We exhibit an algorithm for portfolio selection that asymptotically outperforms the best stock in the market. Let x i = (x i1 ; x i2 ; : : : ; x im ) t denote the performance of the stock market on day i ; where x ij is the factor by which the j-th stock increases on day i : Let b i = (b i1 ; b i2 ; : : : ; b im ) t ; b ij 0; P j b ij = 1 ; denote the proportion b ij of wealth invested in the j-th stock on day i : Then S n = Q n i=1 b t i x i is the factor by which wealth is increased in n trading days. Consider as a goal the wealth S n = max b Q n i=1 b t x i that can be achieved by the best constant rebalanced portfolio chosen after the stock outcomes are revealed. It can be shown that S n exceeds the best stock, the Dow Jones average, and the value line index at time n: In fact, S n usually exceeds these quantities by an exponential factor. Let x 1 ; x 2 ; : : : ; be an arbitrary sequence of market vectors. It will be shown that the nonanticipating sequence ...

Wachter, two anonymous referees, and participants at the Brown Bag Micro Finance Lunch Seminar at the Wharton

Abstract. Mean-variance portfolio analysis provided the first quantitative treatment of the tradeoff between profit and risk. We describe in detail the interplay between objective and constraints in a number of single-period variants, including semivariance models. Particular emphasis is laid on avoiding the penalization of overperformance. The results are then used as building blocks in the development and theoretical analysis of multiperiod models based on scenario trees. A key property is the possibility of removing surplus money in future decisions, yielding approximate downside risk minimization.

The mean-variance formulation by Markowitz in 1950s and its analytical solution by Merton in 1972 paved a foundation for modern portfolio selection analysis in 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 multi-period mean-variance formulation. An efficient algorithm is also proposed in this paper in finding an optimal portfolio policy to maximize a utility function of the expected value and the variance of the terminal wealth. Key Words: Multi-period portfolio selection, multi-period mean-variance formulation, utility function. This research was partially supported by the Research Grants Council of Hong Kong, grant no. CUHK 4130/97E. The authors very much appreciate the constructive comments from Professor Stanley R. Pliska. y Author to whom a...

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Single Premium Deferred Annuities (SPDAs) are investment vehicles, offered to investors by insurance companies as a means of providing income past their retirement age. They are mirror images of insurance policies. However, the propensity of individuals to shift part, or all, of their investment into different annuities creates substantial uncertainties for the insurance company. In this paper we develop a multiperiod, dynamic stochastic program that deals with the problem of funding SPDA liabilities. The model recognizes explicitly the uncertainties inherent in this problem due to both interest rate volatility and the behavior of individual investors. Empirical results are presented with the use of the model for the funding of an SPDA liability stream using government bonds, mortgage-backed securities and derivative products. 2 1 Introduction Interest rate risk in fixed-income markets has, traditionally, been managed using a very simple idea: matching the interest rate se...

Abstract — 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. Index Terms — Markowitz’s mean-variance portfolio selection, discrete-time model, Markov chain, switching diffusion, linearquadratic problem, singular perturbation. I.

We consider a multi-asset investment fund that in the absence of transactions costs and/or taxes would hold assets in constant proportions. The problem is: what trading strategy should be implemented in the presence of transactions costs and/or capital gains taxes? Very frequent trading to maintain the target proportions will incur ruinous transactions costs, whilst infrequent trading will incur significant tracking error relative to the desired returns. Following standard industry practice, the objective is assumed to minimize the expected discounted sum of costs of trading plus the costs resulting from tracking errors. As suggested by the existence results of Akian, Menaldi, and Sulem [1996], the optimal strategy is characterized by a multi-dimensional no-trade region. In contrast with earlier work, we develop a relatively simple means to compute this region and to determine the resulting annual turnover and tracking error of the optimal strategy. Almost surely, the strategy will require trading just one risky asset at any moment, although which asset is traded varies stochastically through time. Compared to the common practice of periodically rebalancing assets to their target proportions, the optimal strategy with the same degree of tracking accuracy will reduce turnover by almost 50%. We show how high trading costs will reduce initial commitments to illiquid markets. Our results are contrasted with the ad hoc approach that reduces expected returns to reflect transactions costs. Capital gains taxes add complexity due to the stochastic evolution of cost bases. We derive the optimal no-trade region and the region requiring tax loss selling. Losses are not immediately realized when there are positive transactions costs, but only when they exceed a critical level. Capital gains taxes lead to lower initial investment levels.

We investigate incentive effects of a typical hedge-fund contract for a manager with power utility. With a one-year horizon, she displays risk-taking that varies dramatically with fund value. We extend the model to multiple yearly evaluation periods and find her risk-taking is rapidly moderated if the fund performs reasonably well. The most realistic approach to modeling fund closure uses an endogenous shutdown barrier where the manager optimally chooses to shut down. The manager increases risk-taking as fund value approaches that barrier, and this boundary behavior persists strongly with multiyear horizons. I.

In this paper, we summarize and add to the evidence on the large and systematic differences in portfolio composition across individuals with varying characteristics, and evaluate some of the theories that have been proposed in terms of their ability to account for these differences. Variation in background risk exposure--from sources such as labor and entrepreneurial income or real estate holdings, and from factors such as transactions costs, borrowing constraints, restricted pension investments and life cycle considerations – can explain some but not all aspects of the observed cross-sectional variation in portfolio holdings in a traditional utility maximizing framework. In particular, fixed costs and life cycle considerations appear necessary to explain the lack of stock market participation by young and less affluent households. Remaining challenges for quantitative theories include the apparent lack of diversification in some unconstrained individual portfolios, and non-participation in the stock market by some households with significant financial wealth.