Asset/Liability Management Models in Insurance and Benchmark Decomposition

1. Asset/Liability Management Models in Insurance and Benchmark Decomposition Alexei A. Gaivoronski and Sergiy Krylov Norwegian University of Science and Technology 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 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 Universita’ degli Studi di Bergamo Corso di dottorato di ricerca

4. Asset/liability management Determine a portfolio investment strategy over time in order to meet a sequence of liability payments in the future • maximize expected utility of wealth or related objective function • maintain competitiveness • maintain adequate reserves and cash levels • meet regulatory requirements Insurance company 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 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 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: 1036 nodes 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 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 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 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) 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 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 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 Universita’ degli Studi di Bergamo Corso di dottorato di ricerca

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

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

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

19. Parametric strategies • Parameters • Parametrization • Problem Universita’ degli Studi di Bergamo Corso di dottorato di ricerca

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

21. Benchmark decomposition • Benchmark • Portfolio optimization problem Universita’ degli Studi di Bergamo Corso di dottorato di ricerca

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

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