The Search for a Better Risk ModelMaking Sense Out of ChaosNick WadeTokyoMarch 1st 2012
OverviewWhile the linear model is prevalent in finance, many of its assumptions are not.While multi-factor risk models are...
More Motivation – A Recent QuoteGARP: “in the past few months volatility has dropped significantly; almost to  the point w...
What is a Factor Model?The main purpose of a factor model is to find a set of common themes that   explain the variability...
How many factors…?The academic consensus seems to be that there is not much difference going   from 5 to 10 to 15 factors....
The	  Linear	  Model	                         N      ⎛ M             ⎞       Rst = ∑ Eit Fit + S st ⎜ = ∑ G jt H jt ⎟ ...
Effect of Model ErrorsMissing factors lead to under estimationSpurious factors add noise  (see Pfleiderer 2005, Scowcroft ...
EvidenceSecurity Returns are not Normal, stationary, i.i.d.   •  Lots of evidence… start with Mandelbrot (1963)   •  But n...
Estimating a Risk Model                                 N                          Rst = ∑ Eit Fit + S st                 ...
PracBcal	  Approaches	  •  There	  are	  three	  common	     approaches:	      –  Observe	  factors	  Fit	  and	  determin...
Estimation methods:The Exogenous or Macro model   seeks to estimate Ei from observing time-series Fit.   Typical factors i...
Pros and Cons of Each Approach	      Approach	                                                    Pros	                   ...
Other ApproachesCombined Models:   •  Northfield Hybrid Model   •  Stroyny (2001)Simultaneous Estimation   •    Black et a...
Simultaneous EstimationRemoving the limitation of binary or membership variables (such as  industry, country, sector, regi...
Hybrid ModelKey Point: Makes the Model AdaptiveCombine macro, micro, and statistical factorsGain the advantages of each, w...
Other Model IssuesTime period – historical data (Scowcroft & Sefton)Frequency – daily, weekly, monthlyReference Day effect...
The Real WorldWe are interested in a practical solution, not just a theoretically interesting  oneSpeculative trading caus...
Non-Stationarity Adjustments (1) •  Non-­‐staBonary	  factor	  return	                                         2          ...
Non-Stationarity Adjustments (2)      What about residuals?      We observe:         •  Serial correlation (not i.i.d.)   ...
Heteroskedasticity and RegimesWe also observe that factor volatility and correlations cluster in “regimes”.Adjust for this...
Turbulence in the marketKritzman (2009):     •  Correlation of US and foreign stocks when both markets’ returns are       ...
The world is very obviously time-varyingNon-stationary volatility (ARCH, GARCH, etc)   •  We spend an heroic amount of tim...
Contemporaneous or Forward-Looking Signals Take a model that has been estimated on purely historical data Find true forwar...
Northfield Asia ex. Japan Risk Models     Tracking Error LH    Tracking Error SH   Linear (Tracking Error SH)6864605652484...
Pause for ThoughtProblem	                                                                     Solu3on	  Fixed	  Factor	  S...
BETTER FACTOR CHOICES           www.northinfo.com
What is a Factor Model?The main purpose of a factor model is to find a set of common themes that   explain the variability...
A Very Common Factor: Country, Industry, Sector, Region…It is pretty much standard practice to take note of    membership ...
A few suggestions for better factorsA factor can be any shared behavior     •  Historical: Semantic clustering (text minin...
Artificial Immune SystemsAt a high level, our immune system consists of two pieces:     •  Innate immunity     •  Learned ...
ISSUES WITH ESTIMATION            www.northinfo.com
Some ProblemsWe are dealing with an evolving data set, not a static one   •  This impacts our common techniques   •  Look ...
Techniques for Evolving DataMost of our favorite tools are designed to fit static data sets where behaviors  are mostly un...
Regression with non-stationary dataTechniques have been developed specifically to allow time-varying sensitivities    •  F...
FLS exampleAn example from Clayton and MacKinnon (2001)The coefficient apparently exhibits structural shift in 1992       ...
Cluster analysis with nonstationary dataGuedalia, London, Werman; “An on-line agglomerative clustering  method for nonstat...
Factor analysis with non-stationary dataDahlhaus, R. (1997). Fitting Time Series Models to Nonstationary   Processes. Anna...
Conclusions – Risk Models Our world changes    •  This requires an adaptive risk model factor structure    •  This require...
RISK MEASURES           www.northinfo.com
Thoughts on Risk MeasuresTracking Error and VaR:   Are impacted by turnover   Ignore uncertainty in the mean   Are both en...
An Unfortunate Truth – with apologies to Al GoreThe Tracking Error ex-ante must, mathematically, be less than the Tracking...
Intra-Horizon Risk - Motivation   Return	                                                                                 ...
Intra-Horizon Risk - Motivation II   Return	                                                  Risk	  Band	                ...
Probability of Loss “Intra Horizon”                   ⎛ (ln(1 + L ) + µT )   Pr I = Pr E + N ⎜                       ⎞(...
Intra-Horizon Risk Multiples                      Average	  VaR	             Maximum	  VaR	                     Average	  ...
Tracking Error in Active ManagementWithin asset management, the risk of benchmark relative performance is   typically expr...
Active Risk Including the MeanA more general conception of the problem would be to think of  active risk as the square roo...
A Rule of ThumbOne way to approach this problem is to consider a binary  distribution for the active return of a manager. ...
A Rule of Thumb 2With this framework, the value of σmean is    σmean = ((1-w) * 4 * αp2).5     Where  w = is the probabili...
The Rule of Thumb and the Information RatioIt is the frequent custom of the asset management industry that the    informat...
A Tale of Two ManagersLet’s make the simplifying assumption that ρ = 0 and  consider two managers, K and L.  Both managers...
Hoisted by One’s Own PetardFor w = .5 we obtain:         For Manager K we get:   σactive  = (2 * .25 * 25 + 25).5 = 6.125%...
Asset CentralityWe can take this idea from Social Network Analysis and apply it to a variety of  contexts:   •  Was Lehman...
Absorption Ratio (Kritzman 2011)On a related note – how tightly connected is the market, or a particular sector?    •  Lo ...
Conclusions – Risk MeasuresTracking error is an inadequate measure of risk for active   managersWe should evaluate risk wi...
Take HomeNorthfield:    •  Forward-looking risk models that utilize implied volatility since 1997    •  adaptive hybrid ri...
REFERENCES             www.northinfo.com
ReferencesAng, A. and Bekaert, G. (1999) ‘International Asset Allocation with time-varying Correlations’,    working paper...
References IIBlack F., Jensen M., Scholes M. “The Capital Asset Pricing Model: some empirical tests” In Jensen M.C., edito...
References IIIR. Kalaba, L. Tesfatsion. Time-varying linear regression via flexible least squares.   International Journal...
References IVLintzenberger R. and Ramaswamy K. “The effects of dividends on common stock prices:    theory and empirical e...
References VPohlson N.G. and Tew B.V. “Bayesian Portfolio Selection: An empirical analysis of the S&P 500 index     1970-1...
References VIOsborne, Jason W. (2003). Effect sizes and the disattenuation of correlation and regression    coefficients: ...
References VIIQian, Edward and Ronald Hua, Active Risk and the Information Ratio,   Journal of Investment Management, Thir...
References VIIISolis, Rafael “Visualizing Stock Mutual Fund Relationships through Social   Network Analysis”, Global Journ...
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The Search for a Better Risk Model - MPT Forum Tokyo March 1st 2012

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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.

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The Search for a Better Risk Model - MPT Forum Tokyo March 1st 2012

  1. 1. The Search for a Better Risk ModelMaking Sense Out of ChaosNick WadeTokyoMarch 1st 2012
  2. 2. OverviewWhile 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 www.northinfo.com
  3. 3. More Motivation – A Recent QuoteGARP: “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!   www.northinfo.com
  4. 4. 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 factorsThe end result is a model of how the portfolio will behave – how much the securities will move together and how much they will behave uniquelyThat can lead us to various useful risk characteristics www.northinfo.com
  5. 5. 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) www.northinfo.com
  6. 6. The  Linear  Model   N ⎛ M ⎞ Rst = ∑ Eit Fit + S st ⎜ = ∑ G jt H jt ⎟ ⎜ ⎟ i =1 ⎝ j =1 ⎠•  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)   www.northinfo.com
  7. 7. Effect of Model ErrorsMissing factors lead to under estimationSpurious 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) www.northinfo.com
  8. 8. EvidenceSecurity 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   www.northinfo.com
  9. 9. Estimating a Risk Model N Rst = ∑ Eit Fit + S st i =1 N N M V p = ∑∑ Ei E jσ iσ j ρi , 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. www.northinfo.com
  10. 10. PracBcal  Approaches  •  There  are  three  common   approaches:   –  Observe  factors  Fit  and  determine   Ei  by  Bme-­‐series  approach   N –  Observe  Eit  and  determine  Fit  by   cross-­‐secBonal  approach   Rst = ∑ Eit Fit + S st –  Assume  N  and  use  staBsBcal   i =1 approach  to  determine  Ei,  then   esBmate  Fi  by  regression   www.northinfo.com
  11. 11. 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-specifiedThe 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-specifiedThe 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 www.northinfo.com
  12. 12. Pros and Cons of Each Approach Approach Pros Cons Fundamental  (micro)   •   Best  for  concentrated  porolios   •   Number  of  factors  is  fixed  model •   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   •  Best  for  diversified  porolios   •  Number  of  factors  fixed  model   •   Factors  are  unchanging   •   No  dependence  on  accounBng  data   •   The  response  of  each  security  to  changes  in   •   Exposure  to  factors  is  staBonary  over  Bme   market/sector/industry/  whatever  to  be  different   unless  e.g.  FLS  is  used   across  securiBes     •   Errors  will  be  in  loadings  (exposures),  thus     diversifiable   • Environment  factors  included  StaBsBcal  model •   Applicable  to  passive  funds  or  to  hedge  a  desk   •   A`ribuBon  of  risk  is  difficult   •   All  correlaBon  is  assumed  informaBon   •   Issues  with  noise  in  data   •   AdapBve:  Captures  new,  or  transient  effects   •   Errors  in  variables     •   Number  of  factors  is  either  pre-­‐specified  or   sample-­‐dependent   www.northinfo.com
  13. 13. Other ApproachesCombined 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) www.northinfo.com
  14. 14. Simultaneous EstimationRemoving 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. www.northinfo.com
  15. 15. Hybrid ModelKey Point: Makes the Model AdaptiveCombine macro, micro, and statistical factorsGain 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 factorsEstimate using time-series approach •  Diversify away estimation error •  best for markets with moderate to low dispersion •  best for portfolios with moderate/high diversification www.northinfo.com
  16. 16. Other Model IssuesTime period – historical data (Scowcroft & Sefton)Frequency – daily, weekly, monthlyReference Day effectForecast Horizon – Rosenberg and Guy (1975)Intra-Horizon riskData – clean, reliable, undisputed, comparable, timely…Asynchronous markets www.northinfo.com
  17. 17. The Real WorldWe are interested in a practical solution, not just a theoretically interesting oneSpeculative trading causes “bubbles”Non-Normality manifests in skew, kurtosis, serial correlation – most of which can be explained by time-varying volatilityLiquidity effects cause serial correlationVolatility and correlation trendJumps occurRegimes occurAny real-world model must accommodate these things www.northinfo.com
  18. 18. Non-Stationarity Adjustments (1) •  Non-­‐staBonary  factor  return   2 ⎛ xi − x ⎞ series  will  lead  to  the  model   V = ∑ ⎜ ⎟ ⎜ n(n − 1) ⎟ underesBmaBng  porolio  risk   ⎝ ⎠ •  Adjust  by  changing  variance   2 calculaBon  to  include  trend   ⎛ xi ⎞ V = ∑ ⎜ ⎜ n(n − 1) ⎟ ⎟ component  of  return   ⎝ ⎠ Adjust Model for the influence of non-stationary factor returns www.northinfo.com
  19. 19. 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 www.northinfo.com
  20. 20. Heteroskedasticity and RegimesWe 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 www.northinfo.com
  21. 21. Turbulence in the marketKritzman (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” www.northinfo.com
  22. 22. The world is very obviously time-varyingNon-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 analysisRead the fine print… www.northinfo.com 22  
  23. 23. 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. www.northinfo.com
  24. 24. Northfield Asia ex. Japan Risk Models Tracking Error LH Tracking Error SH Linear (Tracking Error SH)686460565248444036322824201612 8 4 0 www.northinfo.com
  25. 25. Pause for ThoughtProblem   Solu3on  Fixed  Factor  Structure   Can’t  learn/Missing  Factors   AdapBve  Hybrid  Model    Non-­‐StaBonary  Vol./Corr.   Forward-­‐looking  signals  And  Regimes   e.g.  implieds,  news,   dispersion  Exposures  Fixed  (macro   Simultaneous  esBmaBon  or  model)   e.g.  flexible-­‐least-­‐squares  Membership  variables  e.g.   Simultaneous  esBmaBon  or  all  banks  are  the  same   use  a  Bme-­‐series  model  (fundamental  model)  Serial-­‐correlaBon,  non-­‐ Parkinson  volaBlity  or  normal  security  returns   other  range  measure  Totally  Dependent  on   Use  contemporaneous  or  Historical  Data   forward-­‐looking  signals   www.northinfo.com
  26. 26. BETTER FACTOR CHOICES www.northinfo.com
  27. 27. 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 factorsThe end result is a model of how the portfolio will behave – how much the securities will move together and how much they will behave uniquelyThat can lead us to various useful risk characteristics www.northinfo.com
  28. 28. 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 sectorsProblems: multinational firms, globalization, index dominationSuggestions: •  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). www.northinfo.com
  29. 29. A few suggestions for better factorsA 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 www.northinfo.com
  30. 30. Artificial Immune SystemsAt a high level, our immune system consists of two pieces: •  Innate immunity •  Learned immunityIn our context •  The factors we believe to be useful at t=0 •  Plus the factors the model learns along the wayTune the model •  Criteria for accepting a new factor •  Criteria for archiving / forgetting factors •  Memory length for previously useful factors www.northinfo.com
  31. 31. ISSUES WITH ESTIMATION www.northinfo.com
  32. 32. Some ProblemsWe 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 setsWe 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 www.northinfo.com
  33. 33. Techniques for Evolving DataMost 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 themRecent 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 www.northinfo.com
  34. 34. Regression with non-stationary dataTechniques 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 2 ʹ′ ∑ ( yt − xt βt ) t =1 + λ ∑ (β t +1 − β t ) (β t +1 − β t ) t =1 Minimize  both  sum  of  squared  errors  and  sum  of  squared  dynamic  errors     (coefficient  esBmates)   www.northinfo.com
  35. 35. FLS exampleAn example from Clayton and MacKinnon (2001)The coefficient apparently exhibits structural shift in 1992 www.northinfo.com 35  
  36. 36. Cluster analysis with nonstationary dataGuedalia, London, Werman; “An on-line agglomerative clustering method for nonstationary data” Neural Computation, February 15, 1999, Vol. 11, No. 2, Pages 521-54C. 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 (KDD04), 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 www.northinfo.com 36  
  37. 37. Factor analysis with non-stationary dataDahlhaus, 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. www.northinfo.com
  38. 38. 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. www.northinfo.com
  39. 39. RISK MEASURES www.northinfo.com
  40. 40. Thoughts on Risk MeasuresTracking Error and VaR: Are impacted by turnover Ignore uncertainty in the mean Are both end-of-horizon measuresThe 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 www.northinfo.com
  41. 41. An Unfortunate Truth – with apologies to Al GoreThe 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…? www.northinfo.com
  42. 42. Intra-Horizon Risk - Motivation Return   Risk  Band   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.   www.northinfo.com
  43. 43. Intra-Horizon Risk - Motivation II Return   Risk  Band   Time   www.northinfo.com
  44. 44. Probability of Loss “Intra Horizon” ⎛ (ln(1 + L ) + µT ) Pr I = Pr E + N ⎜ ⎞(1 + L )2 µ σ 2 ⎟ ⎝ σ T ⎠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 www.northinfo.com
  45. 45. Intra-Horizon Risk Multiples Average  VaR   Maximum  VaR   Average  VaR-­‐I   Maximum  VaR-­‐I   Mul3ples   Mul3ples   Mul3ples   Mul3ples         JD   CGMY   FMLS   JD   CGMY   FMLS   JD   CGMY   FMLS   JD   CGMY   FMLS  S&P   1.21   1.33   1.44   1.33   1.37   1.69   1.60   1.39   1.94   1.77   1.50   2.08  500  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   1.37   1.29   1.68   1.39   1.31   1.85   1.61   1.45   2.55   1.67   1.47   2.64  Call   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)   www.northinfo.com
  46. 46. Tracking Error in Active ManagementWithin 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. www.northinfo.com
  47. 47. Active Risk Including the MeanA more general conception of the problem would be to think of active risk as the square root of total active variance σactive = (σmean2 + 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  www.northinfo.com
  48. 48. A Rule of ThumbOne 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.    www.northinfo.com
  49. 49. A Rule of Thumb 2With 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 www.northinfo.com
  50. 50. The Rule of Thumb and the Information RatioIt 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 www.northinfo.com
  51. 51. A Tale of Two ManagersLet’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 = 0.5Manager L is a very aggressive fund that purports to it’s investors that their IR = 3Manager L’s IR is six times as good as Manager K. www.northinfo.com
  52. 52. Hoisted by One’s Own PetardFor 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   www.northinfo.com
  53. 53. Asset CentralityWe 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”? www.northinfo.com
  54. 54. 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 www.northinfo.com
  55. 55. Conclusions – Risk MeasuresTracking error is an inadequate measure of risk for active managersWe 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 timeThe estimation of active risk can be reasonably parameterized either from empirical data for defined manager styles or from a simple “rule of thumb” www.northinfo.com
  56. 56. Take HomeNorthfield: •  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 2011If you’re doing some kind of time-series analysis on financial data you need to keep time-dependence, regimes, and evolving data in mindThere are techniques to conquer all of these challenges, but they’re not the easy ones that come as part of Excel! www.northinfo.com
  57. 57. REFERENCES www.northinfo.com
  58. 58. ReferencesAng, 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 IssueErb, 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 Lavraquad{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. 229-240Jennrich R. (1970) ‘An Asymptotic χ2 Test for the Equality of Two Correlation Matrices’, Journal of the American Statistical Association,Vol. 65, No. 330. www.northinfo.com
  59. 59. References IIBlack 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 2004Hansen L. “Large Sample Properties of Generalized Method of Moments Estimators” Econometrica 50, 1982Heston S. and Rouwenhorst K. G. “Industry and Country Effects in International Stock Returns” Journal of Portfolio Management, Vol 21(3), 1995Hwang 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 www.northinfo.com
  60. 60. References IIIR. 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.Lee, S.B. and Kim, K.J. (1993) ‘Does the October 1987 Crash strengthen the co- Movements among national Stock Markets?’,Review of Financial Economics, Vol. 3, No. 1, pp.89–102.Longin, F. and Solnik, B. (1995) ‘Is the Correlation in International Equity Returns constant: 1960–1990?’, Journal of InternationalMoney and Finance, Vol. 14, No. 1, pp.3–26.Longin, F. and Solnik, B. (2001) ‘Extreme Correlation of International Equity Markets’, The Journal of Finance, Vol. 56, No.2.Mahalanobis, P C (1936). "On the generalised distance in statistics". Proceedings of the National Institute of Sciences of India 2 (1): 49–55. http://ir.isical.ac.in/dspace/handle/1/1268. Retrieved 2008-11-05Nemenman I (2004). Information theory, multivariate dependence, and genetic network inference www.northinfo.com
  61. 61. References IVLintzenberger R. and Ramaswamy K. “The effects of dividends on common stock prices: theory and empirical evidence” Journal of Financial Economics 7, 1979.MacQueen J. “Alpha: the most abused term in Finance” Northfield Conference, Montebello, 2005MacQueen J. and Satchell S. “An Enquiry into Globalisation and Size in World Equity Markets”, Quantec, Thomson Financial, 2001.Mandelbrot B. “The variation of certain speculative prices” Journal of Business, 36. 1963.Markowitz, H.M. “Portfolio Selection” 1st edition, John Wiley, NY, 1959.Marsh T. and Pfleiderer P. “The Role of Country and Industry Effects in Explaining Global Stock Returns”, UC Berkley, Walter A. Haas School of Business, 1997.McElroy M.B., Burmeister E. “Arbitrage Pricing Theory as a restricted non-linear multivariate regression model” Journal of Business and Economic Statistics 6, 1988.Northfield Short Term Equity Risk ModelNorthfield Single-Market Risk Model (Hybrid Risk Model)Pfleiderer, Paul “Alternative Equity Risk Models: The Impact on Portfolio Decisions” The 15th Annual Investment Seminar UBS/Quantal, Cambridge UK 2002. www.northinfo.com
  62. 62. References VPohlson N.G. and Tew B.V. “Bayesian Portfolio Selection: An empirical analysis of the S&P 500 index 1970-1996” Journal of Business and Economic Statistics 18, 2000.Pope Y and Yadav P.K. “Discovering Errors in Tracking Error”. Journal of Portfolio Management, Winter 1994.Rosenberg B. and Guy J. “The Prediction of Systematic Risk” Berkeley Research Program in Finance, Working Paper 33, February 1975.Ross S.A. “The Arbitrage Theory of Capital Asset Pricing” Journal of Economic Theory, 13, 1976.Satchell and Scowcroft “A demystification of the Black-Litterman model: managing quantitative and traditional portfolio construction” Journal of Asset Management 1, 2000.Scowcroft A. and Sefton J. “Risk Attribution in a global country-sector model” in Knight and Satchell 2005 (“Linear Factor Models in Finance”)Scowcroft A. and Sefton J. “Do tracking errors reliably estimate portfolio risk?”. Journal of Asset Management Vol 2, 2001.Shanken J. “The Arbitrage Pricing Theory: Is it testable?” Journal of Finance, 37, 1982.Sharpe W. “Capital Asset Prices: a theory of market equilibrium under conditions of risk” Journal of Finance, 19, 1964.Stroyny A.L. “Estimating a combined linear model” in Knight and Satchell 2005 (“Linear Factor Models in Finance”)Willcox J. “Better Risk Management” Journal of Portfolio Management, Summer 2000. www.northinfo.com
  63. 63. References VIOsborne, Jason W. (2003). Effect sizes and the disattenuation of correlation and regression coefficients: lessons from educational psychology. Practical Assessment, Research & Evaluation, 8(11).Qian, Edward and Ronald Hua. “Active Risk and the Information Ratio”, Journal of Investment Management, Third Quarter 2004.Ramchand, L. and Susmel, R. (1998) ‘Volatility and Cross Correlation across major Stock Markets’, Journal of Empirical Finance, Vol. 5, No. 4, pp.397–416.Ratner, M. (1992) ‘Portfolio Diversification and the inter-temporal Stability of International Indices’, Global Finance Journal, Vol. 3, pp.67–78.Sheedy, E. (1997) ‘Is Correlation constant after all? (A Study of multivariate Risk Estimation for International Equities)’, working paper.Sneath PHA & Sokal RR (1973) Numerical Taxonomy. Freeman, San Francisco.Tang, G.: ‘Intertemporal Stability in International Stock Market Relationships: A Revisit’, The Quarterly Review of Economics and Finance, Vol. 35 (Special), pp.579–593.Sharpe W. F., "Morningstar’s Risk-adjusted Ratings", Financial Analysts Journal, July/August 1998, p. 21-33.Viterbi, A. (1967) ‘Error Bounds for convolutional Codes and an asymptotically Optimum Decoding Algorithm’, IEEE Transactions on Information Theory, Vol. 13, No. 2, pp.260– 269.Watanabe S (1960). Information theoretical analysis of multivariate correlation, IBM Journal of Research and Development 4, 66-82.Yule, 1926. G.U. Yule, Why do we sometimes get nonsense-correlations between time series?. Journal of the Royal Statistical Society 89 (1926), pp. 1–69. www.northinfo.com
  64. 64. References VIIQian, Edward and Ronald Hua, Active Risk and the Information Ratio, Journal of Investment Management, Third Quarter, 2004.Grinold, Richard C., The Fundamental Law of Active Management, Journal of Portfolio Management, Spring 1989.diBartolomeo, Dan, Measuring Investment Skill Using the Effective Information Coefficient, Journal of Performance Measurement, Fall 2008.diBartolomeo, Dan, 2006. Applications of Portfolio Variety, in “Forecasting Volatility” editors S. Satchell and J. Knight. Butterworth-Heineman.De Silva, Harindra, Steven Sapra and Steven Thorley. "Return Dispersion And Active Management," Financial Analyst Journal, 2001, v57(5,Sep/ Oct), 29-42.Ankrim, Ernest M. and Zhuanxin Ding. "Cross-Sectional Volatility And Return Dispersion," Financial Analyst Journal, 2002, v58(5,Sep/Oct), 67-73. www.northinfo.com
  65. 65. References VIIISolis, Rafael “Visualizing Stock Mutual Fund Relationships through Social Network Analysis”, Global Journal of Finance and Banking Issues Vol. 3 No. 3 2009Pascal, Michael C., de Weck, Olivier “Multilayer Network Model for Analysis and Management of Change Propagation” (working paper 2011)Luttrell, S.P. “Adaptive Cluster Expansion: A Multilayer Network for Estimating Probability Density Functions” (working paper 2010)Yenilmez, T, Saltoglu, B, “Analyzing Systemic Risk with Financial Networks during a Market Crash” (presentation March 10th 2011) www.northinfo.com

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