Stochastic Programming for Financial Applications SAMSI Finance - - PowerPoint PPT Presentation

Stochastic Programming for Financial Applications

SAMSI Finance Group Project

Adam Schmidt, Andrew Hutchens, Hannah Adams, Hao Wang, Nathan Miller, William Pfeiffer

Agenda

Portfolio Optimization Our Formulation Our Results and Sensitivity Analysis Limitations and Future Work

Financial Portfolios

“A portfolio is a grouping of financial assets such as stocks, bonds and cash equivalents, as well as their funds counterparts, including mutual, exchange-traded and closed funds.”

• Investopedia

Modern Portfolio Theory

• Illustrates optimal behavior of a single investor

○ Assumes investor is risk-averse

• Describes Expected Return-Variance of Returns Rule (EV Rule)

○ Returns are maximized/risk is reduced through diversification ○ Minimizing variance entails investing in stocks/securities with low covariance

• Efficient frontier

○ Illustrates optimal portfolio & possible combos ○ Highest expected returns for given risk ○ Lowest risk for given expected return

• Limitations

○ Investors will not be both rational and risk-averse ○ Investors may not have the same information ○ Investors don’t understand what returns are possible

Our Conceptual Formulation

Risk

• “The quantifiable likelihood of loss or

less-than-expected returns”

• Return increases with risk
• Variance and covariance can be used

to define the risk of investing in stocks

• Minimize risk or maximize return?

Random Simulation

Random Simulation

Efficient Frontier

Stochastic Programming can find the portfolios along the efficient frontier given a set of stocks.

Random Portfolios Stochastic Program

Stochastic Program Mathematical Formulation

Stochastic Program Implementation in Python 2.7

1. Data Collection

a. Pandas, Datetime Packages b. Google Finance

2. Data Manipulation

a. Calculated Average Daily Return b. Calculated Covariance Matrix c. Used Numpy

3. Optimization

a. Quadratic Program in CVXOPT Package

Results - Risk/Return Matrix

• Optimized over three year period Jan 2014 - Dec 2016

Stocks considered: Apple, Google, Facebook, Kellogg, Dr. Pepper Snapple Group, Kate Spade, Bank of America, Bed Bath and Beyond, Stryker Corp, Ebay, Rocket Fuel

Results

On average...

• The risk of portfolio is

decreasing with increasing number of stocks considered for the portfolio

• The risk of our portfolio is

increasing with increasing minimum allowable return

A Review of Portfolio Success in 2017 (Jan-May)

• Portfolio of 11 different stocks

○

• Dr. Pepper Snapple Group,

Google, Stryker Corp, Kellogg, Kate Spade, Facebook, Bank of America, Bed Bath and Beyond, Rocket Fuel, Ebay, Apple

• Portfolio 1 is least risky
• Portfolio 3 is most risky

Limitations of Our Work

• Do not determine the most optimal period to predict future portfolio success
• Only consider a maximum of 11 stocks
• May not define risk most optimally
• Model only uses past market data to predict future
• Dividends not considered

Future Work

• Determine a more accurate way to choose sample time period

○ Which time period we should work on considering the economic environment ○ How long the sample period should be, using the analysis of how well certain lengths of time have reflected the future in the past

• Measure the risk & return of the portfolio against appropriate benchmarks
• Improve the model by selecting stocks at random

○ Random selection eliminates selection bias

• Add more constraints

○ The weight on each stock could have a upper-limit

• Apply our portfolio method on other investing strategies

○ Based on “best” stock combination, analyse how to decide the investment proportion

Future Work

• Value at Risk (VaR)

○ “VaR provides an estimate, under a given degree of confidence, of the size of a loss from a portfolio over a given time period” ○ We are 95% confident that we will lose no more than $10,000 in the next market day, or, we are 5% confident that we will lose at least $10,000 in the next market day ○ It is possible to compute value at risk using a Monte Carlo Simulation - VaR will be the difference between current stock price and the estimated lowest terminal stock price

• If more time were available, many of these improvements could be

implemented and fully tested in order to “optimize” our optimization method

Closing Remarks

There are many methods one could implement in order to optimize a portfolio. The

• ne we introduced is the method that we could devise, test, and analyze in the

most complete way within the given time period. Thank you!

Questions?

Contributors:

• Adam Schmidt; University of Wisconsin
• Andrew Hutchens; University of Central Florida
• Hannah Adams; University of North Carolina
• Hao Wang; Colorado State University
• Nathan Miller; Robert Morris University
• William Pfeiffer; Georgetown University

Special Thanks: Peter Diao; SAMSI Post Doc

References

Markowitz, Harry. “Portfolio Selection.” The Journal of Finance 7.1 (1952): 77-91. JSTOR. Web. 16 May 2017. “Efficient frontier.” Investopedia. 3 August 2016. Web. 16 May 2017. Rowland, Ron, and Brian Campos. "Is Modern Portfolio Theory Out-of-Date." Investing, Forbes, 19 May 2010, Rowland, R., & Campos, B. (2010, May 19). Is Modern Portfolio Theory Out-of-Date. In Forbes. Accessed 18 May 2017. "Calculating VaR using Monte Carlo Simulation." Finance Train. N.p., 31 Aug. 2011. Web. 18 May 2017. Halls-Moore, Michael. "QuantStart." Value at Risk (VaR) for Algorithmic Trading Risk Management - Part I - QuantStart. QuantStart, 7 July 2014. Web. 18 May 2017. "What Is An Investment Portfolio?" Money Choice. N.p., n.d. Web. 18 May 2017. "Basics of Investing, Financial Concepts." Basics of investing, Financial concepts, Risk return tradeoff. HSBC Global Asset Management, n.d. Web. 18 May 2017.