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FULLTEXT01.Pdf Risk Analysis of Wind Energy Company Stocks Xin Jiang July 2020 Faculty of Techonology Supervisor: Astrid Hilbert Abstract In this thesis, probability theory and risk analysis are used to determine the risk of wind energy stocks. Three stocks of wind energy companies and three stocks of technology companies are gathered and risks are compared. Three different risk measures: variance, value at risk, and conditional value at risk are used in this thesis. Conclusions which has been drawn, are that wind energy company stock risks are not significantly lower than the stocks of other companies. Furthermore, optimal portfolios should include short positions of one or two of the energy companies for the studied time period and under the different risk measures. Key words: Probability theory, risk analysis, variance, value at risk, conditional value at risk. Acknowledgements I would like to thank my supervisor Astrid Hilbert at Linnaeus University for an- swering my questions concerning the thesis and for introducing me to appropriate literature. I also would like to thank Roger Pettersson for teaching me the knowledge of simulation and for showing me the skills to work with real stock price data. Contents 1 Introduction 2 1.1 Background . .2 1.2 Basic Concepts and Tools . .3 1.2.1 Concepts from Probability Theory . .3 1.2.2 Tools from Multivariate Statistics . .4 1.2.3 Concepts from Finance . .8 2 Risk Measures 10 2.1 Variance . 10 2.2 Value at Risk (V aR)........................... 11 2.3 Conditional Value at Risk (CV aR)................... 11 3 Empirical Result of Risk Comparison 12 3.1 Empirical Analysis of Data . 12 3.2 Empirical Results . 14 4 Methodology of Portfolio Optimization 15 4.1 Mean-Variance Optimization . 15 4.2 Value at Risk Optimization . 16 4.3 Conditional Value at Risk Optimization . 16 5 Empirical Results of Portfolio Optimization 19 5.1 Mean-Variance Optimization . 20 5.1.1 Algorithm Analysis . 20 5.1.2 Results of MV Optimization: . 21 5.2 VaR Optimization . 22 5.2.1 Algorithm Analysis: . 22 5.2.2 Results of VaR Optimization: . 23 5.3 CVaR Optimization . 25 5.3.1 Algorithm Analysis: . 25 5.3.2 Results of CVaR Optimization: . 26 6 Conclusion and Discussion 28 7 References 30 8 Appendix 32 1 1 Introduction 1.1 Background Wind energy is a lucrative field for investors. According to the American Wind Energy Association in 2018, wind power industry in the US grew about 9% in 2017, increasing its power generation capacity by more than 7,000 megawatts. In 2017, wind energy accounted for 6.3% of the total US electricity supply. This 6.3% is more important than it looks - enough to give power to nearly 27 million homes. There is still more room for growth at the global level. The International Energy Agency estimates that wind power will be 18% of the worldwide energy production by 2050 (Olson, 2019). Will the new energy company stocks be significantly less risky than other compa- nies? If so, they may want only to raise product prices and infringe on the interests of the people due to rigid demand. Hence, this thesis investigates the stocks of new energy companies to determine if they are less risky than other stocks. This thesis examines stocks of three large wind energy companies: General Elec- tric Company (GE) (Boston, Massachusetts; USA), Vestas Wind Systems A/S (VWDRY) (Aarhus; Denmark), and Pattern Energy Group Inc. (PEGI) (San Fran- cisco, California; USA). For comparison, three stocks of non-energy companies were chosen: Apple (technology company), Amazon (online marketplace), and Facebook (social media). The data for 6 years of daily closing stock prices was obtained from the website of Yahoo Finance. The thesis compares the risks of these 6 stocks with three risk measures: variance, value at risk (VaR), and conditional value at risk (CVaR). Further more, the invest- ment recommendations are given based on the calculation results of the portfolio optimisation. The remainder of this thesis is organised as follows: Section 1 introduces basic con- cepts and tools from probability theory, statistics, and finance which will be used in this thesis. Section 2 reviews preliminaries on risk measures. Section 3 calculates and compares the risk of each stock using different risk measures. Section 4 intro- duces the sample-based portfolio optimisation model with the given risk measures. Section 4 presents the selected statistical results of the sample-based portfolio opti- misation with the different risk measures. Finally, Section 5 presents the conclusions and investment advice. 2 1.2 Basic Concepts and Tools Here, we introduce some basic concepts and tools from probability theory, statistics and finance which will be used in this thesis. 1.2.1 Concepts from Probability Theory Definition 1.1. Let (Ω; F;P ) be a probability space. A random variable X is a Borel measurable function from the sample space Ω to R: X :Ω ! R (Gut, 2005, p.25). Theorem 1. A Borel measurable function of a random variable is a random vari- able, viz., if g is a real, Borel measurable function and X is a random variable, then Y = g(X) is a random variable. Proof. See Gut, 2005, p.28. Proposition 1.1. Suppose that X1;X2; ::: are random variables. The following quantities are random variables: (a) maxfX1;X2g and minfX1;X2g; (b) supn Xn and infn Xn; (c) lim supn!1 Xn and lim infn!1 Xn . Proof. See Gut, 2005, p.28. Definition 1.2. Let X be a real-valued random variable. The distribution func- tion of X is F (X) = P (X ≤ x); x 2 R; The continuity set of F is C(F ) = fx : F (x) is continuous at xg: Whenever convenient we index a distribution function by the random variable it refers to, FX ;FY , and so on (Gut, 2005, p.30). Definition 1.3. Random vectors are elements in the Euclidean space Rn for some n 2 N. An n-dimensional random vector X is a Borel measurable function from 3 the sample space Ω to Rn: n X :Ω ! R Random vectors are here considered as column vectors: 0 X = (X1;X2; :::; Xn) where 0 denotes transpose (i.e., X0 is a row vector). The joint distribution function of X is FX1;X2;:::;Xn (x1; x2; :::xn) = P (X1 ≤ x1;X2 ≤ x2; :::; Xn ≤ xn) for xk 2 R, k = 1; 2; :::; n (Gut, 2005, p.43). Definition 1.4. Let X be a one-dimensional random variable. The (a) moments are EXn, n = 1; 2; :::; (b) central moments are E(X − EX)n, n = 1; 2; :::; The first moment, EX, is the mean. The second central moment is called variance: V arX = E(X − EX)2 (= EX2 − (EX)2): (Gut, 2005, p.62). 1.2.2 Tools from Multivariate Statistics In statistics, the complexities of most phenomena require collecting observations on many variables. In this thesis, I consider the daily returns of six stocks. These observation data are regarded as a 6-variate population. Definition 1.5. A single multivariate observation is the collection of measure- ments on p variables taken on the same item or trial. If n observations have been obtained, the entire data set can be placed in an n × p array (matrix): 0 1 x11 x12 ::: x1p B C B x21 x22 ::: x2p C B C B ·· ::: · C B C Xn×p = B C B ·· ::: · C B C B ·· ::: · C @ A xn1 xn2 ::: xnp 4 Each row of X represents a multivariate observation. Since the entire set of measurements is often one particular realization of what might have been observed, we say that the data is a 'sample' of size n from a p-variate 'population'. The sample then consists of n measurements, each of which has p components (Johnson and Wichern, 2014, pps.111-112). Remark. For a p-dimensional scatter plot, the rows of X represent n points in a p-dimensional space. We can write: 0 1 0 0 1 x11 x12 ::: x1p x1 B C B 0 C B x21 x22 ::: x2p C B x2 C B C B C B ·· ::: · C B · C B C B C Xn×p = B C = B C B ·· ::: · C B · C B C B C B ·· ::: · C B · C @ A @ 0 A xn1 xn2 ::: xnp xn 0 The jth row vector xj, representing the jth observation, contains the coordinates of a point (Johnson and Wichern, 2014, p.112). Definition 1.6. Suppose the data have not been observed, but we intend to collect n sets of measurements on p variables. Before the measurement, their values cannot, in general, be predicted exactly. Consequently, we treat them as random variables. In this context, let the (j; k)-th entry in the data matrix be the random variable Xjk. Each set of measurement Xj on p variables is a random vector, and we have the random matrix 0 1 0 0 1 X11 X12 ::: X1p X1 B C B 0 C B X21 X22 ::: X2p C B X2 C B C B C B ·· ::: · C B · C B C B C Xn×p = B C = B C B ·· ::: · C B · C B C B C B ·· ::: · C B · C @ A @ 0 A Xn1 Xn2 ::: Xnp Xn 5 A random sample can now be defined (Johnson and Wichern, 2014, p.119). 0 0 0 Remark. If the row vectors X1; X2; ·; ·; ·; Xn represent independent observations from a common joint distribution with density function f(x) = f(x1; x2; :::; xp), then X1; X2; ·; ·; ·; Xn are said to form a random sample from f(x). Mathematically, X1; X2; ·; ·; ·; Xn form a random sample if their joint density function is given by the product f(x1)f(x2) ··· f(xn), where f(xj) = f(xj1; xj2; :::; xjp) is the density function for the jth row vector (Johnson and Wichern, 2014, p.119).
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