asset/liability management models in insurance and benchmark decomposition

Alexei.Gaivoronski@iot.ntnu.no June 2001

Universita’ degli Studi di BergamoCorso di dottorato di ricerca

1

Asset/Liability Management Models in Insurance and

Benchmark Decomposition

Alexei A. Gaivoronski and Sergiy Krylov

Norwegian University of Science and Technology

Alexei.Gaivoronski@iot.ntnu.no June 2001

Universita’ degli Studi di BergamoCorso di dottorato di ricerca

2

Contents1. Introduction: ALM model outline2. Approximations:

scenario treesparametric strategies

3. Benchmark decomposition4. Modern risk measures: VaR 5. Solution techniques6. Architecture of software system

for ALM

Alexei.Gaivoronski@iot.ntnu.no June 2001

Universita’ degli Studi di BergamoCorso di dottorato di ricerca

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Literature• D.R. Carino and W. Ziemba (1998)• G. Consigli and M.A.H. Dempster (1998)• A. Consiglio, F. Cocco and S. Zenios (2000)• J. Dupacova, M. Bertocchi and V. Moriggia (1998)• A. A. Gaivoronski and Petter de Lange (1999)• K. Hoyland and S. Wallace (1998)• P. Klaassen (1998)• H. Mausser and D. Rosen (1998)• J. Mulvey and H. Vladimirou (1992)• G. Pflug and A. Swietanowski (1998)• S. Zenios, M. Holmer, R. McKendall and C.

Vassiadou-Zeniou (1998)• W. Ziemba and J. Mulvey (eds.), Worldwide Asset

and Liability Management, Cambridge Univ. Press, 1998

Alexei.Gaivoronski@iot.ntnu.no June 2001

Universita’ degli Studi di BergamoCorso di dottorato di ricerca

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Asset/liability management

• maximize expected utility of wealth or related objective function

• maintain competitiveness• maintain adequate reserves and cash levels• meet regulatory requirements

Determine a portfolio investment strategy over time in order to meet a sequence

of liability payments in the future

Insurance company

Alexei.Gaivoronski@iot.ntnu.no June 2001

Universita’ degli Studi di BergamoCorso di dottorato di ricerca

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Motivation

• Increased interest for adequate risk management from the part of industry

• Integrated ALM models are a challenge– dynamics and uncertainty– complex intertvined structure of

assets/liabilities/regulatory requirement• Approximations to reality are inevitable

– modeling tradeoffs between decision flexibility and representation of uncertainty

• Two main approximation approaches:– scenario trees– parametric strategies

Alexei.Gaivoronski@iot.ntnu.no June 2001

Universita’ degli Studi di BergamoCorso di dottorato di ricerca

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Scenario tree

t=0

t=1

t=2

Each node:• values of risk factors• decisionsHuge amount of nodes:binomial tree with 10 random quantitieseach additional time period multiplies the number of nodes by 1000

Alexei.Gaivoronski@iot.ntnu.no June 2001

Universita’ degli Studi di BergamoCorso di dottorato di ricerca

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Scenario trees• Some important theoretical studies and

applications• Allow rich decision structureBut• Require complex scenario generation

procedures which– reflect dynamics of prices– are sound from the point of view of financial

theory– affordable numericallyPflug & Swietanowski (1998), Hoyland & Wallace (1998)

• Require solution of huge convex optimization problem

Example: 10 assets, one year horizon, one month time step: 1036 nodes

Alexei.Gaivoronski@iot.ntnu.no June 2001

Universita’ degli Studi di BergamoCorso di dottorato di ricerca

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Scenario trees

• easy to represent “mainstream” events, difficult to represent events of relatively small probability

• consequently, difficult to meaningfully utilize modern risk measures like Value-at-Risk

t=0

t=1

Alexei.Gaivoronski@iot.ntnu.no June 2001

Universita’ degli Studi di BergamoCorso di dottorato di ricerca

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Parametrization• select a class of strategies which

represent asset and liability management decision as a function of state which depends on relatively small set of parameters

• optimize the system performance with respect to these parameters

Example: fix mix strategy: parameters - fraction of total asset value invested in a given asset

Scenario optimization• Allows much richer and more adequate

representation of dynamics of risk factors• Allows consideration of small probability

events and, consequently, VaR

Alexei.Gaivoronski@iot.ntnu.no June 2001

Universita’ degli Studi di BergamoCorso di dottorato di ricerca

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Parametrization• optimization problem is of relatively

small sizeBut• decision set is relatively restricted• how to elect good family of strategies is

far from clear• optimization problem is not convex and

may have local minima• estimation of performance necessary for

optimization may be time consuming

Tradeoff between adequate representation of uncertainty and richness of decision set

Alexei.Gaivoronski@iot.ntnu.no June 2001

Universita’ degli Studi di BergamoCorso di dottorato di ricerca

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Combined approach

t=0

t=1

t=2

• scenario tree with decisions on nodes for the first few periods

• parametric strategies on later periods

A.A.Gaivoronski & P. de Lange (1999)

Alexei.Gaivoronski@iot.ntnu.no June 2001

Universita’ degli Studi di BergamoCorso di dottorato di ricerca

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Benchmark decomposition• Objectives:

– reduce the size of the model and yet preserve expressive power

– Permit straightforward utilization of modern risk management approaches, like VaR

• Method: substitute the original large model with sequence of smaller models

• Approach– select benchmark wealth growth process– choose asset portfolio from

performance/risk tradeoff relative to benchmark

– optimize liability part with respect to remaining decisions and performance/risk tradeoff

Alexei.Gaivoronski@iot.ntnu.no June 2001

Universita’ degli Studi di BergamoCorso di dottorato di ricerca

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Top level view of modeling process

• Liability management • Debt/equity structure• Regulatory constraints• Integrated ALM performance

• Selection of portfolio of assets• Portfolio risk management

• benchmark• relative performance/risk tradeoff

Alexei.Gaivoronski@iot.ntnu.no June 2001

Universita’ degli Studi di BergamoCorso di dottorato di ricerca

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Model structure

• Benchmark– market index– wealth growth process– liability growth for products with

guarantees• ALM Model components

– liability process– portfolio rebalancing– cash flow– debts– equity– regulatory constraints– performance objective– decisions

Alexei.Gaivoronski@iot.ntnu.no June 2001

Universita’ degli Studi di BergamoCorso di dottorato di ricerca

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ALM Model• Notations:

• Portfolio rebalancing

t 1, . . . , T timei 1, . . . , Ij 1, . . . , Jk 1, . . . , Kl 1, . . . , L

assets

liabilitiescash inflowsdebts

xit

xit

xit

rit

portfolio

relative return

bought assets

sold assets

xit 1 1 ri

txit xi

t 1 xit 1

xit 1 1 ri

txit

Alexei.Gaivoronski@iot.ntnu.no June 2001

Universita’ degli Studi di BergamoCorso di dottorato di ricerca

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ALM Model, continued• Cash flow

ci

ci

d it

yjt

zkt

vlt

vlt

vlt

w lt

i 1

I

1 ci xi

t i 1

Id i

txit

k 1

Kzk

t l 1

Lvl

t

i 1

I

1 ci xi

t j 1

Jyj

t l 1

Lw l

tvlt

l 1

Lvl

t

buying transaction costs

selling transaction costs

dividends

cash to service liabilities

external cash inflow

current debts

newly acquired debts

repaid debts

debt servicing

Alexei.Gaivoronski@iot.ntnu.no June 2001

Universita’ degli Studi di BergamoCorso di dottorato di ricerca

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ALM Model, continued• Debts

• Equity• Regulatory constraints

– portions of assets

– cash reserves

– debt restrictions

– assets/liabilities ratio

vlt 1 vl

t vlt vl

t

xmt bm

i 1

Ixi

t , m M

x1t C t

l 1

Lvl

t V t

i 1

Ixi

t l 1

Lvl

t eE 1

Tj 1

Jb , tyj

Alexei.Gaivoronski@iot.ntnu.no June 2001

Universita’ degli Studi di BergamoCorso di dottorato di ricerca

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ALM Model, continued• Performance measure

• random quantities

• decisions

• state variables

• strategies

maxx it ,x i

t ,v lt ,v l

t E i 1

Ixi

T l 1

Lvl

T j 1

Jyj

T

t rit , yj

t , zkt , w l

t , d it , C t , b , t

xit , xi

t , vlt , vl

t

t 1, . . . , t

u t xt , vt

xit t 1, xi

t t 1, vlt t 1, vl

t t 1

Alexei.Gaivoronski@iot.ntnu.no June 2001

Universita’ degli Studi di BergamoCorso di dottorato di ricerca

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Parametric strategies• Parameters

• Parametrization

• Problem

xit i

t a, t , u t, xit i

t a, t , u t,vi

t lt a, t , u t, vi

t lt a, t , u t

a A Rn

maxaA E i 1

Ixi

T l 1

Lvl

T j 1

Jyj

T

xit 1 1 ri

txit i

t 1a it 1a

it 1a 1 ri

txit

vlt 1 vl

t lt a l

t a

i 1

I

1 ci i

t a i 1

Id i

txit

k 1

Kzk

t l 1

L l

t a

i 1

I

1 ci i

t a j 1

Jyj

t l 1

Lw l

tvlt

l 1

L l

t a

Alexei.Gaivoronski@iot.ntnu.no June 2001

Universita’ degli Studi di BergamoCorso di dottorato di ricerca

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Fix mix strategy• LP to be solved for each time period

min it , i

t , lt , l

t i 1

I

ci i

t ci i

t l 1

L

lt l

t

1 rqt xi

t it 1 i

t 1

a ix

q 1

I1 rq

t xqt

q 1

I q

t 1 q 1

I q

t 1

it 1 1 ri

txit

vlt l

t lt a l

v q 1

Lvq

t q 1

L q

t q 1

L q

t

i 1

I

1 ci i

t i 1

Id i

txit

k 1

Kzk

t l 1

L l

t

i 1

I

1 ci i

t j 1

Jyj

t l 1

Lw l

tvlt

l 1

L l

t

Alexei.Gaivoronski@iot.ntnu.no June 2001

Universita’ degli Studi di BergamoCorso di dottorato di ricerca

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Benchmark decomposition• Benchmark

• Portfolio optimization problem

R tR t 1 Rt

Q t R t 1R t

P t i 1

I1 r i

tx it

i 1

Ix i

t

a ix x i

t

q 1

Ixq

t

mina xAx Hax, Q

i 1

I ia i

x EQ

i 1

Ia i

x 1, a ix 0

Alexei.Gaivoronski@iot.ntnu.no June 2001

Universita’ degli Studi di BergamoCorso di dottorato di ricerca

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Risk measures• Relative regret

• Value at Risk

• Conditional VaR

Hax, Q max 0, Q i 1

Iria i

x

sup V | P i 1

Iria i

x Q V 1

E i 1

Iria i

x |i 1

Iria i

x Q V

Uryasev & Rockafellar (1999)

Alexei.Gaivoronski@iot.ntnu.no June 2001

Universita’ degli Studi di BergamoCorso di dottorato di ricerca

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General picture

ModelerData

LP-solverNLP solver

File

0102030405060708090

1st Qtr 2nd Qtr 3rd Qtr 4th Qtr

EastWestNorth

Excel,...

MATLAB

XPRESS(XBSL)

Alexei.Gaivoronski@iot.ntnu.no June 2001

Universita’ degli Studi di BergamoCorso di dottorato di ricerca

24

Summary

• asset/liability management by stochastic optimization of simulation model

• curse of dimensionality is beatable by consideration of parametrized policies

• alternative risk measures like VaR can be incorporated in the model

• customization of modern nonlinear optimization tools allow solution of advanced models