Финансовая математика / Финансовая математика - УМК - Зуев-Пыжев - 2002 - 22

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6.

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8

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S = P(1+ ni )

6 = 3 + L

Q

=

3

+

M

23

6

s

)

(

P

GZjZs_gb_

n = S − P

n =

ln(S P)

Q =

OQ(6

3)

Pis

ln(1

+ i)

+

M

P OQ

P

S − P

is =

i

= n S

−1

Pn

P

j = m mn S

P

− 1

P =

S

P =

S

3

=

6

(1

M

23

n

1+ nis

+ i)

+

P

FZl_fZlbq_kdh_

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n =

ln(S P)

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OQ(6

3

)

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_

Pis

ln(1

+ i)

+

M

P OQ

P

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is

=

S − P

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−

1

Pn

P

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P

− 1

P = S (1 − nd

s

)

3 =

6 − G 3

3

=

6

−

I

23

(

)

P

= S

− P

OQ(3 6 )

3 =

!

$ )

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n

Q =

OQ(

Sds

OQ( − G )

−

1

2OQ

2

− P

d

=

S

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3

I =

3

Sn

6

P − 23

s

6

9

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i

j

d

f

glu

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1

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1

is

---

s

s

f

= m

1 + ni

s

1

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ijhp

s

ds

Ijhklu

ds

i

=

---

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1

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1

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− nd

f

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s

j = m mn

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s

s

1 − nds

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(1

n

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(1 + i)

n

−1

i

1

i

is

=

+ i)

d s

=

---

m

1 + i

−1)

d =

f

1 −

m

n

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n

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1

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j

mn

+

j

mn

− 1

+

j

m

− 1

j

j

+

− 1

1

j

m

1

f =

glu

1

=

m

i =

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d =

m

j

is =

m

ds

j

mn

1+

---

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m

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m

1 +

ijhp

n

n 1

m

m

m

Keh`gu

1− (1− d )

n

1

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n

d

1

d

is

=

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=

i =

m

---

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m

1 − d )

n

n

1− d

j = m

1− d

−1

n(1− d )

1−

f

mn

f

mn

1−

f

m

f

1−

1 −

1−

j =

f

m

f

is =

m

1 −

i =

m

−

---

f

mn

ds =

m

f

m

1− f

d = 1− 1

n

−

m

n 1−

1

m

m

m

10

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KD<

Jm[→ KD<→

KD<→

Jm[

1 + ni

glu

SC

= P[1+ n(1− g )is ]

C = P (1+ h100s

)n

(1+ nis )(1+ h100)n −1

ijhp

S

= P K

0

* (1+ ni

s

)*

1

S

r

=

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* (1

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s KD<

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1

r =

n

ν

ν

K1

K0

Ijhklu

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(1

1+ nr

)n

−

1

+ h

is =

100

n

(1+ i)n

)n

SC =P[(1−g)(1+i)n +g]

C = P (1+ h

100

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r = i +

h

+ i

h

S

= P K

0

* (1+ i)n *

1

S

r

=

Pr * (1+ i

)n * K

1

ν

ν

K1

K

KD<

100

0

100

G = P[(1 + i)n

−1]g

i =

1+ r

−1

1

+ h

100

11

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12

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 

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OQ = OQ

65

OQP

OQ

65 OQ

OQ =



6 5

OQP

 5

Q

OQ Q =

Q=

OQ Q =

Q

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5

6 =

6

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6

=

S = P = L

 S= P >

S = P → ∞ 5=6 H

S > P = S>

S> P = S OQPM

6



S > P > P ≠ S

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14

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]h^Z b \_kv ^hoh^ gZijZ\ey_lky gZ m\_ebq_gb_ lhj]h\uo hi_jZpbc		10. <_dk_ev ghfbgZevghc klhbfhklvx luk jm[ mql_g [Zgdhf aZ
\ Hij_^_eblv ^hoh^ghklv wlh]h \b^Z ^_yl_evghklb \ \b^_ ijhklhc ]h^h-		^g_c ^h kjhdZ _]h hieZlu \_dk_e_^Zl_e_f ih ijhklhc mq_lghc klZ\d_ ]h^h-
\hc klZ\db ijhp_glh\ _keb nbgZgkh\Zy hi_jZpby ih\lhjy_lky \ l_q_gb_ ihem-		\uo Q_fm jZ\gZ ^hoh^ghklv wlhc hi_jZpbb \ \b^_ ijhklhc ]h^h\hc ijhp_glghc
]h^Z b ^hoh^Z gZijZ\ey_lky gZ jZa\blb_ lhj]h\uo hi_jZpbc		klZ\db"
4.	Ij_^ijbylb_ _`_^_dZ^gh \deZ^u\Z_l \ p_ggu_ [mfZ]b luk jm[	11. <_dk_ev ghfbgZevghc klhbfhklvx luk jm[ mql_g [Zgdhf aZ
b ijh^Z_l bo gZ ^hjh`_		^g_c ^h kjhdZ _]h hieZlu \_dk_e_^Zl_e_f ih ijhklhc ijhp_glghc klZ\d_
Z Hij_^_eblv ^hoh^ghklv \ \b^_ ijhklhc ijhp_glghc klZ\db ^Zgghc nb-		]h^h\uo Q_fm jZ\_g ^hoh^ hl wlhc hi_jZpbb"
gZgkh\hc hi_jZpbb		12. <_dk_ev ghfbgZevghc klhbfhklvx luk jm[ mql_g [Zgdhf aZ
[ Hij_^_eblv ^hoh^ghklv \ \b^_ ijhklhc ijhp_glghc klZ\db _keb hi_-		^g_c ^h kjhdZ _]h hieZlu \_dk_e_^Zl_e_f ih ijhklhc ijhp_glghc klZ\d_
jZpbb ih\lhjyxlky jZa b \_kv ^hoh^ dZiblZebabjm_lky		]h^h\uo Kdhevdh kj_^kl\ \eh`be [Zgd \ wlm hi_jZpbx"
\ Hij_^_eblv ^hoh^ghklv \ \b^_ ijhklhc ijhp_glghc klZ\db _keb hi_-		13. <_dk_ev ghfbgZevghc klhbfhklvx luk jm[ mql_g [Zgdhf aZ
jZpbb ih\lhjyxlky jZa b ^hoh^Z dZiblZebabjm_lky		^g_c ^h kjhdZ _]h hieZlu \_dk_e_^Zl_e_f ih ijhklhc mq_lghc klZ\d_ ]h^h-
5.	I_gkbhgguc nhg^ gZqbkey_l i_gb aZ g_k\h_\j_f_ggh_ i_j_qbke_-	\uo Q_fm jZ\_g ^bkdhgl wlhc hi_jZpbb"
gb_ \aghkh\ \ k\hc Z^j_k \ jZaf_j_ hl g_mieZq_gghc kmffu aZ dZ`^uc ^_gv
Ij_^ijbylb_ g_ i_j_qbkey_l \aghku \ kmff_ luk jm[ \ l_q_gb_ ^g_c
	15		16

14.GZ kdhevdh ^g_c ^he`_g baf_gblvky kjhd hdhgqZgby k^_edb _keb iZjlg_j ij_^eh`be aZieZlblv gZ f_gvr_ i_j\hgZqZevgZy kmffZ dhgljZdlZ

luk jm[ mjh\_gv ^hoh^ghklb dhgljZdlZ ]h^h\uo"

15.K^_edZ aZdZgqb\Z_lky gZ ^g_c iha`_ q_f ij_^mkfZljb\Zehkv dhgljZdlhf i_j\hgZqZevgh iZjlg_j ij_^eZ]Z_l aZieZlblv gZ f_gvr_ i_j- \hgZqZevgZy kmffZ dhgljZdlZ luk jm[ Q_fm jZ\gZ ^hoh^ghklv dhgljZdlZ \ nhjf_ ijhklhc ]h^h\hc ijhp_glghc klZ\db _keb k^_edZ m^h\e_l\hjy_l h[_ klh- jhgu"

16.GZ kdhevdh ^he`_g baf_gblvky kjhd hdhgqZgby k^_edb _keb iZjl- g_j ij_^eh`be aZieZlblv gZ [hevr_ i_j\hgZqZevgZy kmffZ dhgljZdlZ

luk jm[ mjh\_gv ^hoh^ghklb dhgljZdlZ ]h^h\uo"

17.IZjlg_j ij_^eh`be h[t_^bgblv ieZl_`b ih dhgljZdlm aZdexq_g- ghfm ] \ h^bg Q_fm jZ\_g wlhl ieZl_` _keb h[t_^bgyxlky ieZl_`b

luk jm[ hl ] b luk jm[ hl ] \ ieZl_`

] ijb ^hoh^ghklb dhgljZdlZ \ ]h^h\uo"

18.IZjlg_j ij_^eh`be h[t_^bgblv ieZl_`b ih dhgljZdlm aZdexq_g- ghfm ] \ h^bg Dh]^Z gm`gh k^_eZlv wlhl ieZl_` _keb h[t_^bgyxl- ky ieZl_`b luk jm[ ] b luk jm[ ] \ ieZl_` jZ\guc luk jm[ ijb ^hoh^ghklb dhgljZdlZ \ ]h^h\uo"

19.Hp_gblv \_ebqbgm ieZl_`_c luk jm[ hl ] b luk jm[ hl ] gZ ^Zlm \lhjh]h ieZl_`Z _keb ieZl_`b h[_ki_qb\Zxl ^h- oh^ghklv ]h^h\uo

20.GZclb gZjZs_ggmx kmffm ieZl_`_c luk jm[ hl ] b

luk jm[ hl ] gZ ] _keb ieZl_`b h[_ki_qb\Zxl ^h- oh^ghklv ]h^h\uo

21.GZclb kh\j_f_ggmx klhbfhklv ieZl_`_c luk jm[ hl

] b luk jm[ hl ] gZ ^Zlm i_j\h]h ieZl_`Z _keb ieZ- l_`b h[_ki_qb\Zxl ^hoh^ghklv ]h^h\uo

22.Hp_gblv \_ebqbgm ieZl_`_c luk jm[ hl ] b luk jm[ hl ] gZ ] _keb ieZl_`b h[_ki_qb\Zxl ^hoh^ghklv

]h^h\uo

23.Hp_gblv k ihfhsvx fh^_eb ihklhygghc ]h^h\hc j_glu ihklgmf_- jZg^h kh\j_f_ggmx klhbfhklv ieZl_`_c luk jm[ hl ] b luk jm[ hl ] _keb ieZl_`b h[_ki_qb\Zxl ^hoh^ghklv ]h^h\uo

24.Hp_gblv klhbfhklv ih khklhygbx gZ ] ieZl_`_c

luk jm[ hl ] b luk jm[ hl ] _keb ieZl_`b h[_ki_- qb\Zxl ^hoh^ghklv ]h^h\uo

25.Hp_gblv k ihfhsvx fh^_eb ihklhygghc ]h^h\hc j_glu ihklgmf_- jZg^h gZjZs_ggmx klhbfhklv ieZl_`_c luk jm[ hl ] b luk jm[ hl ] _keb ieZl_`b h[_ki_qb\Zxl ^hoh^ghklv ]h^h\uo

17

26.Ij_^ijbylb_ ih dj_^blm \ kmff_ luk jm[ \aylhfm \ [Zgd_

] k^_eZeh ke_^mxsb_ ieZl_`b ih ih]Zr_gbx hkgh\gh]h ^he]Z

] ±luk j ] luk jm[ ] luk jm[ Hp_gblv kmffm ^he]Z gZ ] _keb \ dj_^blghf ^h]h\hj_ ijhklZy ]h^h- \Zy ijhp_glgZy klZ\dZ khklZ\ey_l b

Z j_Zebah\Zgh©ijZ\beh lhj]h\pZª [ j_Zebah\Zg ZdlmZjguc f_lh^

27.Ij_^ijbylb_ ih dj_^blm \ kmff_ luk jm[ \aylhfm \ [Zgd_

] k^_eZeh ke_^mxsb_ ieZl_`b ih ih]Zr_gbx h[s_c aZ^he`_gghklb

] ± luk jm[ ] luk jm[ ] luk jm[ Hp_gblv kmffm ^he]Z gZ ] _keb \ dj_^blghf ^h]h\hj_ ijhklZy ]h^h\Zy ijhp_glgZy klZ\dZ khklZ\ey_l b

Z j_Zebah\Zgh©ijZ\beh lhj]h\pZª [ j_Zebah\Zg ZdlmZjguc f_lh^

28.Ij_^ijbylb_ ieZgbjm_l ]Zkblv h[smx aZ^he`_gghklv ih dj_^blm \ kmff_ luk jm[ \aylhfm \ [Zgd_ ] ih ijhklhc ]h^h\hc ijhp_gl- ghc klZ\d_ jZ\gufb kmffZfb ih luk jm[ _`_f_kyqgh DZdh_ \j_fy [m^_l kms_kl\h\Zlv aZ^he`_gghklv [Zgdm"

29.Ij_^ijbylb_ ieZgbjm_l ]Zkblv hkgh\ghc ^he] ih dj_^blm \ kmff_

luk jm[ \aylhfm \ [Zgd_ ] ih ijhklhc ]h^h\hc ijhp_glghc klZ\d_ jZ\gufb kmffZfb ih luk jm[ _`_f_kyqgh b dZ`^uc f_kyp \u- ieZqb\Zlv ijhp_glu DZdh_ \j_fy [m^_l kms_kl\h\Zlv aZ^he`_gghklv [Zgdm"

9DHGLJHEV AG:GBC KLM>?GLH<

AZsblZ hlq_lZ ih jZkq_lghfm aZ^Zgbx \uiheg_ggh]h \ ibkvf_gghc nhjf_

AZq_l

<hijhku d aZq_lm \ mklghc nhjf_

1.Ij_^f_l f_lh^ b h[t_dl nbgZgkh\hc fZl_fZlbdb __ f_klh \ dheb- q_kl\_gghf ZgZeba_ nbgZgkh\uo hi_jZpbc

2.Ijhklu_ b pbdebq_kdb_ nbgZgkh\u_ hi_jZpbb iZjZf_lju nbgZg- kh\hc hi_jZpbb

3.DZiblZebaZpby ^hoh^Z dj_^blu b ^jm]b_ nhjfu ^he]Z

4.Ijhp_glu ijhp_glgZy klZ\dZ b ^hoh^ghklv nbgZgkh\hc hi_jZpbb

5.Kj_^gyy ^hoh^ghklv nbgZgkh\uo hi_jZpbc

6.GZjZs_gb_ ih ijhklhc klZ\d_ ijhp_glh\ JZkq_l ijhp_glh\ ^ey djZldhkjhqguo hi_jZpbc

7.Keh`gu_ ijhp_glu

8.FZl_fZlbq_kdh_ ^bkdhglbjh\Zgb_ b [Zgdh\kdbc mq_l

18

9.Baf_jbl_eb ^hoh^ghklb

10.Hij_^_e_gb_ kjhdZ nbgZgkh\hc hi_jZpbb __ klhbfhklguo iZjZ- f_ljh\ b ^hoh^ghklb

11.Hij_^_e_gb_ ^bkdhglZ b gZjZs_gby

12.H[f_ggu_ dmjku \Zexl \ gZqZe_ nbgZgkh\hc hi_jZpbb b ih __ hdhgqZgbb b baf_g_gb_ fgh`bl_ey gZjZs_gby

13.Hij_^_e_gb_ ih khhlghr_gbx dmjkh\ ij_^_evgh]h mjh\gy ^hoh^gh- klb hi_jZpbb

14.Hp_gdZ gZeh]h\uo hlqbke_gbc b __ mq_l \ fh^_eyo nhjfmeZo gZ-

jZs_gby

15.Baf_jbl_eb bgneypbb b __ mq_l \ fh^_eyo gZjZs_gby G_llh b [jmllh klZ\db gZjZs_gby

16.NbgZgkh\u_ hi_jZpbb b dhgljZdlu k^_edb

17.Ijbgpbi nbgZgkh\hc wd\b\Ze_glghklb h[yaZl_evkl\ Wd\b\Ze_gl- ghklv ijhp_glguo b mq_lguo klZ\hd

18.GhfbgZevgu_ b wnn_dlb\gu_ klZ\db

19.Wd\b\Ze_glghklv ieZl_`_c

20.KjZ\g_gb_ ieZl_`_c b bo dhgkheb^Zpby

21.G_j_]meyjgu_ b j_]meyjgu_ ihlhdb ieZl_`_c

22.DeZkkbnbdZpby nbgZgkh\uo j_gl

23.IZjZf_lju j_glu

24.IhklhyggZy j_glZ ihklgmf_jZg^h

25.F_lh^u jZkq_lZ h[h[s_gguo iZjZf_ljh\ ihlhdh\

26.Hij_^_e_gb_ iZjZf_ljh\ j_glu

27.K[ZeZgkbjh\Zgghklv ihlhdh\ ieZl_`_c

28.Dhg\_jkby ihlhdh\

29.IZjZf_lju ^he]hkjhqghc aZ^he`_gghklb b ieZgbjh\Zgb_ __ ih]Z-

r_gby

30.Ih]Zkbl_evguc nhg^

31.Ih]Zr_gb_ qZklbqgufb ieZl_`Zfb

32.:dlmZjguc f_lh^

32.©IjZ\beh lhj]h\pZª

33.Ih]Zr_gb_ jZ\gufb qZklyfb hkgh\gh]h ^he]Z

34.Ih]Zr_gb_ jZ\gufb qZklyfb h[s_c aZ^he`_gghklb

35.Hp_gdZ qbklhc kh\j_f_gghc klhbfhklb bg^_dkZ j_glZ[_evghklb \gmlj_gg_c ghjfu ^hoh^ghklb

19

9,MQ?;GH F?LH>BQ?KDH? H;?KI?Q?GB? DMJK:

Hkgh\gZy ebl_jZlmjZ

Q_lujdbg ? F NbgZgkh\Zy fZl_fZlbdZ Mq_[gbd ± F >_eh

±k

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1. ;ZrZjbg = I GZqZeZ nbgZgkh\hc fZl_fZlbdb F BGNJ:±F

±k

2. ;mo\Zeh\ : < ;mo\Zeh\Z < < B^_evkhg : < NbgZgkh\u_ \uqbke_gby ^ey ijhn_kkbhgZeh\ Ih^ h[s. j_^ : < ;mo\Zeh\Z ± KI[ ;O< I_l_j[mj]

±k be

<Zs_gdh L < FZl_fZlbdZ nbgZgkh\h]h f_g_^`f_glZ ± F I_jki_d- lb\Z±k

<_[_j F Dhff_jq_kdb_ jZkq_lu hl : ^h Y Nhjfmeu ijbf_ju jZkq_- lh\ b ijZdlbq_kdb_ kh\_lu I_j k g_f±F >_eh b K_j\bk±k

?jrh\ X K NbgZgkh\Zy fZl_fZlbdZ \ \hijhkZo b hl\_lZo Mq_[gh_ ihkh[b_±Gh\hkb[bjkd Kb[bjkdh_ kh]eZr_gb_±k

6. DZi_evyg K G E_\dh\bq H : Hkgh\u dhff_jq_kdbo b nbgZgkh\uo jZkq_lh\±Fbgkd GLP©:IBª ±k

7. DZiblhg_gdh < < NbgZgkh\Zy fZl_fZlbdZ b __ ijbeh`_gby Mq_[gh ijZdl ihkh[b_ ^ey \mah\±F IJBHJ±k

8. DZkbfh\Z H X <\_^_gb_ \ nbgZgkh\mx fZl_fZlbdm ZgZeba dj_^blguo b bg\_klbpbhgguo hi_jZpbc±F :gdbe±k

Dh\Ze_\ < < K[hjgbd aZ^Zq ih nbgZgkh\hfm ZgZebam Mq_[. ihkh[b_± F NbgZgku b klZlbklbdZ±k

Dh\Ze_\ < < MeZgh\ < : Dmjk nbgZgkh\uo \uqbke_gbc ± F Nb- gZgku b klZlbklbdZ±k be

Dhebq_kl\_ggu_ f_lh^u nbgZgkh\h]h ZgZebaZ Ih^ j_^ K >` ;jZm- _jZ F I Djbmf_gZ I_j k Zg]e±F BgnjZ F

12. Dhqh\bq ? NbgZgkh\Zy fZl_fZlbdZ L_hjby b ijZdlbdZ nbgZgkh\h [Zgdh\kdbo jZkq_lh\ I_j k k_j[ Ij_^bke ? F Q_lujdbgZ±F NbgZgku b klZlbklbdZ±k

13. EmdZk_\bq B Y :gZeba nbgZgkh\uo hi_jZpbc F_lh^u fh^_eb l_o- gbdZ \uqbke_gbc±F NbgZgku XGBLB±k

14. EmdZrbg X I Hkgh\u nbgZgkh\hc fZl_fZlbdb Mq_[gh_ ihkh[b_ ±

F Ba^ \h FWKB±k

FZeuobg < B NbgZgkh\Zy fZl_fZlbdZ Mq_[. ihkh[b_ ^ey \mah\ ±

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20