Risk Model MethodologiesRisk Model Methodologies

Nick Wade Nick Wade

Northfield Information Services, Northfield Information Services, Inc.Inc.

October/November 2005October/November 2005

MotivationMotivation

• While the linear model is prevalent in While the linear model is prevalent in finance, many of its assumptions are finance, many of its assumptions are not.not.

• While multi-factor risk models are While multi-factor risk models are similarly widely used, the similarly widely used, the assumptions behind their estimation assumptions behind their estimation are likewise not well known (or not are likewise not well known (or not well publicized!)well publicized!)

More Motivation – A Recent More Motivation – A Recent QuoteQuote

• GARP: “GARP: “in the past few months in the past few months volatility has dropped significantly; volatility has dropped significantly; almost to the point where it is below almost to the point where it is below the BARRA estimatesthe BARRA estimates””

GARP Risk Review Issue 16 Jan/Feb 2004

OverviewOverview• Review of the Linear ModelReview of the Linear Model• Review of the assumptions in the Linear ModelReview of the assumptions in the Linear Model• Implications of thoseImplications of those• Review of common approaches to estimating a Review of common approaches to estimating a

risk modelrisk model• Review of the assumptions implicit in those Review of the assumptions implicit in those

techniquestechniques• Implications of those assumptionsImplications of those assumptions• Review of “adjustments” to mitigate the effect of Review of “adjustments” to mitigate the effect of

these assumptionsthese assumptions• Other thoughts on multi-factor modelsOther thoughts on multi-factor models

The Linear ModelThe Linear Model

• Relationship between R and F is linear Relationship between R and F is linear ∀F∀F• There are N common factor sources of returnThere are N common factor sources of return• Relationship Relationship between R and H is linear between R and H is linear ∀H∀H• There is no correlation between F and H There is no correlation between F and H ∀∀ F,H F,H• The distribution of F is stationary, Normal, i.i.d. The distribution of F is stationary, Normal, i.i.d. ∀F∀F• There are M stock-specific sources of returnThere are M stock-specific sources of return• There is no correlation between H across stocksThere is no correlation between H across stocks• The distribution of H is stationary, Normal, i.i.d. The distribution of H is stationary, Normal, i.i.d. ∀H∀H• (Implicitly also the volatility of R and F is stationary)(Implicitly also the volatility of R and F is stationary)

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Implications so FarImplications so Far

• What if the relationship is not linear?What if the relationship is not linear?• What if the Factor Returns or their What if the Factor Returns or their

Variances are not stationary?Variances are not stationary?• What if the Factor Returns or Security What if the Factor Returns or Security

Returns are not Normal, or i.i.d.?Returns are not Normal, or i.i.d.?• What if there is correlation between What if there is correlation between

so-called “stock specific” return so-called “stock specific” return sources across securities?sources across securities?

Any of these will result in Model Error

EvidenceEvidence• Security Returns are not Normal, stationary, Security Returns are not Normal, stationary,

i.i.d.i.i.d.– Lots of evidence… start with Mandelbrot (1963)Lots of evidence… start with Mandelbrot (1963)

• Factor Returns are not Normal, Stationary, iid.Factor Returns are not Normal, Stationary, iid.– We see bubbles, trends, styles, alpha, momentumWe see bubbles, trends, styles, alpha, momentum– Pope and Yadav (1994)Pope and Yadav (1994)

• Completeness – what is N?Completeness – what is N?• Linearity?Linearity?• Security Return Volatility is time-varyingSecurity Return Volatility is time-varying

We need to adjust model to accommodate broken assumptions

Estimating a Risk ModelEstimating a Risk Model

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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.

Practical ApproachesPractical Approaches

• There are three common There are three common approaches:approaches:– Observe factors FObserve factors Fitit and and

determine Edetermine Eii by time-series by time-series approachapproach

– Observe EObserve Eitit and determine F and determine Fitit by cross-sectional approachby cross-sectional approach

– Assume N and use statistical Assume N and use statistical approach to determine Eapproach to determine Eii, , then estimate Fthen estimate Fii by by regressionregression

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Exogenous ModelExogenous Model

• The Exogenous or Macro model The Exogenous or Macro model seeks to estimate Eseeks to estimate Eii from F from Fitit..

• Typical factors include Market, Typical factors include Market, Sector, Oil, Interest-Rates…Sector, Oil, Interest-Rates…– Ross (1976)Ross (1976)– Chen (1986)Chen (1986)

• Model is pre-specifiedModel is pre-specified

Endogenous ModelEndogenous Model

• The Endogenous or Fundamental The Endogenous or Fundamental Model seeks to estimate FModel seeks to estimate Fitit assuming assuming EEitit by regression. by regression.

• Typical factors include E/P, D/E, Typical factors include E/P, D/E, Industry membership, Country Industry membership, Country membership…membership…– King (1966)King (1966)– Rosenberg and Guy (1975) etc.Rosenberg and Guy (1975) etc.

• Model is pre-specifiedModel is pre-specified

Statistical ModelStatistical Model

• Assume NAssume N

• Use Factor Analysis or Principle Use Factor Analysis or Principle Components to estimate EiComponents to estimate Ei

• Use Regression to estimate FitUse Regression to estimate Fit

• EIVEIV

Assumptions with Assumptions with EndogenousEndogenous

• Accounting Statements are trueAccounting Statements are true• Accounting standards are comparable (region etc)Accounting standards are comparable (region etc)• Factors are known and their number unchangingFactors are known and their number unchanging• (Typically) Each security in an industry or market responds (Typically) Each security in an industry or market responds

the same way to changes in the industry or market to the same way to changes in the industry or market to which they belong (Marsh & Pfleider 1997), which they belong (Marsh & Pfleider 1997), andand perhaps perhaps further that each security responds the same way to further that each security responds the same way to changes in the changes in the otherother industries and markets (Scowcroft & industries and markets (Scowcroft & Sefton 2001) Sefton 2001)

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Assumptions with Assumptions with ExogenousExogenous

• Exposure to factors is stationary – is it?Exposure to factors is stationary – is it?• Factors are known and number is unchangingFactors are known and number is unchanging• ……but we are free of accounting statements…but we are free of accounting statements…• ……and we are explicitly allowing the response and we are explicitly allowing the response

of each security to changes in market/ of each security to changes in market/ sector / industry / whatever to be sector / industry / whatever to be differentdifferent across securities…across securities…

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Assumptions with StatisticalAssumptions with Statistical

• There are N factors in the dataThere are N factors in the data

• All correlation is informationAll correlation is information

• Factor returns are estimates based Factor returns are estimates based on estimates (EIV)on estimates (EIV)

• Unique only “up to a rotation”Unique only “up to a rotation”

Estimation Issues: Estimation Issues: ExogenousExogenous• Errors will be in estimation of loadings Errors will be in estimation of loadings

(exposures)(exposures)– Bad for concentrated fundsBad for concentrated funds

• Other errors will be in non-linearity Other errors will be in non-linearity effectseffects

• Other errors will be in non-stationarity Other errors will be in non-stationarity effectseffects

• Missing or spurious factorsMissing or spurious factors

Estimation Issues: Estimation Issues: EndogenousEndogenous• Errors will be in factor returnsErrors will be in factor returns

– and hence in covariance matrixand hence in covariance matrix– and hence not diversifiable: bad for diversified and hence not diversifiable: bad for diversified

fundsfunds• Other errors will be in heterogeneous reaction Other errors will be in heterogeneous reaction

to changes in own or other market / industryto changes in own or other market / industry• Other errors will be in non-linearity effectsOther errors will be in non-linearity effects• Other errors from non-stationarity effectsOther errors from non-stationarity effects• Account statements may be flawed – errors in Account statements may be flawed – errors in

loadingsloadings• Missing or Spurious factorsMissing or Spurious factors

Estimation Issues with Estimation Issues with StatisticalStatistical• All correlation is assumed to be informationAll correlation is assumed to be information• Factor returns are estimated based on estimated Factor returns are estimated based on estimated

factor loadings, compounding error (Errors in factor loadings, compounding error (Errors in Variables!)Variables!)

• Errors from non-linearityErrors from non-linearity• Errors from non-stationarity – de-trend with ARIMA: Errors from non-stationarity – de-trend with ARIMA:

Northfield (1997)Northfield (1997)• Association back to real world effects can be Association back to real world effects can be

difficult, misleading… unique only up to a rotation…difficult, misleading… unique only up to a rotation…• Number of factors is pre-specified or sample-specific Number of factors is pre-specified or sample-specific

if derived from data.if derived from data.• Issues with noise in dataIssues with noise in data• Issues with technique: fitting variance or correlation?Issues with technique: fitting variance or correlation?

Errors in VariablesErrors in Variables

• Happens any time the inputs to Happens any time the inputs to derive the quantity we are trying to derive the quantity we are trying to estimate are themselves based on estimate are themselves based on estimates:estimates:– Fama and MacBeth (1973)Fama and MacBeth (1973)– Lintzenberger & Ramaswamy (1979)Lintzenberger & Ramaswamy (1979)– Shanken (1982)Shanken (1982)

Other ApproachesOther Approaches

• Combined Models:Combined Models:– Northfield Hybrid ModelNorthfield Hybrid Model– Stroyny (2001)Stroyny (2001)

• Simultaneous EstimationSimultaneous Estimation– Black et al (1972)Black et al (1972)– Heston and Rouwenhorst (1994, 1995)Heston and Rouwenhorst (1994, 1995)– Satchell and Scowcroft (2001)Satchell and Scowcroft (2001)– GMM Hansen (1982)GMM Hansen (1982)– McElroy and Burmeister (1988) using NLSUR (which is McElroy and Burmeister (1988) using NLSUR (which is

assymptotically equivalent to ML)assymptotically equivalent to ML)• Bayesian Approach:Bayesian Approach:

– Pohlson and Tew (2000)Pohlson and Tew (2000)– Ericsson and Karlsson (2002)Ericsson and Karlsson (2002)

Hybrid ModelHybrid Model

• Combine macro, micro, and statistical Combine macro, micro, and statistical factorsfactors

• Gain the advantages of each, whilst Gain the advantages of each, whilst mitigating the limitations of eachmitigating the limitations of each– Intuitive, explainable, justifiable observable Intuitive, explainable, justifiable observable

factorsfactors– Minimal dependence on accounting informationMinimal dependence on accounting information– Rapid inclusion of new or transient factorsRapid inclusion of new or transient factors

Simultaneous EstimationSimultaneous Estimation• Removing the limitation of binary or membership variables Removing the limitation of binary or membership variables

(such as industry, country, sector, region etc).(such as industry, country, sector, region etc).– Marsh and Pfleiderer (1997)Marsh and Pfleiderer (1997)– Scowcroft and Satchell (2001)Scowcroft and Satchell (2001)

• Start with an estimate of the exposures (e.g. 1.00 for all Start with an estimate of the exposures (e.g. 1.00 for all companies) use that estimate to solve for the factor return, companies) use that estimate to solve for the factor return, then use that factor return in turn to then use that factor return in turn to re-solvere-solve for a revised for a revised set of exposures, thus converging iteratively on a better set of exposures, thus converging iteratively on a better solution for both Eit and Fit.solution for both Eit and Fit.– Black et al (1972)Black et al (1972)– Heston and Rouwenhorst (1994, 1995)Heston and Rouwenhorst (1994, 1995)– Scowcroft and Satchell (2001)Scowcroft and Satchell (2001)

• Given various limiting restrictions we can ensure that the Given various limiting restrictions we can ensure that the model converges and that it is unique. model converges and that it is unique.

Effect of Model ErrorsEffect of Model Errors

• Non-linearity leads to over/under Non-linearity leads to over/under estimation for different constituencies – estimation for different constituencies – note “blind factors”note “blind factors”

• Non-stationary market/factor variance Non-stationary market/factor variance leads to over/under estimation as model leads to over/under estimation as model struggles to reactstruggles to react

• Non-stationary residual variance leads to Non-stationary residual variance leads to over/under estimation as model struggles over/under estimation as model struggles to reactto react

• Missing factors lead to under estimationMissing factors lead to under estimation• Spurious factors add noiseSpurious factors add noise

Other Model IssuesOther Model Issues

• Time period – historical data Time period – historical data (Scowcroft & Sefton)(Scowcroft & Sefton)

• Frequency – daily, weekly, monthly Frequency – daily, weekly, monthly

• Forecast Horizon – Rosenberg and Forecast Horizon – Rosenberg and Guy (1975)Guy (1975)

• Data – clean, reliable, undisputed, Data – clean, reliable, undisputed, comparable, timely…comparable, timely…

Adjustments for ErrorAdjustments for Error

• More later in the day about coping More later in the day about coping with estimation error, and optimizing with estimation error, and optimizing with it…with it…

• Non-linearity, non-stationarity Non-linearity, non-stationarity adjustmentsadjustments– Momentum: Pope and Yadav (1994)Momentum: Pope and Yadav (1994)– Hwang & Satchell (2001)Hwang & Satchell (2001)

• HeteroskedasticityHeteroskedasticity

Non-Stationarity Adjustments Non-Stationarity Adjustments (1)(1)

• Non-stationary Non-stationary factor return series factor return series will lead to the will lead to the model model underestimating underestimating portfolio riskportfolio risk

• Adjust by changing Adjust by changing variance calculation variance calculation to include trend to include trend component of returncomponent of return

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Non-Stationarity Adjustments Non-Stationarity Adjustments (2)(2)

• OK, so that covers factors… what OK, so that covers factors… what about residuals?about residuals?

• We observe:We observe:– Serial correlation (not i.i.d.)Serial correlation (not i.i.d.)– Bid-ask bounceBid-ask bounce– Non-Normal distributionsNon-Normal distributions

• Parkinson volatilityParkinson volatilityAdjust Model for the influence of non-stationary security returns

HeteroskedasticityHeteroskedasticity

• We also observe that factor volatility We also observe that factor volatility clusters, rises, and falls.clusters, rises, and falls.

• Incidentally, we also see correlations Incidentally, we also see correlations changing over time. Oh boy.changing over time. Oh boy.

• Adjust for this by exponentially weighting Adjust for this by exponentially weighting the return information, or by GARCH, or by the return information, or by GARCH, or by using the implied volatility from option using the implied volatility from option market:market:– Northfield (1997) Short Term ModelNorthfield (1997) Short Term Model– Scowcroft (2005) Scowcroft (2005)

Pause for ThoughtPause for Thought

• What do we really care aboutWhat do we really care about– Securities or Portfolios?Securities or Portfolios?– Historical attribution or Forecasting?Historical attribution or Forecasting?– Hedging a desk? Active long-only fund?Hedging a desk? Active long-only fund?

Historical Attribution or Historical Attribution or ForecastingForecasting

• “…“…estimates of systematic risk that estimates of systematic risk that are ideal for historical evaluation are are ideal for historical evaluation are far from ideal for predictive far from ideal for predictive purposes.” Rosenberg (1975)purposes.” Rosenberg (1975)

Objectives should drive Objectives should drive ModelModel

• Horizon – long and short-term Horizon – long and short-term forecasts?forecasts?

• Approach – do we need to attribute Approach – do we need to attribute risk?risk?

• Alpha versus Beta – so which is it?Alpha versus Beta – so which is it?

• Two moments, or four?Two moments, or four?

Forecast HorizonForecast Horizon

• It should be obvious, but probably It should be obvious, but probably isn’t, that our forecast horizon for our isn’t, that our forecast horizon for our risk model should match our forecast risk model should match our forecast horizon for our alphas, and implicitly horizon for our alphas, and implicitly therefore our holding period.therefore our holding period.– Rosenberg & Guy (1975)Rosenberg & Guy (1975)

What happens if it doesn’t?What happens if it doesn’t?

• TE bias: TE ex-ante will be << TE ex-TE bias: TE ex-ante will be << TE ex-postpost– Lawton-Browne (2000)Lawton-Browne (2000)– Hwang and Satchell Hwang and Satchell

• If portfolio and/or benchmark weights If portfolio and/or benchmark weights are “ex-post stochastic” (dorky way are “ex-post stochastic” (dorky way of saying “they change”) then TE ex-of saying “they change”) then TE ex-a MUST be less than TE ex-p. a MUST be less than TE ex-p. Question is… by how much…Question is… by how much…

Is it Alpha or Beta?Is it Alpha or Beta?

• There are N common factors, and M stock-specific factorsThere are N common factors, and M stock-specific factors• Manager’s search for excess return takes on many forms – Manager’s search for excess return takes on many forms –

but they all translate eventually into two things but they all translate eventually into two things “buy these “buy these securities that have attributes I like”securities that have attributes I like” and and “buy these “buy these securities because I have unanticipated information of securities because I have unanticipated information of some form”some form”. .

• Therefore a lot of the time, managers qualitative or Therefore a lot of the time, managers qualitative or quantitative multi-factor model is forecasting a set of quantitative multi-factor model is forecasting a set of factors for each stock (let’s call them K) that consist of a factors for each stock (let’s call them K) that consist of a shared component and a stock-specific component.shared component and a stock-specific component.

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Alpha or Beta? (2)Alpha or Beta? (2)

• This has an impact on how we make This has an impact on how we make portfolios, and what our expectations portfolios, and what our expectations should be about the risk modelshould be about the risk model– diBartolomeo (1998)diBartolomeo (1998)– MacQueen (2005)MacQueen (2005)

• Condition alphas on factors?Condition alphas on factors?– Adjust factor and residual variances to take Adjust factor and residual variances to take

account of our forecasting ability? (Nobody account of our forecasting ability? (Nobody does this… do they?) Bulsing, Scowcroft and does this… do they?) Bulsing, Scowcroft and Sefton (200x?)Sefton (200x?)

– Pfleiderer (2005). You need to be damn good Pfleiderer (2005). You need to be damn good for it to make any odds…for it to make any odds…

Alpha, Beta, and Portfolio Alpha, Beta, and Portfolio ConstructionConstruction• If your alpha has nothing to do with our observable If your alpha has nothing to do with our observable

common factors, but a common theme, then it common factors, but a common theme, then it must appear in our must appear in our blind factorsblind factors..

• Note that if you do not establish what portion of Note that if you do not establish what portion of your alpha model factors overlaps with the risk your alpha model factors overlaps with the risk model, and constrain systematic effects tightly, model, and constrain systematic effects tightly, this will bias your portfolio, this will bias your portfolio, leading it to load up on leading it to load up on those securities with the best score only on those those securities with the best score only on those alpha factors not shared by the risk modelalpha factors not shared by the risk model. This . This defeats the purpose of a multi-factor alpha defeats the purpose of a multi-factor alpha model…model…

Forecasting: Conditioning Forecasting: Conditioning AlphaAlpha• One could do a regression of alpha One could do a regression of alpha

versus factor exposures to extract versus factor exposures to extract the forecastable part of the factor the forecastable part of the factor return, and thence estimate the truly return, and thence estimate the truly “security specific” part of the alpha.“security specific” part of the alpha.

• This would remove the bias from the This would remove the bias from the alphas.alphas.

• There are firms doing this.There are firms doing this.

Portfolio ConstructionPortfolio Construction

• Now we stick the glorious results of Now we stick the glorious results of our estimation into a mean-variance our estimation into a mean-variance optimizer, and off we go…optimizer, and off we go…

• Straight into an avalanche of new Straight into an avalanche of new assumptions!assumptions!– Are two moments enough? Wilcox Are two moments enough? Wilcox

(2000)(2000)– Markowitz (1959)Markowitz (1959)

Country, Industry, Sector, Country, Industry, Sector, Region…Region…• A useful (I hope) digression into the world of A useful (I hope) digression into the world of

factor selection.factor selection.• It is pretty much standard practice to take note of It is pretty much standard practice to take note of

membership in, or exposure to, one or more membership in, or exposure to, one or more countries or regions, and one or more industries countries or regions, and one or more industries or sectorsor sectors

• Problems: multinational firms, globalization, index Problems: multinational firms, globalization, index dominationdomination– Heston and Rouwenhorst (1994, 1995)Heston and Rouwenhorst (1994, 1995)– Scowcroft and Sefton (2001)Scowcroft and Sefton (2001)– Diermeier and Solnik (2000)Diermeier and Solnik (2000)– MacQueen and Satchell (2001)MacQueen and Satchell (2001)

Country, Industry, Sector, Country, Industry, Sector, Region…(2)Region…(2)• Our three common approaches treat the problem as follows:Our three common approaches treat the problem as follows:

– Exogenous: build e.g. country return index, then estimate security exposure to that index by Exogenous: build e.g. country return index, then estimate security exposure to that index by regressionregression

– Endogenous: assign a membership variable (typically 1 or 0) for each security, then do a Endogenous: assign a membership variable (typically 1 or 0) for each security, then do a regression to estimate factor returnregression to estimate factor return

– Statistical: we don’t have country etc explicitly. No problem! Statistical: we don’t have country etc explicitly. No problem!

• Problem: large multinational companiesProblem: large multinational companies• Markets are becoming more dominated by large-cap firms, which tend to be 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” eventsmultinational, and hence which are a very poor proxy for truly “domestic” events– Exogenous: problem with index constructionExogenous: problem with index construction– Endogenous: problem with heterogeneous responseEndogenous: problem with heterogeneous response

• Suggestions:Suggestions:– Estimate a different kind of index FTSE (1999), Bacon and Woodrow (1999)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 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 foreign sales (Diermeier and Solnik 2000) or by some statistical process (MacQueen and Satchell 2001)Satchell 2001)

– Solve Model iteratively using Heston and Rouwenhorst (1994, 1995) approachSolve Model iteratively using Heston and Rouwenhorst (1994, 1995) approach– Or extensions to that: Scowcroft and Sefton (2001).Or extensions to that: Scowcroft and Sefton (2001).

How many factors…?How many factors…?

• The academic consensus seems to The academic consensus seems to be that there is not much difference be that there is not much difference going from 5 to 10 to 15 factors. In going from 5 to 10 to 15 factors. In other words, 5 do the job.other words, 5 do the job.– Lehmann & Modest (1988)Lehmann & Modest (1988)– Connor & Korajczyk (1988)Connor & Korajczyk (1988)– Roll & Ross (1980)Roll & Ross (1980)

The ResultThe Result

• ““the result need not be equal to the the result need not be equal to the expected value” Rosenberg (1975)expected value” Rosenberg (1975)

ConclusionsConclusions

• We use the linear model because it is We use the linear model because it is tractable and it fits well into MV tractable and it fits well into MV optimization.optimization.

• And, to be fair, because most of the time it And, to be fair, because most of the time it is an excellent approximationis an excellent approximation

• But the assumptions are legion, and need But the assumptions are legion, and need to be clearly stated and properly to be clearly stated and properly understoodunderstood

ReferencesReferences• Black F., Jensen M., Scholes M. “The Capital Asset Pricing Model: some empirical tests” In Jensen Black F., Jensen M., Scholes M. “The Capital Asset Pricing Model: some empirical tests” In Jensen

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combining both analyst and strategy forecasts” UBS Working Paper, 2003.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.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 Connor G and Korajczyck R.A. “Risk and Return in an equilibrium APT: application of a new test

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• Northfield Short Term Equity Risk ModelNorthfield Short Term Equity Risk Model• Northfield Single-Market Risk Model (Hybrid Risk Model) Northfield Single-Market Risk Model (Hybrid Risk Model) • Pfleiderer, Paul “Alternative Equity Risk Models: The Impact on Portfolio Pfleiderer, Paul “Alternative Equity Risk Models: The Impact on Portfolio

Decisions” The 15Decisions” The 15thth Annual Investment Seminar UBS/Quantal, Cambridge UK Annual Investment Seminar UBS/Quantal, Cambridge UK 2002.2002.

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