碳排放权和其他生产投入类似，是很像大宗商品投入的生产要素。然而，参与碳交易的公司只需要在某个时间，比如一年的结尾实现碳排放权限的减量。碳排放的另一个重要特点是不可忽略的固定交易成本。本文试图揭示参与碳交易的企业应该如何优化协调其生产以及碳排放权的采购决策，以尽量减少其在长期成本。在每个阶段，该公司基于已有的排放权，波动的市场价格和库存水平决定其排放权的购买数量和生产量。我们给出了最优的联合决策结构特征: 最优排放权采购欢是状态依赖型的(s，S)型决策。生产决策基本上是状态依赖型的基库守存决策。分解的启发式计算被发现是有效的计算研究。我们解决方法是定义一类具有一类性质的二维的函数，在一个维度上定义类似K-凸函数的性质，并在另一个维度上定义类似凸函数的性质。 / The trading of emissions permits, as an effective market-based approach for emissions control, is becoming widespread over the world. Similar to any other production input, the emissions permit is a factor of production much like commodity inputs. However, a firm participating in emissions trading has only to balance its permits at the end of a time horizon, say one year. Another key feature of permit trading is the existence of non-negligible, fixed, transaction costs. This paper attempts to shed light on how such a firm should optimally coordinate its production and permit purchasing decisions in order to minimize its total cost in the long run. In each period, the firm has to decide on its permit purchase quantities and production levels based on its on-hand permit and inventory levels as well as the market-prevalent permit price. The latter evolves as a Markov process and the firm also faces random demand. We characterize the optimal joint policy structure: an optimal purchase policy is of the state-dependent (s, S) type and an optimal production decision nearly follows a state-dependent base-stock form. A heuristic based on decomposition is found to be effective from computational studies. Our enabling technique is the identification of a class of two-dimensional functions that in one dimension it looks like K-convex and in another it is similar to convex. / Detailed summary in vernacular field only. / Yuan, Quan. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 74-84). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese. / Abstract --- p.i / Abstract in Chinese --- p.ii / Acknowledgements --- p.iii / Contents --- p.vi / List of Tables --- p.viii / List of Figures --- p.ix / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Motivation --- p.1 / Chapter 1.2 --- Purpose of the Work --- p.3 / Chapter 1.3 --- Structure of the Work --- p.4 / Chapter 2 --- Literature Review --- p.6 / Chapter 3 --- Formulations --- p.10 / Chapter 4 --- A New Class of K-convex Functions --- p.13 / Chapter 4.1 --- A Class of K-Convex Functions --- p.13 / Chapter 4.1 --- Convex Functions --- p.13 / Chapter 4.1.2 --- K-convex Functions in R¹ or Z¹ --- p.15 / Chapter 4.1.3 --- K-convex Functions in R[superscript n] or Z[superscript n] --- p.18 / Chapter 4.1.4 --- C²(K) Functions --- p.19 / Chapter 4.2 --- Z¹ Policies --- p.24 / Chapter 4.3 --- Appendix of Chapter 4 --- p.27 / Chapter 5 --- Optimal Policy --- p.43 / Chapter 5.1 --- A Transformation --- p.43 / Chapter 5.2 --- Policy Characterization --- p.44 / Chapter 5.3 --- Appendix of Chapter 5 --- p.48 / Chapter 6 --- Heuristic and Lower Bound --- p.57 / Chapter 7 --- Numerical Study --- p.61 / Chapter 7.1 --- Parameter Setups --- p.61 / Chapter 7.2 --- Non-monotone Trends in Optimal Policy --- p.63 / Chapter 7.3 --- Heuristic Performance --- p.65 / Chapter 8 --- Concluding Remarks --- p.72 / Bibliography --- p.74

This study is a thesis ending a 120 credit masters program in Mathematics with specialization Financial Mathematics and Mathematical Statistics at the Royal Institute of Technology (KTH). The subject of Smart beta is defined and studied in an index fund context. The portfolio weighting schemes tested are: equally weighting, maximum Sharpe ratio, maximum diversification, and fundamental weighting using P/E-ratios. The outcome of the strategies is measured in performance (accumulated return), risk, and cost of trading, along with measures of the proportions of different assets in the portfolio. The thesis goes through the steps of collecting, ordering, and ”cleaning” the data used in the process. A brief explanation of historical simulation used in estimation of stochastic variables such as expected return and covariance matrices is included, as well as analysis on the data’s distribution. The process of optimization and how rules for being UCITS compliant forms optimization programs with constraints is described. The results indicate that all, but the most diversified, portfolios tested outperform the market cap weighted portfolio. In all cases, the trading volumes and the market impact is increased, in comparison with the cap weighted portfolio. The Sharpe ratio maximizer yields a high level of return, while keeping the risk low. The fundamentally weighted portfolio performs best, but with higher risk. A combination of the two finds the portfolio with highest return and lowest risk. / Denna studie är ett examensarbete som avslutar ett 120 poängs mastersprogram i Matematik med inriktning mot Finansiell Matematik och Matematisk Statistik på Kungliga Tekniska Högskolan (KTH). Ämnet Smart beta studeras i kontexten av en indexfond, där de olika testade principerna för viktning i portföljerna är: likaviktad, maximerad Sharpe-kvot, maximerad diversifiering, och fundamental viktning användandes av P/E-tal. Utfallet i testerna utvärderas i ackumulerad avkastning, portföljrisk, kostnad att handla i portföljen, och ett antal mått på fördelningen av tillgångarna. Studien går stegvis igenom processen för att samla in, ordna, och ”tvätta” data. En kort förklaring av historisk simulering, metoden för att estimera stokastiska variabler såsom kovariansmatriser, är inkluderad, såväl som en analys av distributionen av data. Processen för att optimera portföljerna och hur regler för att vara en UCITS-fond kan omformas till optimeringsvillkor beskrivs. Resultaten indikerar att alla utom den mest diversifierade portföljen har högre ackumulerad avkastning än den marknadsviktade portföljen under testperioden. I alla testade fall ökar handelsvolymen liksom marknadspåverkan när en annan strategi än marknadsviktad används. Portföljen med maximerad Sharpe-kvot ger en hög avkastning med bibehållen låg risk. Den fundamentalt viktade portföljen ger bäst avkastning, men med en litet förhöjd risk. Kombinationen av de båda metoderna ger den portföljen med högst ackumulerad avkastning och samtidigt lägst risk under testperioden.