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25503 Investment Analysis Assignment – Part 2

1. (a) Transform the stock prices and index values in the ‘Sample Data’ tab into continuously compounded returns (you do not need to report these in your submission).

(b) Using the resulting returns data, estimate (and report) the vector of expected returns for the twelve stocks and the index. You should also report the variance-covariance matrix for the twelve stocks as well as the variance of the index. The expected returns etc. should be annualised (i.e., in annual units).

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(c) Using the ASX200 index as a proxy for the market portfolio (MP), estimate and report the betas of the twelve stocks.

(d) Decompose the total risk (variance) of each asset into its systematic and unsystematic components, i.e., report all three values (variance, systematic risk, unsystematic risk) along with the diversification ratio (R2) for each stock and the index.

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(e) Assuming risk-free borrowing and lending at rF = 1% per annum, plot the capital market line (CML) and indicate the positions of the twelve stocks as well as the MP.

(f) Plot the security market line (SML), and indicate the positions of the twelve stocks as well as that of the MP. Based on this graph, discuss which stocks look over-valued, and which stocks look under-valued?

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2. (a) Report the weights (in the twelve stocks) of the portfolio whose variance is minimised but whose exposure to the index is exactly one, i.e., that has βP = 1. You should describe in words what you have done in Excel and report the value of your portfolio’s (minimised) variance.

Our purpose is to minimize the variance of the portfolio involving the 12 stocks and at the same time ensure that the beta portfolio is exactly equal to 1. To do this, we first prepare all the values needed for the minimization through solver involving our vector of portfolio weights, omega, input formulas for variance, beta vector and sum of portfolio weights. We then use solver to set the objective for minimization, which is the variance of the portfolio: σ2 (p) = xT*Ω*x. We then set the changes in variables to the portfolio weight vector to find the optimal weights that allow us to achieve the minimum variance for that portfolio. Next is to set the constraints, which are: Beta (p) = xT* = 1 Sum of x's = xT*1 = 1 The optimal weights that minimizes the variance of the portfolio is then solved, receiving the minimum variance of the portfolio of 0.018182.

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(b) Report the weights (in the twelve stocks) of the portfolio that minimises the Root Mean Square Error (RMSE) of the difference in weekly returns between the portfolio and the ASX200 index. More specifically, let r1, …, rT be the vector-valued sample returns of the twelve stocks, for t = 1, …, T weeks. Similarly, let rI,1, …, rI,T denote the sample returns of the index. Then you want to find the vector of portfolio weights that solves the following minimisation problem:

Again you should describe in words what you have done in Excel as well as report the minimum value of the RMSE achieved.

We first create 4 columns: Column 1: This contains the values of the portfolio returns of the 12 stocks for T weeks. This is calculated by multiplying the horizontal vector-valued sample returns of the 12 stocks by the vertical vector-valued portfolio weights which will be subjected to change in solver.

Column 2: This contains the values of the week returns of the ASX 200 index.

Column 3: This contains the differences between column 1 and column 2 which represent the difference between the weekly returns of the portfolio and the returns of the ASX 200 index at T weeks.

Column 4: comlumn 3 squared. When all cells are filled in until T weeks, we then found the RMSE which is

Using solver, we then set this as the objective for minimization as well as setting changes in variables to the vector of portfolio weights. The constraint is set to ensure that the sum of the portfolio weights will equate to 1. Solver then determines the weights and minimized variance subject to the constraint.

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(c) Report the expected return, variance, beta and R2 for your two tracker portfolios constructed above. Which method do you recommend to your boss and why?

We would recommend using tracker 2. This is because using tracker 2 will allow us to generate a greater expected return and a higher variance than tracker 1. The proportion of market risk to the total risk is 91.91% (remaining risk is left undiversified), only slightly less than that of tracker 1.

3. (a) Calculate and report a time-series plot of the tracker portfolio value from June 30, 2017 to April 27, 2018, along with the performance of the ASX200 index, clearly indicating which series is which. You should also normalise the values of both time series so that their values are 100 on June 30, 2017.

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(b) Report the simple annualised return of the tracker portfolio and the ASX200 index over the investment period.

(c) Using the weekly continuously compounded returns for the tracker portfolio and the index, report the beta, R2, and RMSE for the tracker portfolio over the investment period. Comment on how close these values are to the values found for the tracker portfolio ‘in sample’ from Questions 2(b/c).

RMSE: 0.004681 These values of the tracker portfolio is lower than that of 2(b,c). The betas for both portfolios show that the out-of-sample data portfolio is 7% less volatile than the in-sample data portfolio. The R^2 calculation shows that the ASX200 data explains 5% more of the variability of in-sample data portfolio compared to the variability of the out-of-sample data portfolio. The out-ofsample data portfolio has a low RMSE than the in-sample data portfolio.

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