June 2001 Universita’ degli Studi di Bergamo Corso 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

June 2001 Universita’ degli Studi di Bergamo Corso di dottorato di ricerca 2 Contents 1. Introduction: ALM model outline 2. Approximations: scenario trees parametric strategies 3. Benchmark decomposition 4. Modern risk measures: VaR 5. Solution techniques 6. Architecture of software system for ALM

June 2001 Universita’ degli Studi di Bergamo Corso di dottorato di ricerca 3 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

June 2001 Universita’ degli Studi di Bergamo Corso di dottorato di ricerca 4 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

June 2001 Universita’ degli Studi di Bergamo Corso di dottorato di ricerca 5 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

June 2001 Universita’ degli Studi di Bergamo Corso di dottorato di ricerca 6 Scenario tree t=0 t=1 t=2 Each node: values of risk factors decisions Huge amount of nodes: binomial tree with 10 random quantities each additional time period multiplies the number of nodes by 1000

June 2001 Universita’ degli Studi di Bergamo Corso di dottorato di ricerca 7 Scenario trees Some important theoretical studies and applications Allow rich decision structure But Require complex scenario generation procedures which –reflect dynamics of prices –are sound from the point of view of financial theory –affordable numerically Pflug & Swietanowski (1998), Hoyland & Wallace (1998) Require solution of huge convex optimization problem Example: 10 assets, one year horizon, one month time step: nodes

June 2001 Universita’ degli Studi di Bergamo Corso di dottorato di ricerca 8 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

June 2001 Universita’ degli Studi di Bergamo Corso di dottorato di ricerca 9 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

June 2001 Universita’ degli Studi di Bergamo Corso di dottorato di ricerca 10 Parametrization optimization problem is of relatively small size But 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

June 2001 Universita’ degli Studi di Bergamo Corso di dottorato di ricerca 11 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)

June 2001 Universita’ degli Studi di Bergamo Corso di dottorato di ricerca 12 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

June 2001 Universita’ degli Studi di Bergamo Corso di dottorato di ricerca 13 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

June 2001 Universita’ degli Studi di Bergamo Corso di dottorato di ricerca 14 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

June 2001 Universita’ degli Studi di Bergamo Corso di dottorato di ricerca 15 ALM Model Notations: Portfolio rebalancing time assets liabilities cash inflows debts portfolio relative return bought assets sold assets

June 2001 Universita’ degli Studi di Bergamo Corso di dottorato di ricerca 16 ALM Model, continued Cash flow buying transaction costs selling transaction costs dividends cash to service liabilities external cash inflow current debts newly acquired debts repaid debts debt servicing

June 2001 Universita’ degli Studi di Bergamo Corso di dottorato di ricerca 17 ALM Model, continued Debts Equity Regulatory constraints –portions of assets –cash reserves –debt restrictions –assets/liabilities ratio

June 2001 Universita’ degli Studi di Bergamo Corso di dottorato di ricerca 18 ALM Model, continued Performance measure random quantities decisions state variables strategies

June 2001 Universita’ degli Studi di Bergamo Corso di dottorato di ricerca 19 Parametric strategies Parameters Parametrization Problem

June 2001 Universita’ degli Studi di Bergamo Corso di dottorato di ricerca 20 Fix mix strategy LP to be solved for each time period

June 2001 Universita’ degli Studi di Bergamo Corso di dottorato di ricerca 21 Benchmark decomposition Benchmark Portfolio optimization problem

June 2001 Universita’ degli Studi di Bergamo Corso di dottorato di ricerca 22 Risk measures Relative regret Value at Risk Conditional VaR Uryasev & Rockafellar (1999)

June 2001 Universita’ degli Studi di Bergamo Corso di dottorato di ricerca 23 General picture Modeler Data LP-solver NLP solver File Excel,... MATLAB XPRESS(XBSL)

June 2001 Universita’ degli Studi di Bergamo Corso 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