Data 609 Final Project

Data 609 Final Project - Mathematical Modeling Techniques for Data Analytics

Title: Portfolio Optimization

Aim: - Use mathematical models to make a decision for portfolio optimization - Investigating portfolio optimization with expected return on investiment in risk control

Data source:

• Historical stock price data are readily accessible using functions in “Quantmod” package
• The filtered data for this application selects total 9 stock cases from 2017 to August 2018
• The datasets include the date, daily market close price, market volumes, the closing price will be used to make for portfolio optimization

Sections

In the following sections, we use a variety of mathematical tools to perform the following tasks:

• Data loading
• Graphical Exploration
• Compute daily, monthly and yearly return
• Calculate the Mean Variance model by specific stocks
• Use minimax model to optimize portfolio
• Use linear programming techniques to compare the log return
• Use a quadratic programming approach to determine appropriate portfolio weights

1. Loading data

Construct a vector of tickers and gather prices for them using the getSymbols function within quantmod. We will next calculate returns and convert the data to a time series object.

Use Ajusted closed price to Calculate Returns

Calculate returns

## 'getSymbols' currently uses auto.assign=TRUE by default, but will
## use auto.assign=FALSE in 0.5-0. You will still be able to use
## 'loadSymbols' to automatically load data. getOption("getSymbols.env")
## and getOption("getSymbols.auto.assign") will still be checked for
## alternate defaults.
## 
## This message is shown once per session and may be disabled by setting 
## options("getSymbols.warning4.0"=FALSE). See ?getSymbols for details.

## 
## WARNING: There have been significant changes to Yahoo Finance data.
## Please see the Warning section of '?getSymbols.yahoo' for details.
## 
## This message is shown once per session and may be disabled by setting
## options("getSymbols.yahoo.warning"=FALSE).

## pausing 1 second between requests for more than 5 symbols
## pausing 1 second between requests for more than 5 symbols
## pausing 1 second between requests for more than 5 symbols
## pausing 1 second between requests for more than 5 symbols
## pausing 1 second between requests for more than 5 symbols

Graphical Exploration

Use adjusted closed price to plot graph from 2017 until now:

Adjusted price

2. Compute daily, monthly and yearly return

To compare the average yearly return, AMD and AET are the most increasing return yearly.

3.Linear Programming - Mean Variance model

Investors are risk averse in that they prefer higher return for a given level of risk (variance, standard deviation), or they want to minimize risk for a given level of returns, so we go to minimize the variance and maximize the return.

Calculation of Mean Variance model

The average monthly return of the portofolio at the evenly distributed allocation is 6.8 %. After optimization, the average monthly return of portfolio is -0.436 % when the global variance is at minimum 0.048. The maximized monthly return of portfolio is 1.904 % when the global variance is 0.0915.

4.Linear Programming - Minimax Model

The minimax model will maximize return with respect to one of these prior distributions providing valuable insight regarding an investor’s risk attitude and decision behavior.

Calculation of Minimax Model

Average monthly return is 8.6%, After optimization, mininum average loss is 6.49 % when variance is 1e+07.

5. linear programming vs log returns:

Modeling linear vs log returns: Now we are ready to obtain the sample estimates from the returns\(\mathbf{x}_t\)

Daily rebalancing

We will start with a daily rebalancing since we already have the daily returns readily available.

Compute the three corresponding GMV portfolios:

compare the allocations

.

By portfolio allocation, AAPL, AET and APD are shown the most positive in investing value, but it is not significate in difference between log and tranformation.

6. Qudratic programming

• Uniform portfolio: The uniform portfolio allocation

• Global Minimum Variance Portfolio (GMVP): \[ \begin{array}{ll} \underset{\mathbf{w}}{\textsf{minimize}} & \mathbf{w}^T\mathbf{\Sigma}\mathbf{w}\\ {\textsf{subject to}} & \mathbf{1}^T\mathbf{w} = 1\\ & \mathbf{w}\ge\mathbf{0} \end{array}\]tes equal weight to each stock: \(\mathbf{w} = \frac{1}{N}\mathbf{1}\)

• Markowitz portfolio: \[\begin{array}{ll} \underset{\mathbf{w}}{\textsf{maximize}} & \boldsymbol{\mu}^T\mathbf{w} -\lambda\mathbf{w}^T\mathbf{\Sigma}\mathbf{w}\\ {\textsf{subject to}} & \mathbf{1}^T\mathbf{w} = 1\\ & \mathbf{w}\ge\mathbf{0} \end{array}\]

Return-risk tradeoff for all portfolios

.

compare the weight of allocations

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compare the performance:

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Return Performance of different portfolios

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plot the expected return vs the standard deviation along with the efficient frontier

.

Conclusion

Conclusion

We can conclude with the following points:

• To compare the average yearly return, AMD and AET are the most increasing return yearly.
• It seems that using linear returns or log-returns does not make any significant difference for daily, weekly, and monthly returns.
• Mean-variance Markowitz portfolio has larger annualized return and risk, and Global Minimum Variance Portfolio (GMVP) has lower annualized return and risk.