On Heavy-Tailed Probability Distributions and Portfolio Diversification

''Don't put all your eggs in one basket'' is a common saying that applies particularly well to investing. Thus, the concept of portfolio diversification exists and is generally accepted to be a good principle. But is it always and in every situation preferable to diversify one's investments? This Master's thesis explores this question in a restricted mathematical setting. In particular, we will examine the profit-and-loss distribution of a portfolio of investments using such probability distributions that produce extreme values more frequently than some other probability distributions. The theoretical restriction we place for this thesis is that the random variables modelling the profits and losses of individual investments are assumed to be independent and identically distributed. The results of this Master's thesis are originally from Rustam Ibragimov's article Portfolio Diversification and Value at Risk Under Thick-Tailedness (2009). The main results concern two particular cases. The first main result concerns probability distributions which produce extreme values only moderately often. In the first case, we see that the accepted wisdom of portfolio diversification is proven to make sense. The second main result concerns probability distributions which can be considered to produce extreme values extremely often. In the second case, we see that the accepted wisdom of portfolio diversification is proven to increase the overall risk of the portfolio, and therefore it is preferable to not diversify one's investments in this extreme case. In this Master's thesis we will first formally introduce and define heavy-tailed probability distributions as these probability distributions that produce extreme values much more frequently than some other probability distributions. Second, we will introduce and define particular important classes of probability distributions, most of which are heavy-tailed. Third, we will give a definition of portfolio diversification by utilizing a mathematical theory that concerns how to classify how far apart or close the components of a vector are from each other. Finally, we will use all the introduced concepts and theory to answer the question is portfolio diversification always preferable. The answer is that there are extreme situations where portfolio diversification is not preferable

Galerkin procedures for stochastic partial differential equations

Imperial Users onl

Problem equivalence and necessary conditions of relaxed dynamic programming type in optical control

Imperial Users onl

Modelling dynamic stochastic user equilibrium for urban road networks

In this study a dynamic assignment model is developed which estimates travellers' route and departure time choices and the resulting time varying traffic patterns during the morning peak. The distinctive feature of the model is that it does not restrict the geometry of the network to specific forms. The proposed framework of analysis consists of a travel time model, a demand model and a demand adjustment mechanism. Two travel time models are proposed. The first is based on elementary relationships from traffic flow theory and provides the framework for a macroscopic simulation model which calculates the time varying flow patterns and link travel times given the time dependent departure rate distributions; the second is based on queueing theory and models roads as bottlenecks through which traffic flow is either uncongested or fixed at a capacity independent of traffic density. The demand model is based on the utility maximisation decision rule and defines the time dependent departure rates associated with each reasonable route connecting, the O-D pairs of the network, given the total utility associated with each combination of departure time and route. Travellers' choices are assumed to result from the trade-off between travel time and schedule delay and each individual is assumed to first choose a departure time t, and then select a reasonable route, conditional on the choice of t. The demand model has therefore the form of a nested logit. The demand adjustment mechanism is derived from a Markovian model, and describes the day-to-day evolution of the departure rate distributions. Travellers are assumed to modify their trip choice decisions based on the information they acquire from recent trips. The demand adjustment mechanism is used in order to find the equilibrium state of the system, defined as the state at which travellers believe that they cannot increase their utility of travel by unilaterally changing route or departure time. The model outputs exhibit the characteristics of real world traffic patterns observed during the peak, i. e., time varying flow patterns and travel times which result from time varying departure rates from the origins. It is shown that increasing the work start time flexibility results in a spread of the departure rate distributions over a longer period and therefore reduces the level of congestion in the network. Furthermore, it was shown that increasing the total demand using the road network results in higher levels of congestion and that travellers tend to depart earlier in an attempt to compensate for the increase in travel times. Moreover, experiments using the queueing theory based travel time model have shown that increasing the capacity of a bottleneck may cause congestion to develop downstream, which in turn may result in an increase of the average travel time for certain O-D pairs. The dynamic assignment model is also applied to estimate the effects that different road pricing policies may have on trip choices and the level of congestion; the model is used to demonstrate the development of the shifting peak phenomenon. Furthermore, the effect of information availability on the traffic patterns is investigated through a number of experiments using the developed dynamic assignment model and assuming that guided drivers form a class of users characterised by lower variability of preferences with respect to route choice

Low Maintenance Strategies and Challenges within Oregon School Landscapes

Ornamental landscapes require considerable amounts of inputs, including but not limited to irrigation, mowing or pruning, fertilization, and pest management. However, school systems have limited budgets, which reduce their access to resources and labor hours. Therefore, the objective of this project is to identify ground covers that can compete with weeds and maintain aesthetic quality while under minimal maintenance. To explore this objective, a field experiment was initiated in May 2015 at Corvallis, OR. Experimental design was a randomized complete block design with 4 replications. Factors include year (2015 and 2016) and ground cover taxa. Taxa included 3 turfgrasses (Festuca rubra L. ssp rubra ‘Chantilly’, Festuca rubra L. spp. commutata ‘Longfellow II’, Agrostis tenuis Sibth ‘Puritan’) and 7 forb or shrub plants (Vinca minor ‘Illumination’, Cotoneaster dammeri ‘Coral Beauty’, Euonymus fortunei ‘Kawensis’, Juniperus horizontalis ‘Blue Chip’, Herniaria glabra ‘Green Carpet’, Sedum spurium ‘Tri-Color, and Ceanothus glorious ‘Point Reyes’), which were selected using a school system stakeholder group. All plots received daily irrigation for the first 4 months and subsequently discontinued in September 2015. Plots are weeded and fertilized (4.88 g nitrogen m-2) once annually. Results determined that A. tenuis had the highest plant cover (68.1%), followed by F. Rubra ‘Chantilly’ (68.1%) and F. Rubra ‘Longfellow’ (66%), then S. spurium (24%) and J. horizontalis (22.6%). The remaining ground covers all provided less than 7% plant cover. A strong inverse correlation between plant ground cover and weed ground cover was identified in both years (R2= 0.978, R2= 0.948).Keywords: weeds, turfgrass, input, landscape, maintenance, Orego