Mathematical Programming Models for Asset and Liability Management Katharina Schwaiger, Cormac Lucas and Gautam Mitra, CARISMA, Brunel University West London

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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, 800-9.30am
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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

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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

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Pension Fund Cash Flows

• Figure 1 Pension Fund Cash Flows
• Investment portfolio of fixed income and cash

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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

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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

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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

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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

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Linear Programming Model

• Subject to
• Cash-flow accounting equation
• Inventory balance
• Cash-flow matching equation
• PV01 matching
• Holding limits

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Linear Programming Model

• PV01 Deviation-Budget Trade Off

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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

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SP Model Constraints

• Cash-Flow Accounting Equation
• Inventory Balance Equation
• Present Value Matching of Assets and Liabilities
  

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SP Constraints cont.

• Matching Equations
• Non-Anticipativity

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Stochastic Programming Model

• Deviation-Budget Trade-off

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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

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CCP Model

• Cash versus beta

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CCP Model

• SP versus CCP frontier

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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

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Discussion and Future Work

• Generated Model Statistics

LP SP CCP
Obj. Function 1 linear 22 nonzeros 1 linear 13500 nonzeros 1 linear 6751 nonzeros
CPU Time (Using CPLEX10.1 on a P4 3.0 GHZ machine) 0.0625 28.7656 1022.23
No. of Constraints 633 All linear 108681 nonzeros 66306 All linear 2538913 nonzeros 53750 All linear 1058606 nonzeros
No. of Variables 1243 all linear 34128 all linear 20627 6750 binary 13877 linear
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Discussion and Future Work

• Ex post Simulations
• Stress testing
• In Sample testing
• Backtesting

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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.