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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 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

outline
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

problem formulation
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

pension fund cash flows
Pension Fund Cash Flows

Sponsoring Company

Figure 1: Pension Fund Cash Flows

  • Investment: portfolio of fixed income and cash
mathematical models
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
asset liability under uncertainty
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
scenario generation
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
linear programming model
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
linear programming model9
Linear Programming Model
  • Subject to:
    • Cash-flow accounting equation
    • Inventory balance
    • Cash-flow matching equation
    • PV01 matching
    • Holding limits
linear programming model10
Linear Programming Model

PV01 Deviation-Budget Trade Off

stochastic programming model
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
sp model constraints
SP Model Constraints
  • Cash-Flow Accounting Equation:
  • Inventory Balance Equation:
  • Present Value Matching of Assets and Liabilities:
sp constraints cont
SP Constraints cont.
  • Matching Equations:
  • Non-Anticipativity:
stochastic programming model14
Stochastic Programming Model

Deviation-Budget Trade-off

chance constrained programming model
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:
ccp model
CCP Model

Cash versus beta

ccp model17
CCP Model

SP versus CCP frontier

integrated chance constraints
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

discussion and future work
Discussion and Future Work

Generated Model Statistics:

discussion and future work20
Discussion and Future Work
  • Ex post Simulations:
    • Stress testing
    • In Sample testing
    • Backtesting
references
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.