Risk Model Methodologies
- 1. Risk Model Methodologies Nick Wade Northfield Information Services, Inc. October/November 2005
- 2. Motivation 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!)
- 3. 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
- 4. Overview Review of the Linear Model Review of the assumptions in the Linear Model Implications of those Review of common approaches to estimating a risk model Review of the assumptions implicit in those techniques Implications of those assumptions Review of “adjustments” to mitigate the effect of these assumptions Other thoughts on multi-factor models
- 5. The Linear Model 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)
- 6. Implications so Far What if the relationship is not linear? What if the Factor Returns or their Variances are not stationary? What if the Factor Returns or Security Returns are not Normal, or i.i.d.? What if there is correlation between so-called “stock specific” return sources across securities? Any of these will result in Model Error
- 7. Evidence Security Returns are not Normal, stationary, i.i.d. Lots of evidence… start with Mandelbrot (1963) Factor Returns are not Normal, Stationary, iid. We see bubbles, trends, styles, alpha, momentum Pope and Yadav (1994) Completeness – what is N? Linearity? Security Return Volatility is time-varying We need to adjust model to accommodate broken assumptions
- 8. Estimating a Risk Model 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.
- 9. Practical Approaches There are three common approaches: Observe factors F it and determine E i by time-series approach Observe E it and determine F it by cross-sectional approach Assume N and use statistical approach to determine E i , then estimate F i by regression
- 10. Exogenous Model The Exogenous or Macro model seeks to estimate E i from F it . Typical factors include Market, Sector, Oil, Interest-Rates… Ross (1976) Chen (1986) Model is pre-specified
- 11. Endogenous Model The Endogenous or Fundamental Model seeks to estimate F it assuming E it by regression. Typical factors include E/P, D/E, Industry membership, Country membership… King (1966) Rosenberg and Guy (1975) etc. Model is pre-specified
- 12. Statistical Model Assume N Use Factor Analysis or Principle Components to estimate Ei Use Regression to estimate Fit EIV
- 13. Assumptions with Endogenous Accounting Statements are true Accounting standards are comparable (region etc) Factors are known and their number unchanging (Typically) Each security in an industry or market responds the same way to changes in the industry or market to which they belong (Marsh & Pfleider 1997), and perhaps further that each security responds the same way to changes in the other industries and markets (Scowcroft & Sefton 2001)
- 14. Assumptions with Exogenous Exposure to factors is stationary – is it? Factors are known and number is unchanging … but we are free of accounting statements… … and we are explicitly allowing the response of each security to changes in market/ sector / industry / whatever to be different across securities…
- 15. Assumptions with Statistical There are N factors in the data All correlation is information Factor returns are estimates based on estimates (EIV) Unique only “up to a rotation”
- 16. Estimation Issues: Exogenous Errors will be in estimation of loadings (exposures) Bad for concentrated funds Other errors will be in non-linearity effects Other errors will be in non-stationarity effects Missing or spurious factors
- 17. Estimation Issues: Endogenous Errors will be in factor returns and hence in covariance matrix and hence not diversifiable: bad for diversified funds Other errors will be in heterogeneous reaction to changes in own or other market / industry Other errors will be in non-linearity effects Other errors from non-stationarity effects Account statements may be flawed – errors in loadings Missing or Spurious factors
- 18. Estimation Issues with Statistical All correlation is assumed to be information Factor returns are estimated based on estimated factor loadings, compounding error (Errors in Variables!) Errors from non-linearity Errors from non-stationarity – de-trend with ARIMA: Northfield (1997) Association back to real world effects can be difficult, misleading… unique only up to a rotation… Number of factors is pre-specified or sample-specific if derived from data. Issues with noise in data Issues with technique: fitting variance or correlation?
- 19. Errors in Variables Happens any time the inputs to derive the quantity we are trying to estimate are themselves based on estimates: Fama and MacBeth (1973) Lintzenberger & Ramaswamy (1979) Shanken (1982)
- 20. 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)
- 21. Hybrid Model Combine macro, micro, and statistical 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
- 22. 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.
- 23. Effect of Model Errors Non-linearity leads to over/under estimation for different constituencies – note “blind factors” Non-stationary market/factor variance leads to over/under estimation as model struggles to react Non-stationary residual variance leads to over/under estimation as model struggles to react Missing factors lead to under estimation Spurious factors add noise
- 24. Other Model Issues Time period – historical data (Scowcroft & Sefton) Frequency – daily, weekly, monthly Forecast Horizon – Rosenberg and Guy (1975) Data – clean, reliable, undisputed, comparable, timely…
- 25. Adjustments for Error More later in the day about coping with estimation error, and optimizing with it… Non-linearity, non-stationarity adjustments Momentum: Pope and Yadav (1994) Hwang & Satchell (2001) Heteroskedasticity
- 26. Non-Stationarity Adjustments (1) Non-stationary factor return series will lead to the model underestimating portfolio risk Adjust by changing variance calculation to include trend component of return Adjust Model for the influence of non-stationary factor returns
- 27. Non-Stationarity Adjustments (2) OK, so that covers factors… what about residuals? We observe: Serial correlation (not i.i.d.) Bid-ask bounce Non-Normal distributions Parkinson volatility Adjust Model for the influence of non-stationary security returns
- 28. Heteroskedasticity We also observe that factor volatility clusters, rises, and falls. Incidentally, we also see correlations changing over time. Oh boy. Adjust for this by exponentially weighting the return information, or by GARCH, or by using the implied volatility from option market: Northfield (1997) Short Term Model Scowcroft (2005)
- 29. Pause for Thought What do we really care about Securities or Portfolios? Historical attribution or Forecasting? Hedging a desk? Active long-only fund?
- 30. Historical Attribution or Forecasting “…estimates of systematic risk that are ideal for historical evaluation are far from ideal for predictive purposes.” Rosenberg (1975)
- 31. Objectives should drive Model Horizon – long and short-term forecasts? Approach – do we need to attribute risk? Alpha versus Beta – so which is it? Two moments, or four?
- 32. Forecast Horizon It should be obvious, but probably isn’t, that our forecast horizon for our risk model should match our forecast horizon for our alphas, and implicitly therefore our holding period. Rosenberg & Guy (1975)
- 33. What happens if it doesn’t? TE bias: TE ex-ante will be << TE ex-post Lawton-Browne (2000) Hwang and Satchell If portfolio and/or benchmark weights are “ex-post stochastic” (dorky way of saying “they change”) then TE ex-a MUST be less than TE ex-p. Question is… by how much…
- 34. Is it Alpha or Beta? There are N common factors, and M stock-specific factors Manager’s search for excess return takes on many forms – but they all translate eventually into two things “buy these securities that have attributes I like” and “buy these securities because I have unanticipated information of some form” . Therefore a lot of the time, managers qualitative or quantitative multi-factor model is forecasting a set of factors for each stock (let’s call them K) that consist of a shared component and a stock-specific component.
- 35. Alpha or Beta? (2) This has an impact on how we make portfolios, and what our expectations should be about the risk model diBartolomeo (1998) MacQueen (2005) Condition alphas on factors? Adjust factor and residual variances to take account of our forecasting ability? (Nobody does this… do they?) Bulsing, Scowcroft and Sefton (200x?) Pfleiderer (2005). You need to be damn good for it to make any odds…
- 36. Alpha, Beta, and Portfolio Construction If your alpha has nothing to do with our observable common factors, but a common theme, then it must appear in our blind factors . Note that if you do not establish what portion of your alpha model factors overlaps with the risk model, and constrain systematic effects tightly, this will bias your portfolio, leading it to load up on those securities with the best score only on those alpha factors not shared by the risk model . This defeats the purpose of a multi-factor alpha model…
- 37. Forecasting: Conditioning Alpha One could do a regression of alpha versus factor exposures to extract the forecastable part of the factor return, and thence estimate the truly “security specific” part of the alpha. This would remove the bias from the alphas. There are firms doing this.
- 38. Portfolio Construction Now we stick the glorious results of our estimation into a mean-variance optimizer, and off we go… Straight into an avalanche of new assumptions! Are two moments enough? Wilcox (2000) Markowitz (1959)
- 39. Country, 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)
- 40. Country, Industry, Sector, Region…(2) Our three common approaches treat the problem as follows: Exogenous: build e.g. country return index, then estimate security exposure to that index by regression Endogenous: assign a membership variable (typically 1 or 0) for each security, then do a regression to estimate factor return Statistical: we don’t have country etc explicitly. No problem! Problem: large multinational companies Markets are becoming more dominated by large-cap firms, which tend to be multinational, and hence which are a very poor proxy for truly “domestic” events Exogenous: problem with index construction Endogenous: problem with heterogeneous response 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).
- 41. 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)
- 42. The Result “the result need not be equal to the expected value” Rosenberg (1975)
- 43. Conclusions We use the linear model because it is tractable and it fits well into MV optimization. And, to be fair, because most of the time it is an excellent approximation But the assumptions are legion, and need to be clearly stated and properly understood
- 44. References 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
- 45. References II Lintzenberger 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, 2005 MacQueen 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” 1 st 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 Model Northfield Single-Market Risk Model (Hybrid Risk Model) Pfleiderer, Paul “Alternative Equity Risk Models: The Impact on Portfolio Decisions” The 15 th Annual Investment Seminar UBS/Quantal, Cambridge UK 2002.
- 46. References III Pohlson 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.