Y3I>EK11ctoh pecnYI>U11kac11 ou11 ba ypta

0OOuU (coOOa) pou1!l . l\HJlai1JlHK, HHBecTHu,H51 l\HJlHw yYyH l\ H)l,ai1)l,Hp 6OwJl HFHY P nyJl Ma6Jl Jl MaB y)l,. VI ecTHu, 51 l\HJl HaTH acH)l, MabJl M Bal\ My)l,)l, H (n )l,aBpJlap)l, ) yTraHH)l, Kei1 6OwJl HY cyMMa iPn (i – cp cTaBKacH 6yJl , y 6yi1 6OwJl P cyMMara cp H3Jl l\ywHJla)l,H) MHl\)l,Opra y3rapa)l,H. 51bHH 6OwJlaHFHY cyMMa P MabJlyM )l,aBpJlap (n) yTHwH 6HJlaH iPn MHl\)l,Opra OwH6, HaTH a)l,a S (OpTraH cyMMa) MHl\)l,Op x,OcHJl 6yJla)l,H:

S = P + iPn = P(l + in). (4.2.1)

(1 + in) HcpO)l, O)l,)l,Hi1 cp 3Jl 6yi1 OpTTHpMa Kynai1 BYH )l, 6 HOMJl )l, . 0pTTHpHJlraH cyMMa S 6OwJlaHFHY cyMMa Opl\aJl TOnHJl aH; cpOH3 cTaBKacHHH (i) )l, cTaBKa )l,ei1HJla)l, .

4.2.1-YH HcpO)l,a)l,aH 6OwJl P cyMMaHH TOnHw MyMKHH. yH)l, P MHl\)l,OpHHH S Opl\aJl yHH d
cp cTaBKacH (aHTHcHnaTHB cTaBKa) 6yi1 )l, KOHJl i1yJl 6HJl H TOnHw MyMKHH, 51 H:

P = S-ndS = S(1-nd). (4.2.2)

(1-nd) HcpO)l, OpTTHpMa Kynai1 BYH )l,ei1HJl )l, .
0)l,)l,Hi1 cp H3 cTaBKaJl KynpOl\ l\Hcl\ My)l,)l, TJl )l, H x,HcO6Jlap)l, l\yJlJl HJla)l, . Wy MyHOca6aT 6HJl n 6Hp i1 Jl)l, KaM )l,aBpJlap)l, )l, Ba aHTHcHnaTHB cTaBKaJl H aMaJlHeT)l,
l\ JlJl w XycycH51TJl pHHH KypaMH3. eKypcHB cTaBKa x,OJlaTH)l, i MHl\)l,OpH l\ i1H)l, l\ 6yJl l\HJl )l, :

n  ^t
n  ^t

365
_eKH 366

(t – cpOH3HH x, 6Jl KyHJl H cOHH). Arap aHTHcHnaTHB cTaBKa (d) l\yJlJl HJl ,
yH)l,a:

²⁰ UeHHIe 6yMarH: YYe6HHK/nO)l, pe)l,. B.VI KOJl HHKOBa, B.C.TOpKaHOBcKOrO.-2-e H3)l,., nepepa6. H )l, .-M.: HcI H cTaTHcTHKa, 2000.-448 c.

²¹ KHJl51YKOB A.A., 4aJl)l,aeBa Jl.A. PIHOK u,eHHIX 6yMar H 6Hp eBOe )l,eJl : YYe6. nOcO6He. 2-e H3)l,., c H3M.-M.: 3KOHOMHcTb,
2005.-687 c.; PIHOK u,eHHIX 6yMar: YYe6HHK/nO)l, pe)l,. B.A.raJlaHOBa, A.VI. acOBa.-M.: HHaHcI H cTaTHcTHKa, 2005;
ypeHHH A.H. PIHKH npOH3BO)l,HIX cpHHaHcOBIX HHcTpyMeHTOB.-M.: VIHcppa M, 1996.; Wapn Y., AJl KcaH)l,ep r., 3i1Jl . VI ecTHu, .-M.: VIHcp a M, 1999.

n  ^t
360
MypaKKa6 cpOH3Jl . Hp MyHYa MypaKKa6 x,OJl cpOH3Jl HHHr KanHTaJlH3au,H51 HHH x,HcO6ra OJlH6 KypaMH3. YH)l,a Bal\T )l,aBpJlapHra 6OFJlHl\ x,OJl)l,a OpTH6 6OpyBYH cyMMara 3ra 6yJlHwHMH3 MyMKHH,
51 H P i-YH )l,aBpra KeJlH6 L' p_j MHl\)l, OpTa)l,H:
L'P₁=iP;
L'P₂=(P+iP)xi=iP(1+i);
L' P₃ = [P+iP+ip(1+i)]i=iP(1+i)²;
……
L' P_n=iP(1+i)^n-1.

Y x,OJl)l,a OpTTHpHJl H cyMMa:

S = P + iP+ iP(1 + i) + iP(1 + i)² +... + iP(1 + i)^n-1 =

=P+ iP [1+ (1 + i) (1 + i)² + ... + (1 + i)^n-1].
KBa)l, l\aBcJl )l, reOMeTpHK npOrpeccH51 HHr cyMMacHra 3ra 6yJl , MOc paBHw)l, l\yi1 )l, HcpO)l, HH OJl H3:

S P iP ^(1i)
n1
P(1i)ⁿ

(1i) 1 _.
yH)l,a (1 + i)ⁿ KaTTaJl OpTTHpHw Kynai1THpyBYH)l, . WyHra yx,waw MypaKKa6 aHTHcHnaTHB cTaBKa yYyH l\ i1H)l, HcpO)l, OJl MyMKHH:
P=S(1-d)ⁿ
yH)l, OpTTHpHw Kynai1 HpyBYH cHcpaTH)l,a l\ i1H)l, HcpO)l, 6yJl )l, :
1


(1d)ⁿ
Arap n 6yTyH 6yJlMaraH cOH 6yJlca, MacaJlaH, n = k + t (k - 6yTyH i1HJlJlap cOHH, t – KyHJlap cOHH), yH)l,a OpTTHpHJl H S cyMMaHH x,HcO6Jl w yYyH l\yi1H)l,arH HcpO)l, l\yJlJlaHHJl )l,H:

S P(1i)^k x(1ix
t ₎
365

Yw6y HcpO)l, )l, 6yTyH i1HJlJl MO6ai1HH)l, OpTTHpHJl H MypaKKa6 cpOH3Jl 6yi1 a, KyHJl MO6ai1 H)l,a 3ca O)l,)l,Hi1 cpOH3Jl 6yi1 TOnHw MyMKHH.
0pTTHpHJl H S KaTTaJl KeJla aK)l,a OJl Ha)l,HraH cyMMa x,HcO6JlaHa)l,H, P 3ca yHHHr x,O3HprH MHl\)l,OpH 6yJl )l,H, 51 H:
P  ^S
(1i)ⁿ
Arap i1HJlJlap cOHH (n) eTapJlHYa Kyn 6yJlca, yH)l,a x,aM O)l,)l,Hi1 Ba x,aM MypaKKa6 cpOH3Jlap 6yi1HYa x,HcO6Jl HraH x,ap l\ H)l,ai1 cyMMaHHHr x,O3HprH MHl\)l,OpH HOJl HHTHJla)l,H:

S
_limP lim 0

n
^{n}(1i)ⁿ
S

_limP lim 0

n
^{n}1in

y Hcp )l, Kyn i1 JlJlap)l, Kei1 x,ap l\aH)l,ai1 KaTTa cyMMaHH OJlHwJl K x,O3HprH Bal\ )l, HOJl 3cpcp H 6epHwHHH aHrJlaTa)l, . CyMMa)l, OJlHHa)l, aH 3cpcp KeJl aK)l, Te3pOl\ KaMa51)l, , arap)l, x,HcO6JlawJlap MypaKKa6 cp H3Jl 6yi1 Ya OJl 6OpHJlaeTraH 6yJl .

Pea!l, 11oMu11a!l aa Jppe«mua pou1 cmaa«a!lapu.

OH3Jl HH l\ wH6 x,HcO6Jl 6 6OpHw HcTaJl Bal\T )l, BpH)l,a aMaJlra OwHpHJl H MyMKHH. KynpOl\ cpOH3Jl i1HJl)l,a, YOpaK)l,a, Oi1)l, 6Hp MapOTa6a x,HcO6Jl 6 l\ wH6 6OpHJla)l,H. OH3JlapHH i1HJlHra m (m > 1) MapOTa6a x,HcO6Jl HH6 6OpHJlHwH x,OJl HHH KypaMH3. Bal\ r x,ap 6Hp )l, BpHra (51 i1HJl , YOpaKKa, Oi1 ) j cpOH3Jl x,HcO6Jl HH6 6OpHJl )l, . VIx,THepHi1 )l, Bpra x,HcO6Jl HH6 6OpHJla)l,HraH cp H3 cTaBKacHHH peaJl cpOH3 cTaBKacH )l, HOMJlai1 . H3HHHr x,OJl)l, j-peaJl cp H3 cTaBKacH.
TypJl HHBecTHu, OH JlOi1Hx,aJl pHH 6Hp 6HpH 6HJlaH cOJl wTHpHw)l, yJl HHHr )l, pOMa)l,JlHJl HH y3apO cOJlHwTHpHw JlO3HM 6yJla)l,H, Jl HH 6yH)l,ai1 cOJl w yYyH aBBaJl 6Op )l, pOMa)l,Jl HH (cpOH3 cTaBKaJl HH) 6Hp Bal\T )l, BpHra KeJl KepaK. KynHHYa cp 3 cTaBKaJl HH 6Hp i1HJlJlHK )l,aBpra KeJlTHpHJla)l,H. yH)l, cpOH3Jl HH x,HcO6Jl w 6OpHwHHHr TypJl cXeMaJl H l\ JlJl HHJl MyMKHH. 0)l, i1 cpOH3 cTaBKaJl cXeMacH 6yi1 Ya x,HcO6Jl 6OpHJlca, HOMHHaJl cp H3 cTaBKacH )l, HOMJlaHa)l, . MHJlJl cp cTaBKacH MypaKKa6 cpOH3Jl cXeMacH 6yi1 x,HcO6Jl HH6 6OpHJl a, 3cpcp KTHB cp cTaBKacH )l,ei1HJla)l, .
MOc paBHw)l, i_H Ba i₃ - HOMHHaJl Ba 3cpcp KTHB cp 3 cTaBKaJl H )l, 6 6eJlrHJlai1 H3. Bal\ r m
)l, BpJl pH yYyH OpTTHpHJlraH cyMMaHH OJl MyMKHH (O)l,)l,Hi1 cpOH3Jl cXeMacH 6yi1 ):
S₁ P(1mxj)
0pTTHpHJl aH cyMMaHH HOMHHaJl cp H3 cTaBKacH 6yi1 3ca:
S₂ P(1i_n )
S₁ Ba S₂ TeHrJl Jl p)l,aH i_H = m_j TeHrJl HH OJl MH3. MypaKKa6 cpOH3Jl cXeMacH 6yi1 Bal\THHHr m )l,aBpJl pHra OpTTHpHJlraH cyMMa:

1
S P(1j)^m
3cpcp KTHB cpOH3 cTaBKacH 6yi1 6Hp i1HJl OpTTHpHJl cyMMa:
S₂ P(1i₃ ) S_l = S₂ TeHrJl )l, H l\ i1H)l, rHHH OJl 3:

3
i (1j)^m 1
Arap i_H = mj 3KaHJl HHH x,HcO6ra OJl , yH)l, HOMHHaJl Ba 3cpcp cp H3 cTaBKaJl OpacH)l, H 6OFJlHl\Jl HH TOnHw MyMKHH:
i (1^iH )^m 1
3 _m