Investing in commodities may have important benefits for investors but only in the last few decades have they started to think more about this possibility. Furthermore, large investors are more inclined to change their own personal view. Therefore, understanding the benefits that commodities could give to an investment portfolio might alleviate investors’ concerns. Several previous studies, as Belousova and Dorfleitner (2012) suggest, that the commodities with higher benefits are precious metals and gold, in particular. The purpose of our work is to understand which possible benefits are for equity investors and if they are common for certain commodities with different physical characteristics. The first part of our empirical work focuses on the main descriptive statistics of the return distribution (mean, variance, volatility, skewness, kurtosis and correlation) for 8 stock indices and 7 commodity futures. The main goal of this is to understand the differences among the commodities and between the commodities and the stock indices. In the second part of the empirical work, we test the safe-haven and the hedge properties of these commodities on a weekly basis for all of them with stock indices, and we do the same on a daily and monthly basis for only commodities which are negatively correlated on average with the stock indices. In the last part of our work, we combine these 7 commodities, following the principles of Bloomberg Commodity Index (BCOM), in order to create a well-balanced and well-diversified commodity index. Additionally, we create some mixed portfolios using this index and a different stock index every time. After that we look at the volatilities and the returns of these mixed portfolios with different weight combinations. Our main goals in this section are to understand the characteristics of the commodity index in comparison with stock indices and then, finding which weight combinations give the mixed portfolios the optimal risk-return trade off. Understanding which are efficient weights, can lead to conclusions about the weight that commodities should have in a portfolio according to the risk tolerance of the investors.  The research is done considering three time frequencies: daily, weekly and monthly; in line with the ones used by Baur and McDermott (2010). The sample size differs among these three different time basis. In fact, daily data started in January 2007 and the other two time frequencies data began with January 1997. All the time samples ended in March 2016. The results of the first part show that gold is the only commodity with a volatility similar to the stock indices (it also has a higher average return) and that on the daily, weekly and monthly basis. Whereas, the other commodities are much riskier than stock indices since they have higher volatility for all the three time-frequencies analyzed.  The results of the second part suggest that only gold is both a safe-haven and hedging commodity in line with the methodology used by Baur and McDermott (2010), but only for DAX 30 on a weekly basis. Furthermore, our results also show that natural gas is strong hedge in some cases such as natural gas for STI (Singapore) on a monthly basis or gold for Nikkei 225 on daily, weekly and monthly basis. Other commodities are neither safe-haven nor hedge in any case, except for silver which is a safe-haven commodity for DAX 30 and Sensex which at its worst, 1% and 5%, declines in the market respectively. The results of the last part of our work show that all the minimum variance mixed portfolios (the ones with the weights give the lowest risk) - made on a weekly basis - reduce the portfolio volatility and make the portfolio returns higher than the stock indices returns in 5 cases out of 8. Additionally, the results show how investors, who add a well-balanced and well-diversified commodity index to their portfolios, are able to observe several weight combinations and choose the one which suits their risk tolerance. Moreover, our results show that the optimal-weight combinations for commodity weights are lower than 0,5 only for FTSE 100 and S&P 500 (both values are 0,49) and higher than 0,62 but lower than 0,7 for DAX 30, Nikkei 225, Hang Seng, Sensex, SSEC. Furthermore, the optimal weight for STI is 0,54.

Combining multiple forecasts in order to generate a single, more accurate one is a well-known approach. A simple average of forecasts has been found to be robust despite theoretically better approaches, increasing availability in the number of expert forecasts, and improved computational capabilities. The dominance of a simple average is related to the small sample sizes and to the estimation errors associated with more complex methods. We study the role that expert correlation, multiple experts, and their relative forecasting accuracy have on the weight estimation error distribution. The distributions we find are used to identify the conditions when a decision maker can confidently estimate weights versus using a simple average. We also propose an improved expert weighting approach that is less sensitive to covariance estimation error while providing much of the benefit from a covariance optimal weight. These two improvements create a new heuristic for better forecast aggregation that is simple to use. This heuristic appears new to the literature and is shown to perform better than a simple average in a simulation study and by application to economic forecast data.

我們以公平頻寬配置考慮網路上多重等級與多重服務品質的效用函數， 利用權重效用函數提出兩種數學最佳化模型。 這兩個模型的目標都是要尋找權重效用函數總和值的最大值。 本篇論文特別以權重為決策變數， 研究最佳權重的行為模式， 並求得最佳權重分佈公式。 我們發現模型I的總權重效用只看重某個效用值最大的等級， 完全忽略其他效用值較小的等級； 即最大效用函數的最佳權重為1，其他效用較小的最佳權重為0。 在最佳化過程中， 模型II的數值資料呈現出最佳權重架構為：最佳權重中的每個權重均相等，且總和為1。 我們隨後證明這些結果，並利用GAMS軟體來呈現數值資料。 / We propose two mathematical models with weighted utility functions for the fair bandwidth allocation and QoS routing in communication networks which offer multiple services for several classes of users. The formulation and numerical experiments are carried out in a general utility-maximizing framework. In this work, instead of being fixed, the weight for each utility function is taken as a free variable. The objective of this thesis is to find the structure of optimal weights that maximize the weighted sum of utilities of the bandwidth allocation for each class. We solve it by proposing two models in terms of fairness. Model I and II are constructed to compare different choices for optimal weights. For Model I, the structure of optimal weights form a vector which consists of one for a class and zero otherwise. For Model II, the form of optimal weights is that each weight of utility function is equally assigned. The results are proved and illustrated by software GAMS numerically.