The University of SydneySchool of Mathematics and StatisticsComputer ProjectMATH2070/2970: Optimisation and Financial Mathematics Semester 2, 2019Due on Sunday 10th November at 11:59pm on TurnitIn• Submit exactly two files: a pdf with your report and m file with your Matlab code.Report should be pleasant to read and include project formulations, descriptions and outputs(tables, plots, histograms etc), all answers and discussion should be there.Marking will be based on: accuracy, programming and presentation.• Please do not write your name on any sheet.• The deadline is a hard deadline in the sense that in case of a late submission (maximumup to 10 days), you will be deducted 5% of the total marks for each day of delay. Thisis non-negotiable, so make sure you submit in time; a submission on Monday the 11th at12:01am will be an automatic deduction of 5%. It is your responsibility to check that yoursubmission was successful.MATH2070: Do all questions except Question 6.MATH2970: Do all questions.In this computer project you will be analysing real stock market data downloaded from Yahoo!Finance.The file Data_2013_2019.csv which you can download from Ed, contains the daily closing prices ofthe 30 stocks which make up the Dow Jones Industrial Average Index and closing prices of two indexes,Dow Jones Industrial Average Index and S&P 500 Index. Prices are recorded on a (business)-dailybasis between 2/01/2013 and 30/09/2019.There is one particularity with this time series: On 31st of Ausgust 2017 Dow and DuPont merged andwere traded as a new entity DowDuPont, then in 2019 Dow spun off of DowDuPont and was added tothe Dow Jones Industrial Average. Therefore only consider the 29 stocks without Dow (due to a shorttrading history).All prices are in US dollars.Correlations and the covariance matrix1. Export the data into Matlab using csvread and/or readtable. This question investigates thecorrelations of the return rates of the 29 stocks. When analysing return rate data one hasseveral choices. A commonly used variable is the logarithmic change of price or the so called logreturn rate: Let Skt be the price of k-th stock at time t, then consider Ykt = log Skt − log Sk(t−1)(wrt the natural base).(i) Calculate the maximal correlation between the Yk, name and plot the two stock prices associatedwith the highest correlation as a function of time. On the graph present normalisedtime series so that they start from the same value 100 on 2/01/2013.(ii) Calculate the minimal correlation between the Yk, name and plot the two stock prices associatedwith the smallest correlation as a function of time. On the graph present normalisedtime series so that they start from the same value 100 on 2/01/2013.Copyright c 2019 The University of Sydney 1(iii) Visualise the correlation matrices for two subperiods: 1/12/2014–1/09/2016 and 1/09/2016–1/02/2018 (you may use Matlab’s command imagesc). Can you spot differences?Plot the price of Dow Jones Industrial Average in the whole period. Can you relate it toyour observations about the correlation matrices?(iv) Plot the histogram of the correlation coefficients ρij for the two periods from the previouspoint. Comment on your result.Portfolio Theory2. In this section we consider simple return rates, that is Rkt =Skt−Sk(t−1)Sk(t−1), where Skt is the priceof the k-th stock at time t. Carry out the following computational tasks for an unrestrictedoptimal portfolio P∗consisting of the 29 stocks included in the Dow Jones for an agent whowants to invest $200,000 and has a risk aversion parameter t = 0.2.(a) Compute the dollar investment in each of the stocks and the corresponding expected returnand risk of P∗.(b) Illustrate the problem graphically and plot on the same graph in the µσ-plane :(i) The 29 stocks of the Dow Jones.(ii) The minimum variance and efficient frontiers. Use a t-range |t| ≤ 0.35 for your display.(iii) A plot of 1000 random feasible portfolios satisfying |xi| ≤ 20 (for each of the 2代做MATH2070/2970、代写Statistics、M9 stocks)and σi ≤ 0.05 for i = 1, . . . , 1000.You might notice that the random points occupy some region well-separated from theminimum variance frontier (MVF) - comment on this and explain why (This is a/themajor part of the question).(iv) The indifference curve of an investor with t = 0.2 and their optimal portfolio P∗.3. Determine which investors shortsell in the market consisting of the 29 stocks, and which stocksthey shortsell. Are there any stocks which no-one will shortsell or which everyone will shortsell?4. Three funds with different risk profiles: In this question you will divide 29 stocks withrespect to their risk profile into 3 funds. Sort stocks from highest to lowest risk (expressed viavariance or standard deviation). Assuming that each stock has the same contribution to a givenfund, form high-risk fund from the 9 most risky stocks, low-risk fund from the 10 least riskystocks and mid-risk fund from the rest.(a) Compute expected returns and covariance matrix of the 3 funds.(b) Let Pˆ be an unrestricted optimal portfolio consisting of the 3 funds for an agent who wantsto invest $200,000 and has a risk aversion parameter t = 0.2.(i) Compute the dollar investment in each of the stocks and the corresponding expectedreturn and risk of Pˆ.(ii) Plot on the second µσ-plane graph :• The 3 funds.• The minimum variance and efficient frontiers. Use a t-range |t| ≤ 0.35 for yourdisplay.• The indifference curve of an investor with t = 0.2 and their optimal portfolio Pˆ.• The minimum variance frontier and optimal portfolio P∗from Question 2. Comparesolutions P∗ and Pˆ to the two problems based on computations and graphs.2Capital Asset Pricing Model5. Assume that the daily risk free rate in the studied period was 0.002906%. Suppose that Standard& Poor’s 500 Index is the market portfolio (S&P 500 Index prices are included in the data file).Make a new µσ-plane graph showing the risk free asset, market portfolio, and the SecurityMarket Line. Compute the β’s of all relevant assets in this project (29 stocks, 3 funds, andtwo optimal portfolio P∗ and Pˆ from Questions 2 and 4). Plot these assets on the same graph.Identify assets with β’s greater than 1 and lower than 1. Comment on the result and describewhat Portfolio Theory would recommend an investor to do.Log Optimal PortfolioIn this section we consider logarithmic utility maximisation problem. As in Markowitz Portfolio Theory,there is a random vector of returns on n stocks R = [R1, R2, ..., Rn]. The objective is to maximise theexpected logarithmic utility of the final wealth W1, i.e.,where W0 is initial wealth. To ensure well-posedness of the problem there is no-shortselling constraintimposed on feasible portfolio vector x, that is x = [x1, x2, ..., xn] ∈ Rnsatisfies xi ≥ 0 for i = 1, ..., nand Pni=1 xi = 1.Let us introduce R¯ := 1 + R = [1 + R1, 1 + R2, ..., 1 + Rn], and denote the cumulative distributionfunction of R¯ by F, i.e., F(y) := P(R¯ ≤ y). We assume that R¯ is a discrete random variable, andtherefore F is of the form F(y) = Pk:yk≤ypk with pk = P(R¯ = yk) for each yk in a countable set{y1, y2, ...}.The problem of logarithmic utility maximisation can be written asmaximise v(x, F) := ENote that the above expectation is computed with respect to the distribution of R¯ which is uniquelydetermined by F.For a fixed F, a feasible portfolio x∗that achieves the maximum of v(x, F) is called log optimalportfolio, i.e.,(i) v(x, F) is concave in x and linear in F,(ii) v∗(F) is convex,(iii) the set of log optimal portfolios with respect to a fixed F is convex.(b) Prove the following theorem:Theorem: The log optimal portfolio x∗for a fixed distribution F satisfies the followingnecessary and sufficient conditions:Hint: Using 6(a)(i) argue that log optimal portfolio can be characterised locally by anappropriate condition on the directional derivative of v in the direction from x∗to anyother feasible portfolio x.3转自：http://www.3daixie.com/contents/11/3444.html
