the search for a better risk model - mpt forum tokyo march 1st 2012
DESCRIPTION
This presentation discusses the three most common ways to estimate a multi-factor risk model, sheds some light on the numerous assumptions underlying the models, and provides some thoughts about how to address those assumptions to make the models better fit the real world. The Northfield hybrid risk model is discussed. Non-stationary volatility, correlation, clusters in volatility, the use of forward-looking signals such as implied-volatility and cross-sectional dispersion, as well as the use of quantified news information to update risk forecasts are included.TRANSCRIPT
The Search for a Better Risk Model Making Sense Out of Chaos
Nick Wade Tokyo March 1st 2012
www.northinfo.com
Overview While the linear model is prevalent in finance, many of its assumptions are not. While multi-factor risk models are similarly widely used, the assumptions behind
their estimation are likewise not well known (or not well publicized!) • Review the three standard methods of constructing multi-factor risk models, and
detail their assumptions, pro’s, and cons and the situations in which they are least and best suited.
• Discuss recent developments in factor selection and estimation • hybrid models, adaptive models, the use of information from other asset
classes, forward-looking signals, quantified news flows, and ESG signals. • flexible estimation techniques, advances in clustering including semantic
clustering or text-mining, and methods borrowed from social network analysis
• Discuss Risk Statistics [If Time Permits!] • Limitations of tracking error as a measure • Systemic risk
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More Motivation – A Recent Quote GARP: “in the past few months volatility has dropped significantly; almost to
the point where it is below the BARRA estimates”
(GARP Risk Review Issue 16 Jan/Feb 2004)
Clearly risk forecasBng is not easy. We can make risk models more powerful: • include informaBon from other asset classes • include contemporaneous or forward-‐looking signals • make the factor structure of the model adapBve • adjust for real-‐world effects like regimes and “fat tails” • Acknowledge the limitaBons in our risk measures!
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What 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
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How 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)
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The Linear Model
• RelaBonship between R and F is linear ∀F • There are N common factor sources of return • RelaBonship between R and H is linear ∀H • There is no correlaBon between F and H ∀ F,H • The distribuBon of F is staBonary, Normal, i.i.d. ∀F • There are M stock-‐specific sources of return • There is no correlaBon between H across stocks • The distribuBon of H is staBonary, Normal, i.i.d. ∀H • (Implicitly also the volaBlity of R and F is staBonary)
∑ ∑= =
⎟⎟⎠
⎞⎜⎜⎝
⎛=+=
N
i
M
jjtjtstititst HGSFER
1 1
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Effect of Model Errors Missing factors lead to under estimation Spurious factors add noise
(see Pfleiderer 2005, Scowcroft 2006) Non-linearity leads to over/under estimation for
different constituencies – note “blind factors” (UK Super-cap effect)
Non-stationary market/factor variance or residual
variance leads to over/under estimation as model struggles to react (tech bubble)
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Evidence
Security Returns are not Normal, stationary, i.i.d. • Lots of evidence… start with Mandelbrot (1963) • But not too bad for portfolios: Hlawitschka and Stern
(1995) Factor Returns are not Normal, Stationary, iid.
• We know this because we exploit it to make money! • see e.g. tech bubble, trends, styles, alpha,
momentum • Pope and Yadav (1994)
We need to adjust model to accommodate broken assumpBons
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Estimating a Risk Model
( )∑∑ ∑= = =
+=N
i
N
j
M
kkkjijijip SWEEV
1 1 1
2, σρσσ
∑=
+=N
istititst SFER
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.
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PracBcal Approaches
• There are three common approaches: – Observe factors Fit and determine
Ei by Bme-‐series approach – Observe Eit and determine Fit by
cross-‐secBonal approach – Assume N and use staBsBcal
approach to determine Ei, then esBmate Fi by regression
∑=
+=N
istititst SFER
1
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Estimation methods: The Exogenous or Macro model
seeks to estimate Ei from observing time-series Fit. Typical factors include Market, Sector, Oil, Interest-Rates…
• Ross (1976) • Chen (1986)
Model is pre-specified The Endogenous or Fundamental Model
seeks to estimate Fit from observing firm characteristics 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 The Statistical Model
seeks to estimate Ei using Factor Analysis or Principle Components Assume N, or imply from sample data-set Then use Regression to estimate Fit
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Pros and Cons of Each Approach Approach Pros Cons
Fundamental (micro) model
• Best for concentrated por\olios • Number of factors is fixed • Factors are unchanging • Dependent on accounBng statement accuracy and comparability
• Membership factors for industry/country/sector e.g. all banks are the same
• Errors will be in factor returns, hence in covariance matrix, and hence not diversifiable
Macro-‐economic model
• Best for diversified por\olios • No dependence on accounBng data • The response of each security to changes in market/sector/industry/ whatever to be different across securiBes
• Errors will be in loadings (exposures), thus diversifiable
• Environment factors included
• Number of factors fixed • Factors are unchanging • Exposure to factors is staBonary over Bme unless e.g. FLS is used
StaBsBcal model • Applicable to passive funds or to hedge a desk • All correlaBon is assumed informaBon
• AdapBve: Captures new, or transient effects
• A`ribuBon of risk is difficult
• Issues with noise in data • Errors in variables • Number of factors is either pre-‐specified or sample-‐dependent
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Other 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)
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Simultaneous 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.
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Hybrid Model Key Point: Makes the Model Adaptive Combine macro, micro, and statistical factors Gain the advantages of each, whilst mitigating the
limitations of each • Allows securities to have unique exposures to industry, sector,
country (i.e. not all banks are the same) • Intuitive, explainable, justifiable observable factors • Minimal dependence on accounting information • Rapid inclusion of new or transient factors
Estimate using time-series approach • Diversify away estimation error • best for markets with moderate to low dispersion • best for portfolios with moderate/high diversification
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Other Model Issues Time period – historical data (Scowcroft & Sefton) Frequency – daily, weekly, monthly Reference Day effect Forecast Horizon – Rosenberg and Guy (1975) Intra-Horizon risk Data – clean, reliable, undisputed, comparable, timely… Asynchronous markets
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The Real World We are interested in a practical solution, not just a theoretically interesting
one Speculative trading causes “bubbles” Non-Normality manifests in skew, kurtosis, serial correlation – most of which can be
explained by time-varying volatility Liquidity effects cause serial correlation Volatility and correlation trend Jumps occur Regimes occur Any real-world model must accommodate these things
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Non-Stationarity Adjustments (1)
( )∑ ⎟⎟⎠
⎞⎜⎜⎝
⎛
−=
2
1nnxV i
• Non-‐staBonary factor return series will lead to the model underesBmaBng por\olio risk
• Adjust by changing variance calculaBon to include trend component of return
( )∑ ⎟⎟⎠
⎞⎜⎜⎝
⎛
−
−=
2
1nnxxV i
Adjust Model for the influence of non-stationary factor returns
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Non-Stationarity Adjustments (2)
What about residuals? We observe:
• Serial correlation (not i.i.d.) • Bid-ask bounce • Non-Normal distributions
Use Parkinson volatility measure
Adjust Model for the influence of non-stationary security returns
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Heteroskedasticity and Regimes We also observe that factor volatility and correlations
cluster in “regimes”. Adjust for this by exponentially weighting the return
information, or by GARCH, or by using the implied volatility from option market, or cross-sectional dispersion: • Northfield (1997) Short Term Model • Northfield (2007) Near-Horizon Models • Hwang & Satchell (2004) • Scowcroft (2005)
Note: exponential weighting and GARCH are backward-looking naïve, trend-following
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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”
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The 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…
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Contemporaneous or Forward-Looking Signals
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.
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Northfield Asia ex. Japan Risk Models Tracking Error LH Tracking Error SH Linear (Tracking Error SH)
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Pause for Thought Problem Solu3on
Fixed Factor Structure Can’t learn/Missing Factors AdapBve Hybrid Model
Non-‐StaBonary Vol./Corr. And Regimes
Forward-‐looking signals e.g. implieds, news, dispersion
Exposures Fixed (macro model)
Simultaneous esBmaBon or e.g. flexible-‐least-‐squares
Membership variables e.g. all banks are the same (fundamental model)
Simultaneous esBmaBon or use a Bme-‐series model
Serial-‐correlaBon, non-‐normal security returns
Parkinson volaBlity or other range measure
Totally Dependent on Historical Data
Use contemporaneous or forward-‐looking signals
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BETTER FACTOR CHOICES
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What 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
www.northinfo.com
A Very Common Factor: Country, Industry, Sector, Region…
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
Suggestions: • 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).
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A 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 Mitra, Mitra, diBartolomeo (2008)
• Look at instances of occurrence in news, sentiment • Connected:
• 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
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Artificial 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
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ISSUES WITH ESTIMATION
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Some Problems We are dealing with an evolving data set, not a static one
• 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
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Techniques 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
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Regression with non-stationary 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
( ) ( ) ( )∑ ∑=
+
−
=+ −ʹ′−+−
T
ttt
T
tttttt xy
11
1
11
2 ββββλβ
Minimize both sum of squared errors and sum of squared dynamic errors (coefficient esBmates)
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FLS example
An example from Clayton and MacKinnon (2001) The coefficient apparently exhibits structural shift in 1992
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Cluster 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
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Factor analysis with non-stationary 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.
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Conclusions – Risk Models
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 and connections across asset classes
The market is not driven solely by fundamentals • We need to leverage news/perception • We need to explore nuanced relationships beyond bland membership • We need to leverage cross-asset class information
Advances in risk research and in estimation techniques exist to address all of these issues, and are being applied today.
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RISK MEASURES
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Thoughts on Risk Measures Tracking Error and VaR:
Are impacted by turnover Ignore uncertainty in the mean Are both end-of-horizon measures
The events of the last few years show that we need measures of systemic risk
in the markets as a whole - Asset Centrality - Absorption Ratio - Average Weighted Implied Life
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An Unfortunate Truth – with apologies to Al Gore
The Tracking Error ex-ante must, mathematically, be less than the Tracking Error ex-post if the portfolio and/or benchmark turnover during the period. • Lawton-Browne (2000) • Satchell & Hwang (2001)
Question is… by how much…?
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Intra-Horizon Risk - Motivation
Risk Band
Return
Time
Tracking error or VaR are concerned about the likely distribuBon of returns at the end of some investment period. In this example we have scored a goal – the final return is within the risk bands we set ourselves.
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Intra-Horizon Risk - Motivation II
Risk Band Return
Time
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Probability of Loss “Intra Horizon”
Second part never zero or negative • Implies IH loss estimate > EOH estimate (always) • IH P(loss) rises as investment horizon expands, whereas EOH P(loss)
declines as investment horizon expands • => time-diversification argument
( )( ) ( ) 22
11lnPrPr σµ
σµ L
TTLNEI +⎟
⎠⎞
⎜⎝⎛ +++=
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Intra-Horizon Risk Multiples
Average VaR Mul3ples
Maximum VaR Mul3ples
Average VaR-‐I Mul3ples
Maximum VaR-‐I Mul3ples
JD CGMY FMLS JD CGMY FMLS JD CGMY FMLS JD CGMY FMLS
S&P 500
1.21 1.33 1.44 1.33 1.37 1.69 1.60 1.39 1.94 1.77 1.50 2.08
FTSE 1.20 1.21 1.35 1.38 1.44 1.50 1.50 1.38 1.85 1.70 1.55 2.07
Nikkei 1.14 1.11 1.39 1.19 1.25 1.75 1.28 1.27 1.80 1.40 1.34 2.12
ATM Call
1.37 1.29 1.68 1.39 1.31 1.85 1.61 1.45 2.55 1.67 1.47 2.64
Compared to standard Normal VaR JD = Merton’s jump-‐diffusion model CGMY is the two-‐sided pure-‐jump Levy model of Carr, Geman, Madan, and Yor FMLS is the finite-‐moment log-‐stable model of Carr and Wu
(Bakshi and Panayatov)
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Tracking Error in Active Management Within asset management, the risk of benchmark relative performance is
typically expressed by measures such as “tracking error”, which describes the expectation of times-series standard deviation of benchmark relative returns.
This is useful for index fund management, where the expectation of the
mean for benchmark relative return is fixed at zero. The active management case is problematic, as tracking error excludes the
potential for the realized future mean of active returns to be other than the expected value.
All active managers must believe their future returns will be above
benchmark (or peer group average) in order to rationally pursue active management, yet it is axiomatically true that roughly half of active managers must produce below average results.
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Active Risk Including the Mean
A more general conception of the problem would be to think of active risk as the square root of total active variance
σactive = (σmean
2 + σTE2 + 2 * σmean * σTE * ρ).5
Where σmean = uncertainty of the true mean relative to expectation of the mean ρ = correlation between uncertainty and tracking error
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A Rule of Thumb
One way to approach this problem is to consider a binary distribution for the active return of a manager.
We assume that each manager has a benchmark relative return expectation of portfolio alpha αp with a probability w of being correct.
If the manager’s forecast is wrong, they have a probability of (1-w) of realizing –αp.
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A Rule of Thumb 2 With this framework, the value of σmean is
σmean = ((1-w) * 4 * αp2).5
Where
w = is the probability of realizing the expected alpha αp = manager’s expectation of portfolio alpha
For w = .5 we obtain the simple expression σmean = 2.5 * αp
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The Rule of Thumb and the Information Ratio
It is the frequent custom of the asset management industry that the information ratio is used as a proxy for manager skill. IR = (αp/σTE) αp = IR * σTE σmean = ((1-w) * 4 * (IR * σTE ) 2).5
For w = .5 σmean = 2.5 * IR * σTE
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A Tale of Two Managers
Let’s make the simplifying assumption that ρ = 0 and consider two managers, K and L.
Both managers have TE = 5. Manager K is a traditional asset manager that purports to
clients that their IR = 0.5 Manager L is a very aggressive fund that purports to it’s
investors that their IR = 3 Manager L’s IR is six times as good as Manager K.
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Hoisted by One’s Own Petard For w = .5 we obtain: For Manager K we get:
σactive = (2 * .25 * 25 + 25).5 = 6.125%
About 23% greater than original TE, revised IR about .4 For Manager L we get σactive = (2 * 9 * 25 + 25).5 = 20.61%
More than four times the original TE, with adjusted IR = .73
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Asset 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”?
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Absorption 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 independent of systematic volatility) • You could use Implied Correlation and get a similar forward-looking result
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Conclusions – Risk Measures
Tracking error is an inadequate measure of risk for active managers
We should evaluate risk with the broader measure of “active risk” in the spirit of Qian and Hua.
Active risk can be formulated as the aggregate of tracking error and the uncertainty of the mean return over time
The estimation of active risk can be reasonably parameterized either from empirical data for defined manager styles or from a simple “rule of thumb”
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Take Home
Northfield: • Forward-looking 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 • Leveraging cross-asset class relationships to model credit risk 2011
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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REFERENCES
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References Ang, A. and Bekaert, G. (1999) ‘International Asset Allocation with time-varying Correlations’,
working paper, Graduate School of Business, Stanford University and NBER. Banerjee, Arindam; Merugu, Srujana; Dhillon, Inderjit S.; Ghosh, Joydeep (2005). "
Clustering with Bregman divergences". Journal of Machine Learning Research 6: 1705–1749. http://jmlr.csail.mit.edu/papers/v6/banerjee05b.html.
Bertero, E. and Mayer, C. (1989) ‘Structure and Performance:Global Interdependence of Stock Markets around the Crash of October 1987’, London, Centre for Economic Policy Research.
Chesnay, F. and Jondeau, E. (2001) ‘Does Correlation between Stock Returns really increase during turbulent Periods?’, Economic Notes by Banca Monte dei Paschi di Siena SpA, Vol. 30,No. 1, pp.53–80.
Jim Clayton and Greg MacKinnon (2001), "The Time-Varying Nature of the Link Between REIT, Real Estate and Financial Asset Returns" (pdf,6.3M), Journal of Real Estate Portfolio Management, January-March Issue
Erb, C.B., Harvey, C.R. and Viskanta, T.E. (1994) ‘Forecasting international Equity Correlations’, Financial Analysts Journal,pp.32–45.
Jakulin A & Bratko I (2003a). Analyzing Attribute Dependencies, in N Lavra\quad{c}, D Gamberger, L Todorovski & H Blockeel, eds, Proceedings of the 7th European Conference on Principles and Practice of Knowledge Discovery in Databases, Springer, Cavtat-Dubrovnik, Croatia, pp. 229-240
Jennrich R. (1970) ‘An Asymptotic χ2 Test for the Equality of Two Correlation Matrices’, Journal of the American Statistical Association,Vol. 65, No. 330.
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References II
Black F., Jensen M., Scholes M. “The Capital Asset Pricing Model: some empirical tests” In Jensen M.C., editor, “Studies in the Theory of Capital Markets” Praeger, New York, 1972.
Bulsing M., Scowcroft A., and Sefton J., “Understanding Forecasting: A Unified framework for combining both analyst and strategy forecasts” UBS Working Paper, 2003.
Chen N.F. Roll R. Ross S.A. “Economic Forces and the Stock Market” Journal of Business 59, 1986. Connor G and Korajczyck R.A. “Risk and Return in an equilibrium APT: application of a new test methodology” Journal
of Financial Economics 21, 1988. diBartolomeo D. “Why Factor Risk Models Often Fail Active Quantitative Managers. The Completeness Conflict.”
Northfield, 1998. Diermeier J. and Solnik B. “Global Pricing of Equity”, FAJ Vol. 57(4). Ericsson and Karlsson (2002) Fama E. and MacBeth J. “Risk, Return, and Equilibrium: empirical tests” Journal of Political Economy 71, 1973. GARP “Managing Tracking Errors in a Dynamic Environment” GARP Risk Review Jan/Feb 2004 Hansen L. “Large Sample Properties of Generalized Method of Moments Estimators” Econometrica 50, 1982 Heston S. and Rouwenhorst K. G. “Industry and Country Effects in International Stock Returns” Journal of Portfolio
Management, Vol 21(3), 1995 Hwang S. and Satchell S. “Tracking Error: ex ante versus ex post measures”. Journal of Asset Management, vol 2,
number 3, 2001. King B.F. “Market and Industry Factors in Stock Price Behavior” Journal of Business, Vol. 39, January 1966. Lawton-Browne, C.L. Journal of Asset Management, 2001. Lehmann, B. and Modest, D. A. Journal of Financial Economics, Vol. 21, No. 2:213-254
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References III R. Kalaba, L. Tesfatsion. Time-varying linear regression via flexible least squares.
International Journal on Computers and Mathematics with Applications, 1989, Vol. 17, pp. 1215-1245.
Kaplanis, E. (1988) ‘Stability and Forecasting of the Comovement Measures of International Stock Market Returns’, Journal of International Money and Finance, Vol. 7, pp.63–75.
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