# ECON 1095 QUANTITATIVE METHODS IN FINANCE

ECON 1095 QUANTITATIVE METHODS IN FINANCE 150 150 Affordable Capstone Projects Written from Scratch

Essay assignment: ECON 1095 QUANTITATIVE METHODS IN FINANCE

Assignment 2 is and contributes 25% to the assessment of this course.

INSTRUCTIONS

Please up load one (and one only) either word or pdf file.  For the excel sections please just take screen shots of your work 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 also add to word doc.

### QUESTION 1

• Using EXCEL and the formula for the Normal Probability Density Function, graph the function where m equals zero and s equals one (m = 0 and s = 1). What is this function called?
• Do a second normal distribution graph where m and s take on different values (they can take on any values you wish).
• Using the graph data in part (a), take the anti-log of x and re-graph. What is this function called?
• Do a second lognormal distribution graph where m and s take on different values (they can take on any values you wish).

### (1 marks)

QUESTION 2

Interest rates and the share market are often thought to be interrelated.  To look at this you examine the daily S&P200 Index (MKT) and the yield on an Australian Government 10 year bond (RATE) for the 12 month period from 6 July 2017 to 6 July 2018.  This information is available in the Excel file ECON1095 Data Sem 2 2018.xlsx on Canvas.

• To examine the relationship between MKT and RATE graph these two variables against each other. You may do a number of graphs; that is, both as levels and as returns and as XY (scatter) graphs and as line (time series) graphs.  Does it look as though the variables are dependent or independent?
• As a further test, using the appropriate excel function calculate the correlation coefficients between the MKT and RATE. Again, do this both for the levels and the returns. Please interpret these coefficients and offer a brief explanation for the results.
• Using the same data fill in the following table for the number of rises and falls.
MKT

#### RATE

Rise No change Fall Total
Rise
No change
Fall
Total
• Calculate the joint probability distribution and the marginal probability distributions for the data.
• Using your previous answers as a guide, make a decision regarding whether RATE and MKT are dependent or independent?
• Suppose you anticipate a tightening of monetary policy in Australia, therefore you expect RATE increasing in the near future. Calculate the conditional probability distribution for MKT given that RATE has risen.

### (3 marks)

QUESTION 3

An investor is considering putting extra money into the Australian share market; to assist with this decision she analyses the continuous daily returns for MKT from 6 July 2017 to 6 July 2018 using Excel’s Data Analysis/Descriptive Statistics.

• Calculate the 95 % confidence interval for the daily returns for MKT over this period.
• The investor decides that she will only put extra money into the Australian share market if she can rule out negative returns. The average daily returns for the sample period should be above zero.  Therefore the question is, ‘have the daily returns for MKT been far enough above zero for the investor to be confident that they will not go below this level?’  Test to see whether the daily returns on MKT are less than or equal to zero using a level of significance of a = 0.05.  That is, conduct the following test, Ho: m <= 0 , H1: m >0.  Would they invest further in the Australian share market using this rule?
• Test to see whether the daily returns on MKT have a normal distribution using the Jarque-Bera test. This should be done in Excel, by calculating the Jarque-Bera statistic using the formulas from the notes and Excel’s Data Analysis/Descriptive Statistics.  Using the results of this test, comment on the accuracy of the probabilities you calculated in parts (a) and (b) of this question.
• Use the continuous daily returns on MKT to determine whether the rises and falls are independent using a runs test. What are the implications of your findings?

### (4 marks)

QUESTION 4

• Using the same sample period as in QUESTIONS 2 and 3 estimate: MKTt = b0 + b1RATEt. This should be done using the three approaches used in the regression example on Canvas (all should yield the same answer).  That is:
• Use the solver to find the values of b0 and b1 that minimize ei2.
• Ensure that the values of b0 and b1 satisfy the normal equations.
• Use Excel’s regression function to confirm your results in parts (i) and (ii).
• Re-estimate the equation in part (a) using Excel’s regression function, but in the case use continuous returns for MKTt.
• Again re-estimate the same equation using Excel’s regression function, but now use continuous returns for both MKTt and RATEt.
• Discuss the different results you obtained in parts (a), (b) and (c).

### (2 marks)

QUESTION 5

An individual’s utility function is represented by: Their budget constraint is:

That is, their total budget is \$2,000 and the price of X1 is \$6.5 and the price of X2 is \$4.5.  Use the Lagrangian function to find the optimal values of X1 and X2.  That is, find the values of X1 and X2 that maximise utility subject to the budget constraint.  What is the value of l and what is its interpretation?

Check your answer using the solver in 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, = (Xone^0.5)*(Xtwo^0.5) and name this UU.  In cell B9 type the formula for the budget, =6.5*Xone+4.5*Xtwo and name this BB.  Next, go to the solver and set the target cell UU equal to maximum by changing Xone and Xtwo.  Then add the constraint that BB = 2000, then solve.  When given the solver results ask for the sensitivity report as this gives l.

(2 marks)

### QUESTION 6

Using the following matrices:

• State the dimensions of A, B and C.
• Find A.B
• Find B.C
• Find A.C’
• Do (b), (c) and (d) using excel.
• Using excel find C-1, then multiply C-1 by C to find I.

(3 marks)

QUESTION 7 (please use EXCEL for this question).

You believe that concerns about climate change has resulted Australian Oil & Gas stocks being oversold, therefore a portfolio of these types of stocks could present a short term profit opportunity.  Using data in the Excel file ECON1095 Data Sem 2 2018.xlsx, the Raw Prices for constituents and the same sample period used earlier (6 July 2017 to 6 July 2018), follow the instructions below to construct an efficient frontier for the proportions of your funds that need to be allocated to the Australian Oil & Gas stocks.  There are additional instructions in Mathematical Programming notes.

Calculate the average continuous daily returns, then convert to average yearly returns by multiplying each by your sample period (n).  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 the daily 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 =n*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 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 8

One approach to testing the validity of the CAPM is to use the two-pass method.  At the first pass estimate the betas and the variance of the error terms for a number of firms.  This is to be done using the same sample period as in the other questions (6 July 2017 to 6 July 2018) and the daily returns for the Raw Prices for constituents for the Australian Financials firms and the S&P200 Index (MKT).  All of this information is in ECON1095 Data Sem 2 2018.xlsx.  At the second pass use these results to estimate the Security Market Line (SML).  If the model is a good fit this is interpreted as an indication that the CAPM successfully explains the relationship between Risk and Return.

First calculate the continuous returns on all of the Australian Financials shares and the market index.

Note that the cell references in the instructions below are indicative only; they are for a different sample size and the exact cell references you use will depend on how you construct your own worksheets.

At the first pass calculate the systematic risk or betas for all firms with respect the MKT.  Do this by estimating the market model: Rit = b0  +  b1RMt for all firms noting the estimated values of the slopes. To find all the betas go to cell EC64 and enter: =LINEST(EC3:EC62,\$EB3:\$EB62).  Now highlight EC64 and drag it to cell IR64.  To find the average returns for each firm j, the squared beta values and the variances of the error terms for each market model use the following Excel procedures (note: again the cell references are indicative only).

• To find all the average returns go to cell EC65 and enter: =AVERAGE(EC3:EC62)  Then click this cell and drag it to IR65.
• To find the squares of the beta values go to cell EC66 and enter: = EC64^2 Then click this cell and drag it to IR66.
• To find the variances of the error terms in all Market Models use the STEYX function to find their standard deviation and then square this value. Use the STEYX function to enter the Y values and then the X values. To do this, go to cell EC67 and enter the following:             =STEYX(EC3:EC62,\$EB3:\$EB62)^2  Click this cell and drag it to IR67.
• Enter the four sets of values for , E(R), 2 and s2 into a separate worksheet. You can call these three variables BETA, ER, BETASQ and VAR. You may need to transpose rows to columns or columns to rows.  Copy the first five values of each of these variables into your submission so your answers can be checked.

For the second pass estimate the SML:  E(R) = g0 + g1  + uj and evaluate your results.

• What should the values of the intercept and the slope for the SML represent?
• To help determine whether factors other than systematic risk affect expected returns also estimate the following model: E(R) = g0 + g1  + g2 2 + g3s2j + uj
• Interpret the coefficient of determination for both equations, and do the same for the adjusted coefficients of determination.
• Use t-tests to identify the variables which do not have a significant impact on the expected returns, and the variables that do.

(5 marks)

 Use the spreadsheet ‘normal distribution example.xlsx’ on Canvas to do this question.

 Levels means the actual data.

 Do this as continuous returns.

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

 Note that when Excel calculates kurtosis it subtracts 3 from the value.

 There is no need to check the second order conditions.

 Refer to the course notes: Mathematical Programming, Simple and Multiple Regression and Finance Applications for additional help.

 Use a level of significance of a = 0.05 for all tests.

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