Optimum Allocation and Risk Measure in an Asset Liability Management Model for a Pension Fund Via Multistage Stochastic Programming and Bootstrap

Transcription

1 EngOpt International Conference on Engineering Optimization Rio de Janeiro, Brazil, 0-05 June Optimum Allocation and Risk Measure in an Asset Liability Management Model for a Pension Fund Via Multistage Stochastic Programming and Bootstrap Davi Valladão,2, Álvaro Veiga,2 Electrical Engineering Department and 2 Actuarial Department, PUC-Rio, Rio de Janeiro, Brazil Abstract This paper proposes an Asset Liability Management (ALM) multistage stochastic programming model and a new method for measuring and controlling the equilibrium risk of a pension fund in the Brazilian context. According to the Society of Actuaries, ALM can be defined as an ongoing financial management process of formulating, implementing, monitoring, and revising strategies related to assets, future investments, and liabilities in an attempt to achieve an organization s financial objectives, future cash flow needs, and capital requirements, given the risk tolerances and other constraints. This is an optimization problem based on a scenario tree structure called multistage stochastic programming model. A pension fund optimal portfolio is one of the most common ALM applications. The main financial objective of a pension fund is to ensure the benefit payments by investing money from contributions. To do this, the investment policy must assure two conditions: equilibrium and liquidity - long and short term solvency. Several articles in literature have proposed stochastic programming models for optimal investment allocation. However, in all those works, the equilibrium risk is measured at the end of the planning horizon by comparing the final wealth in each scenario to a fixed capital requirement obtained independently from the investment policy adopted, assuming a portfolio of one fixed income contract without risk. We propose a method for measuring and controlling the equilibrium risk that takes into account the fact that the pension fund manager will react to the evolution of the risk factors by rebalancing the portfolio. In order to approximate this effect, we will take the portfolio return scenarios embedded in the stochastic programming model as a sample to bootstrap future observations of the liability discount rate for the period beyond stochastic programming planning horizon. Keywords: ALM, Stochastic Programming, Pension Funds, Solvency Risk, Equilibrium Risk. Introduction The term Asset Liability Management (ALM) designates the practice of managing a business so that decisions and actions taken with respect to assets and liabilities are coordinated. The ALM is a critical activity for any organization that receives and invests money in order to fulfill capital requirements and future cash demands. ALM can be defined as an ongoing sound financial management process of formulating, implementing, monitoring, and revising strategies related to assets, future investments, and liabilities in an attempt to achieve an organization s financial objectives, future cash flow needs, and capital requirements, given the organization s risk tolerances and other constraints. The ALM can take substantially different aspects according to the context in which it is developed. Derivative traders understand assets and liabilities as similar entities, traded in the financial market. On the other hand, a corporate pension fund cannot change its liabilities as easily so the ALM is focused on the investment policy. The main financial objective of a pension fund is to ensure the payment of lifelong benefits by investing money from contributions. To do this, the allocation policy must assure two conditions: equilibrium and liquidity long and short term solvency, respectively. The first condition states that the value of the assets should always be large enough to pay all benefits until the plan extinction. In other words, the solvency capital (the difference between the total asset value and the net present liability value) should be positive. The second condition states that the investment program should provide cash enough to pay benefit and other expenses on time, which means that cash level must always be positive. Solvency capital and cash level are random variables affected both by the investment policy (a decision variable) and by the assets and liabilities future values (other random variables that represent risk factors of the problem). Thus, the risk tolerance of a pension fund can be expressed in terms of a low solvency capital and negative cash probabilities. In this paper, we propose a multistage stochastic programming model for asset liability management (ALM) and a new method for measuring and controlling the equilibrium risk of a pension fund in Brazil. Several articles in literature have proposed stochastic programming models for optimal allocation (Kallberg et al. (982), Cariño and Ziemba (998), Drijver et al. (2000), Kouwenberg (200), Koivu et al., (2007), among others). In all those works, the equilibrium risk can only be measured at end of the planning horizon by comparing the final wealth in each scenario to a fixed capital requirement obtained independently from the investment policy adopted. However, in an ALM problem, the liabilities discount rate should be taken as the portfolio returns (Veiga, 2003), which depend on the stochastic asset returns and the investment policy. In order to solve this problem, we propose an iterative method to measure the equilibrium risk. This method discards the fixed capital requirement approximation and uses the portfolio return scenarios embedded on the stochastic programming to generate, by bootstrap, the future returns that will be used to discount the net benefits flows beyond the stochastic programming planning horizon. The paper is organized as follows. After the introduction, we present the stochastic programming model and all processes to produce the necessary input to the optimization problem. These processes include a stochastic model for economic risk factors, a scenario tree generation method and financial models for the asset and liability involved. After that, the equilibrium risk measuring method will be developed and then some results and conclusions will be presented.

2 Multistage stochastic programming model A multistage stochastic programming model is an optimization problem that is solved simultaneously for all possible sequences of the states of a dynamic system, represented by an event tree. Each node is associated to a possible state of the system and has a unique predecessor and several successors. In this paper, we describe the ALM as a linear optimization model that, given an initial portfolio allocation, defines the capital movements between the asset classes as the decision variables. The classes are stocks, properties, bonds and cash and the model also includes loans, to cover possible cash shortages. The constraints of the problem are the balance equations that compute the incomes and outcomes, considering the decision taken, transaction costs and payments of interests when a loan occurs. There are also regulatory constraints, market liquidity constraints and asset inventory constraints (used to update the amount invested in each class at each period). The objective function is to maximize the final wealth with a penalty for not satisfying capital requirement restriction. The tree structure notation is such that the set of stages is defined as t,..., T and a node related to stage t is defined as n t,..., N t. The event tree chosen has 5 stages and a conditional branching structure of The stages have different lengths, the first and the second have year, the third has three years, the fourth has 5 years and the last one has 0 years. This length structure leads to a 20-year planning horizon. 0X Initial Allocation year year 3 year 5 year 0 year (0 branches) (60 branches) (360 branches) (0 branches) (5760 branches) Figure. Event tree t The optimization problem variables can be classified as decision variables or state variables. The decision variables are the real actions that the fund manager should do, and the state variables are obtained consequently. On the other hand, the coefficients are divided in deterministic (no uncertainty) and stochastic (risk factors with uncertainty). Decisions variables o c i (n t ) = amount bought of asset class i at node n t o v i (n t ) = amount sold of asset class i at node n t o e(n t ) = loan obtained at node n t State variables o a i (n t ) = amount invested in asset class i at node n t o y(n T ) = capital requirement excess at node n T o w(n T ) = capital requirement lack at node n T Deterministic parameters o pe = insolvency penalization o bo = solvency bonus o sp = spread between the borrowing rate and the short term interest rate o ma = maximum stock allocation (%) o ct = transaction cost (%) o cc i = market buying capacity of asset class i o cv i = market selling capacity of asset class i o a i (initial) = initial allocation of asset class i o L * = capital requirement at the end of planning horizon (t = T) Stochastic parameters o l(n t ) = nominal liability cash flow at node n t o r i (n t ) = return of asset class i between nodes n t- e n t

3 2. Objective function The objective function is to maximize the pension fund expected final wealth utility, which penalizes the insolvent scenarios at the end of the planning horizon. The insolvent scenarios are described by an insufficient final wealth to reach the capital requirement y n T = 0 e w n T > 0. On the other hand, the solvent scenarios have a final wealth that exceeds the capital requirement y n T 0 e w n T = 0. Given that pe > bo, the wealth utility at a final node n T is described as a piecewise linear concave function representing the risk-averse pension fund preferences. This assumption also ensures a global maximum of the optimization problem. For the illustrative example, pe = 2 and bo =. u wealt n T = u y n T, w n T = bo. y n T + pe. w n T () N T max z = E u wealt = n T = p. bo. y n T pe. w n T (2) 2.2 Balance constraints The balance constraints determine a consistent portfolio evolution over time. It specifies that the portfolio value is modifies by all cash flow sources such as the asset returns, loan payments and renewals, net benefit payments and transaction costs. + r i n t+. a i n t + e n t+ + sp + r 3 nt+. e n t l n t ct. [c i n t+ + v i n t+ = a i n t+, n t, n t+ as linked nodes, t {0,,, T 2} (3) These constraints have a modified version for the last stage t = T that defines the values of the variables y(n T ) and w(n T ) used on the objective function. The concave characteristic of this function ensures that if y n T > 0 then w n T = 0 and if w n T > 0 then y n T = 0. + r i n t+. a i n t + e n t+ + sp + r 3 nt+. e n t l n t ct. [c i n t+ + v i n t+ = y n T w(n T ), n T, n T as linked nodes () 2.3 Asset inventory constraint The asset inventory constraint specifies that the future value of an asset class i at node n t is equal to the present value of the same asset class adjusted for buying and selling at node n t+. Note that the asset class i =, ie, cash, is not included because there is no meaning in buying or selling cash. a i n t+ = + r i n t+. a i n t + c i n t+ v i n t+, i =,2,3, t {0,,, T 2} (5) a i n 0 = a i initial + c i n 0 v i n 0, i =,2,3 (6) 2. Regulatory constraint for stock allocation The Brazilian law determines that the maximum stock allocation is 50% of the portfolio value. Given that ma = 50% the stock allocation at each node is bounded as follows. a n t ma. a i n t, t =,, T (7) 2.3 Liquidity constraint This constraint represents the fact that large pension funds are not allowed to buy or sell a great amount of an asset class without affecting the respective market prices. So the transactions are bounded by the market capacity. c i n t cc i, t = 0,, T, i =,2,3 (8) v i n t cv i, t = 0,, T, i =,2,3 (9) 3. Stochastic model for economic risk factors The optimization model will give a solution that represents reality if, and only if, the risk factors are appropriately modeled. These risk factors include economic random variables related to the financial market and the economy as a whole. Now, a stochastic model for the economic risk factors will be developed to forecast these random variables over the planning horizon.

4 The stochastic model for economic risk factors chosen is a mean reversion Vector Auto-Regressive based on Dert (998). The variables are chosen to model appropriately the asset returns used as inputs for the optimization problem. The mean vector is specified exogenously giving to the fund manager more sensibility about how the optimal allocation responds to the economic risk factors. This model has quarterly data with a sample representing the Brazilian economy from 996Q2 to 2007Q2. X q μ = α X q μ + ε q, ε q ~N(0, Σ) (0) x jq = ln + y jq, j =,,5 () were, y jq = outputgrowt rate, rental growt rate, inflation rate, interest rate, stock return, j = j = 2 j = 3 j = j = 5 The time series statistics are decrypted in Table. It can be noticed that the historical mean of Brazilian interest rate is too high because of some international crisis (Asia-997 and Russia-998) and the hyperinflation process remnants at the beginning of the sample. For that reason the mean of the variables were estimated exogenously, based on the Brazilian Central Bank expectations, resulting in the following mean vector: μ = (%, %, %, 0%, 2%). Table Statistics, time series 996Q2-2007Q2 x x 2 x 3 x x 5 Mean 2.66% 3.6% 9.0% 8.58% 6.26% Median 2.96%.05% 6.% 7.22%.52% Maximum 3.98%.58% 50.8% 3.% 5.85% Minimum -7.26% -0.36% -6.0%.0% % Std. Dev..57%.6% 9.83%.7% 7.20% Skewness Kurtosis The other coefficients (α and Σ) are estimated using the Ordinary Least Squares method, and are given as follows: Table 2 Covariance matrix, quarterly 996Q2-2007Q2 x x 2 x 3 x x 5 x 0, , , , ,00008 x 2 0, , , , ,00023 x 3 0, , , , , x -0, , , , ,00053 x 5 0, , , , ,03603 Table 3 α coefficient, standard deviation in ( ), sample: 996Q2-2007Q2 x q x 2q- -0. x 3q x q x 5q x q (0.5288) (0.3395) (0.076) (0.7779) ( ) x 2q (0.55) (0.35) (0.0722) (0.7972) (0.0376) x 3q (0.3828) ( ) (0.95) (0.370) (0.0765) x q (0.0998) ( ) (0.030) (0.0697) (0.0222) x 5q ( ) ( ) (0.337) (0.7777) (0.6070). Scenario tree generation method The scenario tree generation method is based on the Adjusted Random Sampling of Kouwenberg (200). Some modifications were introduced in order to take into account the different time intervals between nodes in our event tree. We modify the notation of the equation (0) to make the correspondence between the stage t of the event tree and the quarter q of the model. So, Xt q is the risk factor vector of the quarter q from stage t. X t t q μ = α X q μ + ε t q, ε t q ~N(0, Σ) (2) The first step of the method is to generate a deterministic one-quarter forecast from the beginning of stage t (?) for each predecessor node. X t t q μ = α X q μ (3)

5 The first N t 2 values of ε q t ~N(0, Σ) are randomly generated: ε q t n t ~N 0, Σ, n t =,, N t 2 () In order to guarantee the mean and the other odd central moments as zero, as stated by the Normal distribution, we take the antithetic values. εt q n t + N t 2 ε q t n t, n t =,, N t 2 Another adjustment is made in order to fit the variances of the tree structure and the stochastic model. This adjustment is made for each component j individually. t η jq n t = N t. σ j N t ε q t n t 2 ε q t i, j =,,5, n t =,, N t Finally, these new residuals are used to compute the scenarios the risk factors. Using this new residual ηt q it is calculated the risk factors adjusted values. Xt t q i μ = α X q i μ + ηt q i, i =,, N t (6) This process is repeated for all quarters of stage t computing N t independent scenarios. The last observation of each scenario that belongs to stage t will initialize a set of conditional branches of stage t + restarting all over the process. X t+ t 0 = X lengt t n t (7) (5) (5) 5. Asset pricing model The asset pricing is an important part of ALM process. It consists of transforming the economic risk factors into the asset class returns. The stock return (8) is modeled as the return of stock index, the properties return (9) is modeled as the return on the rental activity, the bonds return (20) is the short term interest rate plus a deterministic spread and, finally the cash return is the short term interest rate (2). Consider a pair of linked nodes n t, n t, the returns are given as follows: r n t = stock index n t stock index n t, (8) r 2 n t = rental activity(n t ) rental activity(n t ) (9) r 3 n t = interest rate(n t ) + spread (20) r 3 n t = interest rate(n t ) (2) Following (8), (9), (20) and (2) the asset returns are calculated using functions of economic risk factors. Then the tree return structure (?) is computed as follows: r n t = exp r 2 n t = exp lengt t lengt t. x 5q q= lengt t lengt t. x 2q q= t n t t n t (22) (23) r 3 n t = exp lengt t lengt t. x q q= t n t + spread (2) r n t = exp lengt t lengt t. x q q= t n t (25)

6 6. Liability model We simulated a pension fund with 0200 participants distributed as 5000 active, retired 5200 pensioners. The participants have all a defined benefit plan to which they contribute, along with the plan sponsor, with 6% of their salary to receive a benefit of 90% of the last salary. Due to the large number of participants, the future net benefit will be very close to their expected values and can be computed by: contribution p, k = 6%. salary p, k. I deat. I retirement (26) benefit p, k = 90%. last_salary p, k. I deat. I retirement (27) E contribution p, k = 6%. salary p, k. q age p. I retirement (28) E benefit p, k = 90%. last_salary p, k. q age p. I retirement (29) The expected values are accumulated for all participants giving: 0200 RF k = E benefit p, k E contribution p, k p= (30) 7. Equilibrium risk measuring method The equilibrium risk is defined as the insolvency probability, i.e., the probability that the pension fund won t meet all obligations until its extinction. In other words, insolvency is a state where the total asset value at a defined instant of time is smaller than the net present value of the fund s liability cash flows, i.e. the technical reserve. The total asset value is easily calculated by the sum of the amount invested in each asset class. On the other hand, the technical reserve calculation is more complicated because the net present value of the fund s liability cash flows needs a discount rate that, following Veiga (2003), should be the fund s portfolio return. In the first years under study, this calculation is implicitly done by the stochastic optimization model. However, the liabilities horizon is, usually, longer than the period chosen to optimize the investment policy. Since the portfolio return and, consequently, the discount rate is known only for these first years, some assumptions are needed for the remaining period. Previous papers in the literature assume a fixed discount rate for this final period mostly based on a regulatory statement of the country under study. Since this choice for the discount rate is not necessarily related to actual future portfolio returns, which depend on the investment policy, it gives rise to arbitrary technical reserve and equilibrium risk measure. This work proposes a new method to better estimate the equilibrium risk. First an optimal solution is obtained with a null capital requirement (L = 0). After that the discount rate distribution is obtained bootstrapping the portfolio return embedded on the stochastic programming model. Then, a sequence of real liabilities cash flows is discounted to the end of the stochastic programming horizon using different sequences of bootstrapped real portfolio return to approximate the technical reserve distribution at the same date. Portfolio return bootstrap Discount rate Figure 2. Bootstrapped discount rate year To form a bootstrap sequence of the returns we choose the returns r(n s )according to the following probabilities. Let S and N be random variables that represent, respectively, the stage and the node to be bootstrapped as a future portfolio return. This process is based on the joint distribution of these two variables (3). P S = s, N = n = P N = n S. P S = s (3)

7 Probability... The conditional distribution of N given S and the marginal distribution of S are described as uniforms as follows: P N = n S = N s (32) P S = s = lengt s T t= lengt t (33) With the stochastic discount rate, the technical reserve distribution can be estimated and consequently a conditional insolvency probability is calculated for each final node of the tree structure. Finally, the insolvency probability at the root node (today) will be the average of all a conditional insolvency probabilities since all scenarios have the same probability. Technical reserve distribution Initial Allocation 0X Figure 3. Equilibrium risk measure Final Wealth ($) Conditional insolvency probability The insolvency probability has to be compared to the fund s risk tolerance to accept the optimal solution. If the equilibrium risk is on an acceptable level then the optimal allocation is defined. But if the equilibrium risk is too high, there are two possibilities to decrease the insolvency probability without changing the initial wealth: raising the insolvency penalization or changing the capital requirement (L ) from zero to one quantile of the technical reserve distribution. To test the possible equilibrium risk control we implemented the latter iterative method that changes the capital requirement. 8. Illustrative example The illustrative example has the objective of comparing two non-arbitrary equilibrium risk measures: the underfunding probability and the insolvency probability. An underfunding state is defined as a negative wealth at the end of the stochastic programming horizon while the insolvency state is described by a deficit at the end of the fund existence. Since the stochastic programming horizon is smaller than the fun existence, the underfunding probability is a low bound approximation for the insolvency probability, underestimating the actual equilibrium risk. The underfunding probability will be calculated as a proportion of the insolvent scenarios and the insolvency probability will be calculated with the bootstrap method proposed in this paper. An iterative method that increases the capital requirement (L ) will also be tested. First, it is proposed to run the whole process with several different initial wealth considering a null capital requirement (Figure ). This example confirms the theoretical result that the underfunding probability is an underestimation of the equilibrium risk, i.e., the insolvency probability. 00,00% 90,00% 80,00% 70,00% 60,00% 50,00% 0,00% 30,00% 20,00% 0,00% 0,00% Underfunding and Insolvency Probabilities Initial Wealth (millions of R$) Underfunding Insolvency Figure. Underfunding and Insolvency probability

8 Probability Second, it is proposed, for different initial wealth, an influential analysis of capital requirement on the insolvency probability. Four cases are considered: Case : Null capital requirement Case 2: Iterative method o Step : Null Capital requirement o Step 2: Capital requirement as the average technical reserve Case 3: Iterative method o Step : Null Capital requirement o Step 2: Capital requirement as the reserve with risk correction (% significance) Case : Capital requirement as a prefixed value (real discount rate: 6% by Brazilian law) highest value It is confirmed that when the capital requirement is increased the insolvency probability is decreased. Figure 5 also shows that the differences between each case are small suggesting that the main factor that influences the equilibrium risk is the initial wealth. 20,00% 8,00% 6,00%,00% 2,00% 0,00% 8,00% 6,00%,00% 2,00% 0,00% Insolvency Probability Initial Wealth (millions of R$) Case Case 2 Case 3 Case Figure 5. Insolvency probability 9. Conclusions This paper proposed an ALM multistage stochastic programming model for pension funds and a new methodology of measuring and controlling the equilibrium risk. The objective was to find the optimal allocation with an appropriately equilibrium risk measure. To do this, the whole process was divided into small parts and each one was described in details. A flowchart (Figure 6) can summarize the whole process. Stochastic model Scenario Tree Generation Liability Model Estimated coefficients Risk Factors Asset Pricing Liability cash flows Stochastic asset returns Optimization Model Optimal Allocation Figure 6. Flowchart Bootstrap Returns inflation Risk Measure Risk Acceptance New Capital Requirement or Penalization Stochastic technical reserve Insolvency probability Final Optimal Allocation The process begins with the estimation of the stochastic model coefficients used to generate the risk factors tree structured scenarios. Financial models use these scenarios to do the asset pricing while the liability cash flows are calculated. The asset returns and the liability cash flows are used as the stochastic programming inputs to find an optimal investment policy with null capital requirement. The optimal portfolio returns are bootstrapped to obtain the technical reserve distribution. Using this result, the insolvency probability is estimated as the average of the conditional insolvency probabilities that are calculated for each final node. If the risk acceptance criteria are satisfied, then the optimal allocation is defined, else another solution is obtained with a higher capital requirement or a higher insolvency penalization.

9 This proposed methodology can give a better estimative of the equilibrium risk involved in a pension fund scheme. The underfunding probability (previous work risk measure) is much smaller than the insolvency probability (this work risk measure). The underfunding probability underestimates the long term risk of a pension fund. It was also tested an iterative method, increasing the capital requirement, to control the equilibrium risk. This approach actually decreased the insolvency probability but it shows just small improvements confirming that the initial wealth is the most important feature to the insolvency probability. 0. References. ANDERSON, R. G.; HOFFMAN, D.; RASCHE, R. H. A Vector Error-Correction Forecasting Model of the U.S. Economy. Working Paper C, FED. Available in: < 2. CARIÑO, D. R.; MYERS, D. H.; ZIEMBA, W. T. Concepts, Technical Issues, and Uses of the Russell-Yasuda Kasai Financial Planning Model. Operations Research, Vol. 6, No.. (Jul. - Aug., 998), pp CARIÑO, D. R.; ZIEMBA, W. T. Formulation of the Russell-Yasuda Kasai Financial Planning Model. Operations Research, Vol. 6, No.. (Jul. - Aug., 998), pp DERT, C. A Dynamic Model for Asset Liability Management for Defined Benefit Pension Funds, in: W.T. ZIEMBA and M. J. MULVEY (eds), Worldwide Asset Liability Modeling - Cambridge University Press (998) 5. DRIJVER, S. J.; HANEVELD, W. K. K.; VLERK M. H. Asset Liability Management modeling using multistage mixed-integer Stochastic Programming. University of Groningen Research Report 00A52, (set.,2000). 6. GÜLPINAR, N.; RUSTEM, B.; SETTERGREN, R. Simulation and optimization approaches to scenario tree generation. Journal of Economic Dynamics & Control, Vol. 28 (200), pp HALL, A. D.; ANDERSON, H. M.; GRANGER, C. W. J. A Cointegration Analysis of Treasury Bill Yields. The Review of Economics and Statistics, Vol. 7, No.. (Feb., 992), pp HILLI, P.; KOIVU, M.; PENNANEN, T. A stochastic programming model for asset liability management of a Finnish pension company. Operations Research, Vol. 52 (Jul., 2007), pp KALLBERG, J. G.; WHITE, R. W.; ZIEMBA, W. T. Short Term Financial Planning under Uncertainty. Management Science, Vol. 28, No. 6. (Jun., 982), pp KOIVU, M.; PENNANEN, T.; RANNE, A. Modeling assets and liabilities of a Finnish pension insurance company: a VEqC approach. Working paper, Helsinki School of Economics, (200).. KOUWENBERG, R. Scenario generation and stochastic programming models for asset liability management. European Journal of Operational Research, Vol.3 (200), pp KUSY, M. I.; ZIEMBA, W. T. A Bank Asset and Liability Management Model. Operations Research, Vol. 3, No. 3. (May - Jun., 986), pp LITTERMAN, R.; SCHEINKMAN, J. Common Factors Affecting Bond Returns. The Journal of Fixed Income, (Jun., 99). MINELLA, A. Monetary Policy and Inflation in Brazil ( ): a VAR Estimation. Working paper series No. 33, Central Bank of Brazil, (Nov., 200) 5. VEIGA FILHO, A. L. Medidas de Risco de Equilíbrio em Fundos de Pensão. In: Antonio M. Duarte Jr., Gyorgy Vargas. (Org.). Gestão de Riscos no Brasil. Rio de Janeiro: Editora Financial Consultoria, 2003, v., p ZIEMBA, W. T. The Stochastic Programming Approach to Asset, Liability and Wealth Management. AIMR Publisher: Mai, 2003