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