Multi-period Portfolio Research Articles

Multiperiod Portfolio Optimization with Terminal Liability: Bounds for the Convex Case

This paper is concerned with an investor trading in multiple securities over many time periods in order to meet an outstanding liability at some future date. The investor is concerned with maximizing the expected profits from portfolio rebalancing under an initial wealth restriction to meet the future liabilities. We formulate the problem as a discrete-time stochastic optimization model and allow asset prices to have continuous probability distributions on compact domains. For the case of Markovian price uncertainty and convex terminal liability, we develop a simplicial approximation, under which bounds on the problem can be computed efficiently. Computations only require evaluating a dynamic programming recursion, which thus, allows its application to problems with a large number of trading periods. The bounds are tight in that they are exact in certain cases. Numerical results are given to demonstrate the computational efficiency of the procedure.

Multiperiod consumption and portfolio decisions under the multivariate GARCH model with transaction costs and CVaR-based risk control

This paper examines the modelling and solution method of complex multiperiod optimal consumption and investment problems with several kinds of constraints. Our work differs from previous results in several ways: typical market imperfections such as short sale constraints, proportional transaction costs are considered simultaneously; the MGARCH model is adopted to provide a satisfactory description of the time-varying behavior of stock returns; the CVaR constraint is introduced to control the wealth loss risk while maximizing the expected utility; it is assumed that the investor wants to maximize the expected utility of both his intermediate consumptions and his terminal wealth; most importantly, the considered problem is described as a stochastic programming problem, which can easily cope with problems with arbitrary utility functions, multiple risky assets and many periods. The derived model can help the investor to find robust consumption-investment decisions. The procedure to solve the resulting nonlinear stochastic optimization problem is discussed in detail. Numerical results show the suitability and promise of our methodology.

Multi-period stochastic optimization models for dynamic asset allocation

Institutional investors manage their strategic asset mix over time to achieve favorable returns subject to various uncertainties, policy and legal constraints, and other requirements. One may use a multi-period portfolio optimization model in order to determine an optimal asset mix. The concept of scenarios is typically employed for modeling random parameters in a multi-period stochastic programming model, and scenarios are constructed via a tree structure. Recently, an alternative stochastic programming model with simulated paths was proposed by Hibiki [Hibiki, N., 2001b. A hybrid simulation/tree multi-period stochastic programming model for optimal asset allocation. In: Takahashi, H. (Ed.), The Japanese Association of Financial Econometrics and Engineering. JAFEE Journal 89–119 (in Japanese); Hibiki, N., 2003. A hybrid simulation/tree stochastic optimization model for dynamic asset allocation. In: Scherer, B. (Ed.), Asset and Liability Management Tools: A Handbook for Best Practice, Risk Books, pp. 269–294], and it is called a hybrid model. The advantage of the simulated path structure compared to the tree structure is to give a better accuracy to describe uncertainties of asset returns. In this paper, we compare the two types of multi-period stochastic optimization models, and clarify that the hybrid model can evaluate and control risk better than the scenario tree model using some numerical tests. According to the numerical results, an efficient frontier of the hybrid model with the fixed-proportion strategy dominates that of the scenario tree model when we evaluate them on simulated paths. Moreover, optimal solutions of the hybrid model are more appropriate than those of the scenario tree model.

Comonotonic Approximations for Optimal Portfolio Selection Problems

We investigate multiperiod portfolio selection problems in a Black & Scholes type market where a basket of 1 riskfree and m risky securities are traded continuously. We look for the optimal allocation of wealth within the class of ’constant mix’ portfolios. First, we consider the portfolio selection problem of a decision maker who invests money at predetermined points in time in order to obtain a target capital at the end of the time period under consideration. A second problem concerns a decision maker who invests some amount of money (the initial wealth or provision) in order to be able to fullfil a series of future consumptions or payment obligations. Several optimality criteria and their interpretation within Yaari’s dual theory of choice under risk are presented. For both selection problems, we propose accurate approximations based on the concept of comonotonicity, as studied in Dhaene, Denuit, Goovaerts, Kaas & Vyncke (2002 a,b). Our analytical approach avoids simulation, and hence reduces the computing effort drastically.

A COMPARISON AMONG PERFORMANCE MEASURES IN PORTFOLIO THEORY

This paper examines some performance measures to be considered as an alternative of the Sharpe Ratio. More specifically, we analyze allocation problems taking into consideration portfolio selection models based on different performance ratios. For each allocation problem, we compare the maximum expected utility observing all the portfolio selection approaches proposed here. We also discuss an ex-post multi-period portfolio selection analysis in order to describe and compare the sample path of the final wealth processes.

Markowitz's Mean-Variance Portfolio Selection With Regime Switching: From Discrete-Time Models to Their Continuous-Time Limits

We study a discrete-time version of Markowitz's mean-variance portfolio selection problem where the market parameters depend on the market mode (regime) that jumps among a finite number of states. The random regime switching is delineated by a finite-state Markov chain, based on which a discrete-time Markov modulated portfolio selection model is presented. Such models either arise from multiperiod portfolio selections or result from numerical solution of continuous-time problems. The natural connections between discrete-time models and their continuous-time counterpart are revealed. Since the Markov chain frequently has a large state space, to reduce the complexity, an aggregated process with smaller state-space is introduced and the underlying portfolio selection is formulated as a two-time-scale problem. We prove that the process of interest yields a switching diffusion limit using weak convergence methods. Next, based on the optimal control of the limit process obtained from our recent work, we devise portfolio selection strategies for the original problem and demonstrate their asymptotic optimality.

Risk Control Over Bankruptcy in Dynamic Portfolio Selection: A Generalized Mean-Variance Formulation

For an investor to claim his wealth resulted from his multiperiod portfolio policy, he has to sustain a possibility of bankruptcy before reaching the end of an investment horizon. Risk control over bankruptcy is thus an indispensable ingredient of optimal dynamic portfolio selection. We propose in this note a generalized mean-variance model via which an optimal investment policy can be generated to help investors not only achieve an optimal return in the sense of a mean-variance tradeoff, but also have a good risk control over bankruptcy. One key difficulty in solving the proposed generalized mean-variance model is the nonseparability in the associated stochastic control problem in the sense of dynamic programming. A solution scheme using embedding is developed in this note to overcome this difficulty and to obtain an analytical optimal portfolio policy.

Advantages of Multiperiod Portfolio Models

A multiperiod portfolio model provides significant advantages over traditional single-period approaches—especially for long-term investors. Such a framework can enhance risk-adjusted performance and help investors evaluate the probability of reaching financial goals by linking asset and liability policies. Multiperiod portfolio models consist of three basic components: a stochastic scenario generator; a policy rule simulator; and an optimization module that identifies non-dominated solutions. Useful applications are in pension planning, insurance risk management, hedge funds, and asset allocation for individual investors.

Generalized Markowitz mean–variance principles for multi–period portfolio–selection problems

The multi–period portfolio–selection problem is formulated as a Markowitz mean–variance optimization problem. It is shown that the single–period Markowitz quadratic programming algorithm can be used to solve the multi–period asset–allocation problem with suitable modifications in the covariance and linear constraint matrices. It is assumed that the number of shares invested in risky assets is deterministic and the amount of money invested in the risk–free asset is random at future trading dates when short sales on assets are allowed. The general covariance matrix in the multi–period setup contains intertemporal correlations between assets, in addition to correlations between assets at all trading dates. Analytical solutions for the optimal trading strategy, which is linear in the risk–aversion parameter, are obtained. The efficient frontier is a straight line in the expected return/standard deviation of the portfolio space. When the dynamics of the risky assets follow geometric Brownian motion, it is shown that the time–zero allocations to the risky assets coincide with those obtained by Merton in the continuous–time framework. When short sales are not allowed on assets, the values of the portfolio at future trading dates may not be conserved. In the modified Markowitz mean–variance formulation, the value of the portfolio at future trading dates is conserved in the expected sense, and the optimal trading strategy is selected so that the deviation is minimized in the least–square sense. The efficient frontier is a parabola in the expected return/total variance space when short–sales are allowed and when the portfolio consists of risky assets only. When short sales are not allowed, it is shown that finding the optimal trading strategy is equivalent to solving the single–period Markowitz quadratic programming problem, with suitable modifications in the covariance and linear constraint matrices. By solving the Karush–Kuhn–Tucker conditions, analytical solutions are obtained for a two–period two–assets case, and the Markowitz mean–variance principle is illustrated by solving a test problem numerically using the revised simplex method.

Growth and utility maximization

The concept of growth maximization as a criterion for multiperiod portfolio selection is generalized. It is shown that relaxing the condition of maximum growth can lead to an inconsistency with the utility maximization concept.

Trend-following hedge funds and multi-period asset allocation

Selected hedge funds employ trend-following strategies in an attempt to achieve superior risk-adjusted returns. We employ a lookback straddle approach for evaluating the return characteristics of a trend-following strategy. The strategies can improve investor performance in the context of a multi-period dynamic portfolio model. The gains are achieved by taking advantage of the funds' high level of volatility. A set of empirical results confirms the advantages of the lookback straddle for investors at the top end of the multi-period efficient frontier.

Value management

Growth maximization as a criterion for multiperiod portfolio selection implies zero consumption before the planning horizon. To allow for intermediate consumption in this paper growth maximization is generalized by the condition that the initial portfolio value follows a given growth pattern. It is shown that a solution can be found by solving an appropriate nonlinear optimization problem. The analysis is carried out under conditions of certainty and uncertainty.

Multiagent Cooperative Search for Portfolio Selection

We present a new multiagent model for the multiperiod portfolio selection problem. A system of cooperative agents divide initial wealth and follow individual worst-case optimal investment strategies from random portfolios, sharing their final profits and losses. The multiagent system achieves better average-case performance than a single agent with the same initial wealth in a simple stochastic market. A further increase in performance is achieved through communication of hints between agents and probabilistic strategy-switching. However, this explicit cooperation is redundant in a market that approximates the Capital Asset Pricing Model, a model of equilibrium stock price dynamics. Journal of Economic Literature Classification Numbers: C63, C73, D83, G11.

Optimal Dynamic Portfolio Selection: Multiperiod Mean‐Variance Formulation

The mean‐variance formulation by Markowitz in the 1950s paved a foundation for modern portfolio selection analysis in a single period. This paper considers an analytical optimal solution to the mean‐variance formulation in multiperiod portfolio selection. Specifically, analytical optimal portfolio policy and analytical expression of the mean‐variance efficient frontier are derived in this paper for the multiperiod mean‐variance formulation. An efficient algorithm is also proposed for finding an optimal portfolio policy to maximize a utility function of the expected value and the variance of the terminal wealth.

Estimating Portfolio and Consumption Choice: A Conditional Euler Equations Approach

This paper develops a nonparametric approach to examine how portfolio and consumption choice depends on variables that forecast time‐varying investment opportunities. I estimate single‐period and multiperiod portfolio and consumption rules of an investor with constant relative risk aversion and a one‐month to 20‐year horizon. The investor allocates wealth to the NYSE index and a 30‐day Treasury bill. I find that the portfolio choice varies significantly with the dividend yield, default premium, term premium, and lagged excess return. Furthermore, the optimal decisions depend on the investor's horizon and rebalancing frequency.

Analytics Underlying the Ag Metallgesellschaft Hedge: Short Term Forwards in a Multi-Period Environment

In a highly publicized example, a marketing and refining subsidiary of AG Metallgesellschaft controlled short term forward positions reportedly equivalent to 160 million barrels of oil, 60 times the daily output of Kuwait. Presumably, the short term positions were taken to hedge oil contracts to customers over an extended period. This paper develops the analytics underlying the hedging of long term flow commitments with short term forward contracts. Its contributions include the determination of minimum variance hedging paths for multi-period flow portfolios and theevaluation of both period-by-period and end-of-horizon volatilities under various hedging schemes. The analysis allows for basis risk, non-zero cost-of-carry, and spot diffusion processes that have a non-zero market price of risk. The results are developed in the context of an efficient market and standard equilibrium pricing models.

Long-Term Portfolio Returns from Timber and Financial Assets

Executive Summary. This study estimates inflation adjusted multiperiod portfolio returns from direct real estate investments in Douglas fir and Southern pine timber stands combined with common stocks, corporate bonds, U.S. Government bonds, and U.S. Government Treasury bills over the fifty-eight-year period 1937-1994. A theoretical timber returns benchmark is created that is highly correlated to historical timber prices. The wealth accumulation one may have realized during the 1937-1994 period from investments in Douglas fir, Southern pine and common stocks is presented. The wealth accumulation is substantial for each of these assets, but the timber assets display both a higher return and higher variability. An empirical risk-return relationship for the financial market investments is developed using a multiperiod portfolio optimization technique. Timber asset returns are then included with security returns as input for the portfolio optimization routine. When the optimization model allowed unrestricted choice among assets, it was common to include timber assets in the portfolio. Timber assets were in some cases the only components of the portfolio. The long-run risk-return results, however, were unfavourable. Employing a strategy of holding a fixed portfolio allocation (%) of timber assets while rebalancing one's holding of financial assets, however, appears favorable. Holding a fixed 10% of a portfolio in timber showed about a 1% higher rate-of-return across portfolios with no increase in risk. Higher timber proportions indicated higher returns, but with higher risk.

Optimal consumption and arbitrage in incomplete, finite state security markets

We study a consistent treatment for both the multi-period portfolio selection problem and the option attainability problem by a dual approach. We assume that time is discrete, the horizon is finite, the sample space is finite and the number of securities is less than that of the possible securities price transitions, i.e. an incomplete security market. The investor is prohibited from investing stocks more than given linear investment amount constraints at any time and he maximizes an expected additive utility function for the consumption process. First we give a set of budget feasibility conditions so that a consumption process is attainable by an admissible portfolio process. To establish this relation, we used an algorithmic approach which has a close connection with the linear programming duality. Then we prove the unique existence of a primal optimal solution from the budget feasibility conditions. Finally, we formulate a dual control problem and establish the duality between primal and dual control problems.

Solving a linear multiperiod portfolio problem by interior-point methodology

In this report we formulate a linear multiperiod programming problem and show how it can be solved by a new interior point algorithm. The conditions of convergence and applicability of the algorithm of centers are explicitly connected to the multiperiod programming problem. For the former, effective application is supported by a set of simplifying conditions stated and proved in the text. For the latter, boundedness and nontriviality under real world conditions is demonstrated, allowing for its solution by the interior point algorithm. The use of a fast interior point algorithm is motivated by some empirical evidence from a Revised Simplex optimizer.

Efficient Combinations of Timber and Financial Market Investments in Single-Period and Multiperiod Portfolios

Abstract This paper presents single-period and multiperiod portfolio expected returns and standard deviations for portfolios constructed from eight sawtimber investments and four financial market investments. Four of the timber investments are from growing timber crops in the South (southern pine, ash, gum, oaks) and four are from growing timber crops in the Midwest (white pine, red pine, aspen, and red oak). The financial market investments are common stocks, corporate bonds, U.S. Government bonds, and U.S. Government Treasury bills. Portfolios are computed two ways: (1) maximizing a quadratic utility function, which is inherently a single-period approach, and (2) maximizing a power utility function, which has been shown to be a suitable multiperiod optimization. The two models provide similar results showing that portfolio diversification across timber species and financial market investments is often appropriate. Risk-tolerant investors may choose only timber investments, while more risk-averse investors will hold both timber and financial market investments. The highest risk-highest return portfolio consists of red pine, while the lowest risk-lowest return portfolio is primarily made up of Treasury bills. This paper shows that as the investment horizon lengthens, the expected compound long-run portfolio return decreases, the standard deviation falls, and the most risky, efficient single-period portfolios are dominated by some less risky multiperiod portfolios. For. Sci. 37(2):461-480.