Transcript Portfolio Modeling with Time Dependent Correlation

Portfolio Modeling with Time
Dependent Correlation Structure
Computational Finance
Rachel Chiu
Sean Zeng
Ricardo Affinito
Sarah 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
Section 1
A BRIEF INTRODUCTION TO STOCKS
AND STOCK DATA
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 selection
2. Market capitalization analysis
3. Portfolio creation
4. Weighing
5. 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: MC  NSO  SP
– 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
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:
N
R p ,t   wt Ri ,t
i 1
where, Ri,t is the simple return of asset i and
N
w
i 1
i
1
Exploratory Data Analysis
25000
Portfolio Sector's Growth
Wind
20000
Solar
Emkt
Oil ML
Oil G
Airl
Auto
15000
10000
5000
Price
Tech
Fina
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
Date
Outline
• A Brief Introduction To Stocks And Stock Data
• Exploratory Data Analysis (EDA)
– Tools and Results
• What Lies Beyond
– Model Introduction
Section 2
EXPLORATORY DATA ANALYSIS
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:
  
mn  E X
n


n
x f ( x)dx
• nth Central Moment of a continuous R.V. X:



mn  E  X   x    ( x   x ) n f ( x)dx
n

• 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)
Mean and Variance
T
• Mean  E[ x]  x  ˆ x 
x
t 1
t
T
– The expected value of a random variable. What is
expected to come based on the information from
the data collected in the past.
• Variance  s
2
 ˆ x
2
1 T
2
xt  x 


T  1 t 1
– The mean subtracted from the random variable,
squared. A measure of the dispersion of values.
• Standard Deviation is the square root of variance.
Skewness & Kurtosis Interpretation
• Skewness : Measurement of distribution symmetry
– Symmetric:
– Right Skewed:
– Left Skewed:
Sx  0
Sx  0
S  Skewness
Sx  0
• Excess Kurtosis : K x*  K x  3 Heavier Tails than Normal Dist?
– Because Kurtosis of the Normal Dist. = 3.
– Positive ( K x*  0 ) means heavy tails (ref. to Normal Dist).
• a.k.a. leptokurtic distribution
– Negative ( K x  0 ) means light tails (ref. to Normal Dist).
*
• a.k.a. platykurtic distribution
Example of Skewness
10
12
14
Distribution of Simple Returns
Group 3 - Renewable NRG (Other)
Stocks = JASO,ESLR,TSL,FSLR,SPWR,YGE,STP
6
4
2
0
Frequency
8
Mean = -0.0004742
SDev = 0.04667
Skewness = 2.861
Mean = -0.0004742
SDev = 0.04667
Skewness = 2.861
-0.2
0.0
0.2
Simple Returns
0.4
0.6
Covariance and Correlation
• Covariance
– Like variance, the measure of the change between
two different variables.
 XY  COV ( X , Y )  E[( X  E[ X ])(Y  E[Y ])]
• Correlation
– Measures the strength of the linear relationship
between two variables.
– Ranges between -1 and 1.
 X ,Y 
COV ( X , Y )
 XY
Multi-Variate Mean Vector & Covariance Matrix
• Consider a Random Vector:
X  X1 , X 2 ,....,X P 
• Mean Vector / Covariance Matrix (Population):
E{X}  μ X  E ( X 1 ),..., E ( X P )

T
Cov(X)  Σ X  E X  μ X X  μ X 
T


provided the expectations exist.
• Mean Vector / Covariance Matrix (Sample):
Sample: {x1 , x 2 ,...,x T }
T
1 T
1
T
ˆ
ˆ
μˆ x   xt
Σˆ x 
(
x

μ
)
(
x

μ
)

t
x
t
x
T t 1
T  1 t 1
Correlation Matrix
ZOLT
JASO
ESLR
TSL
FSLR
SPWR
YGE
STP
COP
CVX
RDS.B
TOT
XOM
APA
CAM
HES
NBL
OXY
CAL
DAL
JBLU
LUV
NWA
AIRLINES
WNDEF
OIL GROWTH
VWSYF
OIL MARKET LEADERS
GCTAF
AMSC
GCTAF
VWSYF
WNDEF
ZOLT
JASO
ESLR
TSL
FSLR
SPWR
YGE
STP
COP
CVX
RDS.B
TOT
XOM
APA
CAM
HES
NBL
OXY
CAL
DAL
JBLU
LUV
NWA
SOLAR
AMSC
WIND
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
0.05
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
-0.04
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
-0.02
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
-0.03
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
0.14
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
0.01
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
0.02
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
0.01
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
0.03
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
0.04
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
-0.05
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
-0.08
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
0.02
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
0.03
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
-0.03
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
-0.06
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
0.05
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
-0.13
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
-0.04
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
-0.07
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
-0.03
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
-0.10
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
0.77
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
0.86
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
0.43
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
0.60
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
Auto-Correlation Function
• Assuming Weakly Stationary Time Series:
COV ( rt , rt l )
COV ( rt , rt l )  l
l 


VAR( rt )
0
VAR( rt )VAR(rt l )
** is the lag-l autocorrelation of rt.
• For a given sample, we can estimate ACF as:
T
 (r  r )(r
ˆ l  t l 1
t
T
t l
2
(
r

r
)
 t
t 1
 r)
;0  l  T  1
ACF Data
• Auto-Correlation Function (ACF)
– The correlation of the data with itself, at different
points in time.
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
Ym1  E[Y | X ]m1  ε m1  X nm β n1  ε m1
T
 i 1..m ~ IID N (0,  2 )
• Objective Function = Sum of Squared Residuals
S  rm1  Ym1  X nm β n1
2
T
2
2
T

• Minimizing… arg min β n1  Ym1  X nm β n1 



T
T
ˆ
• Leads to the Normal Equations:
X Xβ n1  X y
• Solving: βˆ  (X T X)- 1X T y
n1
Quantile Regression
• What is a quantile?
=

F 1{ }  inf{x | F ( x)   }
th Quantile of X.
• Define Loss
for some
(Piecewise Linear) Function
T (u)  u(  I (u  0))
d
d
  (0,1)
Quantile Regression
• Conditional Quantile Function Modeling…
QY ( | x)  X nm β(
 )n1
• Objective Function = Sum of Weighted Differences
T
n
S    |yi  xi b |
i 1
T
for any
  (0,1)
Quantile Regression
Oil G Lag 1, Coefficient vs Quantile
-0.4
-0.02
-0.2
0.0
0.00
0.2
0.02
0.4
Intercept Lag 1, Coefficient vs Quantile
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.6
0.8
1.0
0.8
1.0
0.8
1.0
Wind Lag 1, Coefficient vs Quantile
-0.6
-0.2
0.0
-0.2
0.2
0.4
0.2 0.4 0.6
Oil ML Lag 1, Coefficient vs Quantile
0.4
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.6
EMKT Lag 1, Coefficient vs Quantile
-0.1
-0.10
0.0
0.1
0.00
0.2
0.10
0.3
Solar Lag 1, Coefficient vs Quantile
0.4
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
Quantile Regression
Technology Lag 1, Coefficient vs Quantile
-0.3
-0.03
-0.2
-0.01
-0.1
0.0
0.01
0.1
0.2
0.03
Intercept Lag 1, Coefficient vs Quantile
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.4
0.6
0.8
1.0
Energy Lag 1, Coefficient vs Quantile
-0.2
-0.05
-0.1
0.05
0.0
0.1
0.10 0.15 0.20
Finance Lag 1, Coefficient vs Quantile
0.2
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
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
Section 3
WHAT LIES BEYOND
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).
observed
Yt  P1,t Lt  P2,t M t  P3,t Ht
un-observed
• These random variables vary as a function of time.
We seek building the model based on the empirical
data 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 ([email protected])
Rachel Chiu ([email protected])
Sean Zeng ([email protected])