Mathematics Homework Help

Subject; Statistics

Title
Quantatative methods

Word Count
3 pages, 750 words

Reference Style
Harvard

No of References
0

ECON 1095 QUANTITATIVE METHODS IN FINANCE

INSTRUCTIONS

Please up load one (and one only) either word or pdf file with both your name and student number clearly printed on the first page. Use a file name that gives your identity.For the excel sections just take screen shots to show some of your workings then cut and paste into your document.For the maths sections, if you prefer, you can hand write and scan and add to your document.

QUESTION 1

In all parts of this question the dice that is referred to has a dodecahedron shape.

• If you roll this dice 12 times what is the probability that you will roll three 6s? Work out both by hand and using the EXCEL function for the Binomial distribution.
• If you roll this dice 12 times what is the probability that you will roll more than one 6? Work out both by hand and using the EXCEL function for the Binomial distribution.
• If you roll this dice 12 times what is the mean and the variance of the number of 6s you will throw? Please use the Binomial formulas (Mean = n.p, Variance = n.p.(1 – p)) to work these out and then check using EXCEL and the general formulas for the mean and variance of any probability distribution:
• If you roll this dice 12 times what is the skewness and the kurtosis of the number of 6s you will throw? Please just use EXCEL to work this out.

m = _i P(X _i) and s² =(X_i – m)²P(X_i)

(2 marks)

QUESTION 2

As China is Australia’s largest trading partner it is possible that the share markets in the two countries are dependent.To look at this use data from ECON1095 Data Sem 2 2019.xls on the Int’l share price indices worksheet and examine the relationship between the monthly returns Australia: S&P/ASX 200 (AUS) and China: Shanghai A (CHINA) from 2009 to 2019.

• Using all the available data for AUS and CHINA examine the relationship between these two variables using graphs. You may do a few graphs; that is, both as levels and as returns and as XY and line (time series) graphs. Does it look as though the variables are dependent or independent?
• Calculate the correlation coefficient between the AUS and CHINA (data analysis, correlation). Again, do this both for the levels and returns. Please interpret the coefficient and offer a brief explanation for the result.
• Using the same data fill in the following table for the number of rises and falls^[1].
• Calculate the joint probability distribution and the marginal probability distributions for the data.
• Using your previous answers as a guide, decide whether AUS and CHINA are dependent or independent?
• Due to the trade war between the US and China you expect Chinese share market returns to fall soon. Calculate the conditional probability distribution for AUS given that CHINA has decreased. Explain how this calculation could be used to guide your Australian investment decisions.

AUS CHINA
		Rise	No change	Fall	Total
	Rise
	No change
	Fall
	Total

(3 marks)

QUESTION 3

An individual’s utility function is represented by:U = 100(X₁^0.4).(X₂^0.6)

Their budget constraint is:

That is, their total budget is $10,000 and the price of X₁ is $2 and the price of X₂ is $5. Use the Lagrangian function to find the optimal values of X₁ and X₂ that maximises utility subject to the budget constraint^[2]. What is the value of l and what is its interpretation?

Check your answer using EXCEL. To do this open a new spread sheet and insert names for X₁ and X₂ in cells B2 and C2 and the starting values for these variables of 1 in both cells B3 and C3 and name these cells Xone and Xtwo. In cell B6 type the formula for the utility function, = 100* (Xone^0.4)*(Xtwo^0.6) and name this UU. In cell B9 type the formula for the budget, =2*Xone+5*Xtwo and name this BB. Next, go to the solver and set target cell UU equal to maximum by changing Xone and Xtwo. Then add the constraint that BB = 10,000, then solve. When given the solver results ask for the sensitivity report as this gives l.

(3 marks)

QUESTION 4 (using excel)

You are trying to diversify your investments across different share markets around the world and so want to construct an efficient frontier. Follow the instructions presented below to construct the efficient frontier for the proportions of your funds that need to be allocated to the various markets (there are additional instructions in the Mathematical Programming notes).

Using the file ECON1095 Data Sem 2 2019.xls and the Int’l share price indices worksheet and the entire sample from 2009 to 2019 as a guide, an investor would like to allocate their funds across the shares markets of USA, Japan, Euro area, UK, New Zealand, Hong Kong, Taiwan, Korea, China and Australia in an optimal way by finding the various efficient allocations. An efficient allocation of funds is one in which expected risk is minimised for a given level of expected return. A graph of the possible optimal values of the returns and risk is called the efficient frontier.

Calculate the average continuous monthly returns, then convert the average monthly returns to yearly returns by multiplying each by 12. Transpose this block of cells and name the average returns Ret. Use the covariance command from EXCEL’s Data Analysis Tools to find the variance-covariance matrix for these monthly returns. This matrix is symmetrical, so the missing elements can be easily filled in. Name this matrix Mvac. Convert Mvac into the variance-covariance matrix for yearly returns by highlighting the cells and entering =12*Mvac [Ctrl]+[Shft]+[Enter]. Name this block Vac. Enter the initial guesses for the optimal weights for the shares and name this block of cells Wts. Transpose these weights and name this block Twts. Find the expected return for the portfolio using =MMULT (Wts, Ret) [Ctrl]+[Shft]+[Enter]. Call this cell Pret. Finding the variance of portfolio returns requires three stages. First, highlight the appropriate cells and enter =MMULT (Wts,Vac) [Ctrl]+[Shft]+[Enter]. Name this block Tvac. Second, highlight a single cell and enter =MMULT (Tvac, Twts) [Ctrl]+[Shft]+[Enter]. Call this cell Pvar. Next, find the portfolio risk, or square root of the portfolio variance and name this cell Prsk. To ensure that the portfolio weights sum to one, enter ten 1’s and name this block Unit. To find the expression for the sum of the weights by entering =MMULT (Unit, Twts) [Ctrl]+[Shft]+[Enter]. Name this cell Wtcn.

Using this work sheet and the EXCEL Solver Tool find the minimum risk for the funds allocations for the various expected returns (I suggest performing the exercise for about ten different expected returns, chosen to ensure a solution can be found). In each case you must constrain the weights, so they are non-negative and sum to one. Use these values to graph the Efficient Frontier with risk on the horizontal axis and returns on the vertical axis.

Write a brief report explaining how your portfolio changes as you try different expected returns.

(5 marks)

QUESTION 5[3]

Using your results from QUESTION 4 and the risk-free rate of return of return given in QUESTION 2 of Assignment 1 obtain the equation for the Capital Market line (CML).Next graph the CML against the efficient frontier you calculated in the previous question.Briefly describe the relationship presented in this graph, including an explanation of the meaning and use of the CML.

(2 marks)

QUESTION 6

Again, using the data in ECON1095 Data Sem 1 2018.xlsx the Int’l share price indices worksheet use regression to look at the relationship between returns on Australia: S&P/ASX 200 (AUS) and China: Shanghai A (CHINA) from 2009 to 2019. That is, estimate AUS_t = b₀ + b₁CHINA_t. This should be done using the three approaches used in the example on Canvas (sample regression example.xlsx).All three approaches should yield the same answer.That is:

• Use the solver to find the values of b₀ and b₁ that minimize e_i².
• Ensure that the values of b₀ and b₁ satisfy the normal equations.
• Use Excel’s regression function to confirm your results in parts (a) and (b).
• Next, evaluate your results by:

• Interpreting the coefficient of determination and the F-statistic.
• Conduct a t-test on the slope coefficient.[4]
• Examine and graph the residuals, then comment.
• Briefly comment on the implications of your findings.
• Interpret the coefficient of determination and the F-statistic.
• Use t-tests to identify the variables that do and do not have a significant impact on Y_t.
• Examine and graph the residuals, then comment.
• Briefly comment on your regression findings.

(4 marks)

QUESTION 7

You are trying to find the main drivers of future returns on the Australian share market using information from 2013 to 2019. The data is in ECON1095 Data Sem 2 2019.xls on the Aust share mkt worksheet and the returns on China: Shanghai A from the Int’l share price indices worksheet. This will be used to construct a one period ahead forecasting model of the form:

Y_t = b₀ + b₁X_{1 t-1} + b₂X_{2 t-1} + …… +b_nX_{n t-1}

Where Y_t are the returns on Share price indices: S&P/ASX 200 and the X_{n t-1’}s are the nine other variables in the worksheet as well as the returns on the Chinese share market from the other worksheet. That is, initially you need to consider ten independent variables.Please use returns for all the market indexes and levels for the other variables.That is, use returns for all variables other than for Dividend yield, Price/Earnings ratio, Market capitalization and Average daily turnover of equities – which all should be in levels.

• Calculate the matrix correlation coefficients between the dependent and independent variables, then briefly interpret.
• Estimate the complete multiple regression equation, then:
• Using data from 2013 until the end of 2018 only and an iterative approach construct the best model you can to forecast Y_t.That is, develop a model that only includes the relevant independent variables.
• Use the model you constructed in part (c) to forecast for 2019 and compare your forecasts with the actual values.Please show in both in a table and on a graph.
• Briefly comment on all of your QUESTION 7 findings.

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[1] You will find the IF function in EXCEL very useful when filling in this table.

^[2] There is no need to check the second order conditions.

[3] There are instructions on how to do this in the Mathematical Programming notes.

[4] Use a level of significance of a = 0.05 for this and all other tests.