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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.JPG8/10/2019 Drawbacks of Pearson correlation
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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.svg8/10/2019 Drawbacks of Pearson correlation
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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
The Correlation in Figure 1 from time 1
to time 2, from time 2 to time 3 is 1
The Correlation in Figure 2 from time 1
to time 17 is 0.9203
The Correlation in Figure 2 from time 1
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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1.1 Investments and Correlation
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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1.1 Investments and Correlation
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.xlsx8/10/2019 Drawbacks of Pearson correlation
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1.2 Trading and Correlation
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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1.2 Trading and Correlation
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.xlsx8/10/2019 Drawbacks of Pearson correlation
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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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1.3 Risk Management and Correlation
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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1.3 Risk Management and Correlation
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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1.3 Risk Management and Correlation
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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1.3 Risk Management and Correlation
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.xlsx8/10/2019 Drawbacks of Pearson correlation
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1.4 The Global Financial Crisis and Correlation
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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1.4 The Global Financial Crisis and Correlation
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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1.5 Correlation and Regulation
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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1.5 Correlation and Regulation
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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CVA (Credit Value Adjustment) Approach with WWR (Wrong Way Risk)
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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CVA (Credit Value Adjustment) Approach with WWR (Wrong Way Risk)
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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CVA (Credit Value Adjustment) Approach with WWR (Wrong Way Risk)
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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To learn more and register, visitwww.garp.org/webcast
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