Markowitz principles for multi-period portfolio selection problems with moments of any order

2008 ◽  
Vol 464 (2092) ◽  
pp. 827-854
Author(s):  
Thamayanthi Chellathurai ◽  
Thangaraj Draviam
Keyword(s):  
Portfolio Selection ◽  
Asset Prices ◽  
Efficient Frontier ◽  
Higher Order ◽  
Discrete Version ◽  
Expected Return ◽  
Risky Assets ◽  
Selection Problems ◽  
Merton Problem ◽  
Dynamic Trading

The multi-period portfolio selection problem is formulated as a Markowitz mean–variance optimization problem in terms of time-varying means, covariances and higher-order and intertemporal moments of the asset prices. The crux lies in expressing the number of shares of any particular asset to be transacted on any future trading date, which is a non-anticipative process, as a polynomial of the changes in the discounted prices of all the risky assets. This results in the expected return of the portfolio being dependent on not only the means of the asset prices, but also the higher-order and intertemporal moments, and its variance being dependent on not only the second-order moments, but also the higher-order and intertemporal moments. As illustrations, we study the portfolio selection problems including the discrete version of the Merton problem. It is shown numerically that the efficient frontier obtained from Markowitz's discrete multi-period formulation coincides with that from Merton's continuous-time formulation when the number of rebalancing periods is ‘large’. The effects of dynamic trading, in particular volatility pumping, in comparison with a static single-period model are measured by a non-dimensional number, Dyn( P ) ( P , number of trading periods), which quantifies the relative gain due to dynamic trading. It is sufficient to rebalance the portfolio a few times in order to get more than 95% of the gain due to continuous trading.

scholarly journals A Scalable Algorithm for Sparse Portfolio Selection

2022 ◽  
Author(s):  
Dimitris Bertsimas ◽  
Ryan Cory-Wright
Keyword(s):  
Portfolio Selection ◽  
Minimum Variance ◽  
Outer Approximation ◽  
Approximation Procedure ◽  
Expected Return ◽  
Risky Assets ◽  
Scalable Algorithm ◽  
Second Order Cone ◽  
Real World Problem ◽  
Sparse Portfolio

The sparse portfolio selection problem is one of the most famous and frequently studied problems in the optimization and financial economics literatures. In a universe of risky assets, the goal is to construct a portfolio with maximal expected return and minimum variance, subject to an upper bound on the number of positions, linear inequalities, and minimum investment constraints. Existing certifiably optimal approaches to this problem have not been shown to converge within a practical amount of time at real-world problem sizes with more than 400 securities. In this paper, we propose a more scalable approach. By imposing a ridge regularization term, we reformulate the problem as a convex binary optimization problem, which is solvable via an efficient outer-approximation procedure. We propose various techniques for improving the performance of the procedure, including a heuristic that supplies high-quality warm-starts, and a second heuristic for generating additional cuts that strengthens the root relaxation. We also study the problem’s continuous relaxation, establish that it is second-order cone representable, and supply a sufficient condition for its tightness. In numerical experiments, we establish that a conjunction of the imposition of ridge regularization and the use of the outer-approximation procedure gives rise to dramatic speedups for sparse portfolio selection problems.

CONTINUOUS-TIME MEAN–VARIANCE OPTIMIZATION FOR DEFINED CONTRIBUTION PENSION FUNDS WITH REGIME-SWITCHING

2019 ◽  
Vol 22 (06) ◽  
pp. 1950029
Author(s):  
ZHIPING CHEN ◽  
LIYUAN WANG ◽  
PING CHEN ◽  
HAIXIANG YAO
Keyword(s):  
Brownian Motion ◽  
Portfolio Selection ◽  
Continuous Time ◽  
Optimal Strategy ◽  
Regime Switching ◽  
Efficient Frontier ◽  
Defined Contribution ◽  
Risky Assets ◽  
Mean Variance Portfolio ◽  
Mean Variance

Using mean–variance (MV) criterion, this paper investigates a continuous-time defined contribution (DC) pension fund investment problem. The framework is constructed under a Markovian regime-switching market consisting of one bank account and multiple risky assets. The prices of the risky assets are governed by geometric Brownian motion while the accumulative contribution evolves according to a Brownian motion with drift and their correlation is considered. The market state is modeled by a Markovian chain and the random regime-switching is assumed to be independent of the underlying Brownian motions. The incorporation of the stochastic accumulative contribution and the correlations between the contribution and the prices of risky assets makes our problem harder to tackle. Luckily, based on appropriate Riccati-type equations and using the techniques of Lagrange multiplier and stochastic linear quadratic control, we derive the explicit expressions of the optimal strategy and efficient frontier. Further, two special cases with no contribution and no regime-switching, respectively, are discussed and the corresponding results are consistent with those results of Zhou & Yin [(2003) Markowitz’s mean-variance portfolio selection with regime switching: A continuous-time model, SIAM Journal on Control and Optimization 42 (4), 1466–1482] and Zhou & Li [(2000) Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization 42 (1), 19–33]. Finally, some numerical analyses based on real data from the American market are provided to illustrate the property of the optimal strategy and the effects of model parameters on the efficient frontier, which sheds light on our theoretical results.

scholarly journals Unit trusts and portfolio selection on the Johannesburg Stock Exchange

1982 ◽  
Vol 13 (4) ◽  
pp. 169-175
Author(s):  
K. J. Carter ◽  
J. F. Affleck-Graves ◽  
A. H. Money
Keyword(s):  
Standard Deviation ◽  
Portfolio Selection ◽  
Stock Exchange ◽  
Efficient Frontier ◽  
Expected Return ◽  
Earnings Volatility ◽  
Johannesburg Stock Exchange ◽  
Market Index ◽  
Efficient Portfolios ◽  
Unit Trust

The application of the standard techniques of portfolio selection on the 34 sectors comprising the JSE All Share index is undertaken for the three equal non-overlapping five-year periods between February 1965 and January 1980. Efficient portfolios in each period which carry the same risk as the market index are seen to outperform the market substantially. Portfolios chosen at random to span the efficient frontier in each period reveal the consistent inefficiency of 10 sectors over the 15-year period. Three of these sectors, namely Mining Holding, Mining Houses and Industrial Holding are shown to be favoured in the Association of Unit Trusts portfolio relative to these sectors' proportion of the market. On the presumption that unit trust managers attempt to act efficiently, holding these sectors is only justified if the measure of risk used in the portfolio selection algorithm, namely standard deviation of expected return, is less appropriate than other measures of risk such as earnings volatility. If standard deviation of expected return is a more appropriate measure of risk in the selection of efficient portfolios, it must be concluded that the large sophisticated investors managing the unit trusts act inefficiently.

UNCERTAINTY PORTFOLIO MODEL IN CROSS CURRENCY MARKETS

2010 ◽  
Vol 18 (06) ◽  
pp. 759-777 ◽  
Author(s):  
WEIJUN XU ◽  
WEIDONG XU ◽  
HONGYI LI ◽  
WEIGUO ZHANG
Keyword(s):  
Portfolio Selection ◽  
Asset Allocation ◽  
Efficient Frontier ◽  
Expected Return ◽  
Fuzzy Variables ◽  
Financial Variables ◽  
Fuzzy Environment ◽  
Exchange Risk ◽  
Asset Selection ◽  
Α Level

Owing to the fluctuations in the financial markets, many financial variables such as expected return, volatility, or exchange rate may occur imprecisely. But many portfolio selection models consider precise input of these values. Therefore, this paper studies a multiobjective international asset allocation problem under fuzzy environment. In our portfolio selection model, both of the return risk and the exchange risk are considered. The coefficient matrices in the objectives and constraints and the decision value are considered as fuzzy variables. The calculation of the portfolio and efficient frontier is derived by considering the exchange risk in the fuzzy environment. An empirical study is performed based on a portfolio of six securities denominated in six different currencies, i.e., USD, EUR, JPY, CNY, HKD, and GBP. The α-level closed interval portfolio [Formula: see text] and the fuzzy efficient frontier are obtained with different values of α ∈ (0, 1]. The empirical results indicate that the fuzzy asset selection method is a useful tool for dealing with the imprecise problem in the real world.

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