mathematical programming models for asset and liability management katharina schwaiger, cormac lucas...

Mathematical Programming Models for Asset and Liability Management

Katharina Schwaiger, Cormac Lucas and Gautam Mitra,CARISMA, Brunel University West London

22nd European Conference on Operational ResearchPrague, July 8-11, 2007

Financial Optimisation I, Monday 9th July, 8:00-9.30am

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


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


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


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• 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


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


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

• Inflation rate scenarios are generated using ARIMA models

)()]1)(1[()(

)( yxyx PI

ymerit

xmeritss

Outline

Discussion

Problem Formulation

Scenario Models


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


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


Linear Programming

Chance-Constrained

Programming

Linear Programming ModelPV01 Deviation-Budget Trade Off

Outline

Discussion

Problem Formulation

Scenario Models


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

bxbybz

stbr

stle s

tLPV sbtBPV ,

C

Outline

Discussion

Problem Formulation

Scenario Models


Linear Programming

Chance-Constrained

Programming

SP Model Constraints• Cash-Flow Accounting Equation:

• Inventory Balance Equation:

• Present Value Matching of Assets and Liabilities:

B

bbb

B

bbb yPCxP

11

)1()1(

bbbb yxOz b

stt

st

st

st

stt

B

bb

sbt

B

bb

sbt

brrLPVdevudevo

lerxBPVzBPV

1

11

,

1

,

)1(

)1(

Outline

Discussion

Problem Formulation

Scenario Models


Linear Programming

Chance-Constrained

Programming

SP Constraints cont.

• Matching Equations:

• Non-Anticipativity:

ssB

b

sb

bb

b leLbrxczc 111

111 )(

stt

st

st

B

b

stt

stb

btb

bt lerleLbrrbrxczc 1

11 )1()1()(

sTT

sT

sT

B

b

sTTb

bTb

bT lerleLbrrxczc 1

11 )1()1()(

1, ss

s

1..2, Tts

]1,1[],1[

]1,1[],1[

slesle

sbrsbr

s

Outline

Discussion

Problem Formulation

Scenario Models


Linear Programming

Chance-Constrained

Programming

Stochastic Programming ModelDeviation-Budget Trade-off

Outline

Discussion

Problem Formulation

Scenario Models


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:

t

t 10 tt

st

st

stt

stt

st leAbrLN 11111

t

S

s

st

1

11

1,0st

1..1, Tts

ts,

Ss ...1

Outline

Discussion

Problem Formulation

Scenario Models


Linear Programming

Chance-Constrained

Programming

CCP ModelCash versus beta

Outline

Discussion

Problem Formulation

Scenario Models


Linear Programming

Chance-Constrained

Programming

CCP ModelSP versus CCP frontier

Outline

Discussion

Problem Formulation

Scenario Models


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

01111 st

stt

stt

st

st shortagebrLleA

t

S

s

st Lshortage ˆ

1

ts,

t

Outline

Discussion

Problem Formulation

Scenario Models


Linear Programming

Chance-Constrained

Programming

Discussion and Future Work

Generated Model Statistics:

LP SP CCP

Obj. Function 1 linear22 nonzeros

1 linear13500 nonzeros

1 linear6751 nonzeros

CPU Time(Using CPLEX10.1 on a P4 3.0 GHZ machine)

0.0625 28.7656 1022.23

No. of Constraints

633All linear108681 nonzeros



No. of Variables

1243all linear

34128all linear

206276750 binary13877 linear

Outline

Discussion

Problem Formulation

Scenario Models


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.

mathematical programming models for asset and liability management katharina schwaiger, cormac lucas...

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