Multi-Period Stochastic Programming Models for Dynamic Asset Allocation Norio Hibiki Abstract This paper discusses optimal dynamic investment policies for investors, who make the investment decisions in each of the asset categories over time. We construct the framework integrating stochastic optimization and Monte Carlo simulation for dynamic asset allocation, and we propose the linear programming models using simulated paths to solve a large-scale problem in practice. Linear programming models can be formulated to adopt either a åxed-value rule or a åxed-amount rule instead of the general åxed-proportion rule. These formulations can be simply implemented and solved very fast. Some numerical examples are tested to illustrate the characteristics of the models. These models can be used to improve the trade-oãbetween risk and expected wealth, and we can get interesting results forthe dynamic asset allocation policies. ntroduction The investors need to maximize the expected utility of thereturnsfrom an investment portfolio, orminimize the risk of returns subject to the required expected return. They must decide the optimal investment proportion of the portfolio in securities in order to meet the investors satisfaction. This paper discusses optimal dynamic investment policies for investors, who make the investment decisions in each of the asset categories over time. This problem is called dynamic asset allocation. Theasset allocation decisions are critical for investors with diversiåed portfolios. nstitutional investors must manage their strategic asset mix over time to achieve favorable returns subject to various uncertainties, policy and legal constraints, and other requirements. Two kinds of multi-period stochastic optimization models can be used to solve this problem, as follows; ( stochastic programming model using scenarios, (2 stochastic control(dynamic stochastic programming model using random samples. Critical issues for stochastic modeling involve the handling of uncertainty and decision rules. The decisions have tobe made independent upon knowingactual paths that will occur. Thus we must deåne decision variables and a set of constraints to prevent the optimization model from anticipating the future. n addition, we Faculty of Science and Technology, Keio University 3-4- Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan hibiki@ae.keio.ac.jp need the suécient paths of uncertainties to get a better accuracy with respect to the future possible events. We describe the characteristics of two models in order. The notion of scenarios is typically employed for modeling random parameters in the multi-period stochastic programming(mpsp model, and the path of uncertainty is revealed as a scenario tree. This model is based on the expansion of the decision space, taking into account the conditional nature of the scenario tree. Conditional decisions are made at each node. To assume that a representative set of scenarios is constructed which covers the set of possibilities to a suécient degree, the number of decision variables and constraints appearing in the scenario tree may grow exponentially. (See Mulvey and Ziemba [6, 7] in detail. On the other hand, the stochastic control model is originally proposed as the multi-period portfolio optimization model. The basic framework of this model is proposed by Merton[5] and Samuelson[8]. The stochastic control model can be used to reduce the decision space to a set of policies. n this paper, paths of uncertaintyare revealed as simulated paths with any stochastic process for implementing the dynamic stochastic control numerically. We can get a better accuracy of uncertainty by using simulated paths rather than a scenario tree. However, the investment decisions at each time must be limited to a åxed-proportion rule in general. n addition, though the stochastic control model can be formulated as a stochastic programming model, this is formulated as a non-convex (non-linear programming model, and it is diécult to solve the problem and to ånd the global optimum solution. Two stochastic optimization models involve trading oãçexible decision rules and the accuracy of uncertainty. n this paper, we construct the framework integrating stochastic optimization and Monte Carlo simulation for dynamic asset allocation, and we propose the alternative models using simulated paths, which can be formulated as linear programming models to solve a large-scale problem in practice. Linear programming models can be formulated to adopt either a åxed-value rule or a åxed-amount rule instead of the general åxedproportion rule. These formulations can be simply implemented and solved very fast by using a sophisticated mathematical programming software.
The paper is organized as follows. Section 2 introduces three types of models and their formulations using simulated paths. Section 3 presents numerical examples to the MPSP model with the åxed-amount rule. Section 4 provides some concluding remarks and ourfuture research. 2 The MPSP model using simulated paths 2. Preparation An asset return is a variable parameter, and its process is expressed by a stochastic diãerential equation, or a time series model. We can sample the path of each asset return on each simulation trial. We call it the \simulated path". The example of the simulated path is shown as in Figure. Price Path Figure. : Simulated path Path 2 Path 3 Path 4 Path 5 Next, we show the following three types of models; ç MPSP model with the åxed-proportion rule, 2ç MPSP model with the åxed-value rule, 3ç MPSP model with the åxed-amount rule. These models have the diãerent decision variables, or the diãerent investment decision rule. nvestment proportions are decision variables in the MPSP model with theåxed-proportion rule. The solution totheasset allocation decision of this model provides the recommended proportions. The investment decisions are various over time, however they are åxed on all paths at each time. Therefore, the investment proportions of an risky asset and cash to any path are same. The "åxed-" does not mean \buy and hold strategy", and \constant rebalance strategy". nvestment values are decision variablesin the MPSP model with the åxed-value rule. The investment values of an risky asset to any path are same, however the cash is diãerent in each path. n general, we call it a sample path, however the sample path is also used for the path on the scenario tree, which is also called the scenario path. Therefore we call it the simulated path to distinguish the scenario path with the simulated path. nvestment amounts are decision variables in the MPSP model with the åxed-amount rule. The investment amounts of an risky asset to any path are same, however the cash is diãerent in each path. n general, we decide the investment proportions for the asset allocation decision. Therefore, we start with the MPSP model with the åxed-proportion rule using simulated path. We invest n risky assets and cash. The investment is made at time 0(present, and time T is the planning horizon. 2.2 Modeling with the åxed-proportion rule Notations used in this model are as follows. ( Parameters : number of the simulated path ñ (i : rate of return of asset j of pathiin period t, (j = ;...;n; t = ;...;T ; i = ;...; r (i tä : interest rate of path i in period t (the rate at time tä is used, (t = ;...;T ; i = ;...; W 0 : initial wealth W G : target wealth at the planning horizon W E : required expected wealth at the planning horizon (2 Decision variables w : investment proportion of asset j at time t, (j = ;...;n; t = 0;...;TÄ c t : cash ratio at time t, (t = 0;...;TÄ t : wealth of path i at time t, (t = ;...;T ; i = ;...; q (i : shortfall below the target wealth of path i at the planning horizon, (i = ;...; We formulate the MPSP model with the åxedproportion rule as follows. ( nvestment decision at time 0 : w j0,c 0 w j0 +c 0 = (2 Wealth of path i at time, (i = ;...; 8 9 < ê ë = = j w : j0 + ( +r 0 c 0 ; W 0 (3 Wealth of path i at time t, (t = 2;...;T ; i = ;...; ( ê ë ê ë t = w j;tä + +r (i tä c tä tä (decision at timetä w j;tä +c tä =
(4 Terminal wealth, (i = ;...; ( ê ë ê ë T = jt w j; + +r (i c (5 Objective function The shortfall below the target wealth(the lower partial moment[, 3] at the planning horizon is minimized, subject to the requirement of the minimum expected terminal wealth. minimize X q (i subject to T +q(i ïw G ; (i = ;...; q (i ï 0; (i = ;...; X T ïw E This model is too diécult to ånd the global optimum solutions because Equations for (3 and (4 are nonlinear and non-convex constraints. 2.3 Modeling with the åxed-value rule The investment values are decided instead of the investment proportions in this model. The global optimum solutions can be easily derived in practice, because themodel isformulated as alinearprogrammingmodel. Added notations in this model are as follows; x : investment value of asset j at time t, (j = ;...;n; t = 0;...;TÄ, v 0 : cash at time 0, t : cash of path i at time t, (t = 0;...;TÄ ; i = ;...;. We formulate the MPSP model with the åxed-value rule as follows 2 ; minimize X q (i ; ( subject to (2 x j0 +v 0 =W 0; (3 2 Equations (6 and (7 are originally formulated as follows; X T ïw E; T +q(i ïw G; ê T ë jt ë ê ë x j; + +r (i : ê ë j x j0 + ( +r 0 v 0 ; = x j + ; (i = ;...;; (4 ê ë ê ë x j;tä + +r (i tä tä = x + t ; (t = 2;...;TÄ ; i = ;...;; (5 Ä Å +ñjt xj; + X ê ë +r (i ïw E;(6 ( ê ë ê ë jt x j; + +r (i +q (i ïw G; (i = ;...;; (7 x ï 0; (j = ;...;n; t = 0;...;TÄ ; (8 v 0ï 0; (9 t ï 0; (t = ;...;TÄ ; i = ;...;; (0 q (i ï 0; (i = ;...;; ( where ñ jt = X ñ (i jt. 2.4 Modeling with the åxed-amount rule The investment value multiplies the price by the amount. nvestment value = price (per unitçamount (units price = Ç base value (per unitçamount (units base value = relative price Ç investment base value The investment value can be divided into two parts, the relative price and investment base value. The relative price can be deåned as the ratio of the price to the base value. We can set the base value using the face value, the current value, and so on. New notations are introduced as follows; ö j0 : relative price of asset j at time 0, (j = ;...;n, : relative price of asset j of path i at time t, (j = ;...;n; t = ;...;T ; i = ;...;, z : investment amount of asset j at time t, (j = ;...;n; t = 0;...;TÄ. We formulate the MPSP model with the åxed-amount rule as follows; minimize X q (i ; (2 subject to (3 ö j0z j0 +v 0 =W 0; (4 j (zjäzj0 = ( +r0v0äv(i ; (i = ;...;; (5
ê ë (z Äz j;tä = +r (i tä tä Äv(i t ; (t = 2;...;TÄ ; i = ;...;; (6 ö jt z j; + X ê ë +r (i ïwe; (7 ( ê ë jt z j; + +r (i +q (i ïw G ; (i = ;...;; (8 z ï 0; (j = ;...;n; t = 0;...;TÄ ; (9 v 0ï 0; (20 t ï 0; (t = ;...;TÄ ; i = ;...;; (2 q (i ï 0; (i = ;...;: (22 2.5 Model selection Three MPSP models using simulated paths are not equivalent, while three MPSP models using scenarios are equivalent. Therefore, we can select the model in correspondence with the trading strategy. f we do not have to take the trading strategy with the åxedproportion rule, we should adopt the trading strategy with the åxed-value rule or with the åxed-amount rule because we can easily solve these models. 2.6 Problem size A problem size is as in Table. Table. : Problem size åxed-proportion åxed-value åxed-amount constraints y (T + ( + T + 2 T + 2 decision variables (T + + (n + T (n +T + (n +T + y Non-negative constraints are excluded. The problem size is very large, but it is very sparse. The relationship between the number of path and nonzero elements in the constraints to the MPSP model with the åxed-amount rule is as in Figure 2. 3 Numerical Examples We test the model using numerical examples. Some cases for the MPSP model with the åxed-amount rule are tested 3. Three periods model is solved, and one period is one month in this example. The number of simulated paths is 500. The number of constraints is,535 except non-negative constraints, and the number of decision variables is,58. All of the problems are solved using NUOPT 4. Computing time of this 3 Some cases for the MPSP model with the åxed-value rule are also tested. These results are referred to [4]. 4 NUOPT is the mathematical programming software, and it is developed by Mathematical System, nc. Figure 2: Relationship between the number of path and non-zero elements(n = 3, T = 4 middle-sized problem is about 2 seconds using Windows 98 Personal computer which has 700 MHz CPU and 256MB momory. The number of asset class is four, and we use stock, bond, CB(Convertible Bond, and cash. Simulated paths are generated based on the following data; è stock : Nikko stock performance index (TSE, è bond : Nikko bond performance index, è CB : Nikko CB performance index, è call rate. The rate of return ñ (i is generated as follows. ç The rate of return of assetj derived at timetis nomally distributed with mean ñ and standard deviation õ, and it is generated as follows; ñ (i =ñ +õ " (i ; where " (i is a random sample from a multi-variate standardized normal distribution. 2ç The random variable" (j = 0;...;n; t = ;...;T follows that " N (0; Ü; where Ü is (n + T Ç (n + T correlation matrix. Asset 0 is the interest rate, and ñ (i 0t is the change rate of interest rate. The call rate r (i t is calculated as follows; r (i = r 0 Ç ê r (i t = r (i tä Ç ê ë 0 0t ; ë ; (t = 2;...;TÄ : nitial (relative prices of stock, bond, and CB assume to be for simplicity. The initial call rate is
0.44%. The initial wealth and the target wealth are 00 million yen. We test eight cases where the objective functions and their related constraints to the required expected terminal wealth are various as in Table 2. Table. 2 : Test case case minimization of the risk case 2 minimization of the risk subject to W E = 0; 65 case 3 minimization of the risk subject to W E = 0; 80 case 4 minimization of the risk subject to W E = 0; 95 case 5 minimization of the risk subject to W E = 0; 20 case 6 minimization of the risk subject to W E = 0; 225 case 7 minimization of the risk subject to W E = 0; 240 case 8 maximization of the expected terminal wealth Table 3showstheinvestment amounts, which arethe solutions of this model. n case, because the risk is minimized, investors have more cash and bond rather than stock and CB. While investors have only stock in case 8, because the return is maximized. n case 7, investors have CB and stock because CB and stock are more risky assets than cash or bond. These results are intuitively appealing. We can get interesting results for the dynamic asset allocation policies. Table 4and Table 5show the average investment values, and the average investment proportions, which are calculated using the recommended amounts in Table 3. The expected wealth and LPM value are also computed as in Table 6. The expected terminal wealth is the return measure, and LPM(lower partial moments is the risk measure in this model. We can ånd the trade-oãbetween risk and return in Table 6. Figure3is thecumulativedistribution of theterminal wealth. The terminal wealth on each path is sorted by value, and it can be depicted as in Figure 3. We ånd that the larger the required expected terminal wealth is, the larger the volatility is. We can control the risk and return of the terminal wealth in some degree directly using this model. 4 Conclusion n this paper, we propose the multi-period linear stochastic programming model for the investment decision of the dynamic asset allocation policies. When the decision is the investment proportion, the stochastic programming model proposed in this study has the equivalent formulation to the stochastic control model, or dynamic stochastic programming model. However, this model is diécult to solve and to ånd the global optimum solutions. The models with the åxed-value rule and the åxed-amount rule can be formulated as the linear programming models. Thus, it can be solved very fast for the large scale problem in practice. The model with the åxed-amount rule is examined with numerical test. The trade-oãrelationship between return and risk, and the investment decision to each case is examined. The results are intuitively appealing. n our future research, we must investigate the characteristics of these models using numerical tests. References [] V.S. Bawa and E.B. Lindenberg, Capital Market Equilibrium in a Mean-Lower Partial Moment Framework, Journal of Financial Economics, Vol.5(977, pp.89{200. [2] M.J. Brennan, E.S. Schwartz and R. Lagnado, Strategic Asset Allocation, Journal of Economic Dynamics and Control, 2 (997, pp.377{403. [3] W.V. Harlow, Asset Allocation in a Downside- Risk Framework, Financial Analysts Journal, September-October(99, pp.28{40. [4] N. Hibiki, Multi-Period Stochastic Programming Models for Strategic Asset Allocation, Department of Administration Engineering, Keio University, Technical Report, No.99002, 999, (in Japanese. [5] R.C. Merton, Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case, The Review of Econmics and Statistics, 5 (969, pp.247{257. [6] J.M. Mulvey and W.T. Ziemba, Asset and Liability Allocation in a Global Environment, Chapter 5 in \Handbooks in OR & MS, Vol.9", edited by R.Jarrow et al., 995. [7] J.M. Mulvey and W.T. Ziemba, Asset and LiabilityManagement Systems for Long-Term nvestors: Discussion of the ssues, Chapter in \Worldwide Asset and Liability Modeling", edited by W.T. Ziemba and J.M. Mulvey, 998. [8] P.A. Samuelson, Lifetime Portfolio Selection by Dynamic Stochastic Programming, The Review of Econmics and Statistics, 5 (969, pp.239{246.
Table. 3 : nvestment amounts cash(average stock bond CB 0 2 0 2 0 2 0 2 case 7849.5 7865.2 6984. 0.0 0.0 0.0 383.0 862.2 3075.3 767.5 307.7 0.0 case 2 628.8 606.5 3227.4 0.0 655.7 223.4 2967.8 337.4 6569.0 903.3 36.4 0.0 case 3 3997.9 3856.0 063.0 239.9 002.3 423.0 452.2 5029.8 856.0 240.9 28.6 0.0 case 4 590.5 970.2 92.6 54.2 36.5 504.6 733.2 7899.5 9306.3 564. 0.0 0.0 case 5 0.0 68.7 23.3 864.7 825.2 956.4 5227.0 654.9 6958.4 3908.3 467. 878.8 case 6 0.0 92.9 79.2 984.8 2673.9 283.8 2732.6 3452.3 3694.6 6282.6 3682.5 4847.4 case 7 0.0 86. 20.6 67.8 353.2 909.6 0.0 392.4 926. 8328.2 5909.3 6959.9 case 8 0.0 0.0 0.0 0000.0 0000.0 0000.0 0.0 0.0 0.0 0.0 0.0 0.0 Table. 4 : nvestment values (average cash(average stock(average bond(average CB(average 0 2 0 2 0 2 0 2 case 7849.5 7865.2 6984. 0.0 0.0 0.0 383.0 873.9 33.8 767.5 30. 0.0 case 2 628.8 606.5 3227.4 0.0 66.3 227.3 2967.8 3338.2 665.3 903.3 36.7 0.0 case 3 3997.9 3856.0 063.0 239.9 00.8 430.3 452.2 506.2 8622.8 240.9 29.6 0.0 case 4 590.5 970.2 92.6 54.2 46. 53.3 733.2 7948.8 9423. 564. 0.0 0.0 case 5 0.0 68.7 23.3 864.7 840.7 972.9 5227.0 6582.7 7045.7 3908.3 478.6 908.5 case 6 0.0 92.9 79.2 984.8 2696.6 306.0 2732.6 3473.8 374.0 6282.6 37.5 4924. case 7 0.0 86. 20.6 67.8 3543.0 942.6 0.0 394.8 937.7 8328.2 5955.7 7070.0 case 8 0.0 0.0 0.0 0000.0 0084.7 072.9 0.0 0.0 0.0 0.0 0.0 0.0 Table. 5 : nvestment proportions (average cash(average stock(average bond(average CB(average 0 2 0 2 0 2 0 2 case 78.50% 78.27% 69.7% 0.00% 0.00% 0.00% 3.83% 8.65% 30.83% 7.67% 3.09% 0.00% case 2 6.29% 59.85% 3.94% 0.00% 6.58% 2.25% 29.68% 33.2% 65.8% 9.03% 0.37% 0.00% case 3 39.98% 38.34% 0.5% 2.40% 0.05% 4.25% 45.2% 50.32% 85.24% 2.4%.29% 0.00% case 4 5.9% 9.64%.90% 5.4%.38% 5.07% 73.3% 78.98% 93.03% 5.64% 0.00% 0.00% case 5 0.00%.67% 2.09% 8.65% 8.27% 9.59% 52.27% 65.38% 69.5% 39.08% 4.68% 8.8% case 6 0.00%.9%.75% 9.85% 26.75% 2.85% 27.33% 34.50% 36.90% 62.83% 36.83% 48.49% case 7 0.00%.85% 2.05% 6.72% 35.4% 9.0% 0.00% 3.92% 9.25% 83.28% 59.09% 69.60% case 8 0.00% 0.00% 0.00% 00.00% 00.00% 00.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% Table. 6 : Expected wealth and LPM value expected wealth LPM 0(initial 2 3(terminal case 0000.0 0049.2 0098.0 048.7 0.00 case 2 0000.0 0052.6 006.0 065.0 4.53 case 3 0000.0 0057.6 06. 080.0 4.20 case 4 0000.0 0065. 029. 095.0 26.69 case 5 0000.0 0070.7 040.4 020.0 49.30 case 6 0000.0 0074.8 050.3 0225.0 97.04 case 7 0000.0 0079.6 06.0 0240.0 57.24 case 8 0000.0 0084.7 072.9 0258.3 276.49 Figure. 3 : Cumulative distribution of the terminal wealth