THE X FACTOR: GROUPING SECURITIES, DEFINING "SIMILAR" AND FORMING ESTIMATES - Nick Wade Northfield Information Services 2011
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THE X FACTOR: GROUPING SECURITIES,
DEFINING “SIMILAR” AND FORMING
ESTIMATES
Nick Wade
Northfield Information Services
2011
1WHAT’S THE MAIN POINT?
Most of the models we use feature1:
a fixed factor structure that does not allow for change or evolution in the way that markets
convert information into price change
and are estimated using one of many techniques that assume all the data in the sample
follow some nice, clean, simple rules that bear no resemblance to real life
We contend that risk models:
Need some kind of adaptive structure that allows them to change as new or transient
effects appear
Need adjustment for “regimes” in the data – this impacts asset allocation and “insurance”
Should harness current or forward-looking information as a conditioning input
Should be estimated using a flexible technique that allows for evolution in the underlying
data
We also offer some thoughts about emergent techniques, new candidates for
factors, and directions for future research
1 There are a lot more problems, but the focus for today is JUST the idea that things change over time
2ROADMAP
Introduction to Factor Models
Choices of Factors
Pros and Cons
A Hybrid Approach
Better Factors
Choices of Estimation Methodology
Data Regimes
Better Estimation Methodology
Conclusions
3WHAT IS A FACTOR MODEL?
The main purpose of a factor model is to find a set of
common themes that explain the variability in security
prices that is shared across securities.
Having defined a set of factors, we then look to estimate
the return associated with those factors and the
individual security exposures/sensitivities/betas to
those factors
The end result is a model of how the portfolio will
behave – how much the securities will move together
and how much they will behave uniquely
That can lead us to various useful risk characteristics
4THE LINEAR MODEL
N M
Rst = ∑ Eit Fit + S st = ∑ G jt H jt
i =1 j =1
Relationship between R and F is linear ∀F
There are N common factor sources of return
Relationship between R and H is linear ∀H
There is no correlation between F and H ∀ F,H
The distribution of F is stationary, Normal, i.i.d. ∀F
There are M stock-specific sources of return
There is no correlation between H across stocks
The distribution of H is stationary, Normal, i.i.d. ∀H
(Implicitly also the volatility of R and F is stationary)
5ESTIMATING A RISK MODEL
N
Rst = ∑ Eit Fit + S st
i =1
N N M
V p = ∑∑ Ei E jσ iσ j ρ i , j + ∑ Wkσ 2 (S k )
i =1 j =1 k =1
The Variance of a portfolio is given by the double sum over the
factors contributing systematic or common factor risk, plus a
weighted sum of the stock-specific or residual risks.
6COMMON FACTOR CHOICES
7HOW MANY FACTORS…?
The academic consensus seems to be that
there is not much difference going from 5 to 10
to 15 factors. In other words, 5 do the job.
Lehmann & Modest (1988)
Connor & Korajczyk (1988)
Roll & Ross (1980)
If somebody is suggesting a 90 factor model to
you, tell them to try harder
8ENDOGENOUS MODEL
The Endogenous or Fundamental Model seeks
to estimate Fit assuming Eit by regression.
Typical factors include E/P, D/E, Industry
membership, Country membership…
King(1966)
Rosenberg and Guy (1975) etc.
Model is pre-specified
These models are hugely popular, partly
because we’ve been building them for so long
we’ve become habituated
9EXOGENOUS MODEL
The Exogenous or Macro model seeks to
estimate Ei from Fit.
Typical factors include Market, Sector, Oil,
Interest-Rates…
Ross(1976)
Chen (1986)
Model is pre-specified
10STATISTICAL MODEL
Assume N
Use Factor Analysis or Principle Components to
estimate Ei
Use Regression to estimate Fit
Errors in Variables
11PROS AND CONS OF EACH APPROACH
Approach Pros Cons
Fundamental (micro) • Suitable for concentrated portfolios • Number of factors is fixed thus unchanging
model • Dependent on accounting statement accuracy
• Dependent on accounting standards
comparability
•Membership factors for industry/country/sector
• Errors will be in factor returns, hence in
covariance matrix, and hence not diversifiable
Macro-economic • No dependence on accounting data • Factor number fixed and unchanging
model • Exposure to factors is stationary over time
• The response of each security to changes in
market/sector/industry/ whatever to be different
across securities
• Errors will be in loadings (exposures), thus
diversifiable
Statistical model • All correlation is information • Attribution of risk is difficult
• Captures new, or transient effects • Issues with noise in data
• Adaptive to the market • Errors in variables
• Great for a short-term model • Number of factors is either pre-specified or
sample-dependent
12OTHER APPROACHES
Combined Models:
Northfield Hybrid Model
Stroyny (2001)
Simultaneous Estimation
Black et al (1972)
Heston and Rouwenhorst (1994, 1995)
Satchell and Scowcroft (2001)
GMM Hansen (1982)
McElroy and Burmeister (1988) using NLSUR (which is assymptotically
equivalent to ML)
Bayesian Approach:
Pohlson and Tew (2000)
Ericsson and Karlsson (2002)
13SIMULTANEOUS ESTIMATION
Removing the limitation of binary or membership variables (such as
industry, country, sector, region etc).
Marsh and Pfleiderer (1997)
Scowcroft and Satchell (2001)
Start with an estimate of the exposures (e.g. 1.00 for all companies)
use that estimate to solve for the factor return, then use that factor
return in turn to re-solve for a revised set of exposures, thus converging
iteratively on a better solution for both Eit and Fit.
Black et al (1972)
Heston and Rouwenhorst (1994, 1995)
Scowcroft and Satchell (2001)
Given various limiting restrictions we can ensure that the model
converges and that it is unique.
This is the idea behind the UBS range of models and it’s good. But it’s
not adaptive.
14HYBRID MODEL (NORTHFIELD 1998)
Combine macro, micro, and statistical factors
An observable factor core, and statistical factors
trawling the residual return to find new factors
Gain the advantages of each, whilst mitigating the
limitations of each
Intuitive, explainable, justifiable observable factors
Minimal dependence on accounting information
Rapid inclusion of new or transient factors
15THOUGHTS
Notice we are already trying to allow for an adaptive set of
factors – we will pick this theme up later
In a minute we’ll be allowing for time-dependent volatility
and correlations as well
But are all price movements based on fundamentals?
Apple?
What about news?
What about liquidity impacts?
What about index membership or common ownership?
Exposed to “have to” sales as index weight changes for example
Some securities are held in many funds, high “centrality”
16BETTER FACTOR CHOICES
17A FEW SUGGESTIONS FOR BETTER FACTORS
A factor can be any shared behavior
HISTORICAL: Semantic clustering (text mining)
Dig into everything published on a universe of companies and look
for similarities by phrase comparison etc
PREDICTIVE: News flows
Look at instances of occurrence in news, sentiment
Inference from other asset classes
What does a bond spread change tell us about equity vol?
What about a change in option implied volatility/implied correlation?
Social network analysis
Apply emergent techniques to look at influence within groups,
measures of asset centrality, flow of information, diversification?
Influence: types of network shape
18NEWS AS A FACTOR
Mitra, Mitra, diBartolomeo (2008)
Since Dan’s going into depth later I will restrain
myself – just note that you can use news
incidence/sentiment to condition risk forecasts
19OWNERSHIP DATA
Not a well-explored source of information
Starmine:[see Dirk Renick “Research on sentiment
based smart money”]
Use ownership data to reverse out factor preferences
Funds are attracted to companies that are “like” ones they
already own
Funds exhibit biases toward companies with certain
fundamental characteristics, and these biases change over time
1999 2001 2003 2004 2005 2007
StarMine PriceMo EPS_CAGR3 ROE ROE ROE ROE
LTG ROE Profit Margin Interest Coverage F12m E/P F12m E/P
G5 EPS Profit Margin Interest Coverage F12m E/P Interest Coverage Interest Coverage
Debt/Assets Debt/Assets LTG Profit Margin Profit Margin Profit Margin
Interest Coverage LTG F12m E/P LTG StarMine EQ StarMine PriceMo
20COUNTRY, INDUSTRY, SECTOR, REGION…
A useful (I hope) digression into the world of factor selection.
It is pretty much standard practice to take note of membership
in, or exposure to, one or more countries or regions, and one or
more industries or sectors
Problems: multinational firms, globalization, index domination
Heston and Rouwenhorst (1994, 1995)
Scowcroft and Sefton (2001)
Diermeier and Solnik (2000)
MacQueen and Satchell (2001)
Suggestions:
Estimate a different kind of index FTSE (1999), Bacon and Woodrow (1999)
Split into “global” market and “domestic” market either by some cut off on a variable like foreign sales (Diermeier and
Solnik 2000) or by some statistical process (MacQueen and Satchell 2001)
Solve Model iteratively using Heston and Rouwenhorst (1994, 1995) approach
Or extensions to that: Scowcroft and Sefton (2001).
The real problem is that something as bland as “industry” is one-dimensional and
does not pick up enough nuance about company relationships
21SEMANTIC CLUSTERING
Let’s get rid of Industry Classifications
Why? Are all banks the same? Nope.
Semantic clustering, or text mining can be used
to:
Help predict ratings changes [Starmine]
Help update risk/return estimates [Ravenpack etc]
Measure distance between companies based on
published information [Quid.com]
22SOCIAL NETWORK ANALYSIS
A million followers, and still not winning… [thanks Charlie]
Recent developments in SNA look at the different kinds of
groups and influence between/within groups
Solis (2009) stocks form a “small world” network
“Six degrees of separation” etc.
Measures of “centrality”
Degree centrality
Reach centrality
Flow centrality
Betweeness centrality
Kritzman: Asset Centrality
Google: page rank algorithm
Intuitive groups: index membership, ownership in mutual
funds
23IN CONTEXT
For our purposes, how is a group connected?
Think of localized version of CAPM – which asset
best represents “the market”
Hubs? What degree of connectedness?
Ownership commonality across mutual funds
Why do we care?
Investment strategy, asset allocation
Co-movement (risk), diversification, crowding
Information flow
24ASSET CENTRALITY
We can take this idea from Social Network
Analysis and apply it to a variety of contexts:
Was Lehman too big to fail? Can we quantify it’s
centrality?
Is BHP more influential than Rio, or less with the
Resources sector?
Which sectors are the most/least “democratic”?
Note that this links again with James’ work on
diversification
25ABSORPTION RATIO (KRITZMAN 2011)
On a related note – how tightly connected is the market, or a
particular sector?
Lo (2008)
Yenilmez and Saltoglu (2011)
Absorption ratio quantifies this by looking at the proportion
of variance explained by common themes.
As this number rises, the level of “systemic” risk rises, since
assets are more tightly connected.
This is one requirement for a crash – just add panic
A signal for when to apply costly insurance – e.g. zero-cost collar
You could use Dispersion and get a similar result (but without
allowing for idiosyncratic vol to move indep of syst vol)
You could use Implied Correlation and get a similar forward-
looking result
26ARTIFICIAL IMMUNE SYSTEMS
At a high level, our immune system consists of two
pieces:
Innate immunity
Learned immunity
In our context
The factors we believe to be useful at t=0
Plus the factors the model learns along the way
Tune the model
Criteria for accepting a new factor
Criteria for archiving / forgetting factors
Memory length for previously useful factors
27SINGLE-PASS CLUSTERING AND RELATED
METHODS
Concept: high frequency data in high volumes
presents a storage problem, so need
techniques that can analyze data as it arrives
rather than data-mine.
Issues: it’s a goldfish. No memory.
28ISSUES WITH ESTIMATION
29SOME PROBLEMS
We are dealing with an evolving data set, not a static one
Explore how this impacts our common techniques
Look at more advanced / better techniques to fit evolving data
sets
We are (potentially) dealing with different regimes in the
data, not one uniform set
Look at models that explicitly allow for regime change (not in a
George Bush sense)
We are dealing with complex behavior within groups
For example, some groups play follow the leader
Some groups herd. There is no leader
THOUGHTS: sefton: beta compression, social network analysis,
influence
30THE WORLD IS VERY OBVIOUSLY TIME-VARYING
Non-stationary volatility (ARCH, GARCH, etc)
We spend an heroic amount of time trying to forecast non-
stationary volatility
But we often just ignore it when we calculate correlation, or
perform regression analysis, or run factor analysis (or PCA)
Non-stationary mean (Trend)
We often build models to capture the alpha in momentum,
reversals, and other manifestations of a non-stationary
mean
But we often ignore those when we calculate correlation, or
perform regression analysis, or run factor analysis
Read the fine print…
31SIMPLE ADJUSTMENTS (1)
Non-stationary factor
return series will lead
2
to the model xi − x
V = ∑
underestimating
n (n − 1)
portfolio risk
Adjust by changing
variance calculation to 2
xi
include trend V = ∑
component of return n(n − 1)
Adjust Model for the influence of non-stationary factor returns
32SIMPLE ADJUSTMENTS (2)
What about security volatility?
We observe:
Serialcorrelation (not i.i.d.)
Bid-ask bounce
Non-Normal distributions
Parkinson volatility
Adjust Model for the influence of non-stationary security returns
33CONTEMPORANEOUS OR FORWARD-LOOKING
SIGNALS
You could make the argument that these are “factors” rather than an
estimation approach, but we use them to condition existing factors.
Take a model that has been estimated on purely historical data
Find true forward-looking signals
E.g. option-implied volatility
Find other contemporaneous signals
E.g. dispersion measures, range measures, volume
Adjust the parameters of the “historical” model so that the forecasts of the
model match the signals from “now” and the “future”
Update it daily so that it stays “current”
The Advantage: we have kept the same factor structure but removed the
sole dependency on the past.
34GENERALIZE THE IDEA:
NORTHFIELD “ADAPTIVE NEAR-HORIZON RISK MODELS”
(ANISH SHAH, 2008)
The richest source of information about the future is not the past – increasingly it is consensus
estimates about the future from e.g. option markets, prediction markets…
Take any risk model. e.g. one of our models estimated monthly
Add “flexibility points” and fit to information about current conditions
Adjust for statistical differences between short and long term returns
Many benefits of this approach
Avoid statistical complexities of high frequency data
Keep familiar factor structure
Common factor structure for long and short horizons permits interpolating any
horizon in between
Works with any factor model
35Northfield Asia ex. Japan Risk Models
Tracking Error LH Tracking Error SH Linear (Tracking Error SH)
68
64
60
56
52
48
44
40
36
32
28
24
20
16
12
8
4
0
36DATA WITH REGIMES
37KEY QUESTIONS
Before we get carried away…
What evidence of detectable regimes?
How many regimes?
What kind of model can fit multiple regimes?
Can any of these fit multiple regimes on evolving
data?
i.e. learn new regimes as they appear
38EVIDENCE FOR REGIMES:
TIME-VARYING CORRELATIONS
Increasing attention is being paid to the issue of correlations varying over time:
(stocks) De Santis, G. and B. Gerard (1997), International asset pricing and portfolio
diversification with time-varying risk, Journal of Finance, 52, 1881-1912.
(stocks) Longin, F. and B. Solnik (2001), Extreme correlation of international equity markets,
Journal of Finance, LVI(2), 646-676.
(bonds) Hunter, D.M. and D.P. Simon (2005), A conditional assessment of the relationships
between the major world bond markets, European Financial Management, 11(4), 463-482.
(bonds) Solnik, B., C. Boucrelle and Y.L. Fur (1996), International market correlation and
volatility, Financial Analysts Journal, 52(5), 17-34.
Markov switching model: Chesnay, F and Jondeau, E “Does Correlation Between
Stock Returns really increase during turbulent periods?” Bank of France research
paper.
To date little explored – however, Implied Correlation also seems useful, and more
powerful than historical correlation in forecasting (we saw the same result with
volatility):
Campa, J.M. and P.H.K. Chang (1998), The forecasting ability of correlations implied in foreign
exchange options, Journal of International Money and Finance, 17, 855-880.
39CORRELATION STABILITY
One of the first… Kaplanis (1988): STABLE
Tang (1995), Ratner (1992), Sheedy (1997): STABLE –
although crash of 1987 regarded as an “anomaly”
Bertero and Mayer (1989), King and Wadwhani (1990)
and Lee and Kim (1993): correlation has increased, but
STABLE
Not quite so stable? Erb et al (1994) – increases in bear
markets
Longin and Solnick (1995) – increases in periods of
high volatility
Longin and Solnick (2001) – increases in bear markets
40DETECTING REGIMES
Use Viterbi’s algorithm (Viterbi 1967) to detect
states.
Use Jennrich tests (Jennrich 1970) to decide
whether correlation differences between states
are significant
Mahalanobis Distance (Mahalanobis 1939,
Kritzman 2009)
41MAHALANOBIS DISTANCE
MD is one example of a “Bregman divergence” , a group of distance measures.
Clustering: classifies the test point as belonging to that class for which the Mahalanobis distance is
minimal. This is equivalent to selecting the class with the maximum likelihood.
Regression: Mahalanobis distance and leverage are often used to detect outliers, especially in the
development of linear regression models. A point that has a greater Mahalanobis distance from the
rest of the sample population of points is said to have higher leverage since it has a greater influence
on the slope or coefficients of the regression equation. Specifically, Mahalanobis distance is also
used to determine multivariate outliers. A point can be an multivariate outlier even if it is not a
univariate outlier on any variable.
Factor Analysis: recent research couples Mahalanobis distance with Factor Analysis and use MD to
determine whether a new observation is an outlier or a member of the existing factor set. [Zhang
2003]
MD depends on covariance (S^-1 is the inverse of the covariance matrix), so is exposed to the same
stationarity issues that affect correlation, however as described above it can help us reduce
correlation’s outlier dependence.
42MAHALANOBIS IN ACTION Borrowed from Kritzman: Skulls, financial turbulence, and the implications for risk management. July 2009 43
TURBULENCE IN THE MARKET
Kritzman (2009):
Correlation of US and foreign stocks when both
markets’ returns are one standard deviation above
their mean: -17%
Correlation of US and foreign stocks when both
markets’ returns are one standard deviation below
their mean: +76%
“Conditional correlations are essential for constructing
properly diversified portfolios”
44CORRELATION REGIMES
If it’s not stable, how about Markov switching
models?
Ramchand and Susmel (1998), Chesnay and Jondreau
(2001) – correlation, conditioned on market regime,
increases in periods with high volatility
Ang and Bekaert (1999) – evidence for two regimes; a
high vol/high corr, and a low vol/low corr.
Don’t forget this needs to adapt as our world
changes:
Hidden Markov Experts on evolving data
45FLEXIBLE ESTIMATION TECHNIQUES
46TECHNIQUES FOR EVOLVING DATA
Most of our favorite tools are designed to fit static data
sets where behaviors are mostly unchanged
Neural network, Kalman filter, OLS/GLS regression, PCA,
ICA, factor analysis, variance, correlation… just about all of
them
Recent developments in cluster analysis are
encouraging
Artificial Immune Systems
Single-pass clustering
Regime-switching models e.g. HME etc
[recent] EPCIA
[very recent] HME on evolving data
47REGRESSION WITH NONSTATIONARY DATA
Techniques have been developed specifically to
allow time-varying sensitivities
FLS (flexible least-squares)
FLS is primarily a descriptive tool that allows us to
gauge the potential for time-evolution of exposures
T T −1
′
∑ ( yt − xt β t ) + λ ∑ (β t +1 − β t ) (β t +1 − β t )
2
t =1 t =1
Minimze both sum of squared errors and sum of squared dynamic errors
(coefficient estimates)
48FLS EXAMPLE
An example from Clayton and MacKinnon (2001)
The coefficient apparently exhibits structural shift in 1992
49CLUSTER ANALYSIS WITH NONSTATIONARY DATA
Guedalia, London, Werman; “An on-line agglomerative clustering method for
nonstationary data” Neural Computation, February 15, 1999, Vol. 11, No. 2,
Pages 521-54
C. Aggarwal, J. Han, J. Wang, and P. S. Yu, On Demand Classification of Data
Streams, Proc. 2004 Int. Conf. on Knowledge Discovery and Data Mining (KDD'04),
Seattle, WA, Aug. 2004.
G. Widmer and M. Kubat, “Learning in the Presence of Concept Drift and Hidden
Contexts”, Machine Learning, Vol. 23, No. 1, pp. 69-101, 1996.
Again, there are techniques available to
conquer the problem
50FACTOR ANALYSIS WITH NONSTATIONARY DATA
Dahlhaus, R. (1997). Fitting Time Series Models to Nonstationary
Processes. Annals of Statistics, Vol. 25, 1-37.
Del Negro and Otrok (2008): Dynamic Factor Models with Time-
Varying Parameters: Measuring Changes in International Business
Cycles (Federal Reserve Bank New York)
Eichler, M., Motta, G., and von Sachs, R. (2008). Fitting dynamic
factor models to non-stationary time series. METEOR research
memoranda RM/09/002, Maastricht University.
Stock and Watson (2007): Forecasting in dynamic factor models
subject to structural instability (Harvard).
There are techniques available, and they are
being applied to financial series.
51EVOLVING PRINCIPAL COMPONENT INNOVATION
ANALYSIS (EPCIA)
You want PCA but your factor structure is
changing
EFA (evolving factor analysis): keep adding factors
EFWFA (evolving fixed-window factor analysis): keep
adding new factors, but forget old ones to make
room!
EPCIA allows for new factors to emerge
52LAYER-EMBEDDED NETWORKS
Within our networks, information and interactions
may flow at multiple levels
E.g. Pasquel and de Weck (2011 working paper)
E.g. Luttrell (2010 working paper)
Enter multi-layer modeling
At a high level, a layer-embedded network captures the
effects of multiple interacting processes at different
frequencies or across different groups.
E.g. combining short and long-term alpha signals
Combining global and local factors
Combining pervasive and short-term factors
53SUMMARY THOUGHTS
54CONCLUSIONS
Our world changes
This requires an adaptive risk model factor structure
This requires the ability to accommodate regimes in our risk models,
our portfolio construction, hedging, and asset allocation by
harnessing contemporaneous and forward-looking signals
The market is not driven solely by fundamentals
We need to leverage news/perception
We need to explore nuanced relationships beyond bland
membership
Techniques exist to address all of these issues, and are
being applied today.
55TAKE HOME
Northfield:
risk models that utilize implied volatility since 1997
adaptive hybrid risk models since 1998
risk models utilizing cross-sectional dispersion since 2003
using implied volatility and dispersion in our entire range of
short-horizon adaptive models since 2009
If you’re doing some kind of time-series analysis on
financial data you need to keep time-dependence,
regimes, and evolving data in mind
There are techniques to conquer all of these
challenges, but they’re not the easy ones that come as
part of Excel!
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