mathematical programming models for asset and liability management katharina schwaiger, cormac lucas...
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Mathematical Programming Models for Asset and Liability Management
Katharina Schwaiger, Cormac Lucas and Gautam Mitra,CARISMA, Brunel University West London
11th Conference on Stochastic Programming (SPXI)University of Vienna, Austria 27th August – 31st August 2007
SESSION TA4, Tuesday 28th August, 9.30 am - 11:00 am
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
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
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
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
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
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
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
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Outline
Discussion
Problem Formulation
Scenario Models
Stochastic Programming
Linear Programming
Chance-Constrained
Programming
Salary Curve ExampleSalary Curves:
Outline
Discussion
Problem Formulation
Scenario Models
Stochastic Programming
Linear Programming
Chance-Constrained
Programming
Fund Population ExamplePension Fund Population:
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
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
Outline
Discussion
Problem Formulation
Scenario Models
Stochastic Programming
Linear Programming
Chance-Constrained
Programming
Linear Programming ModelPV01 Deviation-Budget Trade Off
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
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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:
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Outline
Discussion
Problem Formulation
Scenario Models
Stochastic Programming
Linear Programming
Chance-Constrained
Programming
SP Constraints cont.
• Matching Equations:
• Non-Anticipativity:
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Outline
Discussion
Problem Formulation
Scenario Models
Stochastic Programming
Linear Programming
Chance-Constrained
Programming
Stochastic Programming ModelDeviation-Budget Trade-off
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:
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Outline
Discussion
Problem Formulation
Scenario Models
Stochastic Programming
Linear Programming
Chance-Constrained
Programming
CCP ModelCash versus beta
Outline
Discussion
Problem Formulation
Scenario Models
Stochastic Programming
Linear Programming
Chance-Constrained
Programming
CCP ModelSP versus CCP frontier
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
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Outline
Discussion
Problem Formulation
Scenario Models
Stochastic Programming
Linear Programming
Chance-Constrained
Programming
ICCP ModelSP versus ICCP frontier:
Outline
Discussion
Problem Formulation
Scenario Models
Stochastic Programming
Linear Programming
Chance-Constrained
Programming
ICCP ModelSP versus ICCP:
Outline
Discussion
Problem Formulation
Scenario Models
Stochastic Programming
Linear Programming
Chance-Constrained
Programming
ICCP ModelSP versus ICCP:
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
LP SP CCP ICCP
Obj. Function
1 linear22 nonzeros
1 linear13500 nonzeros
1 linear6751 nonzeros
1 linear13500 nonzeros
CPU Time* 0.0625 28.7656 1022.23 56.7344
No. of Constraints
633All linear108681 nonzeros
66306All linear2538913 nonzeros
53750All linear1058606 nonzeros
66201All linear4255363 nonzeros
No. of Variables
1243all linear
34128all linear
206276750 binary13877 linear
34128all linear
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
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.