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

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

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: LP SP
CCP
Obj. Function
1 linear
22 nonzeros
13500 nonzeros
6751 nonzeros
CPU Time
(Using CPLEX10.1 on a P4 3.0 GHZ machine)
0.0625
No. of Constraints
633
All linear
nonzeros
66306
nonzeros
53750
nonzeros
No. of Variables
1243
all linear
34128
20627
6750 binary
13877 linear

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.