Modeling Time-Dependent Correlation Structures in Portfolio Computational Finance

1. Portfolio Modeling with Time Dependent Correlation StructureComputational Finance Rachel Chiu Sean Zeng Ricardo Affinito Sarah Thomas Dr. Katherine Ensor

2. Motivation “Risk management and oversight now focuses too much on the idiosyncratic risk that affects an individual firm and too little on the systematic issues that could affect market liquidity as a whole.  To put it somewhat differently, the conventional risk-management framework today focuses too much on the threat to a firm from its own mistakes and too little on the potential for mistakes to be correlated across firms.” ~ Timothy F. Geithner, President and CEO of the Federal Reserve Bank

3. Objective To develop a mixture model that captures the complex correlations present in a given portfolio

4. Outline • A Brief Introduction To Stocks And Stock Data • Exploratory Data Analysis (EDA) • Tools and Results • What Lies Beyond • Model Introduction

5. Section 1 A Brief Introduction to stocks and STOCK Data

6. Stock Exchange: Key Concepts • Stocks and stock prices • Stock: A share of ownership • Whenever prices match, a trade takes place • Ticker Symbols • Ex. GOOG(Google), RDS-B(Shell), V(Visa), etc. • Returns and adjusted returns • Percent gain or loss in a given period • To accurately calculate returns, adjust for splits and dividends • Portfolios • A collection of stocks invested

7. Data Selection Methodology There are five main steps for selecting the data suitable for this project. Sector selection Market capitalization analysis Portfolio creation Weighing Calculate returns

8. Sector Selection Criteria • History • Past performance and availability of data • Sector characteristics • Startups vs. traditional • Inter-relationship between the sectors • Ex. Oil Market Leaders and Solar -> Positive • Ex. Oil Market Leaders and Airlines -> Negative

9. Sectors Examined • Wind • Solar • Emerging Markets • Oil Market Leaders • Oil Growth • Technology • Finance • Airline • Automotive

10. Market Capitalization (MC) • Seek companies that best represent the performance and characteristics of the chosen sector • Formula: • NSO = Number of Shares Outstanding • SP = Share Price • Market capitalization = the public opinion of a company’s net worth • It helps us select the leaders in a sector

11. Portfolio Creation The stock market is too big; a portfolio limits the scope of the study A portfolio defines a pseudo world for complex correlations

12. Weighing Methods • Weights = Percent of money invested in a sector or a firm • Equally weighted within sectors • Across sectors, for example… • Equally weighted • Optimally weighted (diversify and minimize covariance)

13. Portfolio [Simple Net] Return • Simple Net Return for Portfolio • Consists of N Assets • Simple Weighted Average of Assets • Portfolio P places weight wi on asset I • Simple Return of P at time t is: where, Ri,tis the simple return of asset i and

14. Exploratory Data Analysis

15. Outline • A Brief Introduction To Stocks And Stock Data • Exploratory Data Analysis (EDA) • Tools and Results • What Lies Beyond • Model Introduction

16. Section 2 Exploratory data analysis

17. EDA Statistical Methods/Tools • Population and Sample Moments • Covariance, Correlation, Autocorrelation • Regression Methods (OLS, Quantile)

18. Moments Defined for Continuous R.V.’s • nth Moment of a continuous R.V. X: • nth Central Moment of a continuous R.V. X: • Normal Distributions can be uniquely determined by • the first two moments (mean, variance) • For non-Normal Distributions higher-order moments • are also of interest (skewness, kurtosis)

19. Mean and Variance • Mean • The expected value of a random variable. What is expected to come based on the information from the data collected in the past. • Variance • The mean subtracted from the random variable, squared. A measure of the dispersion of values. • Standard Deviation is the square root of variance.

20. Skewness & Kurtosis Interpretation • Skewness : Measurement of distribution symmetry • Symmetric: • Right Skewed: • Left Skewed: • Excess Kurtosis : Heavier Tails than Normal Dist? • Because Kurtosis of the Normal Dist. = 3. • Positive ( ) means heavy tails (ref. to Normal Dist). • a.k.a. leptokurtic distribution • Negative ( ) means light tails (ref. to Normal Dist). • a.k.a. platykurtic distribution

21. Example of Skewness

22. Covariance and Correlation • Covariance • Like variance, the measure of the change between two different variables. • Correlation • Measures the strength of the linear relationship between two variables. • Ranges between -1 and 1.

23. Multi-Variate Mean Vector & Covariance Matrix • Consider a Random Vector: • Mean Vector / Covariance Matrix (Population): • Mean Vector / Covariance Matrix (Sample): provided the expectations exist. Sample:

24. Correlation Matrix

25. Auto-Correlation Function • Assuming Weakly Stationary Time Series: ** is the lag-l autocorrelation of rt. • For a given sample, we can estimate ACF as:

26. ACF Data • Auto-Correlation Function (ACF) • The correlation of the data with itself, at different points in time.

27. ACF Expanded Might need Autoregressive (AR) Model if empirical auto-correlation is high. For Order Determination of the AR Model the PACF (Partial ACF, a function of the time series’ ACF is used)… [Along with other (likelihood based) criteria such as AIC] For time-varying variance (as opposed to mean) a conditional heteroskedasticity (CH) component must be added to the model proposed.

28. Regression • Ordinary Least Squares Regression • Estimates the conditional mean • Minimizes the sum of squared residuals • Does not show the tail behavior • Quantile Regression • Estimates the Quantiles (percentiles) • Not as affected by outliers • Shows the tail behavior (associations)

29. Ordinary Least Squares (OLS) Regression Analysis • Conditional Mean Function Modeling • Objective Function = Sum of Squared Residuals • Minimizing… • Leads to the Normal Equations: • Solving:

30. Quantile Regression • What is a quantile? • Define Loss (Piecewise Linear) Function = Quantile of X. for some

31. Quantile Regression • Conditional Quantile Function Modeling… • Objective Function = Sum of Weighted Differences for any

32. Quantile Regression

33. Quantile Regression

34. Quantile Regression Benefits • Robust Estimation // Modeling • Better View of Overall Portfolio Distribution as compared to conventional Conditional Mean Modeling. • Explore Sources of Heterogeneity in the Portfolio Response (Observed Return)

35. EDA Results (1) • Distribution of Returns vary in shape from sector to sector. • Traditional Energy (left skewed) • Other Sectors (right skewed) • Correlations: • Stronger within some sectors (e.g. energy, etc.) • Weaker between sectors (e.g. technology & finance, etc.) • Others negatively correlated (e.g. oil & airlines) • Depend on the timeframe inspected (structure change)

36. EDA Results (2) • Autocorrelations: • Low, at the time do not plan to adjust for any sector autocorrelations (AR model) • Regression Methods: • Used as a tools to inspect distributional shape • Portfolio VaR (tails) related to individual security worst case returns (tail behavior).

37. Outline • A Brief Introduction To Stocks And Stock Data • Exploratory Data Analysis (EDA) • Tools and Results • What Lies Beyond • Model Introduction

38. Section 3 What Lies Beyond

39. Incorporating Dynamic Volatility… • Several Modeling Techniques have been developed: • General Autoregressive Conditional Heteroskedastic (GARCH) Models • Regime Switching Approaches • For Incorporating Co-Volatility and External Influences • Multivariate GARCH • Factor MGARCH • Some Plausible Options… • Parametric (MV Normals or weighted MV Normal and Additional MV Distribution) • Non-Parametric

40. Mixture-Modeling (Parametric) • Model Portfolio Daily Returns • Mixture-Model Approach • We observe the portfolio Return at time t • Returns Dist’n (Portfolio) (Yt) is a combination of distributions with different behavior (Lt, Mt, Ht), and with weights constraint (P1,t+P2,t+P3,t=1). • These random variables vary as a function of time. We seek building the model based on the empirical data observed. observed un-observed

41. Looking at Exogenous Predictors… • We are also looking at external predictors to use as part of the model. • Example: Energy – • Commodities Pricing and their association with Energy Stocks. (NYMEX, ETC.) • CPI and PPI relationships to stocks (also other sectors) (BLS) • Data for energy consumption per sectors, etc. (EIA) • Heating/Cooling Degree Days (NCDC) • These factors (data), known to influence certain sectors (supplies, investments) should provide opportunities to build improved models.

42. Questions… We would like to thank… VIGRE, NSF, CoFES Please send any questions to… Ricardo Affinito (Affinito@rice.edu) Rachel Chiu (rchiu@rice.edu) Sean Zeng (Sean.Zeng@rice.edu)