APS 1022 Computational Project 2024

发布时间：2024-06-04

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APS 1022 Computational Project 2024

Instructions: You may do this project in groups of up to two. Use MAT- LAB and its optimization routines. Due June 4nd, 2024 by 5PM.

Introduction

The purpose of this project is to implement and compare several different financial optimization models. Portfolios will be generated using these models and these portfolios will be tested out-of-sample. The optimization models that you will implement are the following.

(1) Mean-variance optimization (MVO)

Use the following version of MVO:

mx(a)xμTx - λxT Qx

subject to eT x = 1

where μ ∈ Rn is the vector of expected returns, Q ∈ Rn ×n is the covariance matrix, and e ∈ Rn is a vector of n ones. Short selling is allowed. λ > 0 is a risk aversion parameter.

(2) Robust mean-variance optimization

You can use box or ellipsoidal uncertainty sets. See slide 16 in the Robust MVO slides posted to see how you should formulate the robust model.

(3) Risk Parity optimization with no short selling.

Use the risk parity model that is easiest to use (the version that uses theta as a variable).

Asset Universe

Our investment universe consists of 20 stocks (n=20) all of which are con- stituents of the S&P 500. The stock symbols are

F (Ford Motor Co.); CAT (Catepillar Inc.); DIS; MCD; KO; PEP; W MT; C; W FC; JPM ,AAPL; IBM; P FE; JNJ; XOM; MRO; ED; T; V Z; and NEM:

Parameter Estimation

(a) Using monthly (last trading day of each month) adjusted closing prices for each stock from Dec-2007 to June-2011 to compute the sample mean and sample variance for each stock. Also, compute sample covariances. Now consider the portfolios generated from MVO, robust MVO, and risk parity by using the estimated parameters. Consider a fourth portfolio of the 20 stocks where the weights are based on market capitalizations (here assume the 20 stocks is the market).

Take each of the four portfolios (weights) generated by using the 4 different methods and using the realized return for each stock for the month of July of 2011 compute the major portfolio quantities for each portfolio for just the month of July 2011 i.e. compute (1) portfolio return (2) portfolio variance and standard deviation and (3) Sharpe ratio. Discuss the results i.e. explain why you think a portfolio did better than another on these portfolio dimensions e.g. why was the Sharpe ratio of a portfolio from a particular strategy better than a portfolio generated by adifferent strategy? and which portfolio did best (worst) on return and why?

Note that for MVO and robust MVO that you have to select a risk aversion value for λ . I would suggest to set this value to the quantity that Idzorek (see Black-Litterman paper and class slides) suggests (in this case the market (mkt) is the group of 20 stocks listed above and you would need to estimate its mean and variance and you need to use a quantity for the risk-free rate.). Be sure to state how you came up with these quantities in your report. Also, for robust MVO do cases for confidence levels of 90% and 95%.

(b) Rolling 6 month horizon test: Now use as the time horizon the last trading date in January 2011 to last trading day of June 2011 and repeat part (a) and test your portfolios for July 2011. Continue with this rolling horizon approach for the next 4 months i.e. your last testing month should be November of 2011.

(c) Plot the cumulative wealth of each portfolio type over the months from June 2011 to November 2011 as well as plotting on another graph the Sharpe ratios of each portfolio type from June 2011 to November 2011. (note: for the Sharpe ratio computations you can ignore using the risk-free rate.)

(d) Discuss the results.

Deliverables

Prepare a formal project report. It should be well written and concise. The report should address the requirements above in written prose and not in a frag- mented manner. Use nice visual aids whenver you can as well as summarizing estimated parameters and computational results in nice tabular form. It is also important to discuss your results in a well written manner and be sure to NOT just dump numbers in the report. Describe advantages and disadvantages of the four di§ering methods. Have a script that executes your models and that generates the graphs so a TA can verify the output of your report. Give clear instructions on how to run your code AND it must be simple and direct as possible. The code must be EASILY run without ANY modification by the TA.

You need to send your actual MATLAB files including the estimated pa- rameters and your report (word or pdf) in a zip file. Send to me via e-mail [email protected].

Name your zip file yourname__APS1022__project2024.zip e.g. Jaylen- Brunson__APS1022__project2024.zip and you must write as the subject of your e-mail APS 1022 project 2024 your name e.g. APS 1022 project 2023 Jaylen Brunson

Failure to name your zip file and submission e-mail subject as instructed may result in a lost project which can inadvertently lead to no marks for your project.