Asset Allocation using Copula-based Multivariate GARCH Models with Value-at-Risk Constraints

ÆÆÆ ???èèèµµµ10384 ÆÆÆÒÒÒµµµ27720071152271 © © ©aaaÒÒÒ ———??? UDC

a

a

a

¬

¬

¬

Æ

Æ

Æ

Ø

Ø

Ø

©

©

©

Ä

Ä

Äu

u

u

ë

ë

ë

¼

¼

¼ê

ê

ê

õ

õ

õ

GARCH

.

.

.

3

3

3

]

]

]

˜

˜

˜¥

¥

¥

A

A

A^

^

^

Asset Allocation using Copula-based Multivariate GARCH

Models with Value-at-Risk Constraints

+

ï

’

•

“6¶

µ

ö

[

œ

Ç

;

’

¶

¡µ

7

K

Æ

Ø©

J

Fϵ

2 0 1 0

c

Ø©

‰

F

ž

mµ

2 0 1 0

c

Æ Çƒ

Fϵ

2 0 1 0

c

‰

F

”

¬

Ì

R

µ

µ

<

µ

2010

c

c

c

厦门大学博硕士论文摘要库

f

f

f€

€

€

Œ

Œ

ή

Æ

ÆÆ

Æ

Æ

Ø

Ø

Ø

©

©

©

M

M

M5

5

5

(

(

(²

²

²

<

¥

Æ

Ø©

´

<

3

“

•

e

,

Õá

¤

ï

Ä

¤

J

"

<

3Ø©

Š

¥

ë•

Ù

¦

‡

<½

8

N®²uL

ïÄ

¤J§

þ

3©

¥

±

·

•

ª

²(I²

§

¿Î

Ü

{

Æ5

‰

Ú

5

f€

Œ

ÆïÄ

)

Æ

â

¹

Ä5

‰

£

Á

1

¤

6"

,

§

T

Æ

Ø©•£

¤

‘K

£

|

¤

ïÄ

¤J§

¼

£

¤

‘K

£

|

¤

²¤½

¢

¿

]

Ï

§3£

¤

¢

¿

¤

"

£

ž

3

±þ

)ÒS

W

‘

K½‘K|

K

I

<½

¢

¿

¶

¡§™

k

d

‘

(

²

S

N

§

Œ±ØŠA

O

(

²

"

¤

(²<£\¶¤µ c F

厦门大学博硕士论文摘要库

f

f

f€

€

€

Œ

Œ

ή

Æ

ÆÆ

Æ

Æ

Ø

Ø

Ø

©

©

©

Í

Í

Í

Š

Š

Š

¦

¦

¦^

^

^(

(

(²

²

²

<Ó

¿f€

Œ

Æ

Š

â

5¥

u<

¬

ÚI

Æ

^

~

6

1

¢–

•

{

6

5½

3Ú

¦

^

d

Æ

Ø©§

¿•

Ì

+

Ü€½

Ù•

½

Å

x

Æ

Ø©£•)

’Ÿ

‡Ú>

f

‡¤§#

NÆ

Ø©

?\f€

Œ

Æã

Ö

,

9

Ùê

â¥

!

/

"

<Ó¿f€

Œ

Æ

ò

Æ

Ø©

\

\

I

Æ

¬!

a

¬

Æ

Ø©

ï

ü

ê

�1

u

¢

§

ò

Æ

Ø©

IK

Ú

Á

‡®?

ч§

æ^K<

!

<½

ö

Ù

§•

ª

Ün

E

›

Æ

Ø©

"

Æ

Ø©

á

u

µ

£

¤

1.

²f€

Œ

Æ

—

”

¬

"

ؽ

—Æ

Ø©§

u

c

F)—

§

)—

·

^þ

ãÇ

"

£

¤

2.

Ø

—

§

·

^þ

ãÇ

"

£

ž

3

±þƒA

)ÒS‹

/K0

½WþƒA

S

N

"

—Æ

Ø©

A

´

®²f€

Œ

Æ

—

”

¬

"

½L

Æ

Ø©§™

²f€

Œ

Æ

—

”

¬

"

½

Æ

Ø©

þ

•

ú

mÆ

Ø©

"

d

(

²

9

ØW

§%

@

•

ú

mÆ

Ø©§

þ

·

^þ

ãÇ

"

¤

(²<£\¶¤µ c F

厦门大学博硕士论文摘要库

Á

Á

Á‡

‡

‡

OÂÃS • - • Ý ´ïÄÝ]|ܯK Ø%"DÚ•{{ü/

b TÝ ð½§Engle£1982¤JÑARCH .± §Œþ©zm©¦^ARCHa

. OžC ^‡• - • Ý " ¡ ˜‡æN´§í S Œ±b

©Ù¼êa.š~k•§ƒcNõÆöb í S Ñl ©Ù½öt©Ù§

ù†7KžmS A ؃ÎÜ"glë ¼ê£Copula¤ VgÚ\7KÆï

Ä+•±5§ ½(¹ í S ©Ù¼êC É~{ü"˜ Æöm©òë

¼êÚGARCH .(Üå5§ïÄÝ]|ÜÀJ¯K"LeeÚLong£2009¤3ƒc ©zJÑ Äuë ¼ê ü GARCH . Ä:þ§JÑ Äuë ¼ê õ GARCH ."T .Ø UÏLõ GARCH•{•x] ÂÃS ƒm ^‡

ƒ'5£Conditional Correlation¤§ÓžŒ±ÏLë ¼ê5Ó¼ÂÃS ƒm š

Õá5£Dependence¤"

©U? Äuë ¼ê£Copula¤ õ GARCH .£CMGARCH¤ §

¿rU? .Ú\ ˜3xdŠ£VaR¤ å ] ˜¯K¥"3¢

yÜ©§^ ¦½|† ®½|êâu T . A ^ J"Äk§ © rU? CMGARCHÚ\] ˜¯K¥" ©O\ LeeÚLong £2009¤JÑ CMGARCHëê . «a§X3 ˜>S©Ù¼êžO\ Skewed Studentta

.§3 ˜ë ¼êžO\ SJCa. "Ùg§ ©U? nØ .§rÄuë

¼ê õ GARCH .Ú\ Campbell £2001¤JÑ ˜3xdŠ£VaR¤

å Ý]|ÜÀJµe¥"U? ëê .Ø U(¹•xÂÃS ©Ù§ „U3½|¡ e1ºxž oÝ]ö"• § ©¦^IS½| ½†®½êâ ?1¢y©Û§¿•âÚO•I' ØÓ . ` "¢y(JL²§ ©U? .3] ˜¥ Ly²w`uÄO .§ …Ta .ÏL n«ÚO•I u §w«T .3½|¡ e1ºxžU•Ð/ oÝ]]7" '…cµë ¼ê§õ GARCH§3xdŠ§] ˜ – I –

厦门大学博硕士论文摘要库

Abstract

Estimating the variance-covariance matrix is the core work in asset allocation. Tradi-tional econometric models assume a constant one-period forecast variance. Engle (1982) proposed the ARCH model to estimate time-varying conditional variance. After then, many researches apply these models to financial data analysis. However, a vast of the previous literature assume that the standardized errorεtfollows normal distribution or student-t dis-tribution, even though it’s not consistent with the well-known asymmetry and excess kurtosis in financial data although. Recent studies employ the popular copulas theory to construct the uncorrelated dependent errors. The copula theory makes it much flexible in selecting the marginal distributions and dependence structure for a joint distribution. Lee and Long (2009) propose the copula-based multivariate GARCH models (CMGARCH) after consid-ering the uncorrected dependent error term that was omitted by the previous studies, these flexible models can capture conditional correlation (by MGARCH) and dependence (by a copula) separately and simultaneously for non-normal multivariate distribution.

This paper contribute to current literature in both theory and application. First, it ap-plies the copula-based multivariate GARCH models in asset allocation. The CMGARCH model in Lee and Long (2009) is the first one that models multivariate GARCH with copula distributions, and can model conditional correlation (by MGARCH) and dependence (by a copula) separately and simultaneously for non-normal multivariate distribution,the current paper tests its usefulness with international stocks data and foreign exchange data. Second, we extend Campbell’s (2001) portfolio selection method into the new class of parametric models, copula-based GARCH models. In this framework, one can fit the non-normal dis-tributed financial data in higher conditional moments, at the same time, it can protect the investment under downside risk by setting a VaR constraints in optimizing process. Thirdly, we estimate different CMGARCH models in the empirical section, and suggest best mod-els that may be well-performed in portfolio section in stock market and foreign exchange market.

Key Words: Copula, Multivariate GARCH, Value-at-Risk, Asset allocation

– III –

8

8

8

¹

¹

¹

Á

Á

Á‡

‡

‡

. . . .

I

Abstract . . . III

1

1

1

˜

˜

˜Ù

Ù

Ù

Ú

Ú

Ú

ó

ó

ó

. . . .

1

1.1 ÀÀÀKKK µµµ . . . 1 1.2 ïïïÄÄÄSSSNNNÚïÚÚïïÄÄÄ¿¿¿ÂÂÂ. . . 3 1.3 ((( SSSüüü . . . 5

1

1

1

Ù

Ù

Ù

©

©

©z

z

z

n

n

n

ã

ã

ã

. . . .

7

1

1

1

n

n

n

Ù

Ù

Ù

Ì

Ì

Ì

‡

‡

‡

ï

ï

ï

Ä

Ä

Ä

•

•

•{

{

{

. . . 11

3.1 ëëë ¼¼¼êêê(Copula) . . . 11 3.2 õõõ GARCH ... . . 12 3.3 ÄÄÄuuuëëë ¼¼¼êêê õõõ GARCH .... . . 13 3.4 [[[444ŒŒŒqqq,,, OOO£££QMLE¤¤¤. . . 14 3.5 333xxxdddŠŠŠ£££VaR¤¤¤. . . 15 3.6 •••```|Ü||ÜÜ --- . . . 16

1

1

1o

o

o

Ù

Ù

Ù

¢

¢

¢

y

y

y

©

©

©Û

Û

Û

. . . 21

4.1 êêêâââ . . . 21 4.2 CMGARCH OOO††† ...ÀÀÀJJJ. . . 23 4.3 ]]] ˜˜˜ . . . 37 – V –

厦门大学博硕士论文摘要库

8 ¹ 4.4 £££ ÿÿÿÁÁÁ . . . 39

1

1

1

Ê

Ê

ÊÙ

Ù

Ù

(

(

(

Ø

Ø

Ø

. . . 43

ë

ë

ë

•

•

•

©

©

©z

z

z

. . . 45

—

—

—

. . . 50

– VI –

厦门大学博硕士论文摘要库

Contents

Abstract . . . III

Chapter 1 Introduction . . . .

1

1.1 Research Background. . . 1 1.2 Research Object . . . 3 1.3 Paper Structure . . . 5

Chapter 2 Literature Review. . . .

7

Chapter 3 Model . . . 11

3.1 Copula . . . 11 3.2 Multiplevariable GARCH. . . 12 3.3 Copula-based MGARCH . . . 13 3.4 QMLE . . . 14 3.5 Value-at-Risk . . . 15

3.6 Optimal Portfolio Weight . . . 16

Chapter 4 Empirical Analysis. . . 21

4.1 Data . . . 21

4.2 CMGARCH Model Estimation and Selection . . . 23

4.3 Asset Allocation. . . 37

4.4 Back Testing . . . 39

– VII –

CONTENTS

Chapter 5 Conclusion . . . 43

Reference . . . 45

Acknowledge . . . 50

– VIII –

厦门大学博硕士论文摘要库

1

1

1˜

˜

˜

Ù

Ù

Ù

Ú

Ú

Ú

ó

ó

ó

1.1

À

À

À

K

K

K

µ

µ

µ

XÛ OÂÃS • - • Ý ´ïÄÝ]|ܯK Ø%"DÚ•{ {ü/b TÝ ð½§¿† ¦^Äu{¤êâ O(J"Engle´ïÄž CS k1ö"Engle31982cÄgJÑARCH .§T .Œ± OžC ^ ‡• - • Ý "Bollerslev£1986¤rT .í2 ˜„/ª§=GARCH ."ƒ §ü GARCH . í2 õ /ª§–d§^žC •{©Û ÂÃS • - • Ý ¯K óä®²Ä ¤."ƒ Œþ ïÄör da•{A^ 7KžmS ¯K ©Û¥"~X§Bollerslev£1986¤3b ƒ'XêØC œ¹e5 Oî³À1NX¥ØÓÀ1ƒm• Ý ( C z¶KronerÚClaessens£1991¤OŽ3õ«À1Œ±ÀJž •`ÅÖ|Ü",

§BaillieÚMyers£1991¤¦^GARCH . OŒmû¬ÏÀ •`éÀ'ǧ

•ÑT'Ç¿šð½ØC "éARCHƒ' .••[ n㧞ë•Bollerslev

£1992§1995¤§BeraÚHiggins£1993¤§Pagan£1996¤§Palm£1996¤§Shephard

£1996¤ÚBauwens £2006¤ "

EngleÚKroner£1995¤JÑBEKKõ GARCH .5( `z§S¥ ^

‡• Ý ½"BEKK-MGARCH . ˜‡"€´éJ)ºˆëê ¹

Â"Bera £1997¤uy3OŽ•`éÀ'Çž§BEKK-MGARCH . Ly

¿ØÐ"Engle£2002¤JÑ Ä ^‡ƒ'5 .£DCC-MGARCH¤§T .Ø

2† Ž^‡• Ý Ht§ ´ Ží S εt • - • Ý "T .Ø

ä ü GARCH (¹5§ …vkDÚõ GARCH .@oE,"A †dÓ

ž§TseÚTsui£2002¤JÑ ŒCƒ'5õ GARCH .£VC-MGARCH¤§d

•{ÏL OÂÃS rt ^‡ƒ'XêÝ 5 OÂÃS žC • - • Ý

"

– 1 –

1.1ÀK µ ±c ïÄöÊH† b í S εtÑl ©Ù½öIO t©Ù", § Œþ©zé7Kêâ¢y©Û (JL²§7KžmS êâäkk¸þ—Úšé ¡ A:§† ©ÙÚt©Ù ÚOA ¿Ø¬Ü§{ü b N´ — . ½†Ø"glë ¼ê£Copula¤ Ú\ 7KÆ+•±5§?nùa¯KC { ü"ë ¼ê£Copula¤nØ1954cÄgÑy3êÆ©z¥§ † 2000cÐâ 7KÆöuy¿m©¦^§8c®²š~•1§ØØ´I „´ISÆâ.ÑkŒ 1©z¦^TnØ©Û7KžmS êâ"ë ¼ênؕ̇ `:´ ïéÜ ©Ùž§é>S©Ù±9éÜ©Ù /ª†ëê ÀJš~(¹"Patton£2004¤ ¦^ë ¼ênØJÑ ˜@õ ëê .§¿A^ Out-of-Sample«mSÝ ]|ÜÀJ ïÄ¥"Patton ïÄ(JL²§3vk ˜ñ˜•› ^‡e§ Äuë ¼ê .ØØ´3²L¿Âþ„´3ÚO¿ÂþÑU •Ð J"JondeauÚRockinger£2006¤rë ¼êÚTGARCH(Üå5 ï .©ÛI

S ¦•ê"Hsu £2008¤æ^Äuë ¼ê GARCH . O•`éÀ'ǧ

¢y(JL²§ØØ´3 S„´3 §Äuë ¼ê GARCH . Ly Ñ`uÄO ." , §·‚J9 ù Äuë ¼ê ARCHa .§3 ï .žÑ¦^ ü GARCH •{üÕ Oz‡ÂÞmS §ù §ù .• O ü ^ ‡• § vk•Ä ^‡ƒ'5½ö^‡ • "Ï•ù .¦^ ë ¼ ê5 ïí S ©Ù¼ê§¤±§ù .••Ä ÂÃS ƒm ^‡ šÕá5 vk•Ä í S ƒm ^‡ƒ'5"¯¤ •§šÕá¿Ø¿›

X؃'5"LeeÚLong£2009¤JÑ ˜«# Äuë ¼ê õ GARCH

.£CMGARCH¤§T .Ø UÏLõ GARCH•{•x] ÂÃS ƒm ^‡ƒ'5£Conditional Correlation¤§ÓžŒ±ÏLë ¼êÓ¼ÂÃS ƒm šÕáA5£Dependence¤§, §Ï•æ^ ë ¼ê •{§3 ï .ž•(¹§ Øüüb ÂÞmS Ñl ©Ù½IO t ©Ù"D Ú õ GARCH .=´T . ˜‡A~"ù«•{1˜grë ¼êÚõ GARCH•{(Üå5" – 2 –

厦门大学博硕士论文摘要库

Degree papers are in the “Xiamen University Electronic Theses and Dissertations Database”. Full texts are available in the following ways:

1. If your library is a CALIS member libraries, please log on http://etd.calis.edu.cn/ and submit requests online, or consult the interlibrary loan department in your library.

2. For users of non-CALIS member libraries, please mail to [email protected] for delivery details.