Our environment deteriorates primarily due to the emissions of carbon dioxide. Scientists and entrepreneurs promote carbon offset to reduce the emissions. Scientists and investors explore the investments of carbon offset. Some scientists encouragingly construct portfolio selection models but do not completely optimize them. Some scientists encouragingly construct capital asset pricing models (CAPM) but do not completely justify them. Under such contexts, this paper proposes a model of multiple-objective portfolio selection (MOPS) and preliminarily explores multiple-objective capital asset pricing models (MOCAPM). By the classical transition from portfolio selection to CAPM, we introductorily conjecture the extended transition from MOPS to MOCAPM. Specifically, we prove mathematical properties for the model. For instance, its minimum-variance surface is convex, and its feasible region is bounded by the convex surface. We examine whether a point on the minimum-variance surface is nondominated. By the properties, we heuristically prove different tangent planes for MOCAPM (instead of the unique tangent line for CAPM). We tentatively hint the conditions for a unique tangent plane. This paper acts as a footstep of the introductory conjecture.

Humans face environmental deterioration. Scholars have identified carbon dioxide as one of the culprits, and they emphasize carbon offset. Researchers are investigating carbon offset investments. Some researchers have encouragingly deployed multivariate variational mode decomposition methods, but they have not fully optimized them. Some researchers have opportunely assessed capital asset pricing models, but they have not fully justified them. We devise multiple-objective portfolio selection models, fully optimize them, and dominate carbon offset indexes. We extend the classical methodology of advancing from portfolio selection to capital asset pricing models into the methodology of advancing from multiple-objective portfolio selection to multiple-objective capital asset pricing models. Specifically, we explore multiple-objective capital asset pricing models by numerically verifying many tangent lines (instead of the traditionally singular tangent line) and suggesting a tangent plane (instead of tangent lines). For multiple-objective zero-covariance capital asset pricing models, we numerically compute a set of zero-covariance portfolios (instead of the traditionally singular zero-covariance portfolio) and suggest picking an advantageous zero-covariance portfolio. We consider the second-level indicators of carbon offset and generalize three-objective portfolio selection to k-objective portfolio selection. As for contributions, first, this paper’s methodology is to logically advance from multiple-objective portfolio selection to multiple-objective capital asset pricing models, whereas the literature typically covers multiple-objective portfolio selection alone and barely covers multiple-objective capital asset pricing models. Second, this paper numerically demonstrates some difficulties and proposes hypothetical solutions in the process of obtaining multiple-objective capital asset pricing models.

Nobel Laureate Markowitz originates portfolio selection as the birth of modern finance. Nobel Laureate Sharpe implements portfolio selection and originates capital asset pricing models. Nobel Laureate Fama also implements portfolio selection and originates zero-covariance capital asset pricing models. After these feats, researchers have gradually realized additional objectives and have promisingly extended portfolio selection into multiple-objective portfolio selection. However, there hardly exists research to leap from multiple-objective portfolio selection to multiple-objective capital asset pricing models (as initiated by Markowitz and Sharpe in finance). Moreover, the extension is basically confined to the branches of mathematics, operations research, optimization, and computer sciences. Many researchers sufficiently review multiple-objective portfolio selection. However, the reviews are extensive. Instead, we intensively criticize and envision the research on multiple-objective portfolio selection from the perspective of capital asset pricing models by crystallizing the research limitations and heralding future directions. Specifically, we emphasize seven research limitations for multiple-objective portfolio optimization, multiple-objective capital asset pricing models, and multiple-objective zero-covariance capital asset pricing models. We also generalize from common three-objective portfolio selection to k-objective portfolio selection. Visually, we orchestrate figures to delineate the complexity. Theoretically, this paper heralds challenging but encouraging future directions. Pragmatically, this paper proposes a formulation for the multiple-objective nature of practical convolution in finance.

Our society is facing serious challenges from global warming and environmental degradation. Scientists have identified carbon dioxide as one of the causes. Our society is embracing carbon offset as a way to field the challenges. The purpose of carbon offset is trying to cancel out the large amounts of carbon dioxide by investing in projects that reduce or remove emissions elsewhere. Examples of carbon offset projects are planting trees, renewable energy projects, and capturing methane from landfills or farms. Not all carbon offset projects are equally effective. In stock markets, investors eagerly pursue carbon offset. Namely, investors favor carbon offset in addition to risk and return when investing. Therefore, investors supervise risk, return, and carbon offset. Investors’ pursuits raise the question of how to model carbon offset for investments. The traditional answer is to adopt carbon offset screening and engineer portfolios by stocks with good carbon offset ratings. However, Nobel Laureate Markowitz emphasizes portfolio selection rather than stock selection. Moreover, carbon offset is composed of multiple components, ranging from business, social, economic, and environmental aspects. This multifaceted nature requires more advanced models than carbon offset screening and portfolio selection. Within this context, we systematically formulate multiple-objective portfolio selection models that include carbon offset. Firstly, we extend portfolio selection and treat carbon offset as a whole. Secondly, we separate carbon offsets into different components and build models to monitor each component. Thirdly, we innovate a model to monitor each component’s expectation and mitigate each component’s risk. Lastly, we optimize the series of models and prove the models’ properties in theorems. Mathematically, this paper makes theoretical contributions to multiple-objective optimization, particularly by proving the consistency of efficient solutions during objective classification and model evolution, describing the structure of properly efficient sets for multiple quadratic objectives, and elucidating the optimization’s sensitivity analyses. Moreover, by coordinating the abstract objective function, our formulation is generalizable. Overall, this paper’s contribution is to model carbon offset investments through multiple-objective portfolio selection. This paper’s methodology is multiple-objective optimization. This paper’s achievements are to provide investors with greater precision and effectiveness than carbon offset screening and portfolio selection through engineering means and to mathematically prove the properties of the model.

Markowitz originates portfolio selection as the birth-place of modern finance. Sharpe originates capital asset pricing models (CAPM) as one quintessence of modern finance. Black and Fama then discovered the existence of a unique zero-covariance portfolio on the minimum-variance frontier. Fama and Roll further proved their CAPM. Recently, researchers have gradually realized additional objectives and extended portfolio selection into multipleobjective portfolio selection. However, there is still limited research to leap from multiple-objective portfolio selection to capital asset pricing models of multiple-objective portfolio selection. In such an area, this paper contributes to the literature as follows: First, we prove the existence of a (whole) curve of the zero-covariance portfolios for a 3-objective model. Second, we extend traditional locating and propose locating a unique portfolio on the curve. Lastly, we extend general k-objective models. This paper acts as a theoretical foothold for accomplishing capital asset pricing models of multiple-objective portfolio selection.

Markowitz [Portfolio selection. Journal of Finance, 7(1), 77-91] originates portfolio selection as the birth of modern finance. After his feat, Markowitz [Foundations of portfolio selection. Journal of Finance, 46(2), 469-477] perceives general considerations in addition to variance and expected return of portfolio selection. However, there is relatively limited research in maximizing the considerations. In such an area, this paper theoretically enriches portfolio selection and makes contribution to the literature. Specifically, we obtain complete efficient sets' piecewise-linear-segment structure by parametric quadratic programming. Only by the structure, in theorems and corollaries, we prove the considerations as piecewise linear functions, maximize the considerations, and dominate stock-market indexes. Our models are general and universally fit numerous scenarios. Practically, we implement our models for the 30 component stocks of Dow Jones Industrial Average and 1937 US stocks of as a comprehensive sample, dominate the average, and can outperform the average out of sample.

Computing efficient sets has long been a topic in multiple-objective optimization and research has made substantial progress. However, there are still limitations in the multiple-objective portfolio selection and optimization areas. Firstly, researchers typically focus on models containing only one quadratic objective. Secondly, few researchers pursue multiple quadratic objectives, but their algorithms could be relatively elusive and it could be a pity that they do not explicitly demonstrate the efficient sets’ structure. Lastly, researchers mostly limit their scope to three objectives. Within this context, this paper makes theoretical contributions to the literature. Operating with multiple quadratic objectives, we analytically derive closed-form formulae for the computation of the properly efficient and weakly efficient sets of problems and demonstrate the efficient sets’ structure in the form of a sequence of pyramids in decision space. Although we are restricted to equality-constraint-only models, our results have implications for general-constraint models. In addition, our methods can be extended to general k -quadratic objective models.

Computing optimal-solution sets has long been a topic in multiple-objective optimization. Despite substantial progress, there are still research limitations in the multiple-objective portfolio optimization area. The optimal-solution sets’ structure is barely known. Public-domain software for even three objectives is absent. Alternatively, researchers scrutinize equality-constraint-only models and analytically resolve them. Within this context, this paper extends these analytical methods for nonnegative constraints and thus theoretically contributes to the literature. We prove the existence of positive elements and negative elements for the optimal-solution sets. Practically, we prove that non-negative subsets of the optimal-solution sets can exist. Consequently, the possible existence endorses these analytical methods, because researchers bypass mathematical programming, analytically resolve, and pinpoint some non-negative optima. Moreover, we elucidate these analytical methods’ alignment with capital asset pricing models (CAPMs). Furthermore, we generalize for k-objective models. In conclusion, this paper theoretically reinforces these analytical methods and hints the optimal-solution sets’ structure for multiple-objective portfolio optimization.

In this paper, we demonstrate a completely new approach for computing cardinality constrained meanvariance efficient frontiers. By cardinality constrained, it is meant that if there is to be investment in a security, it is to be of at least some minimum amount (a buyin threshold), and that there is also a specification on the number of securities to be held in a portfolio (called a cardinality constraint). Whereas the usual strategy, as such problems are NP -hard, is to take the original exact problem and apply heuristics to solve, in this paper the strategy is to perturb the original problem and then apply exact procedures to solve. The advantages of the approach are that the perturbations are tiny, they are only applied to the problem's correlation matrix, and they allow for the accurate computation of cardinality constrained efficient frontiers in problems with up to at least 10 0 0 securities in remarkably little time. Moreover, the simplicity of the approach is such that it can be inserted into existing portfolio management systems without requiring any re-training beyond what a typical portfolio analyst would already know. 1 (c) 2023 Elsevier B.V. All rights reserved.

Researchers traditionally compute isolated points for an efficient frontier and assume a line which passes through a risk-free asset [Formula: see text] and is tangent to the frontier. The tangency plays pivotal roles for the capital asset pricing model (CAPM). However, the assumption may not hold in the presence of kinks (as non-differentiable points) on efficient frontiers. Kinks are detected by parametric-quadratic programming only and not by ordinary portfolio optimization. Up until now, there has been no research to theoretically scrutinize kink properties (especially implications to CAPM) and systematically quantify the nonexistence of the tangency. In such an area, this paper contributes to the literature. In theorems and corollaries, we prove the nonexistence of the tangency and substantiate that expected-return axis is composed of piecewisely connected intervals for which the tangency does not exist and intervals for which the tangency exists. Computationally, we reveal universal existence of kinks (e.g., 0.2 to 8.0 kinks for 5-stock to 1800-stock portfolio selections) and the tangency-nonexistence ratios as about 0.066.

Markowitz formulates portfolio selection and calls the optimal solutions as an efficient frontier. Sharpe initiates Sharpe ratio for frontier portfolios' reward to variability. Finance textbooks assume that there exists a line which passes through a risk-free rate and is tangent to an efficient frontier. The tangent portfolio enjoys the maximum Sharpe ratio. However, the assumption is over-simplistic because we prove that other situations exist. For example, Sharpe ratio itself may not be even well-defined. We comprehensively maximize Sharpe ratio. In such an area, this paper contributes to the literature. Specifically, we identify the other situations by parametricquadratic programming which renders complete efficient frontiers by piecewisehyperbola structure. Researchers traditionally view efficient frontiers by just isolated points. We accomplish handy formulae, so investors can even manually process them. The COVID-19 pandemic is unleashing crises. Unfortunately, there is quite limited research of portfolio selection for COVID. In such an area, this paper contributes to the practice. Specifically, we originate a counter-COVID measure for stocks and integrate it as a constraint into portfolio-selection models. The maximum-Sharpe-ratio portfolio outperforms stock-market indexes in sample. We launch the models for Dow Jones Industrial Average and discover outperformance out of sample.

China has significantly enhanced vegetation coverage and terrestrial carbon sink functions through ecological restoration. However, cropland ecosystems are sensitive to a changing climate over the summertime monsoon transition zone of China (SMTZC), which has implications for carbon cycling. For example, it is unclear how changes in precipitation will affect the cropland ecosystem carbon sinks (CS e ) and carbon sink potential (CS p ), and the mechanisms of tradeoffs that develop between plant and soil organic carbon (SOC) are unclear. Here, we integrated crop yield, total biomass (TB) and water coefficient in a combined field and modelling experimental. We explored the mechanisms of ecosystem carbon cycling and CS p in the SMTZC under different precipitation scenarios. We found that soil organic carbon sink (SOC s ) was strongly correlated with the plant organic carbon sink (POC s ) and discovered. Significant differences in CSp between ecosystems resulting from interannual precipitation. The C 4 (maize, Zea mays ) and C 3 (potato, S olanum tuberosum L ) carbon sinks (CSs) were 68.59, 190.73, 160.37 Mt and 10.21, 30.97, 14.59 Mt for the 3 years, respectively. Precipitation effectively increased TB and yield, but excessive precipitation in them, which was most obvious in C 3 ecosystem ( R 2 > 0.60) and reduced POC s , evident in C 4 ecosystem ( R 2 > 0.16). This study provides data and a scientific basis for increasing CS and achieving carbon neutrality in cropland.

Due to CO2 emissions, humans are encountering grave environmental crises (e.g., rising sea levels and the grim future of submerged cities). Governments have begun to offset emissions by constructing emission-trading schemes (carbon-offset markets). Investors naturally crave carbon-offset options to effectively control risk. However, the research and practice for these options are relatively limited. This paper contributes to the literature in this area. Specifically, according to carbon-emission allowances' empirical distributions, we implement fractal Brownian motions and jump diffusions instead of traditional geometric Brownian motions. We contribute to extending the theoretical model based on carbon-offset option-pricing methods. We innovate the carbon-offset options of Asian styles. We authenticate the options' stochastic differential equations and analytically price the options in the form of theorems. We verify the parameter sensitivity of pricing formulas by illustrations. We also elucidate the practical implications of an emission-trading scheme.

The yield to water use boundary function is a very useful method for estimating attainable yield and yield gap (the difference between attainable yield and actual yield) in water-limited regions. However, the boundary function for potato has not yet to be clearly defined. In this study, we established a boundary function for potato fresh tuber in a climatic transition zone (Longzhong Plateau of China) using a simple crop growth model and validated it by long-term experimental data collected from field experiments and literature. The results demonstrated that the crop model accurately simulated potato fresh tubers weight with an RRMSE of less than 30%, d greater than 0.7, and an RMSE of 5426 kg ha−1. The established boundary function for potato fresh tuber was defined as yield (kg ha−1) =167×(water use [mm]-121), with a plateau yield of 48082 kg ha−1 when water use exceeded 410 mm. Furthermore, the boundary function was found to be perfectly applicable to water use-yield data collected from literature and potato growing season precipitation-yield data from long-term observation. With precipitation data during potato growing season from 1988 to 2017, the boundary function was used to determine the average attainable yield for potato fresh tubers ranged from 10,000 to 48,082 kg ha−1 in the Longzhong Plateau. The boundary function for potato could be used to aid farmers in water-limited areas in determining the most profitable crop management systems.

The COVID-19 pandemic is unleashing crises of humanity, economy, and finance. Portfolio selection is widely recognized as the foundation of modern financial economics. Therefore, it is naturally crucial and inviting to utilize portfolio selection in order to counter COVID-19 in stock markets. We originate a counter-COVID measure for stocks, extend portfolio selection, and construct multiple-objective portfolio selection. Because of the uncertainty in measuring counter-COVID, we perform robust optimization. Specifically, we analytically compute the optimal solutions as a trail of an optimal portfolio due to the change of counter-COVID. We call the trail as mean-parameterized nondominated path. Moreover, the path is a continuous function of the change, so the portfolio relatively mildly varies for the change. In contrast, researchers typically still focus on 2-objective robust illustrations and infrequently explicitly compute the optimal solutions for multiple-objective portfolio optimization. To the best of our knowledge, there is limited research for multiple-objective portfolio selection of COVID and for the robust optimization of multiple-objective portfolio selection. In such an area, this paper contributes to the literature. The implications to fight COVID are that investors minimize risk, maximize return, and maximize counter-COVID in stock markets and that investors ascertain the multiple-objective portfolio selection as relatively robust.

"双碳"目标下,为了应对日益复杂的国际形势及过分依赖石油等传统能源造成的市场不稳定性波动等问题,我国必须加快新能源产业发展,推动能源结构绿色低碳转型.光伏产业在新能源产业中占据重要地位,其发展影响着新能源产业的整体发展状况.目前,光伏产业普遍存在投资成本大、投资风险高、投资期长等问题,而侧重于项目现金流和风险控制的融资租赁能够为光伏项目提供"限制条件少、融资金额足、筹资速度快"的融资方式,有效与光伏项目的融资需求适配.现有文献在探究分析融资租赁与光伏项目的适配性时,多侧重于融资方式的对比选择、风险控制等方面,在研究其融资支持与影响程度上存在一定空白.文章以A公司为例,在对比银行贷款、债券融资、股权融资等融资方式后,通过分析其实施融资租赁前后的实际经营情况和财务状况,发现融资租赁不仅可以为光伏项目带来显著的经济效益,而且可以帮助企业更好地践行绿色低碳循环发展理念、助力"碳中和"目标的实现,具有良好的社会效益.文章对我国光伏企业解决融资问题具有一定的现实指导意义.

The implications to fight COVID are that investors minimize risk, maximize return, and maximize counter-COVID in stock markets and that investors ascertain the multiple-objective portfolio selection as relatively robust.

Green innovation investments have rapidly grown since 2000. Green innovation indexes play important roles and are typically constructed by screening and indexing. However, Nobel Laureate Markowitz emphasizes portfolio selection instead of security selection and accentuates that “A good portfolio is more than a long list of good stocks.” Moreover, the screening-indexing strategies ignore that investors can take green innovation as an additional objective and thus gain additional utility. We consequently construct 3-objective portfolio selection for green innovation in addition to variance and expected return. An efficient frontier of portfolio selection then extends to an efficient surface which is a panorama of the optimal variance, expected return, and expected green innovation. Investors thus fully envisage the trade-offs and enjoy the freedom of choosing preferred portfolios on the surface. In contrast, the screening-indexing strategies inflexibly leave investors with only one point (i.e., the green innovation index). As the originality, we prove in a theorem that there typically exists a curve on the efficient surface so all portfolios on the curve dominate the green innovation index. We test the dominance by component stocks of China Securities Index 300 and obtain affirmative results out of sample. The results still hold in robustness tests. At last, we classify green innovation into categories, further model the categories by general k-objective portfolio selection, and still illustrate the dominance. Consequently, investors can consider and control each category.

投资组合选择中的系统误差与估计误差是决定样本期外绩效的重要因素,其权衡受到资产基数N的影响.本文在变动基数的设定下,将Bootstrapping和样本期外滚动的方法应用到均权重、最小方差组合及其误差修正策略的绩效和尾部风险检验过程中,并在不同的市场状态下进行分组讨论.研究发现:(1)最小方差组合与均权重策略的样本期外夏普比率差异与N存在倒U型的关系.(2)最小方差组合的尾部风险随N的扩大而迅速降低,总体来看最小方差组合的尾部风险低于均权重策略.(3)最小方差组合的换手率与N存在正相关关系,盲目增加投资组合选择中的资产基数会带来无谓损失.研究结果表明,投资者应理性选择资产基数,充分利用最小方差组合带来的分散化收益.

实现双碳目标是一场广泛而深刻的变革.面对双碳目标带来的挑战和变化,近两年来学者进行了多领域的大量研究.梳理我国双碳主题文献,利用CiteSpace软件探究双碳主题研究现状、热点和趋势等,重点指出双碳领域建模研究不足的现状,并创新性地提出双碳目标下多目标决策建模的应用.研究结果有助于明确未来研究方向,为解决双碳目标下我国实际问题提供借鉴.