Time Series Variation in Risk Time Series Variation in Risk Levels and What To Do About ItLevels and What To Do About It

Ghazanfer BaigGhazanfer BaigJune 2006June 2006

1212thth Annual Summer Seminar, NewportAnnual Summer Seminar, Newport

OutlineOutline•• When investors usually think of market When investors usually think of market volatilityvolatility, they , they

are concerned with the times series variation in the are concerned with the times series variation in the returns to securities or market indicesreturns to securities or market indices

•• With respect to active management, of equal or greater With respect to active management, of equal or greater importance is the crossimportance is the cross--sectional dispersion, or sectional dispersion, or variety variety of of returns. returns.

•• Consider the historical trends in variety for a number of Consider the historical trends in variety for a number of major equity marketsmajor equity markets

•• Introduce a simple use of variety statistics that Introduce a simple use of variety statistics that meaningfully improves risk forecastsmeaningfully improves risk forecasts

•• Review the implications of the times series of changes in Review the implications of the times series of changes in variety on various styles of equity managementvariety on various styles of equity management

The Origins of the Variety ConceptThe Origins of the Variety Concept•• SolnikSolnik and and RouletRoulet (2000) examine the dispersion of (2000) examine the dispersion of

country returns as a way of estimating correlations country returns as a way of estimating correlations between marketsbetween markets

•• LiloLilo, , MantegnaMantegna, Bouchard and Potters use the term , Bouchard and Potters use the term VarietyVariety to describe crossto describe cross--sectional dispersion of stock sectional dispersion of stock returnsreturns–– They also define the crossThey also define the cross--sectional dispersion of CAPM alpha as sectional dispersion of CAPM alpha as

idiosyncratic varietyidiosyncratic variety (noted as v(t))(noted as v(t))–– They find that the average correlation between stocks is They find that the average correlation between stocks is

approximately: approximately:

C(t) = 1 / [1 + (vC(t) = 1 / [1 + (v22(t)/r(t)/rmm22(t) ](t) ]

•• diBartolomeo (2000) relates periods of high crossdiBartolomeo (2000) relates periods of high cross--sectional dispersion to positive serial correlation in stock sectional dispersion to positive serial correlation in stock returns (i.e. momentum strategies working)returns (i.e. momentum strategies working)

Global Equities Idiosyncratic Variety Global Equities Idiosyncratic Variety of Local Currency Returnsof Local Currency Returns

Global

0

5

10

15

20

25

30

Jan-

90

Jan-

91

Jan-

92

Jan-

93

Jan-

94

Jan-

95

Jan-

96

Jan-

97

Jan-

98

Jan-

99

Jan-

00

Jan-

01

Jan-

02

Jan-

03

Jan-

04

Jan-

05

Europe Idiosyncratic Variety of Europe Idiosyncratic Variety of Local Currency ReturnsLocal Currency Returns

Europe

0

5

10

15

20

25

Mon

thE

nd

11/3

0/19

90

10/3

1/19

91

9/30

/199

2

8/31

/199

3

7/31

/199

4

6/30

/199

5

5/31

/199

6

4/30

/199

7

3/31

/199

8

2/28

/199

9

1/31

/200

0

12/3

1/20

00

11/3

0/20

01

10/3

1/20

02

9/30

/200

3

8/31

/200

4

7/31

/200

5

UK Idiosyncratic VarietyUK Idiosyncratic Variety

UK

0

5

10

15

20

Jan-

90

Jan-

91

Jan-

92

Jan-

93

Jan-

94

Jan-

95

Jan-

96

Jan-

97

Jan-

98

Jan-

99

Jan-

00

Jan-

01

Jan-

02

Jan-

03

Jan-

04

Jan-

05

US Idiosyncratic VarietyUS Idiosyncratic Variety

US_Single

0

5

10

15

20

25

30

Jan-

90

Jan-

91

Jan-

92

Jan-

93

Jan-

94

Jan-

95

Jan-

96

Jan-

97

Jan-

98

Jan-

99

Jan-

00

Jan-

01

Jan-

02

Jan-

03

Jan-

04

Jan-

05

A Simple Improvement to Risk A Simple Improvement to Risk ForecastingForecasting

•• If there was no crossIf there was no cross--sectional dispersion of sectional dispersion of stock returns, all portfolios would have zero stock returns, all portfolios would have zero tracking errortracking error

•• So lets condition our forecast tracking errors on So lets condition our forecast tracking errors on the level of the level of idiosyncratic varietyidiosyncratic variety

•• Lets try a simple moving average adjustment:Lets try a simple moving average adjustment:

E[TE (adjusted)] = TE * MA[IV,12] / MA [IV,60]E[TE (adjusted)] = TE * MA[IV,12] / MA [IV,60]

Europe Tracking Error BiasEurope Tracking Error Bias

Risk Numbers

0

1

2

3

4

5

6

Jan-9

9Mar-

99May-

99Ju

l-99

Sep-9

9

Nov-99

Jan-0

0Mar-

00

May-00

Jul-0

0

Sep-0

0Nov-

00Ja

n-01

Mar-01

May-01

Jul-0

1

Sep-0

1Nov-

01Ja

n-02

Mar-02

May-02

Jul-0

2

Sep-0

2

Nov-02

Jan-0

3

EstTrkErr RlzTrkErr EstTrkErrAdj

UK Tracking Error BiasUK Tracking Error Bias

Risk Numbers

0

0.5

1

1.5

2

2.5

3

3.5

4

Jan-9

0May-

90

Sep-90

Jan-9

1May-

91

Sep-91

Jan-9

2May

-92

Sep-92

Jan-9

3May-

93

Sep-93

Jan-9

4

May-94

Sep-94

Jan-95

May-95

Sep-95

Jan-9

6

May-96

Sep-96

Jan-97

May-97

Sep-97

Jan-9

8

May-98

Sep-98

Jan-9

9May

-99

Sep-99

Jan-0

0May-

00

Sep-00

Jan-0

1

May-01

Sep-01

Jan-02

May-02

Sep-02

Jan-0

3

EstTrkErr RlzTrkErr EstTrkErrAdj

Global Model Tracking Error BiasGlobal Model Tracking Error Bias

Risk Numbers

0

2

4

6

8

10

12

Jan-9

0

May-90

Sep-9

0Ja

n-91

May-91

Sep-9

1Ja

n-92

May-92

Sep-9

2Ja

n-93

May-93

Sep-9

3Ja

n-94

May-94

Sep-9

4Ja

n-95

May-95

Sep-9

5Ja

n-96

May-96

Sep-9

6Ja

n-97

May-97

Sep-9

7Ja

n-98

May-98

Sep-9

8Ja

n-99

May-99

Sep-9

9Ja

n-00

May-00

Sep-0

0Ja

n-01

May-01

Sep-0

1Ja

n-02

May-02

Sep-0

2Ja

n-03

EstTrkErr RlzTrkErr EstTrkErrAdj

Experiment ResultsExperiment Results

•• The level of average tracking error The level of average tracking error forecast is universally improved for our forecast is universally improved for our UK, Europe, Global and US modelsUK, Europe, Global and US models

•• The key indicator is the time series The key indicator is the time series standard deviation of the average bias standard deviation of the average bias statistic per period across a large set of statistic per period across a large set of portfolios at each time point. This value is portfolios at each time point. This value is reduced an average of about 20%reduced an average of about 20%

Different Active Management Styles Different Active Management Styles Represent Differential Responses to Represent Differential Responses to Price MovementPrice Movement

“Price“Price--sensitive active management sensitive active management strategies can be replicated by option strategies can be replicated by option payoffs”payoffs”

Jarrod Wilcox, Jarrod Wilcox, Better Risk ManagementBetter Risk Management, JPM, , JPM, 20002000

Value and VolatilityValue and Volatility

•• Value approaches are often referred to among Value approaches are often referred to among hedge funds and trading desks as “convergence hedge funds and trading desks as “convergence strategies” as they depend on the convergence strategies” as they depend on the convergence between the market price and a manager’s between the market price and a manager’s definition of the fair price of some security. The definition of the fair price of some security. The greater the noise in the market environment, the greater the noise in the market environment, the more obfuscation and impediments to the more obfuscation and impediments to the convergence process.convergence process.

Momentum and VolatilityMomentum and Volatility

•• Momentum strategies buy stocks on price strength and sell on priMomentum strategies buy stocks on price strength and sell on price ce weakness. This is similar to a Constant Proportion Portfolio weakness. This is similar to a Constant Proportion Portfolio Insurance (Black and Insurance (Black and PeroldPerold, 1992) applied to the cross, 1992) applied to the cross--section of section of stock returns.stock returns.

•• CPPI mimics being long a put option on the underlying asset (pluCPPI mimics being long a put option on the underlying asset (plus a s a long position in the underlying). Option buyers are advantaged long position in the underlying). Option buyers are advantaged when realized volatility is greater than the volatility expectedwhen realized volatility is greater than the volatility expected when when the option was establishedthe option was established

•• If momentum strategies are comparable to being long an option, If momentum strategies are comparable to being long an option, then antithen anti--momentum strategies (value?) must be comparable to momentum strategies (value?) must be comparable to being short an option, so low volatility conditions would be mosbeing short an option, so low volatility conditions would be most t favorablefavorable

Defining Defining VarietyVariety as the as the Basis of StyleBasis of Style•• We could just take the crossWe could just take the cross--sectional dispersion of sectional dispersion of

securities in a particular market on a period by period securities in a particular market on a period by period basisbasis

•• Beta differences will cause crossBeta differences will cause cross--sectional dispersion in sectional dispersion in volatile (market index across time) conditionsvolatile (market index across time) conditions

•• So let us define our dispersion measure as the crossSo let us define our dispersion measure as the cross--sectional standard deviation of alpha (residual returns, sectional standard deviation of alpha (residual returns, net of beta effect)net of beta effect)

•• Think of Think of idiosyncratic varietyidiosyncratic variety as the “excess standard as the “excess standard deviation” (standard deviation of stock returns) minus deviation” (standard deviation of stock returns) minus (the product of the absolute value of the observed (the product of the absolute value of the observed market risk premium times the crossmarket risk premium times the cross--sectional dispersion sectional dispersion of the beta values)of the beta values)

Summing Up the IdeaSumming Up the Idea•• Value strategies should work best in periods of low Value strategies should work best in periods of low

excess crossexcess cross--sectional dispersion of stock returns. sectional dispersion of stock returns. Another way to characterize this is periods when Another way to characterize this is periods when correlations among securities is highestcorrelations among securities is highest

•• Momentum/growth strategies should work best in Momentum/growth strategies should work best in periods of high excess crossperiods of high excess cross--sectional dispersion as they sectional dispersion as they are like being long an option.are like being long an option.

•• StronginStrongin, Petsch, Segal and , Petsch, Segal and SharenowSharenow (2002) find value (2002) find value strategies work best when confined within sector (small strategies work best when confined within sector (small crosscross--sectional dispersion), while growth strategies work sectional dispersion), while growth strategies work best with no sector constraints (high dispersion)best with no sector constraints (high dispersion)

•• This is the basis of definition of the value/momentum This is the basis of definition of the value/momentum factor in many Northfield models (beta to idiosyncratic factor in many Northfield models (beta to idiosyncratic varietyvariety

•• May also explain the “value premium”, see Harvey and May also explain the “value premium”, see Harvey and SiddiqueSiddique (2000)(2000)

Prior Empirical Prior Empirical Test on UK DataTest on UK Data•• Compute the monthly “excess” crossCompute the monthly “excess” cross--sectional standard sectional standard

deviation of stock returns using beta values from the deviation of stock returns using beta values from the Northfield UK Risk modelNorthfield UK Risk model

•• Compute the “GrowthCompute the “Growth--Value” return spread from the Value” return spread from the Salomon Smith Barney UK Primary Market indicesSalomon Smith Barney UK Primary Market indices

•• Data from January 1998 through September 2002Data from January 1998 through September 2002•• Correlation coefficient of .48 with significant T statisticCorrelation coefficient of .48 with significant T statistic•• Comparable results to data for the USComparable results to data for the US•• Captures the build and collapse of the late 1990s Captures the build and collapse of the late 1990s

“bubble” nicely. Consistent with “bubble” nicely. Consistent with DermanDerman (2002)(2002)

Monthly UK Style Returns Versus Excess Dispersion Monthly UK Style Returns Versus Excess Dispersion January 1998 through September 2002January 1998 through September 2002

-8

-6

-4

-2

0

2

4

6

0 5 10 15 20 25

Excess Dispersion

UK

Gro

wth

- U

K V

alue

New Results with Data From 1990 New Results with Data From 1990 through 2005through 2005

•• United StatesUnited States–– Slope coefficient = .161, TSlope coefficient = .161, T--Stat 2.49Stat 2.49

•• GlobalGlobal–– Slope coefficient = .186. TSlope coefficient = .186. T--Stat 3.37Stat 3.37

•• EuropeEurope–– Slope coefficient = .078, TSlope coefficient = .078, T--Stat 1.49Stat 1.49

•• Weaker than in the 1998Weaker than in the 1998--2001 period but still 2001 period but still very significant including the long periods of low very significant including the long periods of low idiosyncratic variety idiosyncratic variety valuesvalues

Monthly US Style Returns Versus Excess Dispersion Monthly US Style Returns Versus Excess Dispersion January 1990 through December 2005January 1990 through December 2005

-15

-10

-5

0

5

10

15

0 5 10 15 20 25 30

Excess Dispersion

US

Gro

wth

- U

S V

alu

e

ConclusionsConclusions•• Idiosyncratic variety is an important aspect of Idiosyncratic variety is an important aspect of

market conditions that has show strong trends market conditions that has show strong trends over timeover time–– Current values are comparable to the early 1990s and Current values are comparable to the early 1990s and

are less than one third the peak values seek during are less than one third the peak values seek during the “tech bubble” period surrounding the year 2000the “tech bubble” period surrounding the year 2000

–– Tracking error forecasts can be improved by including Tracking error forecasts can be improved by including information about varietyinformation about variety

•• Popular equity management styles such as value, growth Popular equity management styles such as value, growth and momentum can be viewed as bets on the future and momentum can be viewed as bets on the future excess dispersion of the crossexcess dispersion of the cross--section of stock returnssection of stock returns–– Risk controls for portfolios defined as style neutral Risk controls for portfolios defined as style neutral

can be viewed as being neutral to future movements can be viewed as being neutral to future movements in the volatility levelin the volatility level

ReferencesReferences

•• diBartolomeodiBartolomeo,Dan. “Recent Variation in the Risk Level of ,Dan. “Recent Variation in the Risk Level of US Equity Securities”. Northfield Working Paper 2000.US Equity Securities”. Northfield Working Paper 2000.

•• Wilcox, Jarrod. “Better Risk Management”, Journal of Wilcox, Jarrod. “Better Risk Management”, Journal of Portfolio Management, 2000.Portfolio Management, 2000.

•• Black, Fischer and Andre F. Black, Fischer and Andre F. PeroldPerold. "Theory Of Constant . "Theory Of Constant Proportion Portfolio Insurance," Journal of Economic Proportion Portfolio Insurance," Journal of Economic Dynamics and Control, 1992, v16(3/4), 403Dynamics and Control, 1992, v16(3/4), 403--426. 426.

•• Harvey, Campbell R. and Harvey, Campbell R. and AkhtarAkhtar SiddiqueSiddique. "Conditional . "Conditional SkewnessSkewness In Asset Pricing Tests," Journal of Finance, In Asset Pricing Tests," Journal of Finance, 2000, v55(3,Jun), 12632000, v55(3,Jun), 1263--1295.1295.

More ReferencesMore References

•• SolnikSolnik, Bruno and Jacques , Bruno and Jacques RouletRoulet. "Dispersion As Cross. "Dispersion As Cross--Sectional Correlation," Financial Analyst Journal, 2000, Sectional Correlation," Financial Analyst Journal, 2000, v56(1,Jan/Feb), 54v56(1,Jan/Feb), 54--61.61.

•• Petsch, Melanie, Steve Petsch, Melanie, Steve StronginStrongin, Lewis Segal and Greg , Lewis Segal and Greg SharenowSharenow. “A . “A Stockpicker’sStockpicker’s Reality Part III: Sector Reality Part III: Sector Strategies for Maximizing Returns to Strategies for Maximizing Returns to StockpickingStockpicking”, ”, Goldman Sachs Research, January 2002Goldman Sachs Research, January 2002

•• DermanDerman, Emanuel, “The perception of time, risk and , Emanuel, “The perception of time, risk and return during periods of speculation,” Quantitative return during periods of speculation,” Quantitative Finance 2(4), Finance 2(4), August 2002, pp. 282August 2002, pp. 282--296.296.

•• LiloLilo, , FabrizioFabrizio, Rosario , Rosario MantegnaMantegna, Jean, Jean--Philippe Bouchard Philippe Bouchard and Marc Potters. “Introducing Variety in Risk and Marc Potters. “Introducing Variety in Risk Management”, WILMOTT December 2002Management”, WILMOTT December 2002