Portfolio Modeling with Time Dependent Correlation Structure

Computational Finance

Rachel ChiuSean Zeng

Ricardo AffinitoSarah Thomas

Dr. Katherine Ensor

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

Objective

To develop a mixture model that captures the complex correlations

present in a given portfolio

Outline

• A Brief Introduction To Stocks And Stock Data

• Exploratory Data Analysis (EDA)– Tools and Results

• What Lies Beyond– Model Introduction

A BRIEF INTRODUCTION TO STOCKS AND STOCK DATA

Section 1

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

Data Selection Methodology

There are five main steps for selecting the data suitable for this project.1. Sector selection2. Market capitalization analysis3. Portfolio creation4. Weighing5. Calculate returns

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

Sectors Examined

• Wind• Solar• Emerging Markets• Oil Market Leaders• Oil Growth

• Technology• Finance• Airline• Automotive

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

SPNSOMC

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

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)

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,t is the simple return of asset i and

N

itittp RwR

1,,

11

N

iiw

Exploratory Data Analysis5

00

01

00

00

15

00

02

00

00

25

00

0

Portfolio Sector's Growth

Date

Pri

ce

2007-09-04 2007-09-27 2007-10-22 2007-11-14 2007-12-10 2008-01-04 2008-01-30 2008-02-25 2008-03-19 2008-04-14 2008-05-07 2008-06-02

Wind

Solar

Emkt

Oil ML

Oil G

Airl

Auto

Tech

Fina

Outline

• A Brief Introduction To Stocks And Stock Data

• Exploratory Data Analysis (EDA)– Tools and Results

• What Lies Beyond– Model Introduction

EXPLORATORY DATA ANALYSISSection 2

EDA Statistical Methods/Tools

• Population and Sample Moments• Covariance, Correlation, Autocorrelation• Regression Methods (OLS, Quantile)

Moments Defined for Continuous R.V.’s

• nth Moment of a continuous R.V. X:

• nth Central Moment of a continuous R.V. X:

dxxfxXEm nn

n )(

dxxfxXEm n

xn

xn )()(

• Normal Distributions can be uniquely determined bythe first two moments (mean, variance)

• For non-Normal Distributions higher-order momentsare also of interest (skewness, kurtosis)

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.

T

xxxE

T

tt

x

1ˆ][

T

ttx xx

Ts

1

222

1

1̂

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

0xS0xS

0* xK

0* xK

0xS

3* xx KK

SkewnessS

Example of Skewness

-0.2 0.0 0.2 0.4 0.6

02

46

81

01

21

4

Distribution of Simple Returns Group 3 - Renewable NRG (Other)

Stocks = JASO,ESLR,TSL,FSLR,SPWR,YGE,STP

Simple Returns

Fre

qu

en

cy

Mean = -0.0004742SDev = 0.04667

Skewness = 2.861

-0.2 0.0 0.2 0.4 0.6

02

46

810

1214

Distribution of Simple Returns Group 3 - Renewable NRG (Other)

Stocks = JASO,ESLR,TSL,FSLR,SPWR,YGE,STP

Simple Returns

Fre

quen

cy

Mean = -0.0004742SDev = 0.04667

Skewness = 2.861

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.

])][])([[(),( YEYXEXEYXCOVXY

YXYX

YXCOV

),(

,

Multi-Variate Mean Vector & Covariance Matrix

• Consider a Random Vector:

• Mean Vector / Covariance Matrix (Population):

• Mean Vector / Covariance Matrix (Sample):

},...,,{ 21 Txxx

TXXX ECov μXμXΣX

)(

T

PX XEXEE )(),...,(}{ 1μX

PXXX ,....,, 21X

provided the expectations exist.

Sample:

Txt

T

txtx

T

ttx T

xT

)ˆ()ˆ(1

1ˆ1ˆ11

μxμxΣ μ

Correlation MatrixAM

SC

GCT

AF

VWSY

F

WN

DEF

ZOLT

JASO

ESLR

TSL

FSLR

SPW

R

YGE

STP

COP

CVX

RDS.

B

TOT

XOM

APA

CAM

HES

NBL

OXY CA

L

DAL

JBLU

LUV

NW

A

AMSC 1.00 0.24 0.27 0.04 0.29 0.36 0.47 0.33 0.34 0.47 0.39 0.37 0.37 0.33 0.24 0.31 0.34 0.36 0.34 0.29 0.35 0.33 0.14 0.11 0.20 0.25 0.05GCTAF 0.24 1.00 0.45 0.02 0.24 0.18 0.25 0.28 0.17 0.29 0.28 0.31 0.21 0.21 0.24 0.32 0.22 0.25 0.24 0.17 0.19 0.20 -0.06 -0.04 -0.03 0.03 -0.04VWSYF 0.27 0.45 1.00 -0.06 0.25 0.27 0.27 0.33 0.27 0.31 0.28 0.33 0.31 0.27 0.30 0.37 0.27 0.25 0.34 0.26 0.24 0.25 0.00 0.01 0.05 0.08 -0.02WNDEF 0.04 0.02 -0.06 1.00 -0.02 0.13 0.03 0.08 0.15 0.12 0.08 0.07 0.00 -0.01 -0.02 -0.01 -0.02 -0.01 0.09 0.04 0.02 0.00 0.01 0.03 0.00 -0.02 -0.03ZOLT 0.29 0.24 0.25 -0.02 1.00 0.29 0.29 0.37 0.36 0.31 0.27 0.29 0.28 0.23 0.21 0.29 0.25 0.27 0.27 0.23 0.26 0.25 0.19 0.18 0.27 0.27 0.14JASO 0.36 0.18 0.27 0.13 0.29 1.00 0.53 0.58 0.70 0.61 0.65 0.59 0.43 0.42 0.29 0.36 0.41 0.41 0.46 0.39 0.41 0.43 0.05 0.06 0.11 0.06 0.01ESLR 0.47 0.25 0.27 0.03 0.29 0.53 1.00 0.46 0.56 0.55 0.50 0.48 0.50 0.42 0.37 0.45 0.46 0.46 0.36 0.37 0.40 0.47 0.06 0.05 0.19 0.14 0.02TSL 0.33 0.28 0.33 0.08 0.37 0.58 0.46 1.00 0.52 0.51 0.61 0.51 0.46 0.43 0.33 0.42 0.44 0.40 0.47 0.45 0.41 0.43 0.05 0.07 0.10 0.09 0.01FSLR 0.34 0.17 0.27 0.15 0.36 0.70 0.56 0.52 1.00 0.57 0.57 0.57 0.45 0.42 0.32 0.37 0.42 0.39 0.42 0.37 0.41 0.44 0.05 0.05 0.09 0.04 0.03SPWR 0.47 0.29 0.31 0.12 0.31 0.61 0.55 0.51 0.57 1.00 0.57 0.63 0.42 0.37 0.33 0.37 0.37 0.42 0.46 0.39 0.42 0.44 0.13 0.08 0.16 0.16 0.04YGE 0.39 0.28 0.28 0.08 0.27 0.65 0.50 0.61 0.57 0.57 1.00 0.61 0.39 0.34 0.33 0.35 0.37 0.37 0.41 0.36 0.42 0.37 0.05 0.01 0.09 0.01 -0.05STP 0.37 0.31 0.33 0.07 0.29 0.59 0.48 0.51 0.57 0.63 0.61 1.00 0.40 0.36 0.34 0.37 0.35 0.38 0.45 0.37 0.35 0.40 -0.08 -0.05 0.07 -0.08 -0.08COP 0.37 0.21 0.31 0.00 0.28 0.43 0.50 0.46 0.45 0.42 0.39 0.40 1.00 0.77 0.24 0.63 0.74 0.71 0.50 0.61 0.57 0.66 0.07 0.05 0.18 0.17 0.02CVX 0.33 0.21 0.27 -0.01 0.23 0.42 0.42 0.43 0.42 0.37 0.34 0.36 0.77 1.00 0.24 0.70 0.81 0.67 0.51 0.60 0.55 0.65 0.09 0.07 0.17 0.19 0.03RDS.B 0.24 0.24 0.30 -0.02 0.21 0.29 0.37 0.33 0.32 0.33 0.33 0.34 0.24 0.24 1.00 0.26 0.19 0.20 0.23 0.22 0.24 0.28 0.00 0.01 0.00 0.08 -0.03TOT 0.31 0.32 0.37 -0.01 0.29 0.36 0.45 0.42 0.37 0.37 0.35 0.37 0.63 0.70 0.26 1.00 0.71 0.58 0.44 0.50 0.47 0.56 -0.02 -0.02 -0.01 0.09 -0.06XOM 0.34 0.22 0.27 -0.02 0.25 0.41 0.46 0.44 0.42 0.37 0.37 0.35 0.74 0.81 0.19 0.71 1.00 0.64 0.48 0.58 0.53 0.63 0.12 0.09 0.21 0.22 0.05APA 0.36 0.25 0.25 -0.01 0.27 0.41 0.46 0.40 0.39 0.42 0.37 0.38 0.71 0.67 0.20 0.58 0.64 1.00 0.54 0.58 0.63 0.62 -0.02 -0.05 0.02 0.05 -0.13CAM 0.34 0.24 0.34 0.09 0.27 0.46 0.36 0.47 0.42 0.46 0.41 0.45 0.50 0.51 0.23 0.44 0.48 0.54 1.00 0.60 0.71 0.64 0.01 -0.03 -0.02 -0.01 -0.04HES 0.29 0.17 0.26 0.04 0.23 0.39 0.37 0.45 0.37 0.39 0.36 0.37 0.61 0.60 0.22 0.50 0.58 0.58 0.60 1.00 0.67 0.73 -0.03 -0.03 0.06 0.01 -0.07NBL 0.35 0.19 0.24 0.02 0.26 0.41 0.40 0.41 0.41 0.42 0.42 0.35 0.57 0.55 0.24 0.47 0.53 0.63 0.71 0.67 1.00 0.72 0.05 0.03 0.08 0.11 -0.03OXY 0.33 0.20 0.25 0.00 0.25 0.43 0.47 0.43 0.44 0.44 0.37 0.40 0.66 0.65 0.28 0.56 0.63 0.62 0.64 0.73 0.72 1.00 0.00 -0.03 0.12 0.08 -0.10CAL 0.14 -0.06 0.00 0.01 0.19 0.05 0.06 0.05 0.05 0.13 0.05 -0.08 0.07 0.09 0.00 -0.02 0.12 -0.02 0.01 -0.03 0.05 0.00 1.00 0.80 0.52 0.67 0.77DAL 0.11 -0.04 0.01 0.03 0.18 0.06 0.05 0.07 0.05 0.08 0.01 -0.05 0.05 0.07 0.01 -0.02 0.09 -0.05 -0.03 -0.03 0.03 -0.03 0.80 1.00 0.45 0.63 0.86JBLU 0.20 -0.03 0.05 0.00 0.27 0.11 0.19 0.10 0.09 0.16 0.09 0.07 0.18 0.17 0.00 -0.01 0.21 0.02 -0.02 0.06 0.08 0.12 0.52 0.45 1.00 0.54 0.43LUV 0.25 0.03 0.08 -0.02 0.27 0.06 0.14 0.09 0.04 0.16 0.01 -0.08 0.17 0.19 0.08 0.09 0.22 0.05 -0.01 0.01 0.11 0.08 0.67 0.63 0.54 1.00 0.60NWA 0.05 -0.04 -0.02 -0.03 0.14 0.01 0.02 0.01 0.03 0.04 -0.05 -0.08 0.02 0.03 -0.03 -0.06 0.05 -0.13 -0.04 -0.07 -0.03 -0.10 0.77 0.86 0.43 0.60 1.00

WIND SOLAR OIL MARKET LEADERS OIL GROWTH AIRLINES

Auto-Correlation Function

• Assuming Weakly Stationary Time Series:

** is the lag-l autocorrelation of rt.

• For a given sample, we can estimate ACF as:

0)(

),(

)()(

),(

l

t

ltt

ltt

lttl rVAR

rrCOV

rVARrVAR

rrCOV

10;)(

))((ˆ

1

2

1

Tlrr

rrrr

T

tt

T

ltltt

l

ACF Data

• Auto-Correlation Function (ACF)– The correlation of the data with itself, at different

points in time.ACF

WIND SOLAR EMKT OILML OILG TECH FINA AIRL AUTOlag1 -0.02535 0.001937 -0.12514 0.098406 0.024246 -0.02548 0.149731 0.030379 -0.01032lag2 0.040051 -0.01604 -0.03283 0.028327 -0.0206 -0.02146 0.019471 -0.05436 0.039133lag3 -0.0392 -0.04285 0.115005 0.008292 -0.03115 -0.01141 -0.04077 0.001591 0.024963lag4 0.017539 -0.02221 -0.12275 -0.02576 -0.0132 -0.00852 -0.05975 -0.02699 0.043242

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.

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)

Ordinary Least Squares (OLS) Regression Analysis

• Conditional Mean Function Modeling

• Objective Function = Sum of Squared Residuals

• Minimizing…

• Leads to the Normal Equations:• Solving:

),0(~

]|[2

..1

11111

N

E

IIDmi

mnT

mnmmm

εβXεXYY

2

11

2

1 nT

mnmmS βXYr

2

111minarg n

Tmnmn

βXYβ

yXβXX TT 1ˆ

n

yXX)Xβ T-1T(ˆ1 n

Quantile Regression

• What is a quantile?

• Define Loss (Piecewise Linear) Function

dd

)1,0(

))0(()( uIuuT

th})(|inf{}{1 xFxF

= Quantile of X.

for some

Quantile Regression

• Conditional Quantile Function Modeling…

• Objective Function = Sum of Weighted Differences

for any

1)|( nT

mnY x )β(XQ

||1

bxyS Tii

n

i

)1,0(

Quantile Regression

0.0 0.2 0.4 0.6 0.8 1.0

-0.0

20.

000.

02

Intercept Lag 1, Coefficient vs Quantile

0.0 0.2 0.4 0.6 0.8 1.0

-0.4

-0.2

0.0

0.2

0.4

Oil G Lag 1, Coefficient vs Quantile

0.0 0.2 0.4 0.6 0.8 1.0

-0.6

-0.2

0.2

0.4

0.6

Oil ML Lag 1, Coefficient vs Quantile

0.0 0.2 0.4 0.6 0.8 1.0-0

.20.

00.

20.

4

Wind Lag 1, Coefficient vs Quantile

0.0 0.2 0.4 0.6 0.8 1.0

-0.1

00.

000.

10

Solar Lag 1, Coefficient vs Quantile

0.0 0.2 0.4 0.6 0.8 1.0

-0.1

0.0

0.1

0.2

0.3

EMKT Lag 1, Coefficient vs Quantile

Quantile Regression

0.0 0.2 0.4 0.6 0.8 1.0

-0.0

3-0

.01

0.0

10.0

3

Intercept Lag 1, Coefficient vs Quantile

0.0 0.2 0.4 0.6 0.8 1.0

-0.3

-0.2

-0.1

0.0

0.1

0.2

Technology Lag 1, Coefficient vs Quantile

0.0 0.2 0.4 0.6 0.8 1.0

-0.2

-0.1

0.0

0.1

Finance Lag 1, Coefficient vs Quantile

0.0 0.2 0.4 0.6 0.8 1.0

-0.0

50.0

50.1

00.1

50.2

0

Energy Lag 1, Coefficient vs Quantile

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)

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)

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).

Outline

• A Brief Introduction To Stocks And Stock Data

• Exploratory Data Analysis (EDA)– Tools and Results

• What Lies Beyond– Model Introduction

WHAT LIES BEYONDSection 3

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

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.

ttttttt HPMPLPY ,3,2,1 observedun-observed

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.

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)