Bayesian Analysis of Stochastic Volatility Model and Portfolio Allocation

A C T A U N I V E R S I T A T I S L O D Z I E N S I S FOLIA OECONOMICA 192, 2005

A n n a P a j o r *

BAYESIAN ANA LY SIS OF STO CH ASTIC VOLATILITY M O D E L A N D PO RTFO LIO ALLOCATION**

Abstract. In this paper we present the multivariate stochastic volatility model based on the Cholesky decomposition. This model and the Bayesian approach is used to model bivariate daily financial time series and construct an optimal portfolio. We consider the hypothetical portfolios consisted o f two currencies that were most important for the Polish economy: the US dollar and the German mark. In the optimization process we used the predictive distributions o f future returns and the predictive covariance matrix obtained from the MSV model.

Keywords: multivariate stochastic volatility model, Bayesian analysis, portfolio allocation, Markov chain Monte Carlo.

JEL Classification: C l l , C l5, C32.

I. INTRODUCTION

T h e p o rtfo lio selection problem under kno w n cov ariance m atrix was originally considered by M arkow itz (1959). His ap p ro ach consisted o f tw o steps. F irst, form ing a set o f efficient portfolios. Second, selecting from the efficient set the one p o rtfo lio w ith the m o st suitable c o m b in atio n o f risk and re tu rn . In this p ap e r the covariance m atrix o f asset re tu rn s is m odeled using a m u ltiv ariate stochastic volatility process. T his a p p ro a c h is based on m odeling variances and covariances as unobserved sto chastic processes. A n earlier B ayesian a p p ro a c h to the po rtfolio selection problem was considered by W inkler and B arry (1975), and P oison and Tew (2000). But none o f th eir m odels considered a form al treatm en t o f stochastic volatility. T he w ork by A gu ilar and W est (2000) introd uced the d yn am ic fa cto r m odels

* Dr (Ph.D., Assistant Professor), Department o f Econometrics, Cracow University of Economics.

** Research supported by the grant from Cracow University o f Economics in the year 2004. 1 would like to thank Jacek Osiewalski for helpful comments and suggestions.

w ith stochastic volatility into dynam ic portfolio selection problem . In this p ap e r in the o p tim izatio n process we use the predictive d istrib u tio n s ol fu tu re retu rn s and the predictive covariancc m atrix o b tain ed from the m ultivariate SV m odel. It allows us to introduce both param eter and predictive uncertain ty into the portfo lio selection problem .

In this p ap e r the bivariate stochastic volatility m odel is used to describe the daily exchange rate o f the G erm an m ark against the P olish zloty and the daily exchange ra te o f the US dollar against the Polish zloty. Based on these tw o currencies we consider the Bayesian portfolio selection problem . In o rd e r to o b tain p osterior and predictive d istrib u tio n s o f the q uan tities o f interest, we use M ark o v chain M o n te C arlo (M C M C ) m eth o d s, m ainly the M etro p o lis-H astin g s algorithm within the G ibb s sam pler to sim ulate from the p o sterio r distrib u tio n .

2. AN MULTIVARIATE STOCHASTIC VOLATILITY PROCESS

Let Xjt d en o te the price o f asset j (or exchange ra te as in o u r application) at tim e I fo r j = 1, 2, ..., n and t = 1, 2, ..., T. T h e vector o f grow th rates У, = (Ук> У2„ •••. Ум)', each defined by the form ula yJ t = 100 l n f o / f y - j ) , is m odeled here using the basic V A R (l) fram ew ork:

(1) У( = 0 + R y ,- i + t = 1. 2> •••» T f T + 1, ..., T + k,

w here T denotes the num ber o f observations used in estim ation, and к is the forecasting horizon. In (1) 6 is an n-dim ensional vector, R is an n x n m atrix

o f p aram eters, an d is a m u ltiv ariate SV process (M SV). O u r m ultivariate stochastic volatility process is based on I say (2002). We assum e th at, conditio­ nally on laten t variable vector 0 , = (<?n,t, <?2 2 ■■■, Чпп,„ Qn.t, •••> 9m,t, Ян,»

<?42.„ ....

4n2.„

....

(introduced later),

ą,

follows a m u ltivariate G a u ssia n d istrib u tio n with m ean vector 0,лх1] and the covarian ce m atrix £,:

ą (| 0 ( ~ N (O lnxl], E,), t e { l , 2, ...

,T,

T + 1, ..., T + k.

Follow ing the definition in T say (2002) we p rop ose for E, to use the C holesky decom position:

(2) x ( = l ( g ( l ; ,

where L t is a low er trian g u lar m atrix with un itary d iag o n al elem ents, G, is a d iag o n al m a trix w ith positive diag onal elem ents:

(3) L , = 1 0 9 2 1 . 1 1 ÍS l.t 432,1 G , = 4 n l , t 4 n 2 , l . . . ." Я п п - 1 , 1 1 Í 1 1 . I 0 0 . . . 0 ■ 0 < ? 2 2 ,( 0 0 0 0 0 0 • • • 0

F o r series qlJt„ ij =j and { ln q jJi(}, sim ilar to the un iv ariate SV, we assum e stan d ard univariate autoregressive processes o f o rd e r one, nam ely

In q jj. t = ľij + (PúIn q j j , t- 1 + 7 = 1, 2, ... n, 4i],i — У у+ PijQv.t-i + a i/lij,t> j, I6 {1> 2, ... n}, 1> Я

i/ü., ~ N (0 ,1 ), tfij't -1- t, s e {1, 2, ..., T + k}, i, j , r, p e { 1, 2, ..., n},

( t ^ s v i j z r s / j j zp), where J. denotes independence.

N o te th a t the positive definiteness o f E, is o b ta in e d by m o delin g In qjj't instead o f qjjtt (the m atrix E, is positive definite if qjJtt > 0 for j = \ , 2, ..., ri). If \ < P i j \ < 1 (i, j — I, 2, ..., n) then {qiJit} an d {In qJJtt} are sta tio n a ry and their m arg inal distrib u tio n s are n o rm al w ith m ean ľy /(l - <Pu) and variance crg/(I - ęfj). U sing p roperties o f th e co nditional m ean, we o b tain

Щ , ) = о (лх1„ E (ą ,ą D =

i for t = s

11лхл] for t Ф s '

w here Í2 is a positive definite m atrix. H ence Ł,, is a m u ltiv ariate w hite noise process. F o r the b ivariate case, we have

(4) E =

P7“’' CTl2,‘l

=

Г1 °T«- °T

' L ^ l . t a 2 2 , t _ _ 4 2 i . , t i J L O <322,t_Jl_

1 Í 2 1 , t

T h e volatility eq u atio n s are

ln 4 n ,f — У н + <Pii ln 9 n , t - i + a iiVu,t>

ln < ? 2 2,t — У22 + <P22 1° 422,«- 1 + a 22r]22,f> 421,1 — У 21 + ^21 421,t - l + a 2 l rl2l.t,

lid

where Ц, = (Пн..иЧ2 1лЧгг,д' and i\, ~ N (0 13x „ , I 3).

U sing e q u a tio n (4), we o btain

i l l , Г ill,l? 2 l,I

/31 l.r^f 21 .t 9ll,I<?21, t<Í22,t

C o nsequently, the co nditional co rrelatio n coefficient betw een £ 1( an d č,2( is tim e-varying an d stochastic:

In this m odel, the conditio n al variances and covariances are th e latent (unobserved) variables. A dditionally the un con d itio n al co varian ce m atrix o f Ł,, exists and takes the form:

w here da = g1_ «,«+2<1~*íT> for i = 1, 2.

T h e SV m odel is able to m odel both th e tim e-varying conditional co rrelatio n coefficients and the variances o f retu rns. T h e first m ultivariate SV m odel proposed in the literatu re by H arvey, ct al. (1994) allowed the variances o f m u ltiv ariate retu rn s to vary over tim e, b u t con strained the co rrelatio n s to be co n stan t. T h e results presented by Osiew alski and Pipień (2004), w ho m ad e a Bayesian com parison o f various b ivariate A R C H -ty p e specifications, indicate th at the assum ptions o f constant conditional covariances o r co rrelatio n s are strongly rejected by the data. P itt and S hephard (1999) p ro posed th e facto r SV m odel, w hich allow s a p arsim o n io u s represen tatio n o f the tim e series evolution o f covariances w hen th e n u m b er o f series being m odelled is very large. Simple m ultiv ariate factor m odels for SV processes

for each t = 1, 2, ..., T + k.

^ u ľ2i / ( l — V2 1 )

have been suggested, bu t n o t applied, by Jac q u ie r et al. (1995, 1999). A practical d raw b ack o f stochastic volatility m odels is the intractab ility of the likelihood function. Because the co ndition al cov arian ce m atrix is an unobserved com ponent and the m odel is no t G aussian, the likelihood function is only available in the form o f a m ultiple integral.

2. BAYESIAN INFERENCE FOR BIVARIATE SV

Let 0 , d en o te the 3-dim ensional vector 0 , = q2i,„ g2 2.1V and let

у/,_! d en o te th e h istory o f the process {y,} up to tim e t — 1. T h e conditional d istrib u tio n o f y t (given the past o f the process the p aram eters and the latent variable vector 0 ,) is b ivariate norm al w ith m ean ц, = 8 - b R y ,-!

and a covariance m atrix L (, th a t we den o te (y i,(,.>>2,ryi&,R,0 t, V 'r-i ~ N (n „

L,). T h e density function is given as follows:

K > > > (- b0 (,O A R,qo) = f N( y M X , ) ) = (2tc)- 1 |2 : t | - ° - 5

ехр{ -0.5(у, - Ц,)ГЕ," ‘ (У, - Ц,)},

w here 0 = ( y i i , y 22,y21,<pl l ,<p22,<p2 l ,cr2ll , a22, a2iy is the v ec to r o f the sto ­ chastic volatility param eters, q0 = ^ n q i i i0\ n q 2z,o, Яи.оУ is treated as an ad d itio n al vector o f param eters and estim ated jo in tly w ith o th er p a ra ­ m eters.

T h e density o f the d a ta (given the p aram eters) is the m ixtu re over 0 = (0 i , © 2, ..., 0 'г У d istribution:

p (y|0,ô,R ,q o) = J p (y |0 ,0 ,R )p (0 |O ,q o)d 0 ,

w here у = (y 1} y 2, ■■■, Ут)' d e n o te s the full d a ta set; p ( y |0 ,6 ,R ,) =

T

Г Ш М щ Д У , ^ n o tatio n f N( ' |n ,,£ t) is used to indicate the density function t= i

o f n o rm al d istrib u tio n w ith a vector o f m eans ц( an d a co varian ce m atrix Ľ,. In the b ivariate case the m arginal likelihood over the p aram eters o f the m odel is defined as a 37" - dim ensional integral. Since th e integral can no t be solved analytically, the m axim um likelihood m eth o d s are co m p u tatio n ally very difficult. In this p ap e r we use th e B ayesian a p p ro a c h to m ak e inference a b o u t the p aram eters and laten t variables o f the SV m odel. T h u s, we trea t the p aram eters as the ran d o m variables and o u r u n ce rtain ty a b o u t them express by specifying a p rio r d istrib u tio n on the p a ra m e te r space, which we d e n o te by p(ô,R ,0,qo). F o r ô and R we assum e th e m u ltiv a ria te

stan d ard ized norm al p rior N ( 0 ,I6), truncated by the restrictio n th a t all eigenvalues o f R lie inside the unit circle. They are ind epen dent with the prio r d istrib u tio n s for the rem aining param eters:

(Уц, <Pij)'~N(Р о Л о 1) I ( - i.i ) (Vij), where p 0 = 0 , A o = 0.01 • I, i, j e { 1, 2}, a f j ~ l G( v 0,s0); where the hyperparam eters are v0 = l , s 0 = 0.005 (cf.

Jacquier, P oison and Rossi 1999)

M ii.o ~ N (z0,p 0), i = 1, 2; Í2i .o ~ N ( z0,/i0), w here z0 = 0, u0 = 100.

T he sym bol IG(v0, s0) denotes the inverse gam m a d istrib u tio n with the m ean and variance .v0/(v0 — 1) and A'o/[(vo — )2( vo — 2)], respectively (then

here the m ean and higher m om ents as well d o n o t exist), I(_ i,d (.) denotes the in d icato r function o f th e interval (-1 , 1). We use the sam e specification o f p rio r d istrib u tio n for (y^, ipy, afj)' as in the u n ivariate SV m odel (cf. P ajo r 2003). T h e prio r for <ри is truncated to force the statio n arity on In qJjit and qlJtt. T his im plies th a t the su p p o rt o f сри is (-1 , 1) - the region o f statio n arity .

T h e jo in t p o ste rio r distribution o f the param eters and laten t variables is given by Bayes theorem :

p(0 ,O ,6 ,R ,q o |y) oc p (y |0 ,0 ,R ,q o)p(0|O ,qo)p(8,R,O,qo)

w here p(0 |O,qo) is a p ro d u ct o f individual density fu n ction o f the elem ents

o f E,:

•n qjjj I Vr- 1 ~ N(yjj + ipjj In qjJit _ !, ajj), j = 1, 2

and <?2i,rlV t - i ~ N ( y2i + (Рг\

U n d er th e Bayesian fram ew ork th e volatility vector 0 consists o f augm ented param eters (this idea was pioneered in M ark o v chain M o n te C arlo by T a n n e r an d W ong 1987). T h u s we augm ent the p aram eter space by the laten t variables and can th ink o f the laten t variables as n uisance param eters th a t are “in teg rated o u t” by th e G ibbs sam pler.

T h e B ayesian paradigm provides a n atu ra l fram ew ork fo r prediction. T h e jo in t predictive density o f the futu re values o f Ľ, an d y, is:

r+ t

p(y/ , 0 / |0 ,e ,S ,R ) = П / » (уЛшД У /иО11 Í i u l y n + P ii I n í n j - b í T i i ) t=T + 1

w here y/ = ( V r + i , У г+2> •••» Ут+к) denotes the vector o f forecasted values,

0 / is a vector o f forecasted unobserved variables. T h is density is also co n d itio n al on som e initial observations, which are n o t show n in o ur n o tatio n .

In o rd e r to generate draw ings from the jo in t p o sterio r d istrib u tio n of unknow n param eters and latent variables we use the G ibbs sam pling, a M onte C arlo M ark o v C hain m ethod. It generates draw ings from a jo in t d istribution (as a stationary distribution) by sequentially sam pling from the full conditional d istrib u tio n s (e.g. G a m e rm an 1997). T h u s, we co n stru c t a G ibb s sam pler w ith lim itin g d is trib u tio n eq u a l to th e jo in t p o s te rio r d is trib u tio n p (0 ,O ,6 ,R q o|y). T he G ibbs sam pler for the u n ivariate stoch astic volatility m odels, presented by Jacq u ier et al. (1994); P ajo r (2003), can be easily generalized to the bivariate ease. H ere we have to d ra w the 3 T + 18- d im entional vector (0,O ,8,R ,qo), which is split in to 3 T + 8 sep arate G ibbs

steps, T+ 8 o f which can be conducted easily, while the rem aining dim ension

require a bit m o re effort. T h e d istrib u tio n o f ß = (ô1,rí l , r l2, S2,r2 í ,r22y given all rem ain in g p aram eters is given as follows:

is the ch aracteristic function o f set A defined as:

Л = { ( f i u ri2tr2u r2i ) e ^ * : all eigenvalues of matrix R lie inside the unite circle}. M atrix X, in and ß* is defined as:

T h e co n d itio n al po sterio r distrib u tio n s o f oy and (yy,Py)' are defined by the follow ing inverse gam m a and tru n cated norm al distribu tio ns:

4. MCMC ALGORITHM FOR BIVARIATE SV MODEL - THE FULL CONDITIONAL DISTRIBUTIONS

(5) P(PI

• ,y ) a c / „ ( ß

I

ß* -,B„

l)

IA(rlur

1 2

,r2i,r

2 2

),

T T

1 У1Г- 1 У 2t-i 0 0 0

0 0 0 1 у и - i y2t - i

with

г

■i*u = s0 + 0.5 £ (ln q,,j - у„ - <ри In , ) 2, t —1

Г

s *y = S 0 + 0 .5 X fay,, - y,j - (»y ду ,,_ j ) 2, i # 7 ,

P((ľy,9»y)'l • ,y)«;/w((ľi;,<í>iy)'|a*ij>'TUA*i])/ (-ia)(í»y). where:

a*ij —

A * y ( W 'y Q y + Ctí2 A 0B 0), A „ y — W ý W y + O y A 0,

W y = (viy , v2y, ..., vry)', v,H = (l,ln g ílr_! ) ', Qij = (ln

qij'i

,

laqiltTy,

vuj = 0,qij.i-i)', Qij = (qtj,i,

•••,

чи.тУ

d >a

i*j.

where byx = tpfj/(rfj+l/u0, bjj2 = (ęa i n q JJ,i - (p)jyjj)ltfj + z 0l u 0, for i,j e {1,2}

and blj2 = (fPiflij, i - <РцУц1(Тц + z0/u0 for i = 2, j = 1.

A ll these conditio n al d istrib u tio n s can be easily draw n . In (6) we have

the inverse gam m a d istrib u tio n and in (5) and (7) we have th e trun cated m u ltiv ariate n o rm al density functions, so th a t rejection sam pling can be applied.

T h e co n d itio n al p o sterio r distrib u tio n s o f the unobserved variables are the following:

(9) p(q2i,rl' ,У) =/jv(<?2i,rlíW a fi>a rI1)>

w here fo r l < i ^ T - l : a, x = ( y u - p , i )2/ q2 2,< + ( 1 + a u =

= ( y It — 1 t)(y 2t — H2t)l 422,1 + (ľ2l0 — 4>21) + P l l f a u t - 1 + 921,1 + l ) ) / CT2 1, a n d

fo r t = T: atl = ( y i t - H u ) 2l q 22.t + l / a l u a,2 = (,Уи ~ Hu) ( y2t ~ +

+ 0/21 +

?>2l921,f-l)/°21-p(.4ij,

o l 1 >y) — / w f a y . o l ^ b y i 1 , b y i ) ,

(1 0)

p (9 n ,tl - . у ) « ---oTs exP

9i l,r 2^2(l n 9 п д ~ st) 2 b

w here fo r l < i < T - l : sf = (уи (1 -? > ii) + PiiOn<Zii.«+i + ln 9 n , t - i ) / ( 1 + P i i ) . = ^ i , / ( l + P1 1), and fo r t = T: s T = y iv + tpl t l n g n ,r _ „

T h e kernel o f density (10) is n o t stan d ard , so in o rd e r to sam ple from (10) we use an accept/reject M etropolis. T h e candid ate generating distribution for the M erto p o lis algorithm is obtained by ap p ro x im a tin g density (10) - the log-norm al kernel in (10) is appro x im ated by an inverse gam m a with the sam e m ean and variance as this log-norm al d istrib u tio n (cf. Jacq uier et al. 1999). T his yields an inverse gam m a d istrib u tio n from which direct draw s are m ad e (the p ro d u c t o f tw o inverse gam m a is still an inverse gam m a). T h e ca n d id a te generating density is:

9 ( 9 i u l ' ) °c 9fi(!t+1)exP{ — 0,lqiUl}, w here <P = V + 0.5; 0, = (<p - 1.5)exp {s, + 0.5<r2} + ( y lt - ц и ) 2/2. 1 — e * n 14 / , 4 1 { - [ ( У и - Ц и ) Ч 2 1 , , - ( У г t - ^ 2 t ) ] 2\ < " > --- --- --- j w here fo r l < i < T - l : s, = (y22(l - <»2 2) + <Pn{\n 9 2 2.1 + 1 + In «2 2,1-1))/ (1 + ^2 2), = cri2/ ( l + <p222), an d fo r £ = T: s T = y 22 + ę 22\ n q 22<T- „ „ 2 _ _2 CT* — CT2 2

-In this case a g en erato r distrib u tio n for the M etro p o lis alg orithm could be an inverse gam m a d istrib u tio n with density function:

9(9 2 2,il ' ) oc q2^ t+ J)exp { - 0 Jq 22>,},

w here

1 — 2e°* CP = l _ é - l + 0 -5;

0, = (<p- 1.5) exp {ä, + 0.5<t*} + [(>»!, - Ми)Я2и ~ (Угг ~ H2t ) Y ß

-We stress th a t a M etropolis chain m ust be used w ithin the G ib b s sam pler. T hus we create the M etropolis subchains within the M ark o v chain constructed by the G ibbs sam pler to sam ple from the n o n -sta n d a rd distribu tio ns.

5. PORTFOLIO ALLOCATION

In this section we present the application o f the B ayesian inference in the p o rtfo lio allocation problem . Let n denote the n u m b er o f assets and let y( = (y lt, y 2„ ..., ynl) be the vector o f the rates o f re tu rn on assets from the period t — 1 to t. A portfolio a t tim e t is defined by a vector w, = (w 1(, w2t, •••, wm)', w here wit is the fraction o f w ealth invested in asset i ( l < i < n ) .

T h e re tu rn on th e p o rtfo lio th a t places weight wit on asset i at tim e t is sim ply a weighted average o f the returns on th e individual assets. T he w eight applied to each re tu rn is the fraction o f the p o rtfo lio invested in th a t asset:

П

K., =

Z

wi<yit = w'fy ,■

i

= i

I f E ( is the m atrix o f covariances o f y, then the variance o f re tu rn on the p o rtfo lio is:

V, = yyf £ , w,.

T h e usual p o rtfo lio co n strain ts gives w1( + w2l + ... + w„, = 1. W e assum e th a t sh o rt sales are allowed and w lt < 0 reflects a sh o rt selling. We consider tw o m ain appro ach es to the portfo lio-optim isatio n process. T h e stan dard ap p ro ach assum es th a t the investor selects the p o rtfo lio w ith m inim um variance (E lton and G ru b e r 1991). T hen the problem for an investor reduces to solve the q u a d ra tic p rogram m ing problem :

m in w j Z t wt subject to w u + w2r + ... + w„, = 1.

",

In th is w ay we o b ta in th e so-called m inim u m v aria n ce p o rtfo lio (the Ł" 1 1

p o rtfo lio th a t has the lowest risk o f any feasible p o rtfo lio s) wMVit = r _ _ , I 2-jf I

l T 1 у

which has a re tu rn R MV t = , (note th a t we co nsider returns on i 'L , 1 1

portfolios, n o t expected returns), and a variance VMv , = -T- ~ j , at time I 2*t I

t, where i is an n x 1 vector o f ones. W e see th a t the m inim um variance p o rtfo lio wMV,t an ^ its variance V u v , d ep en d only o n th e co varian ce m a trix Ľ,.

T h e o th er ap p ro ach assum es th a t an investor w ants to m inim ise the variance o f the p o rtfo lio w ith a given level o f re tu rn R Wit — rt. T his problem reduces to solving the q u ad ratic p rogram m ing problem :

m in tvj I , wt subject to

It is a very sim ple m axim um problem , the solu tio n is

T h en the o ptim um p ro p o rtio n to invest in asset i is

wMV,,t-T h e classic portfolio selection scheme assum es a covariance m atrix and expected re tu rn s a t tim e t to be know n. B ut th e B ayesian inference ap p ro ach natu rally leads to the posterior or predictive d istrib u tio n s o f these qu antities. N o te th a t the m inim um variance p o rtfo lio wMK|I and the m ini­ m um variance portfolio with a given level o f re tu rn wMKrtI are rand om variables as m easu rable functions o f y, and which are ran d o m . T he p o sterio r (or predictive) d istrib utions o f wMK( and wMVr,t o r VMY.t and VMYr%t are induced by the distribution o f y( and E t. T h u s the predictive d istrib u tio n o f m inim um variance portfolio o r wMľ, i( can be used to provide an op tim al portfolio. As the predictive m ean (for wMK, o r wMKr,,) m ay no t exist, we can consider the predictive m edians w ^ Kt, and w°iSvr,t defined respectively by conditions:

It is im p o rta n t to note th a t even for the sim ple case o f n = 2 assets, there is no analytical solution for the optim al p o rtfo lio selection problem w hen we consider the SV m odel. In this case one altern ativ e ap p ro ach is to use a M o n te C arlo m ethod to evaluate the quantiles o f the p osterior (or predictive) distrib u tio n s o f wMVęI and wUVrit and th en find the p o rt­ folio.

W e consider the daily exchange rates o f the G erm an m a rk against the Polish zloty an d the daily exchange rate o f the US d o llar against th e Polish zloty from 6.02.1996 to 8.08.2001. T he d ataset o f the daily logarithm ic

Pr{wMlrift > w^yiif|y} < 0.5 and Pr{wMyiIf ^ w^v.irly} ^ 0.5, Pr{wMK,iJt> w # Kriir|y}<0.5 and Pr{wMVrM ^ w^Vr.i<Iy} < 0.5

0 = 1, 2, ..., n).

g row th rates, у, consisted o f 1390 observations (for each series). T h e d a ta are p lotted in F ig u re 1.

y„ (PLN/USD)

y2, (PLN/DEM)

Fig. 1. Daily return rates on PLN/USD and PLN/DEM (6.02.1996 - 8.08.2001)

T h e re tu rn rates seem to be centred aro u n d zero, w ith chan gin g volatility and the presence o f outliers. T h e sam ple co rrelatio n (equals 0.567) indicates th a t the retu rn s are positively correlated. A s the first grow th rates are used as initial conditions, thus T = 1389 rem aining observations on y, are modeled. All presented results were obtained w ith the use o f th e G ibb s sam p ler using 10s iteratio n s after 5 104 burn-in G ibbs steps. T o check for convergence

o f the G ibbs sam pling, wc have run the procedure several times with different startin g values. T h e results are stable.

6.1. Posterior Results for the Parameters and the Unobserved Variables

T a b le 1 presen ts the p o sterio r m eans and s ta n d a rd d e v iatio n s (in parentheses) o f all param eters o f the M SV m odel. T h e po sterio r m eans o f <Pu arc g reater th an 0.83 with a 90% Bayesian confidence in terv a l1 o f [0.8225, 0.9028] for <pl u [0.9704, 0.9921] for <p22 and [0.7698, 0.8922] for (p2 l . T hese results indicate quite strong persistence in volatility changes. Also, the stationarity conditions imposed on сри arc not binding. T he posterior m eans o f u 22 and o h arc the evidence o f high volatility o f the conditio n al covariancc m atrix. It m eans th a t the SV m odel tries to explain volatility and the large outliers in the d a ta th ro u g h high v a le ts o f the param eters ofj (i, j = 1, 2). In addition, the p o sterio r m ean s and stan dard deviations o f y2i , ip2i , <7 * 1 show significant (non-zero) an d tim e-varying

co rrelatio n between the rates o f returns. T he posterio r m ean o f y2l reported in T a b le 1 is equal to 0.0268 and the 90% Bayesian confidence interval [0.008, 0.048] does not include zero. It shows th a t the d a ta prov id e strong evidence against zero o r co n stan t conditional co rrelatio n coefficient. All elem ents o f the m atrix R are insignificantly different from zero (the absolute values o f the posterio r m eans are lower th an stan d ard deviations), thu s yt does n o t significantly depend on yt i

-Table 1. Posterior means and standard deviations o f the parameters o f the SV model

<52 r u rt i '21 r2i 0.0443 (0.0088) 0.0036 (0.0097) 0.0077 (0.0242) 0.0137 (0.0194) 0.0150 (0.0250) -0.0020 (0.0249) У11 ° n У1 1 <Pn a \ i -0.2179 (0.0438) 0.8647 (0.0244) 0.3452 (0.0602) -0.0316 (0.0132) 0.9824 (0.0065) 0.0302 (0.0099) У21 4>n a h ln<?n,o ^П<?22,0 ^21,0 0.0268 (0.0125) 0.8334 (0.0371) 0.1453 (0.0352) -3.3780 (1.3041) -4.6270 (0.7588) -0.8769 (0.7026)

1 Here the Bayesian confidence interval is not the same as the highest posterior density interval. The quantiles of order 0.05 and 0.95 are the endpoints of a 90% Bayesian confidence interval (cf. Nardari and Scruggs 2003).

E (v L

I У) 5.0 4.0 3.0 2.0 1.0 0.0 £ « , I y ) E ( o 21.1 I y ) 1

_{i 1}

u u

g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g o g g g 3 3 8 8 2 2 3 3 8 8 2 2 3 3 8 8 2 2 3 3 8 8 2 2 3 3 8 8 2 2 3 3 8 g

т т т т т т т л т т т т

y„ (PLN/USD) и и ш и ш ш ш н н ш ш ш ш i i i i l i i i i i i i i i i i i i i i i i i i l i i l i i l i i i g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g 3 3 8 8 2 2 3 3 8 8 2 2 3 3 8 8 2 2 3 3 8 8 2 2 3 3 8 8 2 2 3 3 8 8 I l I I I i i i i i i i i i i l i i i i i H i i l l l I l i í I i Уз , (PLN/DEM ) 2.0 0.0 -2.0 -4.0 -6.0 g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g 3 3 3 8 2 2 3 3 8 8 2 2 3 3 8 8 2 2 3 3 8 8 2 2 3 3 8 8 2 2 3 3 8 8

m m m m m m m n n m m m

1 I

J

i A L A

J

\

iL

t ' r * Г ’ ' ' ’ •» S S S S Ž S S S S S S S g Š S S S S S S S Š S S S S S S S S S S S S 8 5 S S 2 S S S S S 2 = S 3 S S S = S ä S S S S g 3 * g 2 í S ä S S

т т т т т т т т т ш т

E( P, I y ) s s s s s s s s SS253SSS l l l l i l i i

In the to p panel o f F igure 2 we plot the posterio r m ean s o f each of; ,. T he b o tto m panel show s the daily returns o f exchange rates from 5.02.1996 to 8.08.2001. As expected, the p o sterio r m eans o f the co n d itio n al variances for b o th retu rn s have very sim ilar dynam ic variability. W e see th a t th e MSV m odel exhibits volatility clustering and pro duces the volatility peaks for b o th re tu rn s at the sam e times. H ow ever, in the case o f P L N /D E M returns, higher volatility peaks ap p e ar a t aro u n d 7.11.1997 and 1.02.1999. It is induced by high grow th rates o f the P L N /D E M , which d o n o t ap p e ar in the daily g row th rates o f the P L N /U S D .

In F ig u re 2 on the right (b o tto m ) we show the p o sterio r m eans o f the co n d itio n al co rrelatio n coefficient. T hese co rrelatio n s vary m ark ed ly from one d ay to a n o th er. F ro m F igure 2 it is clear th a t in the second period the co rrelatio n is higher and m ore frequently positive th a n in the first period. T h e Bayesian ap p ro ach natu rally leads to th e predictive d istrib utio n o f the co n d itio n al co rrelatio n betw een y u an d y 2t.

Р(Рт+\ I У) 0.10 0.08 0.06 0.04 0.02 0.00 i .|.irfTTtT1|TTTnrííf[TlTllíllŤ|fíllÍllÍ T-OJoqr^tOlO'irrOC^^-OT-CvJCO^intOr^CDO» 1 cp cp cp cp cp cp cp cp cp o o o o o o o o o Р(Рт+2 I y) Р (Р г+ зо I У) 0.10 0.08 0.06 0.04 0.02

0.00_{*—C T )co r"-c£ > L n ^rco c\j^-o *-;c\jm -« im co r-^a o cr>}

1 cp cp cp cp cp cp cji cp cp o o ö ö ö o ö o o

Fig. 3. Histograms o f the predictive distribution of the conditional correlation coefficient (s = 1, 2, 30)

In F ig u re 3 we present the histogram s o f p T+, for s eq ual to 1, 2 and 30. W e see th a t the predictive d istribution o f p T+„ (for s = 1 and s = 2) is highly asym m etric and becom es m ore spread as the forecast horizon grow s. F o r s = 1 and s = 2 m ost o f its prob ab ility m ass is located on the right side ol zero. But for s = 30 p (p r+ » |y ) is alm ost uniform over the interval [ - 1, 1], indicating strong uncertainty a b o u t possible dependence

o f th e grow th rates o f P L N /U S D and P L N /D E M in the future.

6.2. Portfolio selection with M SV model

In this section we re p o rt the results o f building an o p tim al portfolio using the M SV m odel. W e consider the hyp othetical p o rtfo lio s consisted o f tw o currencies: the US d o llar and the G erm an m ark . W e assum e th at there arc no tra n sa c tio n costs and th a t we m ay reallocate zloty to long as well as to sh o rt positions across the currencies. A llocation decisions are m ad e a t tim e T based on the predictive d istrib u tio n for y T + , and 'l t+s

-Table 2. Quantiles o f the predictive distribution o f the minimum variance portfolio (the fraction of wealth invested in the US dollar)

Order _{" W .ir + i WMV ,1T + 2 **W,ir+30}

0.05 -0.045 -0.067 -0.067 0.25 0.439 0.406 0.384 0.5 0.769 0.725 0.604 0.75 1.099 1.033 0.857 0.95 1.825 1.792 1.33 IQR 0.66 0.627 0.473

Table 3. Quantiles o f the predictive distribution o f the variance and return o f the minimum variance portfolio and naive portfolio

Order _{VMV.T + 1} v_{'tiV.T+2 VMV.T+30} n_{M ľ , r+1 RMV,T + 2} ^ м и , г + э о 0.05 0.05 0.04 0.018 -0.7296 -0.6936 -0.5652 0.25 0.1 0.09 0.042 -0.2388 -0.2124 -0.1578 0.5 0.16 0.15 0.084 0.0252 0.0354 0.03 0.75 0.28 0.26 0.165 0.2898 0.282 0.2154 0.95 0.63 0.63 0.486 0.7818 0.7644 0.6204 IQR 0.18 0.17 0.123 0.5286 0.4944 0.3732 True value 0.2665 0.1698 -3.7241

Table 3 (cont.) Order V_{Naive,T + 1} V Naive,'ľ + 2 Vr N a iv e ,T + 3 0 R N aive,14- 1 D Naive,T + 2 R N a iv e ,T + 3 0 0.05 0.07 0.06 0.027 -0.9126 -0.8628 -0.732 0.25 0.14 0.12 0.066 -0.2958 -0.2616 -0.2136 0.5 0.23 0.2 0.126 0.0162 0.0282 0.0228 0.75 0.4 0.38 0.252 0.3222 0.3204 0.2514 0.95 1.02 1.05 0.813 0.906 0.9024 0.7626 IQR 0.26 0.26 0.186 0.618 0.582 0.465 True value 0.565882 0.402003 -3.16305

In T ab le 2 we show the qu antiles o f the predictive d istrib u tio n s o f the m inim um variance p o rtfo lio wMV,r+ J (the elem ents o f wMFiT+, are the fractio n s o f w ealth invested in th e co rresponding currency). T h e optim um p ro p o rtio n to invest in P L N /U S D exchange rate (the m ed ian o f the m arginal predictive d istrib u tio n o f wMK>ir+JI) is equal to 0.769 fo r s = 1, 0.725 for s = 2 an d 0.604 for s = 30. F o r com parison, we also considered th e naive (trivial equally-w eighted) p o rtfo lio allocation wMv,t + s f ° r s = 1, s = 2 and

s = 30. By R-Naive,t+s and Kvaiue.T+s wc d en o te the re tu rn on the naive po rtfo lio and the variance o f this portfolio at tim e T + s, respectively. We see th a t the predictive d istrib u tio n s are very widely dispersed and fat-tailed (cf. F igure 4 an d T ab le 3), th us leaving us w ith considerable ex-ante u n certain ty a b o u t fu tu re re tu rn s on these portfolios. T h e tru e observed ex-post values o f the returns RNaive.r+а are located betw een th e q u an tile of o rd e r 0.75 and the q u an tile o f o rd e r 0.95 (cf. T ab le 3) fo r s = 1 and s = 2, while for s = 30 the tru e value o f the re tu rn R Naive,T+s ‘s below th e q uan tile o f o rd e r 0.05. Sim ilarly, the tru e observed ex-post values o f the returns Rmv,t +s com p u ted using are located betw een th e q u an tile o f order 0.25 and the m ed ian o f R MViT+s for s = 1 and s = 2. H ow ever, fo r s = 30

the tru e value o f the re tu rn Rm v,t + s i-s below th e q u an tile o f o rd er 0.05. T he huge spread and heavy tails o f predictive distributions give unsatisfactory forecasts fo r 30 periods ahead quite u n d erstan d ab le. T h e h istog ram s o f the co n d itio n al variance o f the portfolios indicate decreasing volatility, and the distrib u tio n o f Rmv,t+s becomes slightly sh arp er as the forecast horizon increases. N o te th a t the naive po rtfolio s have higher risk an d low er returns (if the m edians o f the m arginal predictive d istrib u tio n s are trea ted as point forecasts, cf. T ab le 3). T h e m edians o f the predictive d istrib u tio n s o f the variance o f the naive p o rtfo lio are alw ays higher th an the m edians o f the predictive d istrib u tio n s o f the variance o f the m inim um variance p ortfolio. H ow ever, the predictive d istribution s connected w ith th e naive p o rtfo lio are

m o re diffuse - the in terq u artile ranges are longer; (cf. IQ R in T able 3). C o m p arin g the tru e observed ex-post values o f retu rns o f th e naive portfolio and the tru e re tu rn s o f the m inim um variance p ortfo lio , presented in T able 3, we can see th a t the naive portfolio has higher retu rn s. It is im p o rtan t to stress th a t the tru e values o f Rmv,t+s and R-Nawe.r+s for s = 1 and s = 2

lie in the areas o f high predictive density, indicating good forecasting properties o f the SV m odel for sh o rt futu re periods. F o recasting is connected with huge uncertainty, thus the observed values (for d istan t fu tu re periods) far in tails are understan d ab le. T h e distrib u tio n s o f the forecasted value o f ^MVr,T+i and wwvv r+ J, where rT+s = 0.05 are the m ost dispersed and have very thick tails, (cf. T able 4 and F igure 4).

Table 4. Quantiles of the predictive distribution o f the portfolio with the return 0.05 (the fraction o f wealth invested in the US dollar) and quantiles o f the variance of this portfolio

Order _{WM V r , l T + l ' VM ľ r , 1 7 + 2} V MVr,T+ 1 V уМУг.Т + 30 0.05 -4.973 -4.654 -3.015 0.08 0.06 0.027 0.25 -0.144 -fl.l 0.054 0.19 0.16 0.087 0.5 0.769 0.714 0.604 0.41 0.37 0.225 0.75 1.759 1.638 1.176 1.37 1.26 0.813 0.95 6.775 6.225 4.377 31.13 28.9 18.069 IQR 1.903 1.738 1.122 1.18 1.1 0.726 7. CONCLUSIONS

W e presented and applied th e m ultiv ariate stoch astic volatility m odel and the Bayesian appro ach to portfolio selection. T h e m u ltiv ariate SV m odel was used in m odelling the volatility o f the m u ltiv ariate financial tim e series. T h e B ayesian ap p ro ach leads to the predictive d istrib u tio n s o f th e returns and the covariance m atrix, which were included in an optim ization procedure - optim al portfolio. T he Bayesian appro ach leads to a com plete probabilistic d escription o f u ncertainty involved in p aram eters estim atio n , forecasting and the p o rtfo lio constructio n. Presented results indicate huge uncertainty a b o u t possible dependence o f the P L N /U S D and P L N /D E M retu rn s in the future. Sim ilarly, the predictive d istributions o f the optim al po rtfo lio are very spread and have heavy tails. O ur experience leads us to believe th a t the M SV m odel is an appropriate tool for m odeling financial d ata. Obviously, p ractical im plem entation o f the M SV m odel needs extensions, for exam ple to solve large - scale optim al portfolio selection.

0.50 0.45 0.40 0.35 0.3 0 0.25 0.20 0.15 0.10 0.05 o 00 P(Ymv,t+i I У) /'(nw.ir+i I У)

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0.00 fO<OI*10)(OC^OOtnT-(NtOO)fOÍOp,*í;r^ * CNÍfNÍ*7^*7Cp<^q>dbbT-^c\ic\icNÍ 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 P(Vmv.t+io I У)

t :

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Anna Pajor

BAYESOWSKA ANAl.IZA MODELU ZMIENNOŚCI STOCHASTYCZNEJ W OPTYMALIZACJI PORTFELA

(Streszczenie)

W artykule przedstawiono model zmienności stochastycznej, oparty na dekompozycji Choleskiego. Następnie model SV oraz podejście Bayesowskie zostało wykorzystane do modelowania zmienności dwuwymiarowych finansowych szeregów czasowych oraz budowy optymalnego portfela walutowego. Rozważono hipotetyczny portfel, w skład którego wchodzą złotówkowe kursy dwóch walut: dolara amerykańskiego i marki niemieckiej. W procesie optymalizacji portfela wykorzystano predyktywny rozkład stóp zwrotu oraz predyktywny rozkład macierzy warunkowych kowariancji, uzyskany w rozważanym modelu MSV za pomocą metod Monte Carlo (MCMC).