Portfolio Theory: As I Still See It

Markowitz, Harry M.

doi:10.1146/annurev-financial-011110-134602

Cited by 121 publications

(78 citation statements)

References 37 publications

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“…If the expected returns of the assets follow a Gaussian distribution [112] or the investor's utility function is quadratic [41], then the mean-variance criterion is theoretically compatible with the expected utility hypothesis originally introduced by Bernoulli [15] (see [232] for the English translation and see also [93,121,240] for more modern approaches). As pointed out by Markowitz in [176,179], the previous assumptions are sufficient, but not necessary conditions. However, the assumption of Gaussian asset returns might be unrealistic: the probability distribution for the expected returns is generally leptokurtic [191].…”

Section: Computing Mwu Costs/gainsmentioning

confidence: 96%

The multiplicative weights update algorithm for mixed integer nonlinear programming: theory, applications, and limitations

Mencarelli

2018

4OR-Q J Oper Res

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Section: Computing Mwu Costs/gainsmentioning

confidence: 96%

The multiplicative weights update algorithm for mixed integer nonlinear programming: theory, applications, and limitations

Mencarelli

2018

4OR-Q J Oper Res

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“…Hence, the mean-variance portfolio can basically entertain the almost optimal satisfaction of common consumers. Further studies support this approximation; for instances, see [17,24,26,32].…”

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confidence: 90%

Utility-Deviation-Risk Portfolio Selection

Wong¹,

Yam²,

Zheng³

2017

SIAM J. Control Optim.

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Abstract. We here provide a comprehensive study of the utility-deviation-risk portfolio selection problem. By considering the first-order condition for the corresponding objective function, we first derive the necessary condition that the optimal terminal wealth satisfying two mild regularity conditions solves for a primitive static problem, called Nonlinear Moment Problem. We then illustrate the application of this general necessity result by revisiting the non-existence of the optimal solution for the mean-semivariance problem. Secondly, we establish an alternative version of the verification theorem serving as the sufficient condition that the solution, which satisfies another mild condition different from that for necessity, of the Nonlinear Moment Problem is the optimal terminal wealth of the original utility-deviation-risk portfolio selection problem. We then apply this general sufficiency result to revisit the various well-posed mean-risk problems already known in the literature, and to also establish the existence of the optimal solutions for both utility-downside-risk and utility-strictlyconvex-risk problems under the assumption that the underlying utility satisfies the Inada Condition. To the best of our knowledge, the positive answers to the latter two problems have long been absent in the literature. In particular, the existence result in the utility-downside-risk problem is in contrast to the well-known non-existence of an optimal solution for the mean-downside-risk problem. As a corollary, the existence result in utility-semivariance problem allows us to utilize the semivariance as a proper risk measure in the classical portfolio management paradigm.

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“…The starting point for testing our non-stochastic optimization approach for its appropriateness to optimize forest management was Markowitz' (1952Markowitz' ( , 2010 modern portfolio theory (MPT). This stochastic method considers the portfolio return as a random variable and integrates risk as the standard deviation of the expected portfolio return.…”

Section: Short Overview About Existing Literaturementioning

confidence: 99%

A non-stochastic portfolio model for optimizing the transformation of an even-aged forest stand to continuous cover forestry when information about return fluctuation is incomplete

Messerer

Pretzsch

Knoke

2017

Annals of Forest Science

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Abstract& Key message Non-stochastic portfolio optimization of forest stands provides a good alternative to stochastic meanvariance optimization when available statistical data is incomplete. The suggested approach has a theoretical background in the areas of robust optimization, continuous multicriteria decision-making, and fuzzy theory. Resulting robust portfolios only show slight economic losses compared to the efficient frontier of a stochastic optimization. & Context Economic optimization addressing diversification in mixed uneven-aged forest stands is a useful tool for forest planners. & Aims The study aims to compare two approaches for optimizing rotation age cohort portfolios under risk. Rotation age cohorts emerge from age-based regeneration-harvesting operations simulated for two tree species: Picea abies and Fagus sylvatica. & Methods The first optimization approach is a stochastic mean-variance approach. The second is a non-stochastic optimization approach, which has rarely been applied to optimize tree species composition and the distribution of harvested timber over many periods. It aims at relatively good solutions, even if the deviation from the initially assumed return is very high. The objective function for both approaches is sensitive to the selection of various harvesting periods for different parts of the stand. For the stochastic approach, the objective function maximizes the annuitized net present value (economic return) for specific levels of risk by allocating area proportions to harvesting periods and tree species. In the non-stochastic approach, the allocation of area proportions instead minimizes the maximum deviation from the greatest possible economic return among many uncertainty scenarios (non-stochastic approach). & Results Portfolios from both approaches were diverse in rotation age cohorts. The non-stochastic portfolios were more diverse when compared with portfolios from the efficient frontier, which showed the same standard deviation. However, P. abies clearly dominated the non-stochastic portfolios, while stochastic portfolios also integrated beech to a greater extent, but only in very low risk portfolios. The economic losses of the non-stochastic portfolios compared to the efficient frontier of the mean-variance approach lay between 1 and 3% only for different levels of accepted risk. & Conclusion The non-stochastic portfolio optimization over a large uncertainty space is so far uncommon in forest science, yet provides a viable alternative to stochastic optimization, particularly when available data is scarce. However, further research should consider ecological effects, such as increased resistance against hazards of conifers in mixed stands.

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Portfolio Theory: As I Still See It

Cited by 121 publications

References 37 publications

The multiplicative weights update algorithm for mixed integer nonlinear programming: theory, applications, and limitations

The multiplicative weights update algorithm for mixed integer nonlinear programming: theory, applications, and limitations

Utility-Deviation-Risk Portfolio Selection

A non-stochastic portfolio model for optimizing the transformation of an even-aged forest stand to continuous cover forestry when information about return fluctuation is incomplete

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