Optimisation and Financial Mathematics代写 Mathematics代写

The University of Sydney School of Mathematics and Statistics

Optimisation and Financial Mathematics代写 In this computer project you will be analysing real stock market data downloaded from Yahoo!Finance.

Computer Project Optimisation and Financial Mathematics代写

MATH2070/2970: Optimisation and Financial Mathematics Semester 2, 2019

Web Page: http://www.maths.usyd.edu.au/u/IM/MATH2070/

Lecturer: Anna Aksamit and Georg Gottwald

Due on Sunday 10th November at 11:59pm on TurnitIn

Submit exactly two fifiles: a pdf with your report and m fifile 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 (maximum up 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 at 12:01am will be an automatic deduction of 5%. It is your responsibility to check that your submission was successful.

MATH2070: Do all questions except Question 6.Optimisation and Financial Mathematics代写

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 of the 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)-daily basis 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 and were traded as a new entity DowDuPont, then in 2019 Dow spun off of DowDuPont and was added to the Dow Jones Industrial Average. Therefore only consider the 29 stocks without Dow (due to a short trading history).Optimisation and Financial Mathematics代写

All prices are in US dollars.

Correlations and the covariance matrix

1.Export the data into Matlab using csvread and/or readtable.Optimisation and Financial Mathematics代写

This question investigates the correlations of the return rates of the 29 stocks. When analysing return rate data one has several A commonly used variable is the logarithmic change of price or the so called log return rate: Let S_kt be the price of k-th stock at time t, then consider Y_kt = log S_kt log Sk(t−1) (wrt the natural base).

(i)Calculate the maximal correlation between the Y_k, name and plot the two stock pricesassociated with the highest correlation as a function of time. On the graph present normalised time series so that they start from the same value 100 on 2/01/2013.

(ii)Calculate the minimal correlation between the Y_k, name and plot the two stock pricesassociated with the smallest correlation as a function of time. On the graph present normalised time series so that they start from the same value 100 on 2/01/2013.Optimisation and Financial Mathematics代写

(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 spotdifferences?Plot the price of Dow Jones Industrial Average in the whole period. Can you relate it to your observations about the correlation matrices?

(iv)Plotthe histogram of the correlation coefficients ρ_ij for the two periods from the previous point. Comment on your result.

(v)Find the maximal risk (expressed via standard deviation) of the 29 stocks and the most risky stock corresponding to this Calculate the maximal correlation between themostrisky stock and one of the remaining stocks, name and plot these two stock prices as a function of time. On the graph present normalised time series so that they start from the same value 100 on 2/01/2013. (Here work with the entire time series.)Optimisation and Financial Mathematics代写

(vi)Find the minimal risk (expressed via standard deviation) of the 29 stocks and the least risky stock corresponding to this Calculate the maximal correlation between the leastrisky stock and one of the remaining stocks, name and plot these two stock prices as a function of time. On the graph present normalised time series so that they start from the same value 100 on 2/01/2013. (Here work with the entire time series.)

Portfolio Theory Optimisation and Financial Mathematics代写

2.In this section we consider simple return rates,

that isis the price of the k-th stock at time t. Carry out the following computational tasks for an unrestricted optimal portfolio P ^∗ consisting of the 29 stocks included in the Dow Jones for an agent who wants to invest $200,000 and has a risk aversion parameter t = 0.2.

(a)Computethe dollar investment in each of the stocks and the corresponding expected return and risk of P ^∗.

(b)Illustratethe problem graphically and plot on the same graph in the µσ-plane :

(i)The 29 stocks of theDow Jones.Optimisation and Financial Mathematics代写

(ii)Theminimum variance and efficient Use a t-range |t| ≤ 0.35 for your display.

(iii)Aplot of 1000 random feasible portfolios satisfying x_i 20 (for each of the 29 stocks)and σ_i 0.05 for i = 1, . . . , 1000.

You might notice that the random points occupy some region well-separated from the minimum variance frontier (MVF) – comment on this and explain why (This is a/the major part of the question).

(iv)The indifference curve of an investor with t = 0.2 and their optimal portfolio P^∗.

Determinewhich investors shortsell in the market consisting of the 29 stocks, and which stocks they Are there any stocks which no-one will shortsell or which everyone will shortsell?

Optimisation and Financial Mathematics代写

4.Three funds with different risk proftles:

In this question you will divide 29 stocks with respect to their risk profile into 3 funds. Sort stocks from highest to lowest risk (expressed via varianceor standard deviation). Assuming that each stock has the same contribution to a given fund, form high-risk fund from the 9 most risky stocks, low-risk fund from the 10 least risky stocks and mid-risk fund from the rest.

(a)Compute expected returns and covariance matrix of the 3

(b)LetPˆ be an unrestricted optimal portfolio consisting of the 3 funds for an agent who wants to 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 expected returnand risk of Pˆ.Optimisation and Financial Mathematics代写

(ii)Ploton the second µσ-plane graph :

The 3 funds.
The minimum variance and efficient frontiers. Use a t-range t 0.35 for your display.
Theindifference curve of an investor with t = 0.2 and their optimal portfolio Pˆ.
The minimum variance frontier and optimal portfolio P ^∗ from Question Compare solutions P ^∗ and Pˆ to the two problems based on computations and graphs.

Capital Asset Pricing Model Optimisation and Financial Mathematics代写

Assumethat the daily risk free rate in the studied period was 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 Security Market 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 Portfolio

In 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, …, R_n]. The objective is to maximise the expected logarithmic utility of the final wealth W₁, i.e.,

E log(W₁) = E log(W₀x^T (1 + R)) = E log(x^T (1 + R)) + log(W₀)

where W₀ is initial wealth. To ensure well-posedness of the problem there is no-shortselling constraint imposed on feasible portfolio vector x, that is x = [x₁, x₂, …, x_n] ∈ Rn satisfies x_i ≥ 0 for i = 1, …, n and Σn x_i = 1.Optimisation and Financial Mathematics代写

Let us introduce R¯ := 1 + R = [1 + R₁, 1 + R₂, …, 1 + R_n], 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, and therefore F is of the form F (y) = Σk:y ≤y p_k with p_k = P(R¯ = y_k) for each y_k in a countable set y₁, y₂, … .

The problem of logarithmic utility maximisation can be written as

Note that the above expectation is computed with respect to the distribution of R¯ determined by F .

For a fixed F , a feasible portfolio x^∗ that achieves the maximum of v(x, F ) is called log optimal portfolio, i.e.,

(a) Showthat

(i)v(x, F ) is concave in x and linear inF ,

(ii)v^∗(F ) isconvex, Optimisation and Financial Mathematics代写

(iii)theset 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 following necessary and sufficient conditions:

Hint: Using 6(a)(i) argue that log optimal portfolio can be characterised locally by an appropriate condition on the directional derivative of v in the direction from x^∗ to any other feasible portfolio x.

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