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2003 BECTHNK
-
IIPOCTAII
CAHKT-IIETEPGYPTCKOTO YHMBBPCMTBTA Ceu. 5 . I3brn. 2 (Ne1 3 1
MOAEJ136
OGMEHA:
TEOPELSI
CTOXACTEWECICHX
ME HOBO^ LI[EHHOCTH 3KOHOMHKIECXXX EJIAT
UHAEICCOB
B npocmols aodenu o 6 ~ e ~ ~a a, 3 a x ~ a e ~Tame
ofi
<wodenb SlhdZYh (SIMEY - SIMple
EXchange).' paccMa-rpseaeTcs KoHemoe MHoxecTBo G = ( g ,,. . .,gn) pasmmmum Menay
co6oii 3 ~ o ~ o ~ u r e c6naz
~ u x(economic goods), noHwMaeMbrx B caMoM I S I ~ ~ O K O MCMbIcne,
BKmOYliIOJUeM B ce6s BCe B03MOXHMe THl'IbI TOBaPOB, YCJIJ'r, BamOT, UeHHbIX 6 y n T
~. A. ~
~ ~ F U ~ T UZFTO
M , Gnara gl,.
..,g,
, paccMaTpmeMare B paMKax ~ o ~ e r SIMEX,
m
m m x
npOCT?ZbLMU6nazmu, AZIJrfX He AeJUfMbIMH Ha BKLIbI H COW: BCe KOHKpeTHbIe 3IUeMIIJUIpbI Gnara,
memume OAHO H TO m Hamame (NHMR)))
gi , m m xa=iecTBexHo
~ ~
~epa3nmmh~~u.
n p e n n o n a r a m , Z~TOm o e m paccMaTpmaeMbIx npocrarx 6nar 6 e 3 r p m ~ AenFMo
o
H
T. e.
omropomo, w o no3~onrre~
nsniepmb m Gnaro no T ~ ~
K a x m a e ~ outrcane
ii
orn~oute~uil,
rrpennonahimcx, w o m60rl @ m c q o e a ~ r i io 6 s e ~ Gnara g, ~ o x m6 b l ~ b OueHeH
HeorpmraTemtrm x e i i n s m e n ~ ~ b mcnoln
m
qi: qi E R: c R 1 ,R' - naaoxcemo scex
XOBAHOB
Hmo&
Bacrumesm
- np0a. ~ a @ e A p b l~ K O H O M H ~ ~ C ~ii6epuemm
KO~
3KOHOMHqeCKOI'O @ K Y J ~ ~ T CnmY.
~T~
O K O H ~B H1968
~ r.
~ a - r e ~ a m ~ o - ~ e x a ~ u ~ e c4 Ka ~Hy Bn w - e ~ JIT'Y. B 1972 r. 3anlmn KawmaxKylo L I H C C ~ ~ T ~ U H HHB
) aaqnbx~e
n p u m a n n o k M a T e M a m K H - nposeccoe ynpasneHnx JIrY. )$iccep~as~io
Ha concKaHHe y v c n o f t c r e n e H H a o m p a (PH~HKOMa-reuamsecwx HayK sauunur B 1941 r. B BU AH CCCP (Mociaa). k i ~ e oKono
e ~ n ~ y x c Oo ~~~ ~ J T H K O B ~ HHHa ~y sI HXs r x
p a b o r , CpeAH KUTOpblX ABeHaUUaTb M O H O ~ ~ ~ @ HKOJlJleKTHBHbIX
R,
M O H O ~ ~ ~ H@ yH9~e 6 ~ h 1~ ~O C O ~ H n0
# p83nHSHblM
BOnpOCaM S H C T O ~ H I I ~ H K J T W O ~ ~MB'PeMaTHKn, B K J W S U BOOnPOCbl MaTeMa'mSeCKOrO MOAenBpOBaHHR 3KOHOMH9eCUHX
o 6 b e m ~H npoueccoB B ycnoBnxx H e o n p e n e n e n n o m .
0 H.B. XOB~HOB,
2003
~ 6 0 @UKCHPOB~HHW
e
IXOnOlIUiTeJIbHOe KOJTFIeCTBO
m rrpeJ(noJIaraeTCrt,
qi npomro 6nara gi 06~e~1113ae~ca
M OAHO~H~ZIHO
orrpeAeneHHoe nonommmHoe xonwremo
B MOAeJIHS
,
q , = qi. C i, npomru 6 m g , 3aa;l~ae~oe
n o n o m m e m m ~ o @ @ u y u e ~ moo6~t e ~ a
(coeflcient of exchange) C i . Taxm 06pa30~,~os@@Piwem
06Me~aC j ecrb BeJImma, pas~ax
xomecrsy g j /qi 6 m g ,nasaeMomy npn o6mewe 3a e w r m q 6 m a g, .
,
,
COBOK)'IIHOCT~
BCeX KO~@@HIJU~HTOB0 6 ~ e x ao6pa3ye~MampUZfy 0 6 h e ~ a( m a k i ~of
exchange) C = (c, ) , . i , j = I,. ..,n , KOTOPBR 06bri~0rcpemonaram rnpa~mmueuoii,T. e.
,
~AOBJI~XBO~X
COOTHOmeHino
X)~~~
,
,
ci . c,, = ci
,L(JM d J S X
n p e ~ n o n o x e ~ n oe -ipalonmenocm u a q n a , o 6 u e ~ a
C = (c,,)
cynrecreoeaHnn o n ~ o ~ e t m l o@
i i~ H K U H H V ( X ) nHcKpmoro apryMeHTa
{I,.. .,n), -08,
sm m o t
o 6 u e ~ a cy n p n c m s u r
K-HUU~HT
i , j,k
E (1,.
..,n) .'
3 m n a a n e m o npcanonmenHm o
x , npo6eram~qeroMnoxecmo 3Haqe~IIfi
s B u m omomeHwa cil = v(i)/v(j )
c o m e m e y m l q ~ x3 ~ a s e m Rsmti @ ~ H K U H
T oHs.~ e eHueeT
,
u e m cnenyloqasl reopeua
Teopma (o npeikmaenexuu 3 n w e ~ m o snonoxumsnb~otlmpammuexotl Mampuyu. l7ymt. meemcs ~ m d p m ~ r m
AUWplUp
C = ( c ~ E )rPnamUme/BRbLMU 3
-u
6~
@HW
V ( X ) , ~ p m ~ b pdp el H n I
M
W
C,, ,
c!, v(z)/v(j )
mHaaO
=
* ~ l l ~
{I,. . .,n) ,r i e d b h m u ~
m
x
wwm6t.l
, ~ m pC
q
E m
) ~ B ~ ~ L H HC~B~OM~ H
CTB~MH.
nMem
Mecm
OPCBWHOC
surehull
= ( c )~6
m n 6 u a MaTpnm
C = (ci )
~ ( i, V) ( j )
l?KZ%7, vnodbl Ali&d
@YIIUHII
CO-O~CHHC
CYIIlemyOT,
TO
3JWdW#?l
-~c21
od-~m*oll
W
~
~
~
M
A
rn c y w e a o B a m
+~HXUHH
V(X) c
a m d m '~pexn~tneKco8 i , j,k E ( I , .
..,n)
cik.cXI= v ( i ) / v ( k )v(k)/v(
.
j ) = v ( i ) / v ( j )= C y
AOCTaTWlHOCTb
C YKa3bHHbCMH CBO~~CTBBMB.
X
wm p - ~ x a
C = (c,,.)
~~H3HTHBHOC
Ma-ipHUbl.
~
~ ~ O K U ~ Tenepb
M
C)'~eClBOBaHHn@YHI(IZHH V ( X )
dra
~ o m ~ p npwwmrn
a~i
s w e w lu w ~ v .wamzma
m
n o ~ u ~e c~o 6 x o a ~ ~ o crpnHmmenocm
ra
uaqmw
Onp~eJIRIoUlee
. .
I ; ] = 1,..., n. Tbadq
?Tor0
'TPaH3HTHBHOCIH
MIIlpHUbl
~ H K C H K C W M~ ~ O U I J B O J X ~HOMep
H~I~
n onpenemu a y ~ ~ u n Vm( X ) CoornomeHaeu V ( X )
.
JJJM
j
= c,] , x E {I,...,n).
r,s {I,...,n), B BHne ap&H
C,, = (C,, C,, )/c,, H nomyacb npemonaraeuoR r p a m m s a o m m uarpxim ( C,, .C,, = C,, ), n a n p a e u
n c ~ o ~ npcncrasnclutc
oe
c,, = C, /c,, = v ( ~ ) / v ( ,ssm
) poxa3brsae~noenmrwocnycnoslla r p o l u m m o o m
n o n o x m e m ~ o tr m p n a r C = ( c )~anr cywonsoeunu * p w ~ nV ( X ) c yra3aHlol.H s ycnoBHw m o p e m
Tenepb,
npencraaruIR
npo~3~omHblk K O ~ ~ I # ~ H U H ~ HOT~ M C H B C r , ,
,
C B O R C ~ B ~ MAoxa3aH~ILm
H.
rropeMa
caeHern, ebxqnamulnx
O ~ M 6
~ Hm B
B
@axmuec~si
ycraaammeam ~ e o 6 x o n s i m eH nocramsmre ycnosm cyxuecrsoeaxim
Kauecree a ~ e m a m o i ((Mepar
i
~ ~ H H O C T H ) )v(i) eaHHHw1
CameTsrsHH c 3WiIHHblMH nponopumm
v ( x ) ,ncnonb3ye~oronpn aoKa3aTemme .reopeml,
a m n d o e n p o m e 6naro g E G .
,
i -ro Toeapa H 0 6 e ~ n e ~ ~ ~ a m q
ci,, irj = 1,...,n . YL. C ~ M D ~xOe n q O e H m ~ ~ Y H W U I
cnenym, n o
B
mqecree
~ K O (aetremoro
~ O
maapa))
MOWO
TonaKo
-
nps y c n o s I i n cii 1 ), a
ci, c ,, , c m m q n w x b I e O
~ J I ~ M ~ H T M,
T H O C ~ ~ HAwaroHam,
O
C, ' C j i = ci, = 1),
yKZi3hIBm4HM
Ha
TO,
ZIT0
MaTpWa
C = (c, ) XBJUIeTCIl
06pam~ocu~~empuvrroii
Mampuye21 (reciprocal matrix). B a m f o Tame, w o mn6ub1 (crpom)
T P ~ H ~ H T U B H OM
~~
a q H U b l npOITOPLUiOHanbHb1 APYr @Yq,
T O Bb~iBKaeTCfi C O o T H O I J l e ~
'rj
/c, = ' , k
/~,k
5
'I
r /'is
3) nonomw~em~oiI
T
~ p o (G,
h U ,C ) yxa3a-rx
MOAeJIb
SIMm,
WXl
SlMEX
= (G,u, C ) .
~
~
.
= 'kr
/'ks
H
~ma-rprmeii
H O Ro6me~a
~
Bb1IIOnNIIH)LqZIMWCIi
mo6bIx
A;uI
C = (ci ) , i , j = I,. . .,n .
M a r e M a r r p r e c l o a t 06selcroe n o n a o m b m onpegcnner K
MOX€X
6bnb
BbIpmeHO
CNMBOJIWZieCKHM
o q e m p
TO~~CTBOM
azpenrposa~no~,
cuHmemuvecKoM, seKmopHoiu Gnaze (aggregated, synthetic, vectorial good)
~IOCK
Bce
Onor(a3aTem
~KY
~ ( q;j)
, , z = I,. ..,n, n 3 ~ e p m ~ cB1emmuax
U , 6nara
gl,
o ~ ~ o p o m r@ymxmi
n
v+(q;j ) JBKHT B ocxioBe
cnenyrom onpe~ene~nii
6a3osozo azpempoeawtozo 6naza H edu~uybzumepenusz o 6 b e ~ a
azpezupoeanHozo 6naza.
P~CCMOT~ELM
arpernpoBaHHoe 6mm ?J= (q,,...,qn) , 06pa3o~annoerrpocrarm 6nara~a
BLIITO~U~~TCX
AM
d
m A> 0 . BMRBJI-
6a30ebvu azpempoeanHm 6 n m (base
~
aggregated good) a 0603m~a~b
w e e K ~ Kmmop
W = (w],
...,W n ) ( W 1 ... W n = 1). M ~ o x e c m e oW(n) scex 8 0 3 M O W H b Z X 6asoebzx
+ +
n o m e M ~ H O B Oqennocmu
~
(value in exchange, exchangeable value) 6nara pacc~aTpHsaeTcn,traripHMep, yxe y
&&%la Clrc~~a
B 4-ft I-JIUe IYepBofi KHRrR eI-0 TpyAa ( < E ~ C C J X ~ A O B ~ H
0HIIpHpOAe
~
H IIpHnHHaX 6 o r m a HapOAOBD,
sbrmemero B 1776 r. (m.:S m i t h k An Inquiry into the Nature and Causes ofthe Wealth of Nations. Oxford, 1976).
*.
H a n o ~ n m ,urn OJXHO~OAHOR +pulweR m e m T ,
yAOBJle7BOpJiIOlUUANI
BCtX
lHa9e~Id8apQ'MeHTOB X I , .
..,X ,
r E R*, HwBaexn
3TOfi ~ # ~ H U H H H
+p-
f(~,,.
.. , x ~ ) ,
A >0
IUUI ~10601-0
COUIT~OIJI~HHK)
W H ~ a A n e r nBCe
~ TOYm B
m
q' =
'
= (a ' ql ,.. .,A - q, ) , rfie
MHOX~CTB O r r p e A e m e T c s C o o n r o m e m M H
A > 0 . CBII~L
BB~A~HH~IX
R: (q)c Q(n)c:R: c Rn, W ( n )cQ ( n ) .
v + ( q ;j ) = v + ( i l . E ;j ) = A.v+(i?;j ) ,
nerrco ~ a i i rx3~ c o o ~ ~ o m e ~ tq,
i i i= (q, + ... + q n ). w i, i = I , . . .,n. ~ ~ o v r o~ ~~. ~yH T H B H H G
B.pmeHwn
v + ( q ; j ) = ~ j + ( ( q .~. .++ q n ) 5
. ; j ) = ( q , +. . . + q n ) v+(F;j),
.
rne
,...,Cn, .
KO~@@EIIIM~HTOB
0 6 M e ~ aCI,
Hapmy
C
OrfpeAeJIeEWM
~ O B O ~i O
l l ~ ~ O B a H H O r 6mra
0
v+ (5;
j ) , ~ A C r a B J K X I O W ~ X Icociofi
j - m monbsa cCi) MaTp06~e~;1
M ~ H O B O ~UeHHOCrH
~
W E W ( n ) r r p ~ n o ~ o r r ~ l l ra,wmmBHoro
wKqeKca
ssserue~~oe
cpemee a p u a ~ e ~ w e c ~Dor ree M e m o B
H a x o m npaIcmsecKoe n p H M e H t H H C H a B0JUOTHbIX pUHKBX. Ta& HflnpHMep, npX OUeHKe K O ~ @ @ H W ~ H T0~ I6 ~ e eBp0
~ a Ha
K ~ K ~ I O - ~ HH ~ O
H sarwry
~
AJDI n e p a o n a , n p e n n r e c n s y I o W e r o B s e a e H n m m
i3 e a p o n e f t c ~ o f t s a r m n , r B 1999 r., espo
06slreo paCCMa7pHBaeTCX KaK UtpetupOSUHHUR BUJUOmU, o 6 M e H H b I 8 KypC ~ C Y r o p o f inO.Ty'IaCTCs B p e 3 y n b T a T e BbIqHCneHHR
~3eeuce~nozo
Cpeb~e20zwMempurecKozo K ~ @ @ H U H ~ H T O B06~eHan p o c r a r x B~JXKYT, B X O J J X ~ ~ H XB coonsexmylomylo
BtUWlUp
KOp3HHY (CM. HHlXpYKUHIO n0 BblqHCneHHW KYpCa esp0, I l O M e n I e H H y l o Ha k l H T C p H l X - c a # ~ e
paciffc.cornmerce.ubc.ca).&UI BblqHcmeHHs l i l ~HamBaeMoro S&#I~KMUBHOZO hypca 0 6 ~ e (effective
~a
exchange rate)
e e p o m a c e npeanaraeTcn .~cnom3o~a?s~ 3 s e m e ~ ~cpenHee
o e
reoMnpHsecKoe (a H e m s e r n e w o e cpeaxee
B P H @ M ~ ? ~ ~ V ~ C COOTBt7WlB)'fOWHX
KO~)
K O ~ ~ @ H ~ H ~ H0T6O~B e ~ a .
V,
(W;
j ) = Q)" [ w ~
.Q)(c,,) + ... + W,, .Q)(cnj)], rae p,
omtiparmuoman npyr
BO3paWUl(UI
Ha
apyra
- smpmxnom Henpepmum
a p ~ ~ m w r c c ~V+o (P;
e
j) n o n y a e m
06palHUl
la
K
@
~
H
HQ) .
-1
-
r c n a ~ nop fmi o f i R , a Q)
A e R c m m e m ~ o , B3BeXeHHOe C p e X H e e
O T ~ C ~ K (H K O H ~ Y ~ ~ ~ ~ C K O H ~ ~ H
n Le ~I ~m) ~
HenpepbIBHUI @YHKUHII,
1
$ylumm.
o t j o 6 s e ~ ~ omr os e m e H H o r o cpemero
vp (F;
j) npu TOXAeCTBeHHOM
Gnara
q = (q,,. . . q,,) E Rf(iT)
;
noxcmM
n-pocTme 6na1-a g, ,. . . ,g , cyTa pawmwme
HeOTpHqaTeJIbHbIMU W C J I a M U
emHH4d
AeRTeJIbHOCfH,
3MElTKPYeT
MOXCHO
24%
q , ,... ,q,
~JIaJ?e~mklfi B a M a M R
KYmOp
Ha
~
YamHoM
npwMepe,
CJ'IeAYEOL9WM
g,,...,gn
B
%OM
B KOJIWfeCTBaX
Jra
qmopa
w
e m
enw~ma u,
~ O l l O p ~ O H Z U I b H J a I6a30~0fi
X
BaJLEOTe
( J )-
~ ( i fJ;)
~KOHOM&FI~CKO~~
q,,.. .,'9, COOTBeTCTBeHHO,
KOTOPbIX
= (ql ,. . .,qn) E R: ( W ) . T o r ~ a
6a3o~oiis a m o ~ b.E
~ = (w,
,. . . , w n ) ,a
k i ~ ~ e p e ~o bm; % e ~ a
arperqosaiiHm
caMa
-T,
w € w(n).
T ~ K 06pa30~,
~ M
om o 6 3 e ~ aqi rrpocroro 6lrara gi
P,( j ) =
Cy6.bem
(notes, bonds), C ~ M M M
~~~H OHWGUeHHOCTb
.
UesHocTs
uenbmx
C a l Y a e I r p O U e n Y p y . BBeneHHII
o6pa30~.
COOTBeTCTByeT M ~ H O B OUeHHOCTM
~
~T~~IX~OBSU
BaJIK)TlrI
IHO~~
rtemom oa~oZi~ o ~ o~i yi m ~ ppa BbH~a
~ o r m0 6 ~ e ~ w s a e ~ b x e
~ KI o ,n m e m KOTOPJAX WJM~PRIOTCR
T
COOTBeTcTseHHO.
qeACTaBHTb
q, + .. . + qn HOBbIX
B
QibCij
2yr . cr,
EG B
o6.ae~eq,
+ ... + q,
-W, . c1.i
-
n
C w r .crj
nom uemonn i -ro npocmro bnara, ~ 0 3 n a v a e ~ ananee
n
pi, B c y ~ ~ a p ~ o i
uemocru a r p e r w p o B m a o r o Gnara H e x m u c n T OT wcnonasye~oiien~mmxu s ~ e p e ~ m
u, :
npn
TOM
6nam g, M O ~ oOn p e n e n m b C o o T H o m e m i e M cG = V + (F;j ) .
C M ~ ~ H HB W
W A b W a C M a63aye nO3BOJIIim EIwreplTpCTmlpoBaTb m H b f i nOKa3aTeJla
a
o 6 a e ~ aq, = (q, . . . + q , )
V + (q;j ) KaK ulrde~c val, (q, ;j ) ~ e ~ o e oyennoemu
(ibzpaxe~lrozoB edurruy a x u , ) azpenrpoealrlrozo 6naza
(ql,. . .,q, ) ,r r p o n o p ~ o ~ ~ ~ o
~~UOBO
a rM
p e rY
H p o B i 3 E i H O M y 6nary
-
iF = (w2 , . . . , w , ) : . v a l + ( q ,j;) = v, (g;j ) = (q, + . . . + q n ) v. + ( F ;j )
+
Tenepb
MbI
UMeeM,
~ ~ C K O H ~ Y moxecrso
HO~
IIOMHMO
KOHemOrO MHOXeCTBa
KIpOCTbIX
6nar
G = ( g , ,...,g, ) ,
6a3oaba 6nar W(n)= { F = (w,,..., W , ) ) , Kam4oe M Kompm
rrpeAcmSuIeT co602i enwauny u, mMepenwr Bcex arperpsamarx Gnar, nponopwiommmx
m o ~ ~ y~ ~ O B O 6m-y
M Y
AaeT OAHO o6raee
w E W ( n ) .Momo cwaTb, w o 6a30~0e6 n m
= (w,,. ..,w , )
~ ~ ~rrponopwiommxibzx
~
WHOMY
g, AJW Bcex a r p e r n p o ~ aGnar,
E W ( n ) . C K ~ ~ ~ H no3~ome-r
HO~?
BBeCTM B PaCCMOTpeHNe 6ec~o~ewioe
KMR
~ ~ ~ O B O 6nary
M Y
MHOXeCTBO WMeH 6a308b1x aTpelTipOBaHHbU[.6mr, BlClIIOmWee B ce6x H kIMeHa IlPOCTbIX GIIZW:
m R g, 6a30~01-0
a r p e r q B m o r 0 6E W(n) COOTBeTCTBJ'eT WMeHH llpOCTY)rO 6nara
g, ~ G , e c mw, = 1.
3 ~ a nxos@@metrro 6 ~ e mc,, = val+(V;
j) ,
~XW~IB~K)@
Komrrecrao e m m
g, , KOTOpOe o ~ M ~ H ~ U W O T CHIaI e
m Ulii 06'be~a 6830~0r0
arperwposarmom 6nara F E W ( n ) , ~ o m oseecnr ~03@@muern
06snem C,, = l/c,, ,
U, lrpocroro
y~3biBZU0
KO
~m c X X B O
O~M~HUB~~M
HaHemmug
X
en-
2.4,
U, npocroro 6nara
g, .
Ko~@@mpiem
OG~ernCFF, 6a30soro arpernpomoro Gnara
6a3o~oe arperqosannoe
6nam
' = ( W ,...,w )
W,
MeHOBOm ~03@@iUHeH'I'aczEt: cGG,
= c,, , eCJIH
=l;
c,.
= (w, ,...,W , ) x
i
~ o m o onpenemin
czs, = CG, C J V , . Bce BBeneHHare pmee uo*@sme~m o6~eI-mB
CJIyraW
E W(n),
&%€?Ma
6a3pBon>arper~saHHoro6-
@op~ynoA
M C T ~ M
I K~
Yaame
w ,= W ; - 1; czF,= c,,, , KJIU
=c- ' ecm W ; = l .
W l
Cnenye~OTMWHTL, w o menemme ocHoBHr,xe nOxa3aTem v(i;J) , v(qt ;j ) , v+(q;j )
M ~ H O B OueeHH0m-n
~
6na.r (a npomsoAHaIe
OT AEIX
nORa3mnn val+(us ;j ) = V + (F;j ) ,
val, ( q , ;j ) = v+(ij;j ) ) lamepwrcn B emrmax u, Hexmoporo cccma~dapmnozo)~
irpocroro
6nara g, E G . T ~ cxwm
M
U, nrpae-r porn c~emnoiie k ~ u q b(unit
i
of account, numeraire) n p ~
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~
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e m m a M O X ~ ~ ~ ~ c - r y n aHn d o e 6a30soe arpempoaarrsoe 6 n m T
F E W(n).
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emU , q m o r o 6nara giE G
,
~ o m o n p e n e m ~ e m o f nweKca
i
v+(i;R) = c, = l/c,, . Term mercc me~osoii
v + ( q , ; F ) Komecraa q, rrpororo 6mra g, onpenemma @op~ynoii
v+(q*;W)
= *Ciri;.
- Aswm~~blew e K c M ~ H O B O ~ ~uemocra v+ ( q ' ;w ) = (q; + . .. + qi ) . c,.,
ueHHocTu
arpempoaarmoro
82
6nara
q' = (q;,.. .,qi ) E R: (F'
),
~ ~ O p 4 H O H ~ H O6~zO
uo~o~y
SW"'
=c,,
+ ...+ c,,
m
B3BemenHoe
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rti
r*j
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-
-
inT(i), inv, (w)
- -
CI
ci, c k j= c,, non
M ~ IPaccMoTpEiM
J L K I ~ ~ I X i,
m e e cnylraii, KorAa Bce cxoxacrmecme MeHosbre
-
j , k s {1, ..., n) . n o c ~ o n b ~ yme d o g 3 n e ~ e m c,, ,
-
,,,
2
r , s = l ,..., n , r t i , s + i , o ( , =o(,,,
= D l n c i , , r = l ,..., n , r t i ) . * 3 n e ~ e 1 . c,,,
1~
C1
h)
rts, i
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, s g i , c-oi
IIapaMeTpaMH ,u(,), .= E ln F,.,
H
ramxoe myx
2
o(,,,
= D in Fr, , orrpeAemieMsnze m p q n a ~ ~
,, = ElnZ,, = E ( l n 6 , - l n c , ) = pci,,- p ( i , r ,
,u
2
( 1 1)
2
o:,,,= Din Frs = D (In 6,- In Zi,) = qiIr
+ Q , ~ -) ~2 0 ( i ) r.s
(12)
N
norapn+~nsec~n
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N ( p , 0 ),rne ,ii
= ( p l,. ..,,u~
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KOBapHaUHOHHUl MaIp1(9.: oi = COV@~, Y ) - KOBaPHLPM CJl)'Whbl~
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(
= Gi)
N
B M W ~ H
=(
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#
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j
--
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N
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sf =exp(2,q +o:).(expo:
-1).
mi = Ei = exp ( p i + 0:/2),
3necb
pi = E l n F i ,
s i j = exp(,ui + p j +(of + o : ) / 2 ) . ( e x p o i j-1) .
zr,mwr a z
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- =-
Crs
Cri
-
c
-M C T O J I ~ U ~
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,
,,,,= D l n < , = D ( - I n c , ) = o (
o
2
,,,.
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2
(14)
'
H a k n e ~Tenepa ~ o p ~ y r enn
r y Kompapeam o(,,,),, = cov (In Z,. ,In C;, ) norapn@mos
o ,,,,,, = cov (In F r s ln
, Ftu) = cov (ln[c,
- - - = cov (In cis - In c,, ,In c,, - In cir])=
c;' 1,ln[Ziu - F,;' I) =
-
C T ( , ) ~ ~CT(,),~
- o(,.)~,,
+ CT(~),.,
(15)
,
oil = Dln inF(k)= D
mlil = E inF(i)= exp
r,s=l
r+i.s+i
+
s;*] = Dinv"(k)= exp (2,urkl cril)+(expa;, - I)=
&rx Haxowemm K O ~ @ @ H I J H ~ HK0BapHau.m
T~
Memy mmapuammm c r o x a c T m c m H
znF(k) , i ;t k , ~ o c n o ~ m y e ~Np~ynog
cn
cov (inG(l),inF(k))= E[inV(l) inV(k)]- E inF(Z)E inF(k). npozis~e,qelote
H H ~ ~ K C ~ M~enoaog
H
lre~~ocruinS(Z),
( )
[
cov (inV(l),inT(k))= exp punk,+ - - exp pel, + pck,+
Tenepr, 3 ~ m
MaxMamecmie o m i n a m mLll= E inv(I),mcnepca.
4)+
4-k)
,
2
(22)
si1= D inc(l) H
xo3@@awiemb1
KoBapama cov (inT(r),inT(s)) HHsapuaH~m.ixcroxacnrrecwx WeKcoB
2
-
-
r # S , MbI MOXeM JIelTCO ah MaTeMaTFieCKOe OHCHmHHe m+(F)Ef ,4HCIIepCkw s+(w)
EMsapnamoro W m H o r o croxamecicoro WeKca inv, (w)~ e ~ o s ouemocrn
fi
6a30~0r0
arpempoeamom 6 m W = (w,,...,w n) E w (n):
. .
.
yxce o ~ ~ e r a r mB, Bme ~~nbrnunn~~arnu6Hoto.
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onpenemelrroro @pqnoZi
-
Jler~oy 6 e ~ ~ n Bc EMB~~U~HTHOCTEI
x
nocrpoexxoro cmxammeclcoro w e x c a invx(W ;j)
OTHOCHTeJIbHO~b160pam a p H o r 0 Gmra, %Q IIO3BOXlieT er0 0 6 0 3 ~ a ~ a OIlJ'CKU
~b,
HOMep j
U
c-mimpnoro npocroro 6 m g, E G : inVx(iV) = inpx(V;j ) = invx(V;k ) .
n
n
invx(F)=
.
[in~(i)]*
=fi
fi
i=l
i=1
[in~(i;
i)]* =
i=l
r 0[fi$],
?I'
=
i=l
(26)
r=l
ann m a p s i m o m MyremnmwTlisHoro mxacmrecKoro WeKca inFx(F) me~osoii
O r e ~ w om
, , m m a p 1 3 a d ~ y m m r m m m m ~ b IcroxacrmecIUbi
ji
WeKc inTx(F)
mem norHopmanaHoe pacnpeneneHwe (m npomseAeme norxopMarrsKant c v a k m x
2 ~03@@miemoso 6 ~ e ~E,,a , ~ 3 m 1 xB p m m crenewdt) c napaMerpaMa ,u, ( F ) , ox(w) ,
Bemxmm
inCX((w)
no3aonrim
nerxo
xafi-ra
ee
-,
mx(Z) = E znFx(F)H L M C ~ ~ ~ HS:H(iF)
)
= D inTx(F)
.
MaTeMaTmec~oe oxmame
'
TwpKH CTOXaCTEIYeCKHx meKCOB M ~ H o B O~~ ~~ H H O &3KOHOMEWeCIUM 6nar, 3JIeMeHTH
~ o ~ o p oqf at m o ~ 3 n o x e mBNnre, no3~0mern o c r a ~ mB crporoii ~ a ~ e ~ a ~ m e(Pop~e
c~oB
il
p e m pm m e p e c m H tmaixx ~ K O H O M ~ ~ C K H3a~az1,
X
cpem ~ o ~ o p a r~eo6xommo
x
y n o m q m cneAymmHe: 1) 3wam ab16opa <(crairAapTHom6nara>>,ea0 6 Ko-roporo
~ ~
M O ~ -TC
merxioZi em.~qeZi,qmxqeZi porn C T ~ ~ L Z J T ~ HmO ~oOx a mexo~oZiu e m o m
Bcex oc~arra~arx
6 m ; 2) 3mara nocl-poem wexcca n o ~ y n a x m ~ oci ni o c o 6 ~ o m~ o p s m
s
m no oTHorneHmo K m60py TOB~POB B yc~~yr;
-3) 3 m a ~ as ~ 6 0 p aormi~arraxioiiK O P ~ H H ~ I
BeAyWHx Barnor, c I I o c O ~ H O ~kQaTb
~
Porn6 IWfMeHee ~ H C K ~ B~O ~ ~~ 3 BamOTbI;
0 % 4)
onpeaenem CTOIIMOCTH rsammaa B p a ~ ~ MoAem
ax
CAPM n T.A. l o
OCHOBH~I~
TeopeTHreclore nolurmr
H
lIpHiUIWHblC M-M,
CBIP)aHHbleC np0CTOfi MOABJIbiO
O ~ M ~ H(CU MWtnblO
SMEX), o 6 c m m 1 1 . Hanpnruep, B pz.6onrx: X H T p o B T.M., X o B a A o B H.B.Klpocw M O A ~ J Id~~ e s a ocHoeme
:
npemanoxeHm H 6 m d m n e CJIC~CTBHR I! B m . C.-IIe~efi.ymm. 1992. N'26. C. 101-106;X o B a H o B H. B.
0 6 1 . qm
~ p ~ Hr ~ M C ~ ~ H HUOHHOCFH
II
6nar N Ma~epaannr Bcepoccuitc~ofiw o r r ~ e p e ~ u s~n< ~ K O H O M H S ~ C K BHI I a m H
c a ~ n - n e ~ e p 6 y p r cyHaeepcHTem.
~~P
Cn6., 1999. C. 179-,180;H o v a n o v N., K o 1 a r i J., S o k o 1 o v M.
Computing Currency Invariant Indices with an Applicdion to Minimum Variance Currency Baskets // Journal of Economic
Dynamics and Control. 2002. Nc 11. P. 1-21.
noxp06~eeo mKanax m o m e ~ ~CM.,
R Hanp.: C e p r e,eB A.T., K p o x H H B.B.M ~ o g o r mM.,
. 2001;X o siaH o B H.B. M ~ T c M ~ ~O C~H O
~B c~ ITeopuH
K H ~
H ~ M ~ ~ ~ I uaqecma.
HHII
1982;P o t t e r R. The Art of Measurement:
Theory and Practice. Englewood Cliffs (N.J.),1999.
n.,
'
n o ~ m ~e e ~ a e oU~HHOCTH
ft
(value in exchange, exchongeoble value) 6nara pacchtaTpHsaem, HanpnMep, yxe y
&awa CMHTBB 4-R rnaae nep~oRK H H ero
~
Tpyna c(Bccnenosame o npupone H npHqHHax 6 o r a ~ c m aHapOliOB)),
eblnrennxero B 1776 r. (cM.: S m i t h A An Inguiry into the Nature and Causes of the Wealth of Nations. Oxford,1976).
CM.,~anp.:B r o d s k y D. Arithmetic versus Geometric EffectiveExchangeRates /I Weltwirtschaftliches Archiv.
1982.Bd 118.H.3.P.546-562;B u 1 d o r i n i L., M a k r y d a k i s S., T h i m a n n C. The Effective Exchange Rates
of the Euro. Frankfurt am Main, 2002.
'cM., ~ a n p . A
: x H H H K. C p e n ~ n eBeniiqnHhl. M., 1970.
CM., HanP.: c o K o n o B M.B.A ~ c n o ~ a m s e c ~onpeAeneHne
oe
@ ~ H K U ; H O H ~ ~ ~Bnna
H O ~nOo m a r e m ~ e ~ o s o f
UeHHOCTIi 6nara // Marepuanbl MempyHap0~~0k
Hayl1~0fiK O H $ ~ P ~ H U H HQ ~ K O H O M H Y ~ C K B IHaYKa:
I
npo6ne~b1TeOpHH H
Mmononornn)). Cn6., 2002.C. 158-160.
nonpo6~eeo cJyia#~oftsenusme, KaK o6aek Monenn HeonpeneneHHom ~ W ~ H H~ HI ~H ~ H ~ ~ B O ~ K O H O M H ~
napaMelpoB - CM.:X o B a H o B H.B. M a n ~ a m ~ e c ~Monenn
w e pncKa n tieonpenenennomi. Cn6., 1998.CJlylldHbIe
HHJlQKCbI M ~ H O BUeHHOCYli
O#
HCCneAOBUHCb, Hanp.: X H T p 0 B T.M., X 0 B a H 0 B H.B.np0CTU MOnenb 06hfeHa:
paWOMH3HpOBaHHbIeT ~ ~ H ~ H T H MaTpH4bl
B H ~ I ~ K~$~H~IH~
061rleHa//
H T O BB e m . C.-IImp6. YH-Tit. 1994.NE5.C. 94-99;
H o v a n o v N., F e d o t o v Yu., S e r e g i n 1. A stochastic model of commodities exchange N Proceedings of the Third
International Workshop on Simulation. St. Petersburg, 1998.P. 392-394.
'
(cM.,~ a n p . B
: H ro H a K o B H.B., K o n e c o B A.H., X o e a n o B H.B. C m x a c l w s e c ~ ~Monenn
e
AmaMnw neno3mB
// Monennpoea~ue3KOHOMHqeCKHX H COUHaJ'lbHbIX npOUeCCOB. ClT6., 1998. C. 40-68; K 0 H 10 X 0 B C K H fi n.B.
npocrefimarr MynbmnnnKantexiaR mxacntsecrar ~ o ~ t AuHaMaKn
n s
pecypca I/ B e m . C.-nerep6. y ~ - m1998.
.
N2 19.
C.96-102;B n Ul fl a K 0 B R.B. C r o x a q l l e c r a ~Moaenb AHHaMHKH ~LHKOBCKHXAen03HMB DO B O C T ~ ~ ~ O B W L H!/ W I
~ K O H O M H KH~MaTeMamqecKHe M ~ T O2002.
~ I . T. 38. 1. C. 94-104;B r u m m e I h u i s R., C o r d o b a A. Principal
component value at risk11 Int. J. Theor. Appl. Finance. 2000.Vol. 3.Nc.3. P.541-545).
CM.,nanp.: A i t c h i s o n J.; B r o w n J. The Lognormal Distribution. Cambridge, 1969.
'O Bonpoc~, camaeaare c npaMexieHneM o 6 ~ e i i ~ e o p ~ mxacm~ecKHx
w
HmeKcoe ~ c ~ o e o iUeuHom,
i
o 6 c y l r u l a r a r c x , ~ a n p n ~ e p , ~ p a 6 o ~ a x : K o lnIa[pm~. B . , C o x o n o ~M . B . , Q , e n o ~ o e I O . B . , X O B ~ H OH.B.
B
np0CWI MOnenb 06ueaa: noKa3aTem M~HOBO#
QeHHOCTH 6J'lar N B e r n . C.-ne~ep6.YH-Ta. 2001.N9 13. C. 141-147;
K o n a p H Ax. B., C o K o n o B M.B., X o B a H o B H.B. Hcnona30ea~neB Monenn CAPM arperuposan~ofiearnom
MHHHMaRaHOrO pHCKa // M a ~ e p n a mhfe~yHap0AHofi H ~ Y V H O R K O H @ ~ ~ ~ H U I&OHOMHS~CK~R
HH
HayKa: npO6neMbI
ROPHH H Memnonornn~).Cn6., 2002. C. 127-129;X H T p o B T.M., X o B a H o e H.B. npocrarr ona arm 0 6 ~ e e aaHamu
:
AiiHaMHKU no~ynarenb~ofi
C I I O C O ~ H O H~ K Y P C O B O ~ ~CrOHMOCM BBmO7%1 11 B e r n . C.-nmp6. yH-Ta. 1995. N
! 19.
C.90-96;X o B a H o B H.B. C ~ a 6 ~ m ~ cb s1 ee ~ ~ bennHuu.1~e
O C H O B ~ H H ~ $ O ~ M ~ ~ H O H H6Oe~cO
n e r l e ~ n s@HH~XCOBO~KOHOMHS~CKO#
ABRT0JIbHOCIH N Te3uca1nOKJl&QOB 7-R Mex,qwapo~ofiKOH@P~HUHH
( ( P ~ ~ ~ ~ o H s HJ IH~@H~ PUM!~ T H K-B
2000)).Cll6., 2000.C. 95;H o v a n o v N., K o I a r i J., S o k o I o v M. Aggregated world currency of minimal risk I/
Proceedings of the Second International Scientific School ((Modeling and Analysis of Safety and Risk in Complex
Systems)).St. Petersburg, 2002.P. 200-203.
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