Financial Risk Forecasting Chapter 3 Multivariate volatility models · 2020. 8. 10. ·...

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Multivariate volatility EWMA CCC DCC Large problems Estimation comparison BEKK Covid-19

Financial Risk Forecasting

Chapter 3

Multivariate volatility models

Jon Danielsson ©2021London School of Economics

To accompanyFinancial Risk Forecasting

www.financialriskforecasting.com

Published by Wiley 2011

Version 6.0, August 2021



Volatility

• The previous chapter focused on the volatility of a singleasset

• In most we hold a portfolio of assets

• And therefore need to estimate both the volatility of eachasset in the portfolio

• And the correlations between all the assets

• This means that it is much more complicated to estimatemultivariate volatility models than univariate models



The focus of this chapter is on

• EWMA

• Orthogonal GARCH

• CCC and DCC models

• Estimation comparison

• Multivariate extensions of GARCH (MV GARCH andBEKK)



Notation

Σt Conditional covariance matrixYt,k Return on asset k at time t

yt,k Sample return on asset k at time t

yt = yt,k Vector of sample returns on all assets at time ty = yt Matrix of sample returns on all assets and datesA and B Matrices of parameters

R Correlation matrixDt Conditional variance forecast



Data

• Amazon and Google daily stock price from 2006-2015

www.financialriskforecasting.com/data/amzn-goog.csv

p=read . c s v ( ” h t tp : //www.f i n a n c i a l r i s k f o r e c a s t i n g . com/data /amzn−goog . c s v ” )

y=app l y ( l o g ( p [ , 2 : 3 ] ) , 2 , d i f f )p r i n t ( head ( y ) )

http://www.financialriskforecasting.com/data/amzn-goog.csv



R estimation

• It is easy to implement EWMA directly in R

• But no package exists for estimating all the modelsdiscussed below

• We mostly use Alexios Ghalanos’s rmgarch



Other reading

• Many surveys, e.g.

• Multivariate GARCH models for large-scale applications:A survey. Kris Boudt, Alexios Galanos, Scott Payseur,Eric Zivot. Handbook of Statistics. Elsevier

• For large (more than 25 assets) this paper has a proposal

• Engle, R. F., Pakel, C., Shephard, N., and Sheppard, K.(2019). Fitting vast dimensional time-varying covariancemodels.



Multivariate Volatility

Forecasting



• Consider the univariate volatility model:

Yt = σtZt

where Yt are returns, σt is conditional volatility and Zt

are random shocks

• If there are K > 1 assets under consideration, it isnecessary to indicate which asset and parameters arebeing referred to, so the notation becomes more cluttered:

Yt,i = σt,iZt,i

where the first subscript indicates the date and thesecond subscript the asset



Conditional covariance matrix Σt

• The conditional covariance between two assets i and j isindicated by:

Cov(Yt,i ,Ytj) ≡ σt,i j

• In the three-asset case (note that σt,i j = σt,j i):

Σt =

σt,11

σt,12 σt,22

σt,13 σt,23 σt,33



Portfolios

• If w is the vector of portfolio weights

• The portfolio variance is

σ2portfolio = w ′Σw



The curse of dimensionality

• Number of diagonal elements is K and off diagonalelements K (K − 1)/2 so in all

K + K (K − 1)/2

• For two assets it is 2+1, for 3 assets 3+4, for 4 10, etc.

• The explosion in the number of variance and especiallycovariance terms, as the number of assets increases, isknown as the curse of dimensionality

• This is one reason why it is more difficult to estimate thecovariance matrix



Positive semi-definiteness

• For univariate volatility we need to ensure that thevariance is not negative (σ2 ≥ 0)

• And for a portfolio

σ2portfolio = w ′Σw ≥ 0

• So a covariance matrix should be positive semi-definite:

|Σ| ≥ 0

• This can be difficult to ensure



What about multivariate (MV) GARCH?

• For one asset

σ2t+1 = ω + αy2

t + βσ2t

• For two

σ2t+1,1 = ω1 + α1y

2t,1 + β2σ

2t,1 + α2y

2t,2 + β2σ

2t,2 + δ1σ

2t+1,1,2 + γ1yt,1yt,2

σ2t+1,2 = ω2 + α3y

2t,1 + β3σ

2t,1 + α4y

2t,2 + β4σ

2t,2 + δ2σ

2t+1,1,2 + γ2yt,1yt,2

σ2t+1,1,2 = ω3 + α5y

2t,1 + β5σ

2t,1 + α6y

2t,2 + β6σ

2t,2 + δ3σ

2t+1,1,2 + γ3yt,1yt,2

• Or 21 parameters to estimate

• Almost impossible in practice



Numerical issues

• Stationarity is more important for multivariate volatility

• For univariate GARCH model, violation of covariancestationarity doesn’t matter for many applications and

• Does not hinder the estimation process with numericalproblems

• A univariate volatility forecast is still obtained even ifα + β > 1

• This is generally not the case for MV GARCH models



• A parameter set resulting in violation of covariancestationarity might also lead to unpleasant numericalproblems

• Numerical algorithms need to address these problems,thus complicating the programming process considerably

• Problems with multiple local minima, flat surfaces andother pathologies discussed in the last chapter



Instead

• Use some simplification approaches

• Unfortunately they come with significant trade-offs

• So MV estimation is much harder and much less accuratethan univariate estimation

1. EWMA2. CCC3. DCC (perhaps most widely used for portfolios )4. OGARCH (perhaps most widely used for combining

portfolios)5. BEKK (not to be recommended except for very small

problems, K = 2, perhaps K = 3)6. MV GARCH (practically impossible except maybe when

K = 2)7. Composite likelihood



EWMA



EWMA model

• Univariate:σ2t = λσ2

t−1 + (1− λ)y 2t−1

• A vector of returns is

ytK×1

= [yt,1, yt,2, . . . , yt,K ]

• The multivariate EWMA is:

Σt = λΣt−1 + (1− λ)y′

t−1yt−1

with an individual element given by:

σt,i j = λσt−1,i j + (1− λ)yt−1,iyt−1,j



Properties

• The same weight, λ, is used for all assets

• It is pre-specified and not estimated

• The variance of any particular asset only depends on itsown lags



Pros and cons of multivariate EWMA

model

Usefulness:

• Straightforward implementation, even for a large numberof assets

• Covariance matrix is guaranteed to be positivesemi-definite

Drawbacks:

• Restrictiveness:• Simple structure• The assumption of a single and usually non-estimated λ



R estimation

EWMA = mat r i x ( nrow=dim ( y ) [ 1 ] , n c o l =3)lambda = 0.94S = cov ( y )EWMA[ 1 , ] = c (S ) [ c ( 1 , 4 , 2 ) ]f o r ( i i n 2 : dim ( y ) [ 1 ] )

S = lambda∗S+(1− lambda ) ∗y [ i −1 ,] %∗% t ( y [ i −1 , ] )

EWMA[ i , ] = c (S ) [ c ( 1 , 4 , 2 ) ]EWMArho = EWMA[ , 3 ] / s q r t (EWMA[ , 1 ] ∗EWMA[ , 2 ] )

Be careful with the matrix multiplications, getting thetransposes right (y[i-1,] %*% t(y[i-1,]) and nott(y[i-1]) %*% y[i-1]



Constant Conditional

Correlation

Models



Steps

• We usually start by removing the mean — de-meanreturns

• Separate out correlation modelling from volatilitymodelling

1. correlation matrix2. variances

• Model volatilities with GARCH or some standard method

• The correlation matrix can be static (CCC) or dynamic(DCC)



Definitions

• Let Dt be a diagonal matrix where each element is thevolatility of each asset

Dt,ii = σt,i , i = 1, . . . ,K

Dt,ij = 0, i 6= j

• Let R or Rt be the constant or time varying correlationmatrix, respectively

• We will employ a two-step procedure and estimate Dt andRt separately

• Then combine them to get the conditional covariancematrix, Σt



Conditional variance

• Use univariate GARCH (or some method) to estimate thevariance of each asset separately, so for asset i

σ2t,i = ωi + αiy

2t−1,i + βiσ

2t−1,i

• Note the parameters will be different for each asset

• Dt is then created by putting these into the diagonalelements

Dt,ii = σt,i , i = 1, . . . ,K



Getting the correlations

• After fitting the volatilities, divide them into the returnsto get residuals

ǫt,i =yt,i

Dt,i

• The vector of residuals at time t is then

ǫtK×1

= D−1t yt

• And the matrix of residuals for all assets and time is

ǫT×K



Correlations of the residuals

• We want the correlations of the residuals

• For two different assets they are

Corr(ǫi ,t , ǫj ,t)

• Since mean is zero and variance one, the K × K

correlation matrix is simply

R := Cov(ǫ) =ǫǫ′

T



Constant conditional correlations (CCC)

• We now have the constant matrix R and the time varyingmatrix Dt

• Combine these two to create the conditional variancematrix

Σt = Dt RDt



Pros and cons

Pros • Guarantees the positive definiteness of Σt ifR is positive definite, as it is by definition

• Simple model, easy to implement• Since matrix Dt has only diagonal elements,we can estimate each volatility separately

Cons • The assumption of correlations beingconstant over time is at odds with the vastamount of empirical evidence supportingnonlinear dependence



Dynamic Conditional

Correlation

Models



Dynamic conditional correlations (DCC)

• DCC model is an extension of the the CCC

• Let the correlation matrix Rt be time dependent

• The obvious way to do that is to do it like GARCH model

Rt = a + bǫt−1ǫ′

t−1 + cRt−1

• That will not work since it does not ensure |Σt | > 0, i.e.positive definite



Conditions for positive definiteness

• Dt is positive definite by construction since all elements inDt are positive or zero

• We can then ensure |Σt | > 0 if

• All elements in Rt are ≤ 1 and ≥ −1



So need more steps

• We decompose Rt into two parts

• One drives the dynamics — call that Qt

• The other re-scales the dynamics to ensure each elementis between -1 and +1 — call that Zt

• SoRt = ZtQtZt



• Decompose Rt into

• Qt is a positive definite matrix that dives the dynamics

• Zt re-scales Qt to ensure each element in it, the |qt,ij | < 1

• All that is needed is to make each diagonal element of Zt

be one over the corresponding diagonal element in Qt

Zt =

1/√qt,11 0 · · · 00 1/

√qt,22 · · · 0

......

......

0 · · · 0 1/√qt,KK



• The have Qt follow a GARCH type process (ARMA) typeprocess

Qt = (1− ζ − ξ)R + ζǫt−1ǫ′

t−1 + ξQt−1

• R is the (K × K ) unconditional covariance matrix of ǫ

• ζ and ξ are parameters

• Parameter restrictions:• Positive definiteness ζ, ξ > 0• Stationarity ζ + ξ < 1



Pros and cons of DCC

Pros • Large covariance matrices can be easily beestimated

Cons • Parameters ζ and ξ are constants• So the conditional correlations of all assetsare driven by the same underlying dynamicsas the two parameters are same for all assets



Large problems



Large problems

• Even a medium-sized financial institutions will havehundreds of thousands or millions of types of assets —what is known as risk factors

• Estimating the covariance matrix for the entire institutionis effectively impossible using methods like EWMA orDCC

• Instead, we can split the risk factors up intosubcomponents

• Estimate the covariance of each

• And then combine them back



For example

Bank

US

Bonds

Commodities

EquitiesS&P 100 Index

S&P 400 Mid Cap Index

China

Bonds

Equities

Brazil

Bonds

Equities



Simplifying the problem

• Suppose you are willing to assume a single correlationbetween all Brazilian assets and all US assets

• And you are willing to assume that the correlationbetween US bonds and US equities is the same for allbonds and equities

• And you are willing to assume that the correlationbetween US large-cap and small-cap stocks is the samefor all stocks

• Then it is easy to calculate the covariance matrix for theentire bank

• Use perhaps DCC for US small caps and other portfolios

• And combine them by the constant correlation



Consider an example...• Your financial institution has the following securities:




σ2E




σ2E σ2

B




σ2E σ2

B σ2E




σ2E σ2

B σ2E σ2

B




σ2E σ2

B σ2E σ2

B

where red is UK, blue is Germany, E denotes EQUITYand B is BOND




σ2E σ2

B σ2E σ2

B


• Individual covariance matrices of your securities can becombined into a covariance matrix for the entire financialinstitution:




σ2E σ2

B σ2E σ2

B



σ2E




σ2E σ2

B σ2E σ2

B



σ2E

σ2B




σ2E σ2

B σ2E σ2

B



σ2E

σ2B

ρE ,B




σ2E σ2

B σ2E σ2

B



σ2E

σ2B

ρE ,B

σ2E




σ2E σ2

B σ2E σ2

B



σ2E

σ2B

ρE ,B

σ2E

ρE ,E ρB,E




σ2E σ2

B σ2E σ2

B



σ2E

σ2B

ρE ,B

σ2E

ρE ,E ρB,E

σ2B




σ2E σ2

B σ2E σ2

B



σ2E

σ2B

ρE ,B

σ2E

ρE ,E ρB,E

σ2B

ρB,B ρE ,B



Estimation Comparison



Prices

2000 2005 2010 2015

20

40

60

80

100

120

140

160

180

200 MSFT

IBM



(b) Returns

2000 2005 2010 2015

−15 %−10 %−5 %

0 %5 %

10 %15 % MSFT

2000 2005 2010 2015

−15 %

−10 %

−5 %

0 %

5 %

10 %IBM



Correlation estimateswith average correlation 49%

2000 2005 2010 2015

−20%

0%

20%

40%

60%

80%

EWMA



Correlation estimateswith average correlation 49%

2000 2005 2010 2015

−20%

0%

20%

40%

60%

80%

EWMA

DCC



• Let’s focus on 2008, the midst of the 2007-2009 crisis,when the correlations of all stocks increaseddramatically...



Prices

Jan Mar May Jul Sep Nov Jan

20

40

60

80

100

MSFT

IBM



Returns


−10 %

−5 %

0 %

5 %

10 %

15 % MSFT


−10 %

−5 %

0 %

5 %

10 %

15 % IBM



Correlationswith average correlation 49%


20%

40%

60%

80%

EWMA



Correlationswith average correlation 49%


20%

40%

60%

80%

EWMA

DCC



BEKK



The BEKK model

• An alternative to the MV-GARCH models

• The matrix of conditional covariances is Σt

• A function of the outer product of lagged returns andlagged conditional covariances

• Each pre-multiplied and post-multiplied by a parametermatrix

• Results in a quadratic function that is guaranteed to bepositive semi-definite



• The two-asset, one-lag BEKK(1,1,2) model is defined as:

Σt = ΩΩ′ + A′Y ′

t−1Yt−1A+ B ′Σt−1B

or:

Σt =

(

σt,11 σt,12

σt,12 σt,22

)

=

(

ω11 0ω21 ω22

)(

ω11 0ω21 ω22

)

′

+

(

α11 α12

α21 α22

)

′(

Y 2t−1,1 Yt−1,1Yt−1,2

Yt−1,2Yt−1,1 Y 2t−1,2

)(

α11 α12

α21 α22

)

+

(

β11 β12

β21 β22

)

′(

σt−1,11 σt−1,12

σt−1,12 σt−1,22

)(

β11 β12

β21 β22

)



• The general BEKK(L1, L2,K ) model is given by:

Σt = ΩΩ′ +

K∑

k=1

L1∑

i=1

A′

i ,kY′

t−iYt−iAi ,k

+

K∑

k=1

L2∑

j=1

B ′

j ,kΣt−jB j ,k

• The number of parameters in the BEKK(1,1,2) model isK (5K + 1)/2

• 11 in two asset case• 24 in three asset case• 42 in four asset case



Pros and cons

Pros • Allows for interactions between differentasset returns and volatilities

• Relatively parsimonious

Cons • Parameters hard to interpret• Many parameters are often found to bestatistically insignificant, which suggests themodel may be overparametrized

• Can only handle a small number of assets



Multivariate volatility analysis

• Pick 7 stocks

• Some that did quite well in the crisis

• Others that performed really badly

• And yet others that performed average

• Use DCC to get the conditional correlations

• Normalize the prices of all to start at one at thebeginning of the sample



Normalized prices

2014 2016 2018 2020

2

4

6

8

10

12

14ATVI

UAL

AMZN

CVX

GS

MCD

CLX



Returns

2014 2016 2018 2020

−0.3

−0.2

−0.1

0.0

0.1

0.2

ATVI

UAL

AMZN

CVX

GS

MCD

CLX



Amazon

• Amazon prices jumped by 8.5% at the end of January

• Because of really good fourth-quarter results

• Note how that affects the conditional correlations



SP-500

Jan Mar May Jul

−0.2

0.0

0.2

0.4

0.6

AMZN CVX

GS − UAL

UAL − MCD

ATVI − AMZN

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