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Asset/Liability Management Models in Insurance and
Benchmark Decomposition
Alexei A. Gaivoronski and Sergiy Krylov
Norwegian University of Science and Technology
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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)
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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)
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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)
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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