Stochastic programming approach to ALM in Finnish pension insurance companies p.1/36

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1 Stochastic programming approach to ALM in Finnish pension insurance companies Aktuaaritoiminnan kehittämissäätiön syysseminaari Teemu Pennanen Helsinki School of Economics Stochastic programming approach to ALM in Finnish pension insurance companies p.1/36

2 The problem The company has several billion euros in assets and is facing a situation where a large number of people is retiring at around The problem is to find an asset management/bonus transfer strategy that 1. covers the uncertain future liabilities, 2. satisfies the statutory restrictions, 3. is optimal (safe, profitable,... ) in the long run. Stochastic programming approach to ALM in Finnish pension insurance companies p.2/36

3 The problem Essential characteristics of the problem The planning horizon is years. Uncertainties: Investment returns, cash-flows, liabilities. Dynamics: Decisions are made sequentially under increasing information. This is an optimization problem. Stochastic programming approach to ALM in Finnish pension insurance companies p.3/36

4 At stage variable Stochastic Programming, observe the value of a random and make a decision. The decision will be a function of A realization of the sequences results in a cost. and. The problem is to find a strategy that minimizes subject to given constraints Stochastic programming approach to ALM in Finnish pension insurance companies p.4/36

5 Stochastic Programming 1. Model the problem 1.1 Model the optimization problem 1.2 Model the uncertainty 2. Solve the model 2.1 Discretize the model 2.2 Solve the discretized model Stochastic programming approach to ALM in Finnish pension insurance companies p.5/36

6 Stochastic Programming 1. Model the problem 1.1 Model the optimization problem 1.2 Model the uncertainty 2. Solve the model 2.1 Discretize the model 2.2 Solve the discretized model More efficient (optimal) than duration/convexity matching or pure simulation approaches. Much more general than the better known approaches of stochastic optimal control or dynamic programming. Stochastic programming approach to ALM in Finnish pension insurance companies p.6/36

7 Stochastic Programming 1. Model the problem 1.1 Model the optimization problem (convex analysis) 1.2 Model the uncertainty (stochastics, econometrics) 2. Solve the model 2.1 Discretize the model (variational analysis) 2.2 Solve the discretized model (math. programming) More efficient (optimal) than duration/convexity matching or pure simulation approaches. Much more general than the better known approaches of stochastic optimal control or dynamic programming. Stochastic programming approach to ALM in Finnish pension insurance companies p.7/36

8 1.1 Model the optimization problem Here, the aim is to specify so that stochastic factors decision variables objective constraints,,,, all important features of the real problem are captured and are convex. Stochastic programming approach to ALM in Finnish pension insurance companies p.8/36

9 1.1 Model the optimization problem Assets are considered as aggregate investment classes: cash, bonds, stocks, real estate, loans. The stochastic factors are, return on asset, dividend on asset!, cash flows from period " over period, technical reserves at time # over period,.,. Stochastic programming approach to ALM in Finnish pension insurance companies p.9/36

10 1.1 Model the optimization problem The decision variables are $ & ' ( holdings in asset purchases in asset sales in asset total wealth at time + in period at time, at time,, %, $ * ) transfers to bonus reserve at time solvency capital at time,, Stochastic programming approach to ALM in Finnish pension insurance companies p.10/36

11 % Model the optimization problem Inventory, budget, portfolio and transaction constraints: ' %& $ $ ' %&. $ -, $ " % 3 ' / * 2 ). +, % 1 & %0/ * ) ". $!, * % ) ' 3 / * 2 ). + -, %54 & 1 %/ * ) ( 287 $ ( 6 ( & ( ' Stochastic programming approach to ALM in Finnish pension insurance companies p.11/36

12 9 < ; > Solvency capital: 1.1 Model the optimization problem # ( Solvency border: = $ $ > = * =) 9% $ < * ; ) : where,, and are given parameters. Stochastic programming approach to ALM in Finnish pension insurance companies p.12/36

13 ? 1.1 Model the optimization problem Objective is to maximize expected utility: maximize # + : B 7 A@ subject to the above constraints and the requirement that the decision process + ( &' $ is adapted to the flow of information, i.e. the decision made at time only depends on the values of the stochastic process up to time + ( ' & $ # "!. Stochastic programming approach to ALM in Finnish pension insurance companies p.13/36

14 Stochastic Programming 1. Model the problem 1.1 Model the optimization problem 1.2 Model the uncertainty 2. Solve the model 2.1 Discretize of the model 2.2 Solve the discretized model Stochastic programming approach to ALM in Finnish pension insurance companies p.14/36

15 # "! K DC F 1.2 Model the uncertainty We express all the stochastic factors seven economic variables,,, in terms of = short term interest rate, = real estate price index, = long term bond yield, ED P LNMO = rental index, = stock price index, = wage index, GHJI = dividend index, whose development will be modeled by an econometric model. Stochastic programming approach to ALM in Finnish pension insurance companies p.15/36

16 ]] R W 1.2 Model the uncertainty T R QSR Asset class 0 ^_ ` WXZ ]UV \ RJ[ WYXZ UUV Cash ]-nr mz R W \ UmZl ikj h g ced_5f a\b \ R[ ced_ \b Bonds ]nr iprq_ o_ g iprq_5f \ U l g o_5f o_ g o_f Stocks h nr }zz { R_ ]yx tvuw s_ g R_5f tvuw a\ U l s_5f g s_ s_5f g Real estate ]-nr mz R W \ R[ \ UmZl Loans 1 where is the length of period in years and denotes the average duration of the company s bond portfolio.!s~ 4 depend on # The cash-flows and the technical reserves the population forecasts and the wage index. " Stochastic programming approach to ALM in Finnish pension insurance companies p.16/36

17 Ÿž ž are independent normally distributed with zero mean and ž 1.2 Model the uncertainty The quarterly development of Š Œ ƒ ƒ! ˆ ƒ 9 ƒ ' will be described with a VEqC model: %š., %0 Ž, Ž, and,, are parameters. The innovations Ÿ Ÿ0 ž., œ Ž where Ÿ0 Ÿ0 ž ž,, Ÿ Ÿ0. variance matrix Stochastic programming approach to ALM in Finnish pension insurance companies p.17/36

18 Stochastic Programming 1. Model the problem 1.1 Model the optimization problem 1.2 Model the uncertainty 2. Solve the model 2.1 Discretize of the model 2.2 Solve the discretized model Stochastic programming approach to ALM in Finnish pension insurance companies p.18/36

19 2 Solve the model Since has an infinite support, the space of possible strategies is infinite-dimensional. Stochastic optimal control or dynamic programming do not work here. We proceed by discretization and numerical computation. Stochastic programming approach to ALM in Finnish pension insurance companies p.19/36

20 & & 2.1 Discretize of the model Denote the stochastic factors by. We replace by a finitely supported measure ǻ ) where ǻ. is the unit mass on a given scenario, and Then the problem becomes finite-dimensional convex program. Constructing appropriately, these finite-dimensional approximations converge to the original problem as the number of scenarios is increased. Stochastic programming approach to ALM in Finnish pension insurance companies p.20/36

21 2.2 Solve the discretized model The problem now is to solve a (very large) finite-dimensional optimization problem. We use AMPL and MOSEK (an interior point solver for general convex problems). One could also use more efficient solvers tailored for stochastic programs over finite scenario trees. Stochastic programming approach to ALM in Finnish pension insurance companies p.21/36

22 Stochastic Programming 1. Model the problem 1.1 Model the optimization problem 1.2 Model the uncertainty 2. Solve the model 2.1 Discretize of the model 2.2 Solve the discretized model Stochastic programming approach to ALM in Finnish pension insurance companies p.22/36

23 3. Implementation MODEL DISCRETIZATION SOLUTION Optimization model AMPL Time series model QMC MOSEK Stochastic programming approach to ALM in Finnish pension insurance companies p.23/36

24 A discretized process (a) Short rate % (b) Bond rate % (c) Equity index (d) Dividend yield index Stochastic programming approach to ALM in Finnish pension insurance companies p.24/36

25 A discretized process (e) Real estate index (f) Rental index (g) Yearly cash flows (h) Technical reserves Stochastic programming approach to ALM in Finnish pension insurance companies p.25/36

26 z Optimal first stage portfolios as a function of z real estate, stocks, bonds, cash, loans With target zones Without target zones Stochastic programming approach to ALM in Finnish pension insurance companies p.26/36

27 Out-of-sample simulation Compare the performance of Stochastic Programming strategies with Fixed-Mix, Portfolio Insurance strategies. 325 out-of-sample scenarios over a 20 year time horizon Portfolio re-balancing every year i.e. FM and PI portfolios are rebalanced according to the given rules and SP problems are solved with respect to the current values of the stochastic parameters along the simulated scenario. Stochastic programming approach to ALM in Finnish pension insurance companies p.27/36

28 Out-of-sample simulation ±# vs. bankruptcy probability Stochastic programming approach to ALM in Finnish pension insurance companies p.28/36

29 Out-of-sample simulation Development of ±# over the simulation period (i) Stochastic Programming (j) Portfolio Insurance Stochastic programming approach to ALM in Finnish pension insurance companies p.29/36

30 : ± Out-of-sample simulation Distribution of at the beginning of the second period Stochastic Programming Portfolio Insurance Stochastic programming approach to ALM in Finnish pension insurance companies p.30/36

31 References Koivu M., Pennanen T. and Ranne A., Modeling assets and liabilities of a Finnish pension insurance company: a VEqC approach, Scandinavian Actuarial Journal, to appear. Hilli P., Koivu M., Pennanen T. and Ranne A., A stochastic programming model for asset liability management of a Finnish pension insurance company, Annals of Operations Research, to appear. Stochastic programming approach to ALM in Finnish pension insurance companies p.31/36

32 Another example: a pension fund The problem is the same as above except that there is no bonus payments to policy holders, there is no loans to policy holders, the company supports the fund, it is required that for all ³ ² : ±, stocks are divided into Finnish and international ones, instead of building a time series model for the nominal values we model real values and include inflation in the time series model. Stochastic programming approach to ALM in Finnish pension insurance companies p.32/36

33 Another example: a pension fund Initial and the optimized portfolio. 100% 90% 80% 70% 60% 50% 40% Real Stocks_Int Stocks_Fi Cash Bond 30% 20% 10% 0% Initial Optimized Stochastic programming approach to ALM in Finnish pension insurance companies p.33/36

34 Another example: a pension fund The optimal portfolio at the beginning of the second period as a function of. ±#.. 100% 80% Portfolio 60% 40% Real Stocks_Int Stocks_Fi Cash Bond 20% 0% Stochastic 0.56 programming approach to ALM in Finnish pension 0.76 insurance companies p.34/36

35 Another example: a pension fund The optimal portfolio at the beginning of the second period as a function of. ±# 100% 80% Portfolio 60% 40% Real Stocks_Int Stocks_Fi Cash Bond 20% 0% Stochastic programming approach to ALM in Finnish pension insurance companies p.35/36

36 µ Articles etc. can be found at the web-site of Stochastic Programming and its Applications in Finance at Helsinki School of Economics. *Funded by Academy of Finland Stochastic programming approach to ALM in Finnish pension insurance companies p.36/36