post modern portfolio theory

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Market Crashes and Modeling Market Crashes and Modeling Volatile Markets Volatile Markets Prof. Svetlozar (Zari) T.Rachev Prof. Svetlozar (Zari) T.Rachev Chief-Scientist, FinAnalytica Chief-Scientist, FinAnalytica Chair of Econometrics, Statistics and Mathematical Finance Chair of Econometrics, Statistics and Mathematical Finance School of Economics and Business Engineering School of Economics and Business Engineering University of Karlsruhe University of Karlsruhe Department of Statistics and Applied Probability Department of Statistics and Applied Probability University of California, Santa Barbara University of California, Santa Barbara Thalesian seminar , London - December 2, 2009 Thalesian seminar , London - December 2, 2009

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Page 1: Post Modern Portfolio Theory

Market Crashes and Modeling Volatile Market Crashes and Modeling Volatile MarketsMarkets

Prof. Svetlozar (Zari) T.RachevProf. Svetlozar (Zari) T.Rachev Chief-Scientist, FinAnalytica Chief-Scientist, FinAnalytica

Chair of Econometrics, Statistics and Mathematical Finance Chair of Econometrics, Statistics and Mathematical Finance School of Economics and Business Engineering School of Economics and Business Engineering University of Karlsruhe University of Karlsruhe Department of Statistics and Applied Probability Department of Statistics and Applied Probability University of California, Santa Barbara University of California, Santa Barbara

Thalesian seminar , London - December 2, 2009Thalesian seminar , London - December 2, 2009

Page 2: Post Modern Portfolio Theory

Post Modern Portfolio TheoryPost Modern Portfolio Theory

Page 3: Post Modern Portfolio Theory

MPT “translation” for Volatile MarketsMPT “translation” for Volatile Markets

Normal (Gaussian) Distributions

– Correlation– Sigmas– Sharpe Ratios– BS Option Pricing– Markowitz Optimal

Portfolios

Fat-tailed Distributions

– Tail & Asymmetric Dependence– Expected Tail Loss– STARR Performance– Tempered-Stable Option Pricing– Fat-tail ETL Optimal Portfolios

Old World Real World

Page 4: Post Modern Portfolio Theory

Models MapModels Map

Page 5: Post Modern Portfolio Theory

Agenda

• The Fat-tailed Framework– Univariate model (single asset)

• Subordinated models• Stable model

– Dependence– Risk and Performance measures

• Applications– Option pricing - Some extension of the main fat-tailed

model: Tempered Stable models– Modeling market crashes– Risk monitoring– Portfolio management and optimization

Page 6: Post Modern Portfolio Theory

Fat-tail Modeling FrameworkFat-tail Modeling Framework

Page 7: Post Modern Portfolio Theory

Phenomena of Primary Market Drivers - 1Phenomena of Primary Market Drivers - 1

• Univariate level– Time-varying volatility– Fat-tails– Asymmetry– Long-range dependence (intra-day)

DJ Daily returns

Page 8: Post Modern Portfolio Theory

Fat-tailedFat-tailed

Subordinator (g(W)) < 1Subordinator (g(W)) < 1

ZWgWX

NZZWgX

tTZtTZtX

:

)1,0(,)(:

))(()()( “On the days when no new information is available,

trading is slow and the price process evolves slowly. On days when new information violates old expectations,

trading is brisk, and the price process evolves much faster”.

Clark (1973)

Emp.

Page 9: Post Modern Portfolio Theory

Subordinator > 1

Page 10: Post Modern Portfolio Theory

Stable FamilyStable Family

1.5)

Positive skewed densities

( 0)

Symmetric densities

(

Rich history in probability theoryKolmogorov and Levy (1930-1950), Feller (1960’s)

Long known to be useful model for heavy-tailed returnsMandelbrot (1963) and Fama (1965)

Page 11: Post Modern Portfolio Theory

Fat Tails Study: 17,000+ factorsFat Tails Study: 17,000+ factors

May 2007

14%

4%

76%

6%

Normal Vol Clust Enhanced Normal

Stable Vol Clust Enhanced Stable

Dec 20087%0%3%

90%

Normal Vol Clust Enhanced Normal

Stable Vol Clust Enhanced Stable

85%, 95%, 97.5%, and 99% VaR tested

Page 12: Post Modern Portfolio Theory

Fat Tails Study: Factors BreakdownFat Tails Study: Factors Breakdown

Factors Tested Number Percentage

Equities 8346 48.5%

CDS Spreads 7803 45.3%

Interest Rates 528 3.1%

Implied Volatilities 518 3.0%

Currencies 12 0.1%

Total 17207 100.00%

Page 13: Post Modern Portfolio Theory

Alpha Tail Parameter:Alpha Tail Parameter: Varies Across Assets & TimeVaries Across Assets & Time

• Important to:– Distinguish tail risk contributors and diversifiers– Changes in the market extreme risk

S&P 500 alpha

1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

1.9

1.95

2

15/06/2000 15/06/2001 15/06/2002 15/06/2003 15/06/2004 15/06/2005 15/06/2006 15/06/2007 15/06/2008

after removing GARCH

Page 14: Post Modern Portfolio Theory

Tail parameter Alpha for 41 indices after removing GARCH effect/May 15th 2009/

1.6 1.65 1.7 1.75 1.8 1.85 1.9 1.95 2

MSCI Japan JPY

HK HANG SENG

RU RTS INDEX

US NASDAQ COMPOSITE

MSCI WRLD/Energy USD

S&P GSCI Energy Index

MSCI France EUR

UK FTSE 100

US DOW JONES INDUS. AVG

MSCI United Kingdom GBP

US S&P 500

JP NIKKEI 225

IN BSE SENSEX 30

MSCI China CNY

MSCI Russia USD

MSCI India INR

FR CAC 40

US RUSSELL 2000

MSCI Hong Kong HKD

DE DAX

MSCI Germany EUR

MSCI Japan JPY

HK HANG SENG

RU RTS INDEX

US NASDAQ COMPOSITE

MSCI WRLD/Energy USD

S&P GSCI Energy Index

MSCI France EUR

UK FTSE 100

US DOW JONES INDUS. AVG

MSCI United Kingdom GBP

US S&P 500

JP NIKKEI 225

IN BSE SENSEX 30

MSCI China CNY

MSCI Russia USD

MSCI India INR

FR CAC 40

US RUSSELL 2000

MSCI Hong Kong HKD

DE DAX

MSCI Germany EUR

There is NO universal tail index!

Page 15: Post Modern Portfolio Theory

Phenomena of Primary Market Drivers - 2Phenomena of Primary Market Drivers - 2

• Tail Dependence

Zero tail dependenceGaussian copula

Page 16: Post Modern Portfolio Theory

Dependence ModelsDependence ModelsAsymmetric Dependence

Page 17: Post Modern Portfolio Theory

Dependence ModelsDependence Models

Modeling of Extreme Dependency in market crashes is critical for taking correct investment decisions

Bi-variate Normal

Fat-tailed indicesGaussian Copula

Fat-tailed indicesFat-tailed copula

Observed returns in Q3 1987

))(),...,((),...,( 111 nnn xFxFCxxF

F is the multivariate cdf, C is the copula function and Fi are the one-dimensional cdf.

Page 18: Post Modern Portfolio Theory

Risk & Performance MeasuresRisk & Performance Measures

Downside risk penaltyand upside reward

Symmetric risk penalty

frSHARPE

ETL

rSTARR

qrrEETRVaRrrEETL

f

)|()|( 11

ETL

ETRRatioR

Downside risk penalty

Page 19: Post Modern Portfolio Theory

Why not Normal ETL?Why not Normal ETL?

-0.2 -0.1 0.0 0.1 0.2

05

10

15

20

25

30

OXM DAILY RETURNS

1% STABLE ETL vs. NORMAL VAR AND ETL: $1M OVERNIGHT

Normal VaR = $47K

Normal ETL = $51K

Stable ETL = $147K

STABLE DENSITYNORMAL DENSITY

Page 20: Post Modern Portfolio Theory

SummarySummary

• Fat-tailed world is a complex one:– GARCH is not enough– Fat-tails are not enough– Copula choice is important– Fat-tails change across assets and across time– Beware of pseudo-fat-tailed models– Fat-tailed ETL as a risk measure is important

Page 21: Post Modern Portfolio Theory

Application 1 – Option pricing Application 1 – Option pricing Stable and Tempered Stable DistributionsStable and Tempered Stable Distributions

Page 22: Post Modern Portfolio Theory

Tempered Stable Models IntroductionTempered Stable Models Introduction

• The stable model does not allow for unique equivalent martingale measure

• Take a stable model and make the very end of the tails lighter (still much heavier than the Gaussian)

• All moments exist

• No-arbitrage option pricing exists

Page 23: Post Modern Portfolio Theory

Tempered StableTempered Stable

Page 24: Post Modern Portfolio Theory

Tempered StableTempered Stable

Page 25: Post Modern Portfolio Theory

Map of Tempered Stable DistributionsMap of Tempered Stable Distributions

Rapidly Decreasing

Tempered Stable(RDTS)

Smoothly Truncated

Stable(STS)

Kim-Rachev(KR)

ClassicalTempered

Stable(CTS)

NormalTempered

Stable(NTS)

ModifiedTempered

Stable(MTS)

Page 26: Post Modern Portfolio Theory

Incorporating GARCH EffectIncorporating GARCH Effect

other tempered stable models

Page 27: Post Modern Portfolio Theory

Is GARCH Enough? … No!Is GARCH Enough? … No!

• QQ plots between the empirical residual and innovation distributions for daily return /data for IBM/

Page 28: Post Modern Portfolio Theory

Option Prices and GARCH Models

where N is the number of observation, is the n-th price determined by thesimulation, and is the n-th observed price.

SPX Call Prices (April 12, 2006)

Page 29: Post Modern Portfolio Theory

Model UniverseModel Universe

• We studied the full spectrum of tractable (infinitely divisible) models• We see that Stable ARMA-GARCH is the best choice to model primary risk drivers• We propose a form of tempered stable (RDTS) for option pricing

Page 30: Post Modern Portfolio Theory

The Option Pricing Models UniverseThe Option Pricing Models Universe

Page 31: Post Modern Portfolio Theory

Application 2 - Modeling Market CrashesApplication 2 - Modeling Market Crashes

Page 32: Post Modern Portfolio Theory

Daily Returns: S&P 500 Index

Page 33: Post Modern Portfolio Theory

Crash Probability: Black Monday

On October 19 (Monday), 1987 the S&P 500 index dropped by 23%. Fitting the models to a data series of 2490 daily observations ending with October 16 (Friday), 1987 yields the following results:

Page 34: Post Modern Portfolio Theory

Crash Probability: U.S. Financial Crisis

On the September 29 (Monday),2008 the S&P 500 index dropped by 9%. Fitting the models to a data series of 2505 daily observations ending with the September 26 (Friday), 2008 yields the following results:

Page 35: Post Modern Portfolio Theory

S&P BacktestS&P Backtest

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.152

00

3-1

1-1

1

20

04

-01

-07

20

04

-03

-04

20

04

-04

-30

20

04

-06

-28

20

04

-08

-24

20

04

-10

-20

20

04

-12

-16

20

05

-02

-11

20

05

-04

-11

20

05

-06

-07

20

05

-08

-03

20

05

-09

-29

20

05

-11

-25

20

06

-01

-23

20

06

-03

-21

20

06

-05

-17

20

06

-07

-13

20

06

-09

-08

20

06

-11

-06

20

07

-01

-02

20

07

-02

-28

20

07

-04

-26

20

07

-06

-22

20

07

-08

-20

20

07

-10

-16

20

07

-12

-12

20

08

-02

-07

20

08

-04

-04

20

08

-06

-02

20

08

-07

-29

20

08

-09

-24

20

08

-11

-20

20

09

-01

-16

20

09

-03

-16

20

09

-05

-12

Return normal GARCH VaR 99% fat tail GARCH VaR 99% fat tail GARCH ETL 99%

Page 36: Post Modern Portfolio Theory

Application 3 – Risk MonitoringApplication 3 – Risk Monitoring

Page 37: Post Modern Portfolio Theory

Backtest ExampleBacktest Example

• Long-short stock portfolio• 99% VaR backtest was run from 8/1/2007 to 5/15/2008

(206 days)• 250 rolling window used to fit the models• Models:

– Historical method– Normal method

• Constant Volatility• EWMA for Cov matrix

– Asymmetric Stable with Copula• Constant Volatility• Volatility Clustering

Page 38: Post Modern Portfolio Theory

Model ComparisonModel Comparison

• Quantitative - Number of exceedances– Average - must be on average 2– Number of exceedances above 4

/95% CI is 0-4/– Checked on portfolio and industry level

• Qualitative– Visual check of VaR evolution vs returns

Historical 3.25 24%

Normal 99 7.03 76%

Normal 99 EWMA 3.90 42%

Asym Stable Fat-tail Copula 1.64 3%

Asym Stable Fat-tail Copula Volatility Clustering 2.27 6%

Av. # of exceedances

% Industries VaR rejected

Page 39: Post Modern Portfolio Theory

Fat-tailed VaR with constant volatility provides long-term equilibrium VaR

Fat-tailed VaR with volatility clustering provides dynamic short-term view of the tail risk (VaR)

Both are important!

Returns

Normal 99

Normal 99 EWMA

Asym Stable Fat-tail Copula

Asym Stable Fat-tail Copula Vol Clustering

Risk BacktestRisk Backtest

Page 40: Post Modern Portfolio Theory

Application 4 – Portfolio Management and Application 4 – Portfolio Management and OptimizationOptimization

Page 41: Post Modern Portfolio Theory

Portfolio Risk BudgetingPortfolio Risk Budgeting

• Marginal Contribution to RiskStandard Approach: St Dev

( ) cov( , )i i Pi

P P

r rMCTR

Ωw

Pi i P

i P

w MCTR

w Ωw

ww

ppii

i rVaRrrEw

ETLMCETL

|

The expression for marginal contribution to ETL is

and the resulting risk decomposition:

pi

ppiii

ii rETLrVaRrrEwMCETLw |

ETL:

Page 42: Post Modern Portfolio Theory

Portfolio OptimizationPortfolio Optimization

• Flexibility in problem types, a very general formulation is

where the first ETL is of a tracking-error type, the second one measures absolute risk and l ≤ Aw ≤ u generalizes all possible linear weight constraints

If future scenarios are generated, there are two choices: • Linearize the sample ETL function and solve as a LP• Solve as a convex problem

uAwl

ew

ts

Erwrwrrw

T

TTb

T

w

1

..

ETLETL min 21

Page 43: Post Modern Portfolio Theory

SummarySummary

• Modeling Fat-tailed world is a complex taskBUT crucial for:

• Option pricing– Explaining volatility smile– Identifying statistical arbitrage opportunities

• Crash warning indicators– Helps identify changes in the market structure faster

• Risk monitoring– Realistic understanding of risk and its evolution

• Portfolio construction and optimization– Achieve higher risk-adjusted returns

Page 44: Post Modern Portfolio Theory

Q&A…

Thank you!