1. Mathematical Programming Models for Asset and Liability Management Katharina Schwaiger, Cormac Lucas and Gautam Mitra, CARISMA, Brunel University West London 11 th Conference on Stochastic Programming (SPXI) University of Vienna, Austria 27t h August – 31 st August 2007 SESSION TA4, Tuesday 28 th August, 9.30 am - 11:00 am

2. Outline Discussion Problem Formulation Scenario Models Stochastic Programming Linear Programming Chance- Constrained Programming 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. Outline Discussion Problem Formulation Scenario Models Stochastic Programming Linear Programming Chance- Constrained Programming 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. Outline Discussion Problem Formulation Scenario Models Stochastic Programming Linear Programming Chance- Constrained Programming Pensions: Introduction Two broad types of pension plans: defined- contribution and defined-benefit pension plans Defined-contribution plan: benefit depends on accumulated contributions at time of retirement Defined-benefit plan: benefit depends on length of employment and final salary We consider only defined-benefit pension plans

5. Outline Discussion Problem Formulation Scenario Models Stochastic Programming Linear Programming Chance- Constrained Programming Pension Fund Cash Flows Figure 1: Pension Fund Cash Flows Investment: portfolio of fixed income and cash Sponsoring Company

6. Outline Discussion Problem Formulation Scenario Models Stochastic Programming Linear Programming Chance- Constrained Programming 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 – Ex ante decision by Integrated Chance- Constrained Programming All models are multi-objective: (i) minimise deviations (PV01 or NPV) between assets and liabilities and (ii) reduce initial cash required

7. Outline Discussion Problem Formulation Scenario Models Stochastic Programming Linear Programming Chance- Constrained Programming 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

8. Outline Discussion Problem Formulation Scenario Models Stochastic Programming Linear Programming Chance- Constrained Programming 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 Pension Fund Population is determined using standard UK mortality tables

9. Outline Discussion Problem Formulation Scenario Models Stochastic Programming Linear Programming Chance- Constrained Programming Salary Curve Example Salary Curves:

10. Outline Discussion Problem Formulation Scenario Models Stochastic Programming Linear Programming Chance- Constrained Programming Fund Population Example Pension Fund Population:

11. Outline Discussion Problem Formulation Scenario Models Stochastic Programming Linear Programming Chance- Constrained Programming 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

12. Outline Discussion Problem Formulation Scenario Models Stochastic Programming Linear Programming Chance- Constrained Programming Linear Programming Model Subject to: – Cash-flow accounting equation – Inventory balance – Cash-flow matching equation – PV01 matching – Holding limits

13. Outline Discussion Problem Formulation Scenario Models Stochastic Programming Linear Programming Chance- Constrained Programming Linear Programming Model PV01 Deviation-Budget Trade Off

14. Outline Discussion Problem Formulation Scenario Models Stochastic Programming Linear Programming Chance- Constrained Programming 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

15. Outline Discussion Problem Formulation Scenario Models Stochastic Programming Linear Programming Chance- Constrained Programming SP Model Constraints Cash-Flow Accounting Equation: Inventory Balance Equation: Present Value Matching of Assets and Liabilities:

16. Outline Discussion Problem Formulation Scenario Models Stochastic Programming Linear Programming Chance- Constrained Programming SP Constraints cont. Matching Equations: Non-Anticipativity:

17. Outline Discussion Problem Formulation Scenario Models Stochastic Programming Linear Programming Chance- Constrained Programming Stochastic Programming Model Deviation-Budget Trade-off

18. Outline Discussion Problem Formulation Scenario Models Stochastic Programming Linear Programming Chance- Constrained Programming 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:

19. Outline Discussion Problem Formulation Scenario Models Stochastic Programming Linear Programming Chance- Constrained Programming CCP Model Cash versus beta

20. Outline Discussion Problem Formulation Scenario Models Stochastic Programming Linear Programming Chance- Constrained Programming CCP Model SP versus CCP frontier

21. Outline Discussion Problem Formulation Scenario Models Stochastic Programming Linear Programming Chance- Constrained Programming 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

22. Outline Discussion Problem Formulation Scenario Models Stochastic Programming Linear Programming Chance- Constrained Programming ICCP Model SP versus ICCP frontier:

23. Outline Discussion Problem Formulation Scenario Models Stochastic Programming Linear Programming Chance- Constrained Programming ICCP Model SP versus ICCP:

24. Outline Discussion Problem Formulation Scenario Models Stochastic Programming Linear Programming Chance- Constrained Programming ICCP Model SP versus ICCP:

25. Outline Discussion Problem Formulation Scenario Models Stochastic Programming Linear Programming Chance- Constrained Programming Discussion and Future Work Generated Model Statistics: * : Using CPLEX10.1 on a P4 3.0 GHz machine LPSPCCPICCP Obj. Function 1 linear 22 nonzeros 1 linear 13500 nonzeros 1 linear 6751 nonzeros 1 linear 13500 nonzeros CPU Time*0.062528.76561022.2356.7344 No. of Constraints 633 All linear 108681 nonzeros 66306 All linear 2538913 nonzeros 53750 All linear 1058606 nonzeros 66201 All linear 4255363 nonzeros No. of Variables 1243 all linear 34128 all linear 20627 6750 binary 13877 linear 34128 all linear

26. Outline Discussion Problem Formulation Scenario Models Stochastic Programming Linear Programming Chance- Constrained Programming Discussion and Future Work Ex post Simulations: – Stress testing – In Sample testing – Backtesting

27. 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.