drawbacks of pearson correlation

8/10/2019 Drawbacks of Pearson correlation

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Presented by:Gunter Meissner, Ph.D.University of Hawaii and CEO, Cassandra Capital Management

Email: [email protected]

Tuesday, Oct 21, 2014

GARP Webcast

Correlation Risk and Whyit is Critical in Finance


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After a lectureship in mathematics and statistics at the Economic Academy Kiel,

Gunter Meissner PhD joined Deutsche Bank in 1990, trading interest rate futures,

swaps, and options in Frankfurt and New York. He became Head of Product

Development in 1994, responsible for originating algorithms for new derivatives

products, which at the time were Index Amortizing Swaps, Lookback Options, and

Quanto Options and Bermuda Swaptions. In 1995/1996 Gunter was Head of

Options at Deutsche Bank Tokyo. From 1997 to 2007 he was Professor of Finance

at Hawaii Pacific University and from 2008 to 2013 Director of the financial

engineering program at the University of Hawaii. Currently, Gunter is President of

Derivatives Software Founder and CEO of Cassandra Capital Management, and

Adjunct Professor of Mathematical Finance at NYU-Courant.

Gunter Meissner has published numerous papers on derivatives and is a frequent

speaker at conferences and seminars. He is author of 5 books, including his 2014

book on Correlation Risk Modeling and Management An Applied Guide

including the Basel III Correlation Framework (John Wiley).

Gunter Meissner, Cassandra Capital Management


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Motivation

3

correlation, while being one of the most ubiquitous

concepts in modern finance and insurance, is also one of themost misunderstood concepts. (Embrechts et al. 1999)

I think correlation modeling is basically at the stage volatilitymodeling was about 15 years ago (Vladimir Piterbarg)

Correlationsalways increase in stressed markets(John Hull)


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Contents Overview

Basics: What are Financial Correlations?

1.1 Investments and Correlation

Model: The Impact of Correlation in the CAPM model

1.2 Trading and Correlation

Model: Dispersion trading is a play on Correlation!

1.3 Risk Management and Correlation

Model: Deriving the Impact of Correlation on VaR (Value at Risk)

1.4 The Global Financial Crisis and Correlation

1.5 Correlation and Regulation

1.5 Regulation and Correlation


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What are Financial Correlations? Three Interpretations:

1) In Trading Practice: The term correlation is typically used quite loosely for the co-movement ofassets in time.

2) In Financial Theory: The term correlation is often defined narrowly, only referring to the linearPearson correlation model, as in Cherubini et al (2004), Nelsen (2006) or Gregory (2010).

The Correlation Haters Club :

Nassim Taleb refers to this narrow definition:

Everything that has to do with correlation, is charlatanism

3) Broader Definition in Financial Theory: The term correlation is also often applied togenerally describe dependencies, as in the terms credit correlation, default correlation orcopula correlation, which are quantified by non-Pearson models as in Heston (1993), Lucas(1995), or Li (2000).
http://en.wikipedia.org/wiki/File:Taleb_mug.JPG


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The heavily criticized Pearson Correlation Model

Y (Price of AAPL)

X (Profit Margin)

0

1

2

3

4

(X)^Y

To find , we minimize the sum of the squared error terms:(X)^Y

n

1i

2imin

dX

dY1


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The Heavily Criticized Pearson Correlation Model

1YX

XYCOV1-XY
http://en.wikipedia.org/wiki/File:Correlation_examples2.svghttp://en.wikipedia.org/wiki/File:Correlation_examples2.svghttp://en.wikipedia.org/wiki/File:Correlation_examples2.svghttp://en.wikipedia.org/wiki/File:Correlation_examples2.svg


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Main Limitations of the Pearson Correlation Model

1) The Pearson correlation model measures the linearassociation between variables.

As a result, non-linear relationships as Y=X2, cannot be evaluated!

3) The Pearson correlation coefficient is not robust i.e. it is time frame sensitive

0%

5%

10%

15%

20%

25%

30%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

X Y

0%

5%

10%

15%

20%

25%

30%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

X Y

Figure 1 Figure 2

The Correlation in Figure 1 from time 1

to time 17 is -0.8291


to time 2, from time 2 to time 3 is 1


to time 17 is 0.9203


to time 2, from time 2 to time 3 is -1

2) For example, a dependence (as in Y=X2), can result in correlation!

(see spreadsheet www.dersoft.com/Dependence and Correlation.xlsm)

zero

Source Wilmott 2013


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Main Limitations of the Pearson Correlation Model

4) Outliers are over-weighted and can the results

n

1i

2)_xi(x1n

1Variance

5) Nonsense Correlation possible: E.g. we will find a positive correlation

consumption of Organic Food and Autism

distort

6) Linear correlation measures are only natural dependence measures if the jointdistribution of the variables is elliptical.

7) The variances of the sets X and Y have to be finite. However, for distributions with

strong kurtosis, for example the student-t distribution with v2, the variance is infinite.

8) In contrast to the Copula approach, which is invariant to strictly increasing

transformations, the Pearson correlation approach is typically not invariant to

transformations.


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Conclusion: Can We Apply the Pearson Correlation Model in Finance?

For the reasons mentioned, the application of Pearson Correlation

approach in Finance is questionable.

Only if we have a large data set, which is outlier-free, approximately

linear (or linearized), and causally related, can the Pearson model serve

as an approximation for the association between financial variables.

More advanced correlation concepts as Correlating Brownian motions

(Heston 1993), Multivariate Copulas (Li 2000, Albanese 2010),Stochastic Correlations (Buraschi et al 2010, Lu Meissner 2013) are the

better choice for most financial data.


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XYYXYwXw22

Y

2

Y

w2

X

2

X

wXY

The negative relationship of with respect to nin equation (3) comes from

P

P

0XY

XY

We can write:

...)

)(

,)(

(

nnf

P

P

(3)or

(4)

From equation (4), we see that

},{ YXnfor

))

)()(

,((

nnationDiversificf

P

P


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Example 1.1

Figure 1.3: The negative relationship of the portfolio return - portfolio risk ratio P/Pwith respect to thecorrelation of the assets in the portfolio.

Year Asset X Asset Y Return of asset X Return of asset Y2008 100 200

2009 120 230 20.00% 15.00%

2010 108 460 -10.00% 100.00%

2011 190 410 75.93% -10.87%

2012 160 480 -15.79% 17.07%

2013 280 380 75.00% -20.83%

Average 29.03% 20.07%

it follows:

www.dersoft.com/Investment and Correlation.xlsx
http://www.dersoft.com/Investment%20and%20Correlation.xlsxhttp://www.dersoft.com/Investment%20and%20Correlation.xlsxhttp://www.dersoft.com/Investment%20and%20Correlation.xlsxhttp://www.dersoft.com/Investment%20and%20Correlation.xlsx


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Every major investment bank and hedge fund has Correlat ion Desks.

Many Correlation dependent Products and Strategies are traded:

- Correlation Swaps

- Correlation dependent Options as Exchange Options Payoff = max(0, S2-S1)

- Dispersion Trading

ji

1N

1i

N

ij

jwiw2

N

1i

2i

2i

w2I

ij

Okane dake des(Its only money) Japanese saying


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The price of Multi-asset options, also called Rainbow options, or Mountain range

options depends critically on Correlation between the Assets!

For all options above, except one, we have V : Option value0

V

- Option on the better of two. Payoff = max (S1, S2)

- Option on the worse of two. Payoff = min (S1, S2)

- Call on the maximum of two. Payoff = max [0, max(S1,S

2)K]

- Exchange option (as imbedded in a convertible bond). Payoff = max (0, S2S1)

- Spread call option. Payoff = max [0, (S2S1) - K]

- Option on the better of two or cash. Payoff = max (S1, S2, cash)

- Dual strike call option. Payoff = max (0, S1-K1, S2-K2)

- Portfolio of basket option.

n

1i

ii 0K,SnPayoff where niis the weight of assets i


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Dispersion TradingA Play on Correlation

Long Dispersion trading is selling options on an index (e.g. S&P) and

buying options on individual stocks in the index, and vice versa.

Dispersion trading is a play on correlation (between the assets in the index)!

So Dispersion trading is effectively Correlation trading!


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Dispersion Trading Example

Figure 1:Low Correlation ij

Scenario 1: We have an index I of 20 stocks, which have performed as in Figure 1:

In Figure 1, the (Standard moves), i.e. I= 0

A long dispersion trade of long options (straddles) on the stocks 15 and short options(straddles) on the index I would be a successful trade!

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

S

t

a

n

d

a

rd

m

o

v

e

Stock number

Index Performance


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Long Dispersion Trading Scenarios

Scenario 2: We have an index I of 20 stocks, which have performed as in Figure 2:

Our long dispersion trade of long options (straddles) on the stocks 15 and short options(straddles) on the index I is now a disaster.

A short dispersion trade of short options (straddles) on the stocks 15 and long options(straddles) on the index I would have been the correct trade.

Figure 2High Correlation ij

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

S

t

a

n

d

ar

d

m

o

v

e

Stock number

Index Performance


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Dispersion TradingWhy is it a Play on Correlation?

From Stats 101, we remember:

XY2COVYVarXVarXYVar

Generalizing for N assets, for our index I, with Var 2we have

ijji

1N

1i

N

ij

jwiw

N

1i

22i

2i

w2I

Solving for the correlation ij between the N asset in the index I, we get

ji

1N

1i

N

ij

jwiw2

N

1i

2i

2i

w2I

ij

The CBOE publishes the ICJ, ICK Correlation indexes, which consist of 50 stocks, tracking the S&P 500. These indexesare designed to reflect the average correlation of the 50 stocks, i.e. the bid and ask vols are averaged. So ijaverage.

(2)

(1)

See File www.dersoft.com/Dispersion.xlsx

)N1,...,i,If(ij

Equation (2) shows the general concept:

(4*)
http://www.dersoft.com/Dispersion.xlsx%20or%20File%20%E2%80%98Dispersion.xlsxhttp://www.dersoft.com/Dispersion.xlsx%20or%20File%20%E2%80%98Dispersion.xlsx


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Dispersion TradingWhy is it a Play on Correlation?

From equation (1)

ijji1N

1i

N

ij

jwiwN

1i

22i

2i

w2I

we derive 0ij

2I

ji

1N

1i

N

ij

jwiw2

N

1i

2i2iw2I

ij

we derive 02

i

ij

or 0

ij

2i

From equation (2)

0ij

2I

and

0ij

2i

tell us that

If we expect an increase in ij buy options (straddles) on I and sell options (straddles) on individual

stocks i (short dispersion) [typically in a Recession]

Vice versa, if we expect an decrease in ij sell options (straddles) on I and buy options (straddles) on

individual stocks i (long dispersion) [typically in an Expansion]

The equations


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1.3 Risk Management and CorrelationHow is Risk Quantified?

1) Volatility

(Standard deviation of Returns)

XYYXYwXw22Y

2Yw

2X

2XwXY

2) Sharpe ratio(Risk adjusted Return):

For a 2-asset portfolio:

For a n>2 asset portfolio P: vChP

P

rPPS

3) Value at Risk (parametric): XPVaR

4) Expected Shortfall (parametric): VaR]L|E[LES VaRES

5) Extreme Value Theory:1/

u)(x1

n

unx)F(L

Extension: Sortino ratio:Returns)(P

rPP

*Snegative

(1)


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VARP = P x

P= vCh

h is the horizontal vector of invested amounts (price x quantity)v is the vertical vector of invested amounts (price x quantity)C is the covariance matrix

The impact or Correlation in the VaR model


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22cov21cov

12cov11covC

Example 1.1: What is the 10-day VaR for a 2-asset portfolio with a correlation coefficient of0.7, daily standard deviation of returns of asset 1 of 2%, asset 2 of 1%, and $10 mio investedin asset 1 and $5mio invested in asset 2, on a 99% confidence level?

cov11 = 111 1 1 x 0.02 x 0.02 = 0.0004

cov12 = 211 2

0.7 x 0.02 x 0.01 = 0.00014

cov21 = 122 1 0.7 x 0.01 x 0.02 = 0.00014

cov22 = 222 2 1 x 0.01 x 0.01 = 0.0001

The Covariance matrix for a 2-asset portfolio is


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vChP

Ch(10 5)

00010000140

00014000040

..

.. = (10x0.0004+5x0.00014 10x0.00014+5x0.0001) = (0.0047 0.0019)

vC)h( %..x.x.. 6550019050047010

5100019000470

%77.23%65.5vChP

VARP = P x

for a 99% confidence level, = normsinv(0.99) = 2.3264 and for a x=10 day time horizon,

VARP = 0.2377 x 2.3264 x =10 1.7486

Applying


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1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

V

A

R

Corrleation

VAR with respect to correlation

Figure 1.6: VaR of the two-asset portfolio of example 1.1 with respect to correlation

between asset 1 and asset 2.

see Model at www.dersoft.com/2-asset VaR.xlsx
http://www.dersoft.com/2-asset%20VaR.xlsxhttp://www.dersoft.com/2-asset%20VaR.xlsxhttp://www.dersoft.com/2-asset%20VaR.xlsxhttp://www.dersoft.com/2-asset%20VaR.xlsx


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Copulas (Sklar 1959, Vasicek 1987, Li 2000)

Founder: Abe Sklar 1959

An Introduction to Copulas Roger Nelson 1998, second ed. 2006

Vasicek 1987 derives a one-factor Gaussian CVaR:

1

X1N(PD(T))1NNT)V(X,

applied in Basel IIsIRB approach

David Li 2000 On Default Correlation: A Copula Function Approach

Copula Milestones

Later more

mapping, which results in an abscise value of a standard normal dist

]M(t));n(Q-1N(t)),...,B(Q

-1N(t)),A(Q-1[NnM(t)]nQ(t),...,BQ(t),AC[Q

Copula Methods in Finance Cherubini et al 2004


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Can we blame the Copula Correlation Model for the Global Financial Crisis?

Recipe for Disaster: The Formula that killed Wall Street Wired Magazine (2009)

Wall Street Wizards Forgot a Few Variables New York Times (2009)

The reason for the Global Financial Crisis can be summed up in one word:

Greed

Resulting in irresponsible Overinvesting and Risk-taking:

AIG hat sold 500 billion inCDSs. Their risk management strategy was

Icelands banks had borrowed and invested 10 times the national GDP

In 2007, the US debt ratio was 470% of national income!!!

Confessions of a Risk Manager The Economist (2008)


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Can We Blame the Copula Correlation Model for the Global Financial Crisis?

Models are not perfect. That doesnt man they are not useful Robert Merton

A major problem in the global financial crisiss was inadequate Calibration!

Models as VaR, CVaR, Copulas to value CDOs, were fed benign vol and correlation data!

If a model is fed wrong input data, it cant be expected that is produces correct results!

Garbage in, garbage out!!

Models are now stress tested, required and supervised by Basel III, Fed, ECB..

Naturally, we need general mindfulness about financial models and not trust them uncritically:

(See also Chapter 3.1 in Correlation book)

David Li: The most dangerous thing is when people believe everything that comes out of it


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An Overview

Figure 12.4: CVA (Credit Value Adjustment) and WWR (Wrong Way Risk) in the Basel III framework.

Source: Moodys Analytics 2011.


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Basel II (and III) applies the OFGC (One-factor Gaussian Copula)Correlation Model to derive CVaR(Credit Value at Risk). Too simplistic???

Basel III applies Correlations between Credit exposure = f(Market) and

Credit risk, i.e. wrong-way risk (WWR) to derive CVA(Credit Value Adjustment)

Basel III recognizes credit exposure, which is hedged (e.g. with CDS)

and allows two Correlation Concepts for Double-Default

More on CVaR later

More on Basel IIIs two Double Default Approaches later

More on CVA later

Market RiskCredit Risk


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CVA (Credit Value Adjustment) Approach WWR (Wrong Way Risk)

in the Basel Accord

What is CVA (Credit Value Adjustment)?

Definition: CVA is a specific capital charge to address counterparty risk

However, CVA is typically defined narrower referring to counterparty risk in

Derivativestransactions.


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CVA (Credit Value Adjustment) Approach with WWR (Wrong Way Risk)

in the Basel Accord

Why CVA?

Basel committee: 2/3 of the credit risk losses during the global financial crisiswere caused by CVA volatility rather than actual defaults

AIG had sold close to $500 billion in CDSs!!! Needed bailout of $180 billion! When Lehman defaulted in September 2008, it had 1.5 million derivative

transactions with 8,000 different counterparties...

Trading and Hedging CVA

Financial Institution do not want to pay CVA. Therefore the vast majority of

financial institution has CVA desks, where CVA is traded and hedged.


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in the Basel AccordBasics

CVAa,c = f (D+

a,c, PDc)

Market risk Credit risk

CVAa,c: Credit Value Adjustment of entity a with respect to the counterparty c

D+

a,c: Netted, positive derivatives portfolio value of entity a with counterparty cPDc: Default probability of counterparty c

(12.11)

Market risk or Market price changes determine the Credit Exposure.

E.g. Bank a has a long put on a bond of Greek Bond BGbought fromcounterparty c. If B

G Credit exposure of a with respect to c, since D+

a,c

Intuitive way to look at WWR: When credit exposure and credit risk both

tend to increase together (are positively correlated).

C A (C i A j ) A i ( i )


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in the Basel Accord

What is Wrong Way Risk (WWR)?

Two types of WWR exist

1) Generalwrong-way risk exists when the probability of default of a counterparty is

positively correlated with general market risk factors (BCBS 2003)

An example of general WWR is a bond:

i

PDc

Higher credit exposure

Higher credit risk

B(12.7)

Intuitive way to look at WWR: When credit exposure and credit risk both tend to

increase together (are positively correlated).

CVA (C di V l Adj ) A h i h WWR (W W Ri k)


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in the Basel Accord

The second type of WWR is specific WWR

A bank is exposed to speci f icwrong-way risk (WWR) if future exposure

to a specific counterparty is positively correlated with the counterpartysprobability of default (BCBS 2011)

We can formulize specific WWR as

PDc

D

a,c

0 (12.13))

Where D+a,cis the netted positive Derivatives value of a with respect to thecounterparty c and PDcis the prob of default of a counterparty c.

Credit risk

Credit exposure


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An Example of Specific WWR:

Investor andCDS buyer i

Fixed CDS spread s

Guarantor gi.e. CDS seller

Reference assetof obligor o

coupon k$M million

Payout of $M(1-R)million in case of

default of obligor o(12.8)

Specific WWR exists if there is a positive correlation between the obligor o and the guarantor g:

Higher credit exposure

Higher credit risk

PV(CDS) for i

P of payoff

(PDo PDg)

What if o and g are identical??? See model CDS with default correlation.

(12.9)


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Critical Appraisal of Basels CVA

ProsIs necessary in light of the 2007-2009 crisis

Cons

Basels =1.2 to 1.4, while conservative (banks report 1.07 to 1.1), is simplistic.More rigorous WWR correlation models are being developed (Hull, Meissner)

The WWR correlation approach is also necessary. Basels factor =1.2 to 1.4,which is multiplied to CVA if WWR exists,is a simple way to deal with WWR.

CVA needs inputs of PD until end of the exposure, which can be up to 30years. PD data unreliable for this time frame.


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Concluding Summary

Correlations are ubiquitous in Finance, but not very well understood.

The most popular approach, the Pearson Correlation model has

significant limitations in Finance!

Correlations are especially critical in Risk Management, especially

Credit Risk Management, as seen in the global financial crisis, since

Correlations change (typically increase) in stressed markets.

Correlation modeling is in the beginning stages. Several promising

approaches as stochastic Correlation models are emerging.


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Upcoming GARP Webcasts

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2015 FRM Program: A First Look

Tuesday, Nov 11, 2014 Time: 11:00 am EDT | 4:00 pm BST | 11:00 pm HKT

When Risk Data Is Sparse: Modeling Emerging Market Credit RiskTuesday, December 2, 2014 Time: 11:00 am EDT | 4:00 pm BST | 11:00

pm HKT

To learn more and register, visitwww.garp.org/webcast


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drawbacks of pearson correlation

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