Multistep value-at-risk and expected shortfall forecasting using copula-GARCH models

Multistep value-at-risk and expected shortfallforecasting using copula-GARCH models

Eugene Levin

October 12, 2012

Eugene Levin Multistep value-at-risk and expected shortfall forecasting using copula-GARCH modelsOctober 12, 2012 1 / 23

Presentation Structure

1. Motivation

2. Models

3. Model evaluation

4. Data and results

5. Conclusion


Motivation

The problem consists of forecasting value at risk and expectedshortfall of a portfolio of returns

VaRt,α|Ψt−1 = inf{rt,p : Fp,t(rt,p|Ψt−1) ≥ α}ESt,α|Ψt−1 = E (rt,p|Ψt−1, rt,p ≤ VaRt(α))

The idea is to incorporate the dependence structure of returns whilecapturing the information about the individual distributions ofportfolio instruments

Literature often considers single period forecasts

Motivating papers include Jondeau & Rockinger (2006), Gerlach etal. (2012) and Uryasev and Rockafellar (2000)


Motivation

Value at risk vs Expected shortfall

Figure : VaR vs ES


Copula-GARCH models

Copula framework requires a choice of the dependence structure andchoice of univariate margin models

Models for margins are residuals from a GARCH process

2 distributions are considered for the GARCH residuals - student-tand Hansen’s skewed-t

A copula function represents the joint CDF in terms of uniformmargins Fi (Xi ) such that

F (X1,X2, ...,Xn) = C (F (X1),F (X2), ...,F (Xn))


Copula-GARCH models

Two copula models are considered - Gaussian copula and t-copula

Why no asymmetric dependence?

Time-varying correlations are examined using the dynamic of aDCC(1, 1) model (Engle (2002))

Rt is the correlation matrix that is being forecasted

Rt−1 is estimated using IFM method - either from a Gaussian copulaor a t-copula

Qt = (1−M∑

m=1

αm −N∑

n=1

βn)Q̄ +M∑

m=1

αm(εt−mε′t−m) +

N∑n=1

βnQt−n

Rt = Q ∗−1t QtQ∗−1t

where Q∗t contains diagonal elements of Q where the typical element isqij = σij


Models summary

Table : Models’ summary

(G-t-c) Gaussian copula with student-t margins (constant)(G-sk-t-c) Gaussian copula with Hansen’ skewed-t margins (constant)(T-t-c) t-copula with student-t margins (constant)(T-sk-t-c) t-copula with Hansen’s skewed-t margins (constant)(G-t-DCC) Gaussian copula with student-t margins (DCC)(G-sk-t-DCC) Gaussian copula with Hansen’ skewed-t margins (DCC)(T-t-DCC) t-copula with student-t margins (DCC)(T-sk-t-DCC) t-copula with Hansen’s skewed-t margins (DCC)

Estimation of copula parameters is done using the IFM methodproposed by Joe & Xu (1996)

Estimation of DCC and GARCH parameters is done using QMLE


Model evaluation

Model evaluation is done in terms of how often the value at risk isgreater than the actual portfolio return or rp,t < VaRt,α - in this caseIt = 1, otherwise It = 0

Proportion of violations (vRate) should be close to α in a model thataccurately forecasts VaR

For each step ahead forecasts we get a vector I consisting of 0’s and1’s - now we can test for statistical validity of the unconditional andconditional coverage - Christoffersen (1998)

An alternative test is used as well - out-sample dynamic quantile testproposed by Engle and Manganelli (2004) - to test for no correlationin lagged values of I and the VaR forecasts themselves


Model evaluation

When evaluating in terms of expected shortfall, we can find thequantiles corresponding to an ESα and treat the forecasted ES as aninterval forecast

This allows to evaluate in a similar way to VaR, comparing the vRateto the one hypothesised

We can also perform the same tests as for VaR if we know thequantile

In my data set, assuming a student-t distribution, an equally weightedportfolio had an average degrees of freedom of ν = 10.89

ES 5% corresponds to the 1.84% quantile

ES 1% corresponds to the 0.35% quantile


Data and results

Data consists of returns for international stock indices - ASX200,HSI, DAX, FTSE 100, NASDAQ, NEIKEI 225

The total data time frame used → from 17/10/2001 to 27/08/2012 -total of 2777 observations (adjusted for public holidays)

In sample estimation was done using a rolling window of 1765observations

Total out of sample forecasts = 1012

Why this data set? Potential problems?


Data and results

NASDAQ prices

0 500 1000 1500 2000 2500 3000800

1000

1200

1400

1600

1800

2000

2200

2400

2600

2800NASDAQ in and out−of−sample periods

NASDAQ in−sample periodNASDAQ out−of−sample period

Figure : NASDAQ prices


Table : 1 step ahead vRate under equal weights

Equal weights, 5% VaR Equal weights 1% VaR

G-t-c 0.0731 0.0208G-sk-t-c 0.0692 0.0148

T-t-c 0.0652 0.0158T-sk-t-c 0.0692 0.0138

G-t-DCC 0.0731 0.0198G-sk-t-DCC 0.0692 0.0158

T-t-DCC 0.0741 0.0168T-sk-t-DCC 0.0648 0.0158


Table : 1 step ahead vRate under ES minimised weights

ES5, 5% VaR ES5, 1% VaR

G-t-c 0.0919 0.0277G-sk-t-c 0.0919 0.0237

T-t-c 0.086 0.0267T-sk-t-c 0.0929 0.0247

G-t-DCC 0.087 0.0306G-sk-t-DCC 0.0899 0.0217

T-t-DCC 0.0939 0.0287T-sk-t-DCC 0.0899 0.0267


Table : 1 step ahead vRate under MVP weights

MVP 5% VaR MVP, 1% VaR

G-t-c 0.087 0.0287G-sk-t-c 0.084 0.0237

T-t-c 0.0899 0.0287T-sk-t-c 0.087 0.0237

G-t-DCC 0.086 0.0277G-sk-t-DCC 0.0879 0.0237

T-t-DCC 0.0879 0.0267T-sk-t-DCC 0.0868 0.0237


Equal weights, 5% VaR vRate

1 2 3 4 5 6 7 8 9 100.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

Steps ahead forecast

vR

ate

5% vRate − Equal weights

G−t−cG−sk−t−cT−t−cT−sk−t−cG−t−DCCG−sk−t−DCCT−t−DCCT−sk−t−DCCExpected vRate

Figure : VaR vs ESEugene Levin Multistep value-at-risk and expected shortfall forecasting using copula-GARCH modelsOctober 12, 2012 15 / 23

Equal weights, 1% VaR vRate

1 2 3 4 5 6 7 8 9 100

0.01

0.02

0.03

0.04

0.05

0.06

Steps ahead forecast

vR

ate

1% vRate − Equal weights

G−t−c

G−sk−t−c

T−t−c

T−sk−t−c

G−t−DCC

G−sk−t−DCC

T−t−DCC

T−sk−t−DCC

Expected vRate


ES minimised weights, 5% VaR vRate

1 2 3 4 5 6 7 8 9 100

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Steps

vR

ate

5% vRate − ES minimised weights

G−t−c

G−sk−t−c

T−t−c

T−sk−t−c

G−t−DCC

G−sk−t−DCC

T−t−DCC

T−sk−t−DCC

Expected vRate


ES minimised weights, 1% VaR vRate

1 2 3 4 5 6 7 8 9 100

0.02

0.04

0.06

0.08

0.1

0.12

Steps

vR

ate

1% vRate − ES minimised weights

G−t−c

G−sk−t−c

T−t−c

T−sk−t−c

G−t−DCC

G−sk−t−DCC

T−t−DCC

T−sk−t−DCC

Expected vRate


MVP weights, 5% VaR vRate

1 2 3 4 5 6 7 8 9 100.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Steps

vR

ate

5% vRate − MVP minimised weights

G−t−c

G−sk−t−c

T−t−c

T−sk−t−c

G−t−DCC

G−sk−t−DCC

T−t−DCC

T−sk−t−DCC

Expected vRate


Table : ES forecast evaluation

EQ weights 5% ES quantile level 1% ES quantile level

G-t-c 0.0326 0.0069G-sk-t-c 0.0237 0.003

T-t-c 0.0247 0.003T-sk-t-c 0.0217 0.001

G-t-DCC 0.0346 0.0079G-sk-t-DCC 0.0247 0.004

T-t-DCC 0.0336 0.0044T-sk-t-DCC 0.033 0.0043

Correct level 0.0184 0.0035


Table : ES evaluation - ES minimised weights

ES weights 5% ES quantile level 1% ES quantile level

G-t-c 0.0435 0.0158G-sk-t-c 0.0385 0.0089

T-t-c 0.0415 0.0119T-sk-t-c 0.0415 0.0089

G-t-DCC 0.0435 0.0168G-sk-t-DCC 0.0425 0.0109

T-t-DCC 0.0415 0.0099T-sk-t-DCC 0.0405 0.0089



Table : MVP weights - ES forecast evaluation

MVP weights 5% ES quantile level 1% ES quantile level

G-t-c 0.0425 0.0158G-sk-t-c 0.0385 0.0099

T-t-c 0.0405 0.0109T-sk-t-c 0.0375 0.0079

G-t-DCC 0.0435 0.0188G-sk-t-DCC 0.0385 0.0109

T-t-DCC 0.0415 0.0099T-sk-t-DCC 0.0405 0.089



Conclusion

In this presentation I did not include:

Risk-vs-return properties of the portfolio

Whether or not the ES forecasts correspond to the realised ES

How the models performed in the non-crisis period


Multistep value-at-risk and expected shortfall forecasting using copula-GARCH models

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