HOMEWORK I: PORTFOLIO MANAGEMENT

Calculations and optimization should be done preferably in Matlab (although Excel is also acceptable). You are welcome to use the Matlab Tutorial provided on my webpage as well as formulas weve used in class (included in Lecture_2.m on my webpage) for guidance.

The project should have a summary of all results separately (for example in a MS Word file) from the programs themselves (printed from a Matlab Editor .m file). Both results and programs should be submitted in HARD COPIES IN CLASS ON THE DUE DATE .

DUE: February 14 (in class)

This homework uses the data you obtained for the preliminary homework due the first day of class. Section A below summarizes what data you had to extract. It is provided here for your convenience, you have already done this part.Section B below you have already done, Im providing it as a check that you have the correct data.Your homework should start with Section I.A.Import the data into Matlab: a)You should have the monthly value-weighted return of Food, Oil, Cars,Finan from July 1926 to December 2013 into a matrix R.b)The returns should be in decimals (i.e. 0.0209 not 2.09%). c)You should have three variables in your workspace: R = a 1050x4 matrix of industry portfolio returns dates = a 1050x1 vector of dates names = a 1x4 vector of industry portfolio namesB.This part you already did, so Im providing it to make sure you have the correct data to start with. You will need those to plot your industries and optimize your portfolio. These are the mean returns (returns), standard deviations, and variance-covariance matrix of returns. Hint: make sure that your returns are correct and in decimals! They should be:returns =mean(R) = 0.0100 0.0108 0.0114 0.0102

st.deviations = std(R) = 0.0481 0.0610 0.0784 0.0688covariances =cov(R) = 0.0023 0.0018 0.0026 0.00270.0018 0.0037 0.0028 0.00280.0026 0.0028 0.0061 0.00420.0027 0.0028 0.0042 0.0047II.Including the risk-free rate (you can combined the risk free rate with only one industry)i.If the investor decides to allocate his money between the risk-free rate and one of the portfolios, which portfolio would he choose and why? ii.Plot the best capital allocation line (you still need to show the other portfolios and the previous indifference curve in the same graph to see how things change).iii.How much should this investor allocate to the risk-free asset and how much to the preferred industry portfolio to maximize his utility? (Assume the borrowing rate is the risk-free rate.)iv.What is his utility at this optimal allocation? Has his utility improved due to the existence of a risk-free rate? v.Plot the investors new best indifference curve (you still need to show the other portfolios in the same graph, as well as the best capital allocation line, and the old indifference curve to see the change. Hint, use the hold on function).

III. Efficient Frontier of RISKY assets (WITHOUT the risk-free rate)a.No Short Selling or Leveraged Positions Allowedi.Plot the efficient frontier based on all portfolios.

(Make sure you plot all portfolios in the same graph, so use hold on after youve run the command below and then plot the portfolios. Youll plot more things on this graph). %This plots the efficient frontierfrontcon(returns,covariances,100) % This gives you the risk, returns, and weights along the efficient frontier [PortRisk, PortReturn, PortWts] = frontcon(returns,covariances,100)

ii.Find (numerically) the optimal portfolio for the investor (the one with the highest utility) and report its weights. (Calculate the utility for all portfolios along the frontier: you have the risk and return of all points on the frontier. Then find the maximum utility point and see what its weights are.)

iii.What is the utility of the investor at this optimal portfolio?iv.Plot the indifference curve across the optimal portfolio. (The graph needs to show all industry portfolios, the efficient frontier, and the indifference curve!)

b.Allowing for short selling (weights up to -100% in any stock) or leveraged positions (weights up to 200% in any stock)v.Plot the new less constrained efficient frontier based on these portfolios. (ignore any margin requirements and margin fees) (This new frontier needs to be on the same graph as in a.)[PortRisk, PortReturn, PortWts] = frontcon(returns,covariances,100,[],[MinWeights,MaxWeights])Then plot the new efficient frontier (using the outputs above) on the OLD graph so you can compare how the frontier has expanded.

vi.Find the new optimal portfolio (i.e. optimal weights) for the investor. (Find it numerically as before). vii.What is the utility of this investor at this optimal portfolio?viii.Plot the indifference curve across the optimal portfolio. (Make sure you still show on the same graph all portfolios, the old frontier and indifference curve, and the new frontier and indifference curve).

ix.Has the utility of the investor increased by removing the no-short-sale investment constraints?

IV. Optimal OVERALL portfolio: combining the risk-free asset with the optimal RISKY portfolioi.Plot the new efficient frontier with the risk free rate based on the portfolios Assume short selling and leveraged positions are allowed as above.(Start a new graph, stop holding on to old one in section III).Hint:Inputs for function:[RiskyRisk, RiskyReturn, RiskyWts,RiskyFraction, OverallRisk,OverallReturn] = portalloc (PortRisk, PortReturn, PortWts,RisklessRate, BorrowRate, RiskAversion)Note that you need the output of frontcon as inputs in portallocTo get just the graph:portalloc(s_p,r_p,w_p,Rf,Rf,A);

To get the coordinates of the points on the graph:[RiskyRisk,RiskyReturn,RiskyWts,RiskyFraction,OverallRisk,OverallReturn] = portalloc (s_p,r_p,w_p,Rf,Rf,A)

ii.What are the weights of the different industries in the optimal RISKY portfolio?iii.What is the return and standard deviation of the optimal RISKY portfolio?iv.What is the asset-allocation (i.e. weight of the risky asset) in this investors optimal OVERALL portfolio?v.What is the risk and standard deviation of the investors optimal OVERALL portfolio?vi.What is the utility of the investor at this optimal portfolio?vii.Summary: Now compare the utility that the investor had under the following investment options:1.Just one industry portfolio:2.Just one industry portfolio combined with the risk free rate3.A portfolio of all industry portfolios with NO borrowing and short-selling.4.A portfolio of all industry portfolios with the more relaxed weight constraints5.A portfolio of all industry portfolios with the risk free rate (and more relaxed weight constraints).(Youve already calculated these, now just put them in a table for comparison and point which investment option is the best).Note: Im looking for the numerical solutions/answers throughout the homework. You dont need to estimate anything by hand, just locate the answer in the matrices/vectors.