5. Results

Transcription

1 A stochastic programming model for asset liability management of a Finnish pension company Λ Petri Hilli, Matti Koivu, Teemu Pennanen Helsinki School of Economics Antero Ranne Mutual Pension Insurance Company Ilmarinen 1. Introduction 2. The optimization model 3. Stochastic factors 4. Numerical solution Λ Full paper available at

2 5. Results

3 Asset Liability Management The problem is to cover uncertain future liabilities of a large Finnish pension company by dynamic investment strategies under complicated constraints. The planning horizon is years. Uncertainties: Investment returns, cash-flows,... Dynamics: Decisions are made sequentially under incomplete information; when choosing the investment portfolio for a year, one should consider different possible realizations of the stochastic factors and the fact that one will update the portfolio also in the future.

4 Stochastic programming At stage t = 0; 1;:::;T, observe the value of a random variable ο t and make a decision x t. The decision x t will be a function of (ο 1 ;:::;ο t ). A realization of the sequences ο = (ο 1 ;:::;ο T ) and x = (x 1 ;:::;x T ) incurs a cost f(x 1 ;:::;x T ;ο 1 ;:::;ο T ): The problem is to minimize x2n E P f[x(ο);ο]; where N denotes the subspace of nonanticipative functions.

5 The optimization model Assets J = fcash, bonds, stocks, property, loansg. Inventory constraints h 0;j = h 0 j + p 0;j s 0;j h t;j = R t;j h t 1;j + p t;j s t;j t = 0;:::;T 1; where h 0 j = initial holdings in asset j, R t;j = return on asset j over period [t 1;t] are parameters, and p t;j = purchases in asset j at time t, s t;j = sales in asset j at time t, h t;j = holdings in asset j in period [t; t + 1] are decision variables.

6 Budget constraints X + c j2j(1 p j )p 0;j + H 1» X (1 c s j)s 0;j + F 0 ; j2j X + c j2j(1 p j )p t;j + fi t H t 1» X (1 c s j)s t;j j2j + X D t;j h t 1;j + F t ; j2j where c p j = transaction costs for buying asset j, c s j = transaction costs for selling asset j, fi t = length of period [t 1;t] in years, D t;j = dividend on asset j over period [t 1;t], F t = cash flows in period [t 1;t] are parameters and H t = yearly bonuses to be paid during [t; t + 1] are decision variables.

7 Portfolio constraints where l j w t» h t;j» u j w t ; w t = X h t;j for t = 0;:::;T 1 j2j is the total wealth and l j, u j are parameters. At the horizon, w T = X T;j + D T;j )h T 1;j + F T fi T H T 1 : j2j(r Transaction constraints p t;j» fi t b p j w t t = 0;:::;T 1; s t;j» fi t b s j w t t = 0;:::;T 1; where b p j, bs j are parameters.

8 Statutory restrictions are described in terms of technical reserves L t = present value" of future pension expenditure, solvency capital C t = w t (1 c G )L t ; solvency border B t = X s X a m j h t;j + b j2j j;k2j ff j;k h t;j h t;k ; where c G, a, b, m and ff are parameters. A company is declared bankrupt if C t» 0. The amount of bonuses is required to satisfy H t» 0:03 maxfc t B t ; 0g:

9 Objective function The Ministry of Social Affairs and Health defines four zones according to which companies' solvency situation is classified: C t =B t 2 [2; 1) : target C t =B t 2 [1; 2) : below target C t =B t 2 [0; 1) : crisis C t =B t 2 ( 1; 0) : bankrupt. We define the shortfall variables: SF t;1 2B t C t t = 1;:::;T 1; SF t;2 B t C t + H t =0:03 t = 1;:::;T 1; SF t;3 C t t = 1;:::;T:

10 For t = 0;:::;T 1, the state of the company will be measured by 3X u(c t ;B t ;H t ;L t ) = C t =L t z=1 fi z SF t;z =L t +u b (H t =L t ); where fi z are parameters and u b is a nondecreasing concave function. At stage T, the utility is measured by u T (C T ;L T ) = C T =L T fi 3 SF T;3 =L T : The objective function in our model is the discounted expected utility 8 9 < E P T X 1 = d : t u(c t ;B t ;H t ;L t ) + d T u T (C T ;L T ) ; ; t=1 where d t is the discount factor for stage t.

11 Problem summary Stochastic parameters: R t;j = return on asset j over period [t 1;t], D t;j = dividend on asset j over period [t 1;t], F t = cash flows from period [t 1;t], L t = technical reserves at time t, Decision variables: h t;j = holdings in asset j from period t to t + 1, p t;j = purchases in asset j at time t, s t;j = sales in asset j at time t, w t = total wealth at time t, H t = transfers to bonus reserve at time t, C t = solvency capital at time t, B t = solvency border at time t, SF t;z = shortfall from zone z at time t.

12 np maximize E P T 1 o t=1 d tu(c t ;B t ;H t ;L t )+d T u T (C T ;L T ) X j2j X j2j (1 + c p j )p 0;j + H 1» (1 + c p j )p t;j + fi t H t 1» h0;j = h 0 j + p 0;j s0;j; h t;j = R t;j h t 1;j + p t;j s t;j ; p t;j ;s t;j 0; w t = X j2j X j2j X j2j (1 c s j )s 0;j + F0; (1 c s j )s t;j + h t;j ; l j w t» h t;j» u j w t ; p t;j» fi t b p j w t; s t;j» fi t b s jw t ; C t = w t (1 c G )L t ; B t a X j2j m j h t;j + b X j2j s X SF t;1 2B t C t ; SF t;2 B t C t + 100=3H t ; SF t;3 C t ; for all t = 1;:::;T 1; j 2 J; w T = X j2j C T = w T (1 c G )L T ; SF T;3 C T ; (h; b; s; w; H;C; B; SF ) 2 N : D t;j h t 1;j + F t ; j;k2j ff j;k h t;j h t;k ; (R T;j + D T;j )h T 1;j + F T fi T H T 1:

13 The stochastic factors We express all the stochastic factors in terms of seven economic variables: sr = short term interest rate, br = long term bond yield, S = stock price index, Div = dividend index, P = property price index, Rent = rental index, W = wage index, whose development will be modelled by an econometric model. The cash-flows F t and technical reserves L t depend on the population forecasts and the wage index W.

14 Return and dividend will be expressed as Asset class Cash Bonds Stocks R t (1 + srt 1 )(1 + sr t ) fi t 1+brt 1 1+br t S t S t 1 P t P t 1 Property Loans 1 DM Asset class D t Cash 1 1 Bonds 2 (br t 1 + br t )fi t 1 Stocks 2 ( Div t 1 S + Div t t 1 S )fi t t 1 Property 2 (Rent t 1 P t 1 + Rent t P t ) 0:03 Loans 1 2 (br t 1 + br t )fi t, 2 fi t where fi t is the length of period t in years and D M denotes the average duration of the company's bond portfolio.

15 The quarterly development of 2 3 ln sr q ln br q ln S q x q = ln Div q ln P 6 q 4 7 ln Rent q 5 ln W q will be described with a Vector Equilibrium Correction (VEqC) model of the form ffix q = k X i=1 A i ffix q i + ff(fi 0 x q 1 μ) + ffl q ; where A i 2 R 7 7, fi 2 R 7 l, μ 2 R l, ff 2 R 7 l, ffix q := x q x q 1 ffi with ffi 2 R 7, and ffl q are independent normally distributed random variables with zero mean and variance matrix ± 2 R 7 7.

16 Writing the VEqC-model in the companion form μx q = μ Aμxq 1 + μc + μffl q ; we find that the vectors μx t in the formulas for R t and D t satisfy μx t = μ A 4fi t μx t 1 + X4fi t i=1 μa 4fi t i μc + e t ; where e t = P 4fi t μ i=1 A 4fit i μffl i is normally distributed with zero mean and the variance matrix μ± 4fit = X4fi t i=1 μa 4fi t i ± ( A μ T ) 4fi t i : 0 0

17 Numerical solution of the optimization model We discretize the model by approximating the distribution of the stochastic factors by a finite distribution in the form of a scenario tree. This results in a finite-dimensional optimization problem which is solved by an interior point solver.

18 INPUT COMPUTER SYSTEM OUTPUT Multiperiod stochastic optimization model Econometric model -Assets - Wage index Solution 1 Scenario generator Solver - Optimal strategy -Statistics -Graphics Data - AMPL -Mosek - Statistics - Graphics -Market data Liability model - Expert information -Cashflows - Technical reserves Data - Initial values - Population forecasts Figure 1: Stochastic optimization system

19 (a) Short rate % (b)bondrate% (c) Stock index (d) Dividend index (e) Property index (f) Rental index 2

20 property stocks bonds short rate loans 3 Initial SP1 SP2 SP3 SP4 SP5 Figure 2: Initial portfolio h 0 and the optimal portfolios corresponding to the parameter values in Table??.

21 (b) H t=l t. (a) C t=l t. 4

22 Sobol MC Figure 4: Convergence of the optimal value Branches per node

23 Fix Mix SP Bankruptcy probability Average C/L

24 (b) Fixed-mix (a) Stochastic programming