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D @0AG5B0 4>E>4=>AB8 F5==KE 1C<03. >4E>4, ?@8 :>B>@>< D0:B>@ 2@5<5=8 83@05B @5H0NICN @>;L, =07K205BAO: A 2@5<5==>9; B AB0B8G5A:89; C 48=0<8G5A:89; D AB0B8AB8G5A:89. @>F5=BK 2 D8=0=A>2KE @0AG5B0E: A MB> 4>E>4=>ABL, 2K@065==0O 2 2845 45AOB8G=>9 4@>18; B MB> 01A>;NB=0O 25;8G8=0 4>E>40 >B ?@54>AB02;5=8O 45=53 2 4>;3 2 ;N1>9 53> D>@<5; C ?>:07K20NB, A:>;L:> 45=56=KE 548=8F 4>;65= 70?;0B8BL 705<I8: 70 ?>;L7>20=85 2 B5G5=85 >?@545;5==>3> ?5@8>40 2@5<5=8 100 548=8F ?5@2>=0G0;L=>9 AC<<K 4>;30; D MB> %. @>F5=B=0O AB02:0 MB>: A >B=>A8B5;L=K9 ?>:070B5;L, E0@0:B5@87CNI89 8=B5=A82=>ABL =0G8A;5=8O ?@>F5=B>2; B 01A>;NB=0O 25;8G8=0 4>E>40 >B ?@54>AB02;5=8O 45=53 2 4>;3 2 ;N1>9 53> D>@<5; C AB02:0, 70D8:A8@>20==0O 2 2845 >?@545;5==>3> G8A;0 2 D8=0=A>2KE :>=B@0:B0E; D >B=>H5=85 AC<<K ?@>F5=B=KE 45=53 : 25;8G8=5 AAC4K.  :0G5AB25 548=8FK 2@5<5=8 2 D8=0=A>2KE @0AG5B0E ?@8=OB: A 3>4; B :20@B0;; C <5AOF; D 45=L. 0@0I5=85 MB>: A ?@>F5AA C25;8G5=8O :0?8B0;0 70 AG5B ?@8A>548=5=8O ?@>F5=B>2; B 1078A=K9 B5<? @>AB0; C >B=>H5=85 =0@0I5==>9 AC<<K : ?5@2>=0G0;L=>9 AC<<5 4>;30; D 42865=85 45=56=>3> ?>B>:0 >B =0AB>OI53> : 1C4CI5<C. >MDD8F85=B =0@0I5=8O MB>: A >B=>H5=85 AC<<K ?@>F5=B=KE 45=53 : 25;8G8=5 ?5@2>=0G0;L=>9 AC<<K; B >B=>H5=85 =0@0I5==>9 AC<<K : ?5@2>=0G0;L=>9 AC<<5; C >B=>H5=85 ?5@2>=0G0;L=>9 AC<<K : 1C4CI59 25;8G8=5 45=56=>9 AC<<K; D >B=>H5=85 ?@>F5=B>2 : ?@>F5=B=>9 AB02:5. 84K ?@>F5=B=KE AB02>: 2 7028A8<>AB8 >B 8AE>4=>9 107K: A ?>AB>O==0O, A;>6=0O; B ?@>AB0O, ?5@5<5==0O; C ?@>AB0O, A;>6=0O; D ?>AB>O==0O, ?5@5<5==0O. $8:A8@>20==0O ?@>F5=B=0O AB02:0 MB>: A AB02:0, =587<5==0O =0 ?@>BO65=88 2A53> ?5@8>40 AAC4K; B AB02:0, ?@8<5=O5<0O : >4=>9 8 B>9 65 ?5@2>=0G0;L=>9 AC<<5 4>;30; C AB02:0, 70D8:A8@>20==0O 2 2845 >?@545;5==>3> G8A;0 2 D8=0=A>2KE :>=B@0:B0E; D >B=>H5=85 AC<<K ?@>F5=B=KE 45=53 : 25;8G8=5 AAC4K.  ;020 2. ?5@0F88 =0@0I5=8O   HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/2.1.ru.shtml" 2.1. @>ABK5 ?@>F5=BK  HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/2.2.ru.shtml" 2.2. !;>6=K5 ?@>F5=BK  HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/2.3.ru.shtml" 2.3. -:2820;5=B=>ABL AB02>: 8 70<5=0 ?;0B5659  HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/test2.ru.shtml" "5ABK 4;O ?@>25@:8 CA2>5=8O ?@>945==>3> <0B5@80;0  2.1. @>ABK5 ?@>F5=BK   HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/2.1.1.ru.shtml" 2.1.1. $>@<C;0 ?@>ABKE ?@>F5=B>2  HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/2.1.2.ru.shtml" 2.1.2. 0AG5B ?@>F5=B>2 A 8A?>;L7>20=85< ?@>F5=BKE G8A5;  HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/2.1.3.ru.shtml" 2.1.3. 5@5<5==K5 AB02:8  HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/2.1.4.ru.shtml" 2.1.4. ?@545;5=85 A@>:0 AAC4K 8 25;8G8=K ?@>F5=B=>9 AB02:8 2.1.1. $>@<C;0 ?@>ABKE ?@>F5=B>2  0AA<>B@8< ?@>F5AA =0@0I5=8O (accumulation), B.5. >?@545;5=8O 45=56=>9 AC<<K 2 1C4CI5<, 8AE>4O 87 7040==>9 AC<<K A59G0A. -:>=><8G5A:89 A<KA; >?5@0F88 =0@0I5=8O A>AB>8B 2 >?@545;5=88 25;8G8=K B>9 AC<<K, :>B>@>9 1C45B 8;8 65;05B @0A?>;030BL 8=25AB>@ ?> >:>=G0=88 MB>9 >?5@0F88. 45AL 845B 42865=85 45=56=>3> ?>B>:0 >B =0-AB>OI53> : 1C4CI5<C. 5;8G8=0 FV ?>:07K205B 1C4CICN AB>8<>ABL "A53>4=OH=59" 25;8G8=K PV ?@8 7040==>< C@>2=5 8=B5=A82=>AB8 =0G8A;5=8O ?@>F5=B>2 i.  INCLUDEPICTURE "http://irbis.asu.ru/mmc/econ/u_finmath/ris03.gif" \* MERGEFORMATINET  8A. 3. >38:0 D8=0=A>2>9 >?5@0F88 =0@0I5=8O@8 8A?>;L7>20=88 ?@>ABKE AB02>: ?@>F5=B>2 ?@>F5=BK (?@>F5=B=K5 45=L38) >?@545;ONBAO 8AE>4O 87 ?5@2>=0G0;L=>9 AC<<K 4>;30. !E5<0 ?@>ABKE ?@>F5=B>2 ?@54?>;0305B =587<5==>ABL 107K, A :>B>@>9 ?@>8AE>48B =0G8A;5=85 ?@>F5=B>2. 7 >?@545;5=8O ?@>F5=B>2 =5 B@C4=> 70<5B8BL, GB> ?@>F5=BK (?@>F5=B=K5 45=L38) ?@54AB02;ONB A>1>9, ?> ACB8, 01A>;NB=K5 ?@8@>ABK: I = FV - PV, 0 ?>A:>;L:C 1070 4;O 8E =0G8A;5=8O O2;O5BAO ?>AB>O==>9, B> 70 @O4 ;5B >1I89 01A>;NB=K9 ?@8@>AB A>AB028B 8E AC<<C 8;8 ?@>872545=85 01A>;NB=KE ?@8@>AB>2 =0 :>;8G5AB2> ;5B AAC4K: I = (FV - PV) n = [(FV - PV) / PV " PV] n = i " PV " n, 345 i = (FV - PV) / PV ?> >?@545;5=8N ?@>F5=B=>9 AB02:8. "0:8< >1@07><, @07<5@ >68405<>3> 4>E>40 7028A8B >B B@5E D0:B>@>2: >B 25;8G8=K 8=25AB8@>20==>9 AC<<K, >B C@>2=O ?@>F5=B=>9 AB02:8 8 >B A@>:0 D8=0=A>2>9 >?5@0F88. ">340 =0@0I5==CN AC<<C ?> AE5<5 ?@>ABKE ?@>F5=B>2 <>6=> 1C45B >?@545;OBL A;54CNI8< >1@07><: FV = PV + I = PV + i " PV " n = PV (1 + i " n) = PV " k=, 345 k=  :>MDD8F85=B (<=>68B5;L) =0@0I5=8O ?@>ABKE ?@>F5=B>2. 0==0O D>@<C;0 =07K205BAO "D>@<C;>9 ?@>ABKE ?@>F5=B>2". >A:>;L:C :>MDD8F85=B =0@0I5=8O ?@54AB02;O5B A>1>9 7=0G5=85 DC=:F88 >B G8A;0 ;5B 8 C@>2=O ?@>F5=B=>9 AB02:8, B> 53> 7=0G5=8O ;53:> B01C;8@CNBAO. "0:8< >1@07><, 4;O >1;53G5=8O D8=0=A>2KE @0AG5B>2 <>6=> 8A?>;L7>20BL D8=0=A>2K5 B01;8FK, A>45@60I85 :>MDD8F85=BK =0@0I5=8O ?> ?@>ABK< ?@>F5=B0<. @8<5@ 1. !C<<0 2 @07<5@5 2'000 @C1;59 40=0 2 4>;3 =0 2 3>40 ?> AE5<5 ?@>AB>3> ?@>F5=B0 ?>4 10% 3>4>2KE. ?@545;8BL ?@>F5=BK 8 AC<<C, ?>4;560ICN 2>72@0BC. 5H5=85: 0@0I5==0O AC<<0: FV = PV (1 + n " i ) = 2'000 (1 + 2 " 0'1) = 2'400 @C1. 8;8 FV = PV " k= = 2'000 " 1,2 = 2'400 @C1. !C<<0 =0G8A;5==KE ?@>F5=B>2: I = PV " n " i = 2'000 " 2 " 0,1 = 400 @C1. 8;8 I = FV - PV = 2'400 - 2'000 = 400 @C1. "0:8< >1@07><, G5@57 420 3>40 =5>1E>48<> 25@=CBL >1ICN AC<<C 2 @07<5@5 2'400 @C1;59, 87 :>B>@>9 2'000 @C1;59 A>AB02;O5B 4>;3, 0 400 @C1;59  "F5=0 4>;30". !;54C5B 70<5B8BL, GB> ?>4>1=K5 7040G8 =0 ?@0:B8:5 2AB@5G0NBAO @54:>, ?>A:>;L:C : ?@>ABK< ?@>F5=B0< ?@81530NB 2 A;CG0OE: 2K40G8 :@0B:>A@>G=KE AAC4, B.5. AAC4, A@>: :>B>@KE ;81> @025= 3>4C, ;81> <5=LH5 53>, A >4=>:@0B=K< =0G8A;5=85< ?@>F5=B>2; :>340 ?@>F5=BK =5 ?@8A>548=ONBAO : AC<<5 4>;30, 0 ?5@8>48G5A:8 2K?;0G820NBAO.  B5E A;CG0OE, :>340 A@>: AAC4K <5=55 3>40, ?@>8AE>48B <>48D8:0F8O D>@<C;K: 0) 5A;8 A@>: AAC4K 2K@065= 2 <5AOF0E (  ), B> 25;8G8=0 n 2K@0605BAO 2 2845 4@>18: n =  / 12, B>340 2A5 D>@<C;K <>6=> ?@54AB028BL 2 2845: FV = PV (1 +  / 12 " i); I = PV "  / 12 " i; k= = 1 +  / 12 " i. @8<5@ 2. 7<5=8< CA;>28O ?@54K4CI53> ?@8<5@0, A=8782 A@>: 4>;30 4> 6 <5AOF52. 5H5=85: 0@0I5==0O AC<<0: FV = PV (1 +  / 12 " i) = 2'000 (1 + 6/12 " 0'1) = 2'100 @C1. 8;8 FV = PV " k= = 2'000 " 1,05 = 2'100 @C1. !C<<0 =0G8A;5==KE ?@>F5=B>2: I = PV "  / 12 " i = 2'000 " 6/12 " 0,1 = 100 @C1. 8;8 I = FV - PV = 2'100 - 2'000 = 100 @C1. "0:8< >1@07><, G5@57 ?>;3>40 =5>1E>48<> 25@=CBL >1ICN AC<<C 2 @07<5@5 2'100 @C1;59, 87 :>B>@>9 2'000 @C1;59 A>AB02;O5B 4>;3, 0 ?@>F5=BK  100 @C1;59. 1) 5A;8 2@5<O 2K@065=> 2 4=OE (t), B> 25;8G8=0 n 2K@0605BAO 2 2845 A;54CNI59 4@>18: n = t / T, 345 t  G8A;> 4=59 AAC4K, B.5. ?@>4>;68B5;L=>ABL A@>:0, =0 :>B>@K9 2K40=0 AAC40; T  @0AG5B=>5 G8A;> 4=59 2 3>4C (2@5<5==0O 1070). BAN40 <>48D8F8@>20==K5 D>@<C;K 8<5NB A;54CNI89 284: FV = PV (1 + t / T " i ); I = PV " t / T " i; k= = 1 + t / T " i. 45AL 2>7<>6=K A;54CNI85 20@80=BK @0AG5B0: @5<5==CN 107C ( T ) <>6=> ?@54AB028BL ?>-@07=><C: CA;>2=> A>AB>OICN 87 360 4=59.  MB>< A;CG05 @5GL 845B >1 >1K:=>25==>< (ordinary interest), 8;8 :><<5@G5A:>< ?@>F5=B5; 27OBL 459AB28B5;L=>5 G8A;> 4=59 2 3>4C (365 8;8 366 4=59).  MB>< A;CG05 ?>;CG0NB B>G=K9 ?@>F5=B (exact interest). '8A;> 4=59 AAC4K ( t ) B0:65 <>6=> ?>-@07=><C >?@545;OBL: CA;>2=>, 8AE>4O 87 B>3>, GB> ?@>4>;68B5;L=>ABL ;N1>3> F5;>3> <5AOF0 A>AB02;O5B 30 4=59, 0 >AB02H85AO 4=8 >B <5AOF0 AG8B0NB B>G=>,  2 @57C;LB0B5 ?>;CG0NB B0: =07K205<>5 ?@81;865==>5 G8A;> 4=59 AAC4K; 8A?>;L7CO ?@O<>9 AG5B 8;8 A?5F80;L=K5 B01;8FK ?>@O4:>2KE =><5@>2 4=59 3>40, @0AAG8BK20NB D0:B8G5A:>5 G8A;> 4=59 <564C 40B0<8,  2 MB>< A;CG05 ?>;CG0NB B>G=>5 G8A;> 4=59 AAC4K.  HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/2.1.1.ru.shtml" \l "note2#note2" 2>>> "0:8< >1@07><, 5A;8 2@5<O D8=0=A>2>9 >?5@0F88 2K@065=> 2 4=OE, B> @0AG5B ?@>ABKE ?@>F5=B>2 <>65B 1KBL ?@>872545= >4=8< 87 B@5E 2>7<>6=KE A?>A>1>2: 1K:=>25==K5 ?@>F5=BK A ?@81;865==K< G8A;>< 4=59 AAC4K, 8;8, :0: G0AB> =07K20NB, "35@<0=A:0O ?@0:B8:0 @0AG5B0", :>340 ?@>4>;68B5;L=>ABL 3>40 CA;>2=> ?@8=8<05BAO 70 3604=59, 0 F5;>3> <5AOF0  70 30 4=59. -B>B A?>A>1 >1KG=> 8A?>;L7C5BAO 2 5@<0=88, 0=88, (25F88. 1K:=>25==K5 ?@>F5=BK A B>G=K< G8A;>< 4=59 AAC4K, 8;8 "D@0=FC7A:0O ?@0:B8:0 @0AG5B0", :>340 ?@>4>;68B5;L=>ABL 3>40 CA;>2=> ?@8=8<05BAO 70 360 4=59, 0 ?@>4>;68B5;L=>ABL AAC4K @0AAG8BK205BAO B>G=> ?> :0;5=40@N. -B>B A?>A>1 8<55B @0A?@>AB@0=5=85 2> $@0=F88, 5;L388, A?0=88, (259F0@88. ">G=K5 ?@>F5=BK A B>G=K< G8A;>< 4=59 AAC4K, 8;8 "0=3;89A:0O ?@0:B8:0 @0AG5B0", :>340 ?@>4>;68B5;L=>ABL 3>40 8 ?@>4>;68B5;L=>ABL AAC4K 15@CBAO B>G=> ?> :0;5=40@N. -B>B A?>A>1 ?@8<5=O5BAO 2 >@BC30;88, =3;88, !(. '8AB> D>@<0;L=> 2>7<>65= 8 G5B25@BK9 20@80=B: B>G=K5 ?@>F5=BK A ?@81;865==K< G8A;>< 4=59 AAC4K,  => >= ;8H5= M:>=><8G5A:>3> A<KA;0. ?>;=5 5AB5AB25==>, GB> 2 7028A8<>AB8 >B 8A?>;L7>20=8O :>=:@5B=>9 ?@0:B8:8 =0G8A;5=8O ?@>ABKE ?@>F5=B>2 8E AC<<0 1C45B @07;8G0BLAO ?> 01A>;NB=>9 25;8G8=5. ;O C?@>I5=8O ?@>F54C@K @0AG5B0 B>G=>3> G8A;0 4=59 D8=0=A>2>9 >?5@0F88 ?>;L7CNBAO A?5F80;L=K<8 B01;8F0<8 ?>@O4:>2KE =><5@>2 4=59 3>40 ( HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/pril1.ru.shtml" @8;>65=85 1), 2 :>B>@KE 2A5 4=8 2 3>4C ?>A;54>20B5;L=> ?@>=C<5@>20=K. ">G=>5 :>;8G5AB2> 4=59 ?>;CG05BAO ?CB5< 2KG8B0=8O =><5@0 ?5@2>3> 4=O D8=0=A>2>9 >?5@0F88 87 =><5@0 ?>A;54=53> 4=O D8=0=A>2>9 >?5@0F88. @8<5@ 3. !C<<0 2 <;= @C1. ?>;>65=0 2 10=: 18 D52@0;O =5 28A>:>A=>3> 3>40 8 2>AB@51>20=0 25 45:01@O B>3> 65 3>40. !B02:0 10=:0 A>AB02;O5B 35% 3>4>2KE. ?@545;8BL AC<<C =0G8A;5==KE ?@>F5=B>2 ?@8 @07;8G=>9 ?@0:B8:5 8E =0G8A;5=8O. 5H5=85: 5@<0=A:0O ?@0:B8:0 =0G8A;5=8O ?@>ABKE ?@>F5=B>2: @5<5==0O 1070 ?@8=8<05BAO 70 360 4=59, T = 360. >;8G5AB2> 4=59 AAC4K:  HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/2.1.1.ru.shtml" \l "note3#note3" 3>>> t = 11 (D52@0;L) + 30 (<0@B) + 30 (0?@5;L) + 30 (<09) + 30 (8N=L) + + 30 (8N;L) + 30 (023CAB) + 30 (A5=BO1@L) + 30 (>:BO1@L) + + 30 (=>O1@L) + 25 (45:01@L) - 1 = 305 4=59 !C<<0 =0G8A;5==KE ?@>F5=B>2: I = P " t / T " i = 2'000'000 " 305/360 " 0,35 = 593'055,55 @C1. $@0=FC7A:0O ?@0:B8:0 =0G8A;5=8O ?@>F5=B>2: @5<5==0O 1070 ?@8=8<05BAO 70 360 4=59, T = 360. >;8G5AB2> 4=59 AAC4K: t = 11 (D52@0;L) + 31 (<0@B) + 30 (0?@5;L) + 31 (<09) + 30 (8N=L) + + 31 (8N;L) + 31 (023CAB) + 30 (A5=BO1@L) + 31 (>:BO1@L) + + 30 (=>O1@L) + 25 (45:01@L) - 1 = 310 4=59 > B01;8F0< ?>@O4:>2KE =><5@>2 4=59 2 3>4C ( HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/pr1il1.ru.shtml" @8;>65=85 1) <>6=> >?@545;8BL B>G=>5 G8A;> 4=59 D8=0=A>2>9 >?5@0F88 A;54CNI8< >1@07><: t = 359 - 49 = 310 4=59. !C<<0 =0G8A;5==KE ?@>F5=B>2: I = P " t / T " i = 2'000'000 " 310/360 " 0,35 = 602'777,78 @C1. =3;89A:0O ?@0:B8:0 =0G8A;5=8O ?@>F5=B>2: @5<5==0O 1070 ?@8=8<05BAO 70 365 4=59, T = 365. >;8G5AB2> 4=59 AAC4K 15@5BAO B>G=K<, t = 310 4=59. !C<<0 =0G8A;5==KE ?@>F5=B>2: I = P " t / T " i = 2'000'000 " 310/365 " 0,35 = 594'520,55 @C1. 0: 284=>, @57C;LB0B D8=0=A>2>9 >?5@0F88 2> <=>3>< 7028A8B >B 2K1>@0 A?>A>10 =0G8A;5=8O ?@>ABKE ?@>F5=B>2. >A:>;L:C B>G=>5 G8A;> 4=59 2 1>;LH8=AB25 A;CG052 1>;LH5 ?@81;865==>3> G8A;0 4=59, B> 8 ?@>F5=BK A B>G=K< G8A;>< 4=59 AAC4K >1KG=> ?>;CG0NBAO 2KH5 ?@>F5=B>2 A ?@81;865==K< G8A;>< 4=59 AAC4K.  ?@0:B8G5A:>< A<KA;5 MDD5:B >B 2K1>@0 B>3> 8;8 8=>3> A?>A>10 7028A8B >B 7=0G8B5;L=>AB8 AC<<, D83C@8@CNI8E 2 D8=0=A>2>9 >?5@0F88.  INCLUDEPICTURE "http://irbis.asu.ru/mmc/images/dot.gif" \* MERGEFORMATINET   HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/2.1.1.ru.shtml" \l "rev2#rev2" <<<2=8<0=85: ?@8 >?@545;5=88 ?@>4>;68B5;L=>AB8 D8=0=A>2>9 >?5@0F88 40B0 2K40G8 8 40B0 ?>30H5=8O AG8B0NBAO 70 >48= 45=L.  HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/2.1.1.ru.shtml" \l "rev3#rev3" <<<35 701C4LB5: 45=L 2K40G8 8 45=L 2>72@0I5=8O AAC4K AG8B0NBAO 70 >48= 45=L. 2.1.2. 0AG5B ?@>F5=B>2 A 8A?>;L7>20=85< ?@>F5=B=KE G8A5;   10=:>2A:>9 ?@0:B8:5 @07<5I5==K9 =0 4;8B5;L=>5 2@5<O :0?8B0; <>65B 2 B5G5=85 MB>3> ?5@8>40 2@5<5=8 87<5=OBLAO, B.5. C25;8G820BLAO 8;8 C<5=LH0BLAO ?CB5< 4>?>;=8B5;L=KE 27=>A>2 8;8 >BG8A;5=89. "0:8< >1@07><, ?@8 >1A;C6820=88 AG5B>2 10=:8 AB0;:820NBAO A =5?@5@K2=>9 A5BLN ?>ABC?;5=89 8 @0AE>4>20=85< A@54AB2 8 =0G8A;5=8-5< ?@>F5=B>2 =0 ?>AB>O==> <5=ONICNAO AC<<C.  MB>9 A8BC0F88 2 10=:>2A:>9 ?@0:B8:5 8A?>;L7C5BAO ?@028;>: >1I0O =0G8A;5==0O 70 25AL A@>: AC<<0 ?@>F5=B>2 @02=0 AC<<5 ?@>F5=B>2, =0G8A;5==KE =0 :064CN 87 ?>AB>O==KE =0 =5:>B>@>< >B@57:5 2@5<5=8 AC<<. -B> :0A05BAO 8 4515B>2>9, 8 :@548B>2>9 G0AB8 AG5B0. 07=8F0 ;8HL 2 B><, GB> :@548B>2K5 ?@>F5=BK 2KG8B0NBAO.  B0:8E A;CG0OE 4;O @0AG5B0 ?@>F5=B>2 8A?>;L7C5BAO <5B>48:0 @0AG5B0 A 2KG8A;5=85< ?@>F5=B=KE G8A5;: :064K9 @07, :>340 AC<<0 =0 AG5B5 87<5=O5BAO, ?@>872>48BAO @0AG5B "?@>F5=B=>3> G8A;0" 70 ?5@8>4, 2 B5G5=85 :>B>@>3> AC<<0 =0 AG5B5 1K;0 =587<5==>9. @>F5=B=>5 G8A;> 2KG8A;O5BAO ?> D>@<C;5: @>F5=B=>5 G8A;> = = (!C<<0 =0 AG5B5 " ;8B5;L=>ABL ?5@8>40 2 4=OE) / 100 = = (PV " t) / 100 ;O >?@545;5=8O AC<<K ?@>F5=B>2 70 25AL A@>: 8E =0G8A;5=8O 2A5 "?@>F5=B=K5 G8A;0" A:;04K20NBAO, 8 8E AC<<0 45;8BAO =0 ?>AB>O==K9 45;8B5;L, :>B>@K9 =>A8B =0720=85 "?@>F5=B=K9 :;NG" 8;8 48287>@, >?@545;O5<K9 >B=>H5=85< :>;8G5AB20 4=59 2 3>4C : 3>4>2>9 ?@>F5=B=>9 AB02:5: I = @>F5=B=KE G8A5; : >AB>O==K9 45;8B5;L, 345 >AB>O==K9 45;8B5;L = @>4>;68B5;L=>ABL 3>40 2 4=OE / >4>20O AB02:0 ?@>F5=B>2 = T / i  HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/2.1.2.ru.shtml" \l "note4#note4" 4>>> @>F5=BK, 2KG8A;O5<K5 A 8A?>;L7>20=85< 48287>@0, @0AAG8B0==>3> 8AE>4O 87 365 4=59 2 3>4C, 1C4CB <5=LH5, G5< ?@>F5=BK ?> 48287>@C, 345 :>;8G5AB2> 4=59 2 3>4C ?@8=OB> 70 360, ?>MB><C ?@8 >1A;C6820=88 :>=:@5B=>3> :;85=B0 2A5340 8A?>;L7C5BAO >48= 87 48287>@>2. 5B>48:0 A 8A?>;L7>20=85< ?@>F5=B=KE G8A5; ?> A2>59 ACB8 O2;O5BAO ?>A;54>20B5;L=K< ?@8<5=5=85< D>@<C;K ?@>ABKE ?@>F5=B>2 4;O :064>3> 8=B5@20;0 ?>AB>O=AB20 AC<<K =0 AG5B5: I = I1 + I2 + I3 = P1 " t1 / T " i + P2 " t2 / T " i + P3 " t3 / T " i @8<5@ 4. @8 >B:@KB88 A15@530B5;L=>3> AG5B0 ?> AB02:5 28% 3>4>2KE, 20 <0O 1999 3>40 1K;0 ?>;>65=0 AC<<0 2 @07<5@5 1'000 @C1;59, 0 5 8N;O =0 AG5B 4>102;5=0 AC<<0 2 500 @C1., 10 A5=BO1@O A=OB0 A> AG5B0 AC<<0 2 750 @C1., 0 20 =>O1@O AG5B 1K; 70:@KB. A?>;L7CO ?@>F5=B=K5 G8A;0 >?@545;8BL AC<<C =0G8A;5==KE ?@>F5=B>2 ?@8 CA;>288, GB> 10=: 8A?>;L7C5B "35@<0=A:CN ?@0:B8:C". 5H5=85: !@>: E@0=5=8O AC<<K 2 1'000 @C1. A>AB028; 46 4=59, B>340 @>F5=B=>5 G8A;> 1 = (1'000 " 46) / 100 = 460; A@>: E@0=5=8O AC<<K 2 @07<5@5 1'500 @C1. A>AB028; 66 4=59, >B:C40 @>F5=B=>5 G8A;> 2 = (1'500 " 66) / 100 = 990; A@>: E@0=5=8O C<5=LH5==>9 4> 750 @C1. AC<<K A>AB028; 70 4=59: @>F5=B=>5 G8A;> 3 = (750 " 70) / 100 = 525 8287>@ = 360 / 28 = 12,857 !;54>20B5;L=>, AC<<0 =0G8A;5==KE ?@>F5=B>2 70 ?5@8>4 459AB28O A15@530B5;L=>3> AG5B0 A>AB028B: I = (460 + 990 + 525) / 12,857 = 153,61 @C1. >6=> ?@>25@8BL ?@028;L=>ABL ?@>872545==KE =0<8 @0AG5B>2, 8AE>4O 87 ACB8 ?@>F5=B>2: I = 1'000 " 46 / 360 " 0,28 + 1'500 " 66 / 360 " 0,28 + 750 " 70 / 360 " 0,28 = 153,61 @C1. 0: 2848<, @57C;LB0B 2KG8A;5=89 B>B 65 A0<K9. 2.1.3. 5@5<5==K5 AB02:8  !B02:0 ?@>F5=B>2 =5 O2;O5BAO 70ABK2H59 =0 25G=K5 2@5<5=0 25;8G8=>9, ?>MB><C 2 D8=0=A>2KE >?5@0F8OE, 2 A8;C B5E 8;8 8=KE ?@8G8=, ?@54CA<0B@820NBAO 48A:@5B=> 87<5=ONI85AO 2> 2@5<5=8 ?@>F5=B=K5 AB02:8. 0?@8<5@, =0;8G85 8=D;OF88 2K=C6405B A>1AB25==8:0 45=53 ?5@8>48G5A:8 20@L8@>20BL ?@>F5=B=>9 AB02:>9.  B0:8E A;CG0OE =0@0I5==CN AC<<C >?@545;ONB, 8A?>;L7CO A;54CNICN D>@<C;C: FV = PV " (1 + n1 " i1 + n2 " i2 + & + nk " ik), 345 k  :>;8G5AB2> ?5@8>4>2 =0G8A;5=8O; nk  ?@>4>;68B5;L=>ABL k-3> ?5@8>40; ik  AB02:0 ?@>F5=B>2 2 k->< ?5@8>45. @8<5@ 5. :;04 2 AC<<5 5'000 @C1. 1K; ?>;>65= 2 10=: 25 <0O =5 28A>:>A=>3> 3>40 ?> AB02:5 35% 3>4>2KE, 0 A 1 8N;O 10=: A=878; AB02:C ?> 2:;040< 4> 30% 3>4>2KE 8 15 8N;O 2:;04 1K; 2>AB@51>20=. ?@545;8BL AC<<C =0G8A;5==KE ?@>F5=B>2 ?@8 0=3;89A:>9 ?@0:B8:5 8E =0G8A;5=8O. 5H5=85: >;8G5AB2> 4=59 4;O =0G8A;5=8O ?@>F5=B>2 ?> ?5@2>=0G0;L=> 459AB2CNI59 ?@>F5=B=>9 AB02:5 2 @07<5@5 35% 3>4>2KE @0AAG8BK205BAO B>G=> 8 A>AB02;O5B 37 4=59, 0 ?> 87<5=5==>9 AB02:5 30% 3>4>2KE  14 4=59. BAN40 25;8G8=0 ?@>F5=B>2 1C45B @02=0: I = 5'000 " (37 / 365 " 0,35 + 14 / 365 " 0,30) = 234,93 @C1. "0:8< >1@07><, ?@8 70:@KB88 AG5B0 :;85=B 4>;65= ?>;CG8BL ?@>F5=B>2 2 AC<<5 234,93 @C1. 2.1.4. ?@545;5=85 A@>:0 AAC4K 8 25;8G8=K ?@>F5=B=>9 AB02:8   ;N1>9 ?@>AB59H59 D8=0=A>2>9 >?5@0F88 2A5340 ?@8ACBAB2CNB G5BK@5 25;8G8=K: A>2@5<5==0O 25;8G8=0 (PV), =0@0I5==0O 8;8 1C4CI0O 25;8G8=0 (FV), ?@>F5=B=0O AB02:0 (i) 8 2@5<O (n). =>340 ?@8 @07@01>B:5 CA;>289 D8=0=A>2>9 A45;:8 8;8 55 0=0;875 2>7=8:05B =5>1E>48<>ABL @5H5=8O 7040G, A2O70==KE A >?@545;5=85< >BACBAB2CNI8E ?0@0<5B@>2, B0:8E :0: A@>: D8=0=A>2>9 >?5@0F88 8;8 C@>25=L ?@>F5=B=>9 AB02:8. 0: ?@028;>, 2 D8=0=A>2KE :>=B@0:B0E >1O70B5;L=> D8:A8@CNBAO A@>:8, 40BK, ?5@8>4K =0G8A;5=8O ?@>F5=B>2, ?>A:>;L:C D0:B>@ 2@5<5=8 2 D8=0=A>2>-:><<5@G5A:8E @0AG5B0E 83@05B 206=CN @>;L. 4=0:> 1K20NB A8BC0F88, :>340 A@>: D8=0=A>2>9 >?5@0F88 ?@O<> 2 CA;>28OE D8=0=A>2>9 A45;:8 =5 >3>2>@5=, 8;8 :>340 40==K9 ?0@0<5B@ >?@545;O5BAO ?@8 @07@01>B:5 CA;>289 D8=0=A>2>9 >?5@0F88. 1KG=> A@>: D8=0=A>2>9 >?5@0F88 >?@545;ONB 2 B5E A;CG0OE, :>340 8725AB=0 ?@>F5=B=0O AB02:0 8 25;8G8=0 ?@>F5=B>2. A;8 A@>: >?@545;O5BAO 2 3>40E, B> n = (FV - PV) : (PV " i), 0 5A;8 A@>: A45;:8 =5>1E>48<> >?@545;8BL 2 4=OE, B> ?>O2;O5BAO 2@5<5==0O 1070 2 :0G5AB25 A><=>68B5;O: t = [(FV - PV) : (PV " i)] " T. @8<5@ 6. 0 A:>;L:> 4=59 <>6=> 40BL 2 4>;3 1'000 4>;;0@>2, 8AE>4O 87 8% 3>4>2KE, 5A;8 2>72@0I5==0O AC<<0 1C45B A>AB02;OBL 1'075 4>;;0@>2? 5H5=85: AE>4O 87 D>@<C;K A@>:0 4>;30 4;O ?@>ABKE ?@>F5=B>2, A;54C5B: 4;O >1KG=KE ?@>F5=B>2 t = [(FV - PV) : (PV " i)] " T = = [(1'075 - 1'000) : (1'000 " 0,08) " 360 = 338 4=59; 4;O B>G=KE ?@>F5=B>2 t = [(FV - PV) : (PV " i)] " T = = [(1'075 - 1'000)/(1'000 " 0,08) " 365 = 342 4=O. "0:8< >1@07><, AC<<0 2 1'000 4>;;0@>2 <>65B 1KBL ?@54>AB02;5=0 =0 A@>: 2 342 4=O, 5A;8 2 CA;>28OE D8=0=A>2>9 >?5@0F88 1C45B 8A?>;L7>20= B5@<8= "B>G=K5 ?@>F5=BK", 0 ?> C<>;G0=8N 8;8 8A?>;L7>20=8N B5@<8=0 ">1K:=>25==K5 ?@>F5=BK", A@>: AAC4K A>:@0I05BAO 4> 338 4=59. 5>1E>48<>ABL >?@545;5=8O C@>2=O ?@>F5=B=>9 AB02:8 2>7=8:05B 2 B5E A;CG0OE, :>340 >=0 2 O2=>< 2845 2 CA;>28OE D8=0=A>2>9 >?5@0F88 =5 CG0AB2C5B, => AB5?5=L 4>E>4=>AB8 >?5@0F88 ?> 7040==K< ?0@0<5B@0< <>6=> >?@545;8BL, 2>A?>;L7>202H8AL A;54CNI8<8 D>@<C;0<8: i = (FV - PV) : (PV " n) = [(FV - PV) : (PV " t)] " T. @8<5@ 7.  :>=B@0:B5 ?@54CA<0B@8205BAO ?>30H5=85 >1O70B5;LAB2 G5@57 120 4=59 2 AC<<5 1'200 4>;;0@>2, ?@8 ?5@2>=0G0;L=>9 AC<<5 4>;30 1'150 4>;;0@>2. ?@545;8BL 4>E>4=>ABL >?5@0F88 4;O :@548B>@0 2 2845 ?@>F5=B=>9 AB02:8. 5H5=85: 0AAG8BK205< 3>4>2CN ?@>F5=B=CN AB02:C, 8A?>;L7CO D>@<C;C ">1K:=>25==>3> ?@>F5=B0", ?>A:>;L:C 2 CA;>28OE A45;:8 =5B AAK;:8 =0 "B>G=K9 ?@>F5=B": i = [(FV - PV) : (PV " t)] " T = = [(1'200 - 1'150) : (1'150 " 120)] " 360 = 0,13 "0:8< >1@07><, 4>E>4=>ABL D8=0=A>2>9 >?5@0F88 A>AB028B 13% 3>4>2KE, GB> A>>B25BAB2C5B 25AL<0 2KA>:>4>E>4=>9 D8=0=A>2>9 >?5@0F88, B.:. >1KG=> 4>E>4=>ABL ?>4>1=KE >?5@0F89 :>;51;5BAO >B 2% 4> 8%. 2.2. !;>6=K5 ?@>F5=BK   HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/2.2.1.ru.shtml" 2.2.1. $>@<C;0 A;>6KE ?@>F5=B>2  HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/2.2.2.ru.shtml" 2.2.2. -DD5:B82=0O AB02:0 ?@>F5=B>2  HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/2.2.3.ru.shtml" 2.2.3. 5@5<5==0O AB02:0 ?@>F5=B>2  HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/2.2.4.ru.shtml" 2.2.4. 5?@5@K2=>5 =0G8A;5=85 ?@>F5=B>2  HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/2.2.5.ru.shtml" 2.2.5. ?@545;5=85 A@>:0 AAC4K 8 25;8G8=K ?@>F5=B=>9 AB02:8 2.2.1. $>@<C;0 A;>6=KE ?@>F5=B>2   D8=0=A>2>9 ?@0:B8:5 7=0G8B5;L=0O G0ABL @0AG5B>2 2545BAO A 8A?>;L7>20=85< AE5<K A;>6=KE ?@>F5=B>2. @8<5=5=85 AE5<K A;>6=KE ?@>F5=B>2 F5;5A>>1@07=> 2 B5E A;CG0OE, :>340: ?@>F5=BK =5 2K?;0G820NBAO ?> <5@5 8E =0G8A;5=8O, 0 ?@8A>548=ONBAO : ?5@2>=0G0;L=>9 AC<<5 4>;30. @8A>548=5=85 =0G8A;5==KE ?@>F5=B>2 : AC<<5 4>;30, :>B>@0O A;C68B 107>9 4;O 8E =0G8A;5=8O, =07K205BAO :0?8B0;870F859 ?@>F5=B>2; A@>: AAC4K 1>;55 3>40. A;8 ?@>F5=B=K5 45=L38 =5 2K?;0G820NBAO A@07C ?> <5@5 8E =0G8A;5=8O, 0 ?@8A>548=ONBAO : ?5@2>=0G0;L=>9 AC<<5 4>;30, B> 4>;3, B0:8< >1@07><, C25;8G8205BAO =0 =52K?;0G5==CN AC<<C ?@>F5=B>2, 8 ?>A;54CNI55 =0G8A;5=85 ?@>F5=B>2 ?@>8AE>48B =0 C25;8G5==CN AC<<C 4>;30: FV = PV + I = PV + PV " i = PV " (1 + i)  70 >48= ?5@8>4 =0G8A;5=8O; FV = (PV + I) " (1 + i) = PV " (1 + i) " (1 + i) = PV " (1 + i)2  70 420 ?5@8>40 =0G8A;5=8O; >BAN40, 70 n ?5@8>4>2 =0G8A;5=8O D>@<C;0 ?@8<5B 284: FV = PV " (1 + i)n = PV " k= , 345 FV  =0@0I5==0O AC<<0 4>;30; PV  ?5@2>=0G0;L=0O AC<<0 4>;30; i  AB02:0 ?@>F5=B>2 2 ?5@8>45 =0G8A;5=8O; n  :>;8G5AB2> ?5@8>4>2 =0G8A;5=8O; k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i). ">340 1078A=K9 B5<? @>AB0 70 25AL ?5@8>4, 8AE>4O 87 ?>AB>O==>3> B5<?0 ?@8@>AB0, 8<55B 284: (1 + i)n . 078A=K5 B5<?K @>AB0 8;8 :>MDD8F85=BK (<=>68B5;8) =0@0I5=8O, 7028AOI85 >B ?@>F5=B=>9 AB02:8 8 G8A;0 ?5@8>4>2 =0@0I5=8O, B01C;8@>20=K 8 ?@54AB02;5=K 2  HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/pril2.ru.shtml" @8;>65=88 2. -:>=><8G5A:89 A<KA; <=>68B5;O =0@0I5=8O A>AB>8B 2 B><, GB> >= ?>:07K205B, G5<C 1C45B @02=0 >4=0 45=56=0O 548=8F0 (>48= @C1;L, >48= 4>;;0@ 8 B.?.) G5@57 n ?5@8>4>2 ?@8 7040==>9 ?@>F5=B=>9 AB02:5 i.  HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/2.2.1.ru.shtml" \l "note5#note5" 5>>> @0D8G5A:0O 8;;NAB@0F8O A>>B=>H5=8O =0@0I5==>9 AC<<K ?> ?@>ABK< 8 A;>6=K< ?@>F5=B0< ?@54AB02;5=0 =0 @8AC=:5 4.  INCLUDEPICTURE "http://irbis.asu.ru/mmc/econ/u_finmath/ris04.gif" \* MERGEFORMATINET  8A. 4. 0@0I5=85 ?> ?@>ABK< 8 A;>6=K< ?@>F5=B0<.0: 284=> 87 @8AC=:0 4, ?@8 :@0B:>A@>G=KE AAC40E =0G8A;5=85 ?> ?@>ABK< ?@>F5=B0< ?@54?>GB8B5;L=55, G5< ?> A;>6=K< ?@>F5=B0<; ?@8 A@>:5 2 >48= 3>4 @07=8F0 >BACBAB2C5B, => ?@8 A@54=5A@>G=KE 8 4>;3>A@>G=KE AAC40E =0@0I5==0O AC<<0, @0AAG8B0==0O ?> A;>6=K< ?@>F5=B0< 7=0G8B5;L=> 2KH5, G5< ?> ?@>ABK<. @8 ;N1>< i, 5A;8 0 < n < 1, B> (1 + ni) > (1 + i)n ; 5A;8 n > 1, B> (1 + ni) < (1 + i)n ; 5A;8 n = 1, B> (1 + ni) = (1 + i)n . "0:8< >1@07><, 4;O ;8F, ?@54>AB02;ONI8E :@548B: 1>;55 2K3>4=0 AE5<0 ?@>ABKE ?@>F5=B>2, 5A;8 A@>: AAC4K <5=55 3>40 (?@>F5=BK =0G8A;ONBAO >4=>:@0B=> 2 :>=F5 3>40); 1>;55 2K3>4=>9 O2;O5BAO AE5<0 A;>6=KE ?@>F5=B>2, 5A;8 A@>: AAC4K ?@52KH05B >48= 3>4; >15 AE5<K 40NB >48=0:>2K9 @57C;LB0B ?@8 ?@>4>;68B5;L=>AB8 ?5@8>40 >48= 3>4 8 >4=>:@0B=>< =0G8A;5=88 ?@>F5=B>2. @8<5@ 8. !C<<0 2 @07<5@5 2'000 4>;;0@>2 40=0 2 4>;3 =0 2 3>40 ?> AB02:5 ?@>F5=B0 @02=>9 10% 3>4>2KE. ?@545;8BL ?@>F5=BK 8 AC<<C, ?>4;560ICN 2>72@0BC. 5H5=85: 0@0I5==0O AC<<0 FV = PV " (1 + i)n = 2'000 " (1 + 0'1)2 = 2'420 4>;;0@>2 8;8 FV = PV " k= = 2'000 " 1,21 = 2'420 4>;;0@>2, 345 k= = 1,21 ( HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/pril2.ru.shtml" @8;>65=85 2). !C<<0 =0G8A;5==KE ?@>F5=B>2 I = FV - PV = 2'420 - 2'000 = 420 4>;;0@>2.  HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/2.2.1.ru.shtml" \l "note6#note6" 6>>> "0:8< >1@07><, G5@57 420 3>40 =5>1E>48<> 25@=CBL >1ICN AC<<C 2 @07<5@5 2'420 4>;;0@>2, 87 :>B>@>9 2'000 4>;;0@>2 A>AB02;O5B 4>;3, 0 420 4>;;0@>2  "F5=0 4>;30". >AB0B>G=> G0AB> D8=0=A>2K5 :>=B@0:BK 70:;NG0NBAO =0 ?5@8>4, >B;8G0NI89AO >B F5;>3> G8A;0 ;5B.  A;CG05, :>340 A@>: D8=0=A>2>9 >?5@0F88 2K@065= 4@>1=K< G8A;>< ;5B, =0G8A;5=85 ?@>F5=B>2 2>7<>6=> A 8A?>;L7>20=85< 42CE <5B>4>2: >1I89 <5B>4 70:;NG05BAO 2 ?@O<>< @0AG5B5 ?> D>@<C;5 A;>6=KE ?@>F5=B>2: FV = PV " (1 + i)n, n = a + b, 345 n  ?5@8>4 A45;:8; a  F5;>5 G8A;> ;5B; b  4@>1=0O G0ABL 3>40. A<5H0==K9 <5B>4 @0AG5B0 ?@54?>;0305B 4;O F5;>3> G8A;0 ;5B ?5@8>40 =0G8A;5=8O ?@>F5=B>2 8A?>;L7>20BL D>@<C;C A;>6=KE ?@>F5=B>2, 0 4;O 4@>1=>9 G0AB8 3>40  D>@<C;C ?@>ABKE ?@>F5=B>2: FV = PV " (1 + i)a " (1 + bi). >A:>;L:C b < 1, B> (1 + bi) > (1 + i)a, A;54>20B5;L=>, =0@0I5==0O AC<<0 1C45B 1>;LH5 ?@8 8A?>;L7>20=88 A<5H0==>9 AE5<K. @8<5@.  10=:5 ?>;CG5= :@548B ?>4 9,5% 3>4>2KE 2 @07<5@5 250 BKA. 4>;;0@>2 A> A@>:>< ?>30H5=8O G5@57 420 3>40 8 9 <5AOF52. ?@545;8BL AC<<C, :>B>@CN =5>1E>48<> 25@=CBL ?> 8AB5G5=88 A@>:0 709<0 42C<O A?>A>10<8, CG8BK20O, GB> 10=: 8A?>;L7C5B 35@<0=A:CN ?@0:B8:C =0G8A;5=8O ?@>F5=B>2. 5H5=85: 1I89 <5B>4: FV = PV " (1 + i)n = 250 " (1 + 0,095)2,9 = 320,87 BKA. 4>;;0@>2. !<5H0==K9 <5B>4: FV = PV " (1 + i)a " (1 + bi) = = 250 " (1 + 0,095)2 " (1 + 270/360 " 0,095) = = 321,11 BKA. 4>;;0@>2. "0:8< >1@07><, ?> >1I5<C <5B>4C ?@>F5=BK ?> :@548BC A>AB02OB I = S - P = 320,87 - 250,00 = 70,84 BKA. 4>;;0@>2,  HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/2.2.1.ru.shtml" \l "note7#note7" 7>>> 0 ?> A<5H0==><C <5B>4C I = S - P = 321,11 - 250,00 = 71,11 BKA. 4>;;0@>2. 0: 284=>, A<5H0==0O AE5<0 1>;55 2K3>4=0 :@548B>@C. 2.2.2. -DD5:B82=0O AB02:0 ?@>F5=B>2  5@8>4 =0G8A;5=8O ?> A;>6=K< ?@>F5=B0< =5 2A5340 @025= 3>4C, >4=0:> 2 CA;>28OE D8=0=A>2>9 >?5@0F88 C:07K205BAO =5 AB02:0 70 ?5@8>4, 0 3>4>20O AB02:0 A C:070=85< ?5@8>40 =0G8A;5=8O  =><8=0;L=0O AB02:0 ( j ). ><8=0;L=0O AB02:0 (nominal rate)  3>4>20O AB02:0 ?@>F5=B>2, 8AE>4O 87 :>B>@>9 >?@545;O5BAO 25;8G8=0 AB02:8 ?@>F5=B>2 2 :064>< ?5@8>45 =0G8A;5=8O, ?@8 =0G8A;5=88 A;>6=KE ?@>F5=B>2 =5A:>;L:> @07 2 3>4. -B0 AB02:0 2>-?5@2KE, =5 >B@0605B @50;L=>9 MDD5:B82=>AB8 A45;:8; 2>-2B>@KE, =5 <>65B 1KBL 8A?>;L7>20=0 4;O A>?>AB02;5=89. A;8 =0G8A;5=85 ?@>F5=B>2 1C45B ?@>872>48BLAO m @07 2 3>4, 0 A@>: 4>;30  n ;5B, B> >1I55 :>;8G5AB2> ?5@8>4>2 =0G8A;5=8O 70 25AL A@>: D8=0=A>2>9 >?5@0F88 A>AB028B N = n " m BAN40 D>@<C;C A;>6=KE ?@>F5=B>2 <>6=> 70?8A0BL 2 A;54CNI5< 2845: FV = PV " (1 + j / m)N = P " (1 + j /m)mn , 345 j  =><8=0;L=0O 3>4>20O AB02:0 ?@>F5=B>2. @8<5@ 9. 7<5=8< CA;>28O ?@54K4CI53> ?@8<5@0, 2254O 565:20@B0;L=>5 =0G8A;5=85 ?@>F5=B>2. 5H5=85: >;8G5AB2> ?5@8>4>2 =0G8A;5=8O: N = m " n = 4 " 2 = 8 0@0I5==0O AC<<0 A>AB028B: FV = PV " (1 + j / m)mn = 2'000 " (1 + 0,1 / 4 )8 = 2'436,81 @C1. !C<<0 =0G8A;5==KE ?@>F5=B>2: I = FV - PV = 2'436,81 - 2'000 = 436,81 @C1. "0:8< >1@07><, G5@57 420 3>40 =0 AG5B5 1C45B =0E>48BLAO AC<<0 2 @07<5@5 2'436,81 @C1., 87 :>B>@>9 2'000 @C1. O2;O5BAO ?5@2>=0G0;L=>9 AC<<>9, @07<5I5==>9 =0 AG5B5, 0 436,81 @C1.  AC<<0 =0G8A;5==KE ?@>F5=B>2. 0@O4C A =><8=0;L=>9 AB02:>9 ACI5AB2C5B MDD5:B82=0O AB02:0 (effective rate), 87<5@ONI0O B>B @50;L=K9 >B=>A8B5;L=K9 4>E>4, :>B>@K9 ?>;CG5= 2 F5;>< 70 3>4, A CG5B>< 2=CB@83>4>2>9 :0?8B0;870F88. -DD5:B82=0O AB02:0 ?>:07K205B, :0:0O 3>4>20O AB02:0 A;>6=KE ?@>F5=B>2 405B B>B 65 D8=0=A>2K9 @57C;LB0B, GB> 8 m-@07>2>5 =0@0I5=85 2 3>4 ?> AB02:5 j / m: (1 + i)n = (1 + j / m)m " n, A;54>20B5;L=>, i = (1 + j / m)m - 1. 7 D>@<C;K A;54C5B, GB> MDD5:B82=0O AB02:0 7028A8B >B :>;8G5AB20 2=CB@83>4>2KE =0G8A;5=89. 0AG5B MDD5:B82=>9 AB02:8 O2;O5BAO <>I=K< 8=AB@C<5=B>< D8=0=A>2>3> 0=0;870, ?>A:>;L:C 55 7=0G5=85 ?>72>;O5B A@02=820BL <564C A>1>9 D8=0=A>2K5 >?5@0F88, 8<5NI85 @07;8G=K5 CA;>28O: G5< 2KH5 MDD5:B82=0O AB02:0 D8=0=A>2>9 >?5@0F88, B5< (?@8 ?@>G8E @02=KE CA;>28OE) >=0 2K3>4=55 4;O :@548B>@0. @8<5@ 10. 0AAG8B05< MDD5:B82=CN AB02:C 4;O D8=0=A>2>9 >?5@0F88, @0AA<>B@5==>9 2 ?@54K4CI5< ?@8<5@5, 0 B0:65 4;O 2:;040 ?@8 565<5AOG=>< =0G8A;5=88 ?@>F5=B>2 ?> 3>4>2>9 AB02:5 10%. 5H5=85: -DD5:B82=0O AB02:0 565:20@B0;L=>3> =0G8A;5=8O ?@>F5=B>2, 8AE>4O 87 10% 3>4>2KE, A>AB028B: i = (1 + j / m)m - 1 = (1 + 0,1 / 4)4 - 1 = 0,1038. -DD5:B82=0O AB02:0 565<5AOG=>3> =0G8A;5=8O ?@>F5=B>2 1C45B @02=0: i = (1 + j / m)m - 1 = (1 + 0,1 / 12)12 - 1 = 0,1047. "0:8< >1@07><, 3>4>20O AB02:0, M:2820;5=B=0O =><8=0;L=>9 AB02:5 ?@>F5=B>2 2 @07<5@5 10% 3>4>2KE ?@8 565<5AOG=>< =0G8A;5=88 ?@>F5=B>2, A>AB028B 10,47% ?@>B82 10,38% A 565:20@B0;L=K< =0G8A;5=85< ?@>F5=B>2. '5< 1>;LH5 ?5@8>4>2 =0G8A;5=8O, B5< 1KAB@55 845B ?@>F5AA =0@0I5=8O. ;O >1;53G5=8O @0AG5B>2 <>6=> ?>;L7>20BLAO B01;8F0<8 :>MDD8F85=B>2 =0@0I5=8O A;>6=KE ?@>F5=B>2, => 2=8<0B5;L=> A;548BL 70 A>>B25BAB285< 4;8=K ?5@8>40 =0G8A;5=8O 8 ?@>F5=B=>9 AB02:8 70 MB>B 65 ?5@8>4. 0?@8<5@, 5A;8 ?5@8>4>< =0G8A;5=8O O2;O5BAO :20@B0;, B> 2 @0AG5B0E 4>;6=0 8A?>;L7>20BLAO :20@B0;L=0O AB02:0. 2.2.3. 5@5<5==0O AB02:0 ?@>F5=B>2  5>1E>48<> >B<5B8BL, GB> >A=>2=0O D>@<C;0 A;>6=KE ?@>F5=B>2 ?@54?>;0305B ?>AB>O==CN ?@>F5=B=CN AB02:C =0 ?@>BO65=88 2A53> A@>:0 =0G8A;5=8O ?@>F5=B>2. 4=0:>, ?@54>AB02;OO 4>;3>A@>G=CN AAC4C, G0AB> 8A?>;L7CNB 87<5=ONI85AO 2> 2@5<5=8, => 70@0=55 70D8:A8@>20==K5 4;O :064>3> ?5@8>40 AB02:8 A;>6=KE ?@>F5=B>2.  A;CG05 8A?>;L7>20=8O ?5@5<5==KE ?@>F5=B=KE AB02>:, D>@<C;0 =0@0I5=8O 8<55B A;54CNI89 284:  INCLUDEPICTURE "http://irbis.asu.ru/mmc/econ/u_finmath/f01.gif" \* MERGEFORMATINET 345 ik  ?>A;54>20B5;L=K5 2> 2@5<5=8 7=0G5=8O ?@>F5=B=KE AB02>:; nk  4;8B5;L=>ABL ?5@8>4>2, 2 B5G5=85 :>B>@KE 8A?>;L7CNBAO A>>B25BAB2CNI85 AB02:8. @8<5@. $8@<0 ?>;CG8;0 :@548B 2 10=:5 =0 AC<<C 100'000 4>;;0@>2 A@>:>< =0 5 ;5B. @>F5=B=0O AB02:0 ?> :@548BC >?@545;5=0 2 10% 4;O 1-3> 3>40, 4;O 2-3> 3>40 ?@54CA<>B@5=0 =04102:0 : ?@>F5=B=>9 AB02:5 2 @07<5@5 1,5%, 4;O ?>A;54CNI8E ;5B 1%. ?@545;8BL AC<<C 4>;30, ?>4;560ICN ?>30H5=8N 2 :>=F5 A@>:0 709<0. 5H5=85: A?>;L7C5< D>@<C;C ?5@5<5==KE ?@>F5=B=KE AB02>:: FV = PV " (1 + i1)n1 " (1 + i2)n2 " & " (1 + ik)nk = = 100'000 " (1 + 0,1) " (1 + 0,115) " (1 + 0,125)3 = = 174'632,51 4>;;0@>2 "0:8< >1@07><, AC<<0, ?>4;560I0O ?>30H5=8N 2 :>=F5 A@>:0 709<0, A>AB028B 174'632,51 4>;;0@0, 87 :>B>@KE 100'000 4>;;0@>2 O2;ONBAO =5?>A@54AB25==> AC<<>9 4>;30, 0 74'632,51 4>;;0@0  ?@>F5=BK ?> 4>;3C. 2.2.4. 5?@5@K2=>5 =0G8A;5=85 ?@>F5=B>2  A5 A8BC0F88, :>B>@K5 <K 4> A8E ?>@ @0AA<0B@820;8, >B=>A8;8AL : 48A:@5B=K< ?@>F5=B0<, ?>A:>;L:C 8E =0G8A;5=85 >ACI5AB2;O5BAO 70 D8:A8@>20==K5 ?@><56CB:8 2@5<5=8 (3>4, :20@B0;, <5AOF, 45=L, G0A). > =0 ?@0:B8:5 =5@54:> 2AB@5G0NBAO A;CG08, :>340 ?@>F5=BK =0G8A;ONBAO =5?@5@K2=>, 70 A:>;L C3>4=> <0;K9 ?@><56CB>: 2@5<5=8. A;8 1K ?@>F5=BK =0G8A;O;8AL 5654=52=>, B> 3>4>2>9 :>MDD8F85=B (<=>68B5;L) =0@0I5=8O 2K3;O45; B0:: k= = (1 + j / m)m = (1 + j / 365)365 > ?>A:>;L:C ?@>F5=BK =0G8A;ONBAO =5?@5@K2=>, B> m AB@5<8BAO : 15A:>=5G=>AB8, 0 :>MDD8F85=B (<=>68B5;L) =0@0I5=8O AB@5<8BAO : e j:  INCLUDEPICTURE "http://irbis.asu.ru/mmc/econ/u_finmath/f02.gif" \* MERGEFORMATINET 345 e H" 2,718281, =07K205BAO G8A;>< -9;5@0 8 O2;O5BAO >4=>9 87 206=59H8E ?>AB>O==KE <0B5<0B8G5A:>3> 0=0;870. BAN40 <>6=> 70?8A0BL D>@<C;C =0@0I5==>9 AC<<K 4;O n ;5B: FV = PV " e j " n = P " e  " n !B02:C =5?@5@K2=KE ?@>F5=B>2 =07K20NB A8;>9 @>AB0 (force of interest) 8 >1>7=0G0NB A8<2>;>< , 2 >B;8G85 >B AB02:8 48A:@5B=KE ?@>F5=B>2 ( j ). @8<5@. @548B 2 @07<5@5 =0 100 BKA. 4>;;0@>2 ?>;CG5= A@>:>< =0 3 3>40 ?>4 8% 3>4>2KE. ?@545;8BL AC<<C ?>4;560I53> 2>72@0BC 2 :>=F5 A@>:0 :@548B0, 5A;8 ?@>F5=BK 1C4CB =0G8A;OBLAO: 0) >48= @07 2 3>4; 1) 5654=52=>; 2) =5?@5@K2=>. 5H5=85: A?>;L7C5< D>@<C;K 48A:@5B=KE 8 =5?@5@K2=KE ?@>F5=B>2: =0G8A;5=85 >48= @07 2 3>4 FV = 100'000 " (1 + 0,08)3 = 125'971,2 4>;;0@>2; 5654=52=>5 =0G8A;5=85 ?@>F5=B>2 FV = 100'000 " (1 + 0,08 / 365)365 " 3 = 127'121,6 4>;;0@>2 =5?@5@K2=>5 =0G8A;5=85 ?@>F5=B>2 FV = 100'000 " e0,08 " 3 = 127'124,9 4>;;0@>2. @0D8G5A:8 87<5=5=85 =0@0I5==>9 AC<<K 2 7028A8<>AB8 >B G0AB>BK =0G8A;5=8O 8<55B A;54CNI89 284:  INCLUDEPICTURE "http://irbis.asu.ru/mmc/econ/u_finmath/ris05.gif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n = [log (FV / PV)] / [log (1 + i)] = [log (FV / PV) ] / [log(1 + j / m)m]; AB02:0 A;>6=KE ?@>F5=B>2:  INCLUDEPICTURE "http://irbis.asu.ru/mmc/econ/u_finmath/f03.gif" \* MERGEFORMATINET .@8<5@. 'B> 2K3>4=55: C25;8G5=85 2:;040 2 B@8 @070 70 B@8 3>40 8;8 46% 3>4>2KE? 5H5=85: "0:>3> @>40 7040G8 ?@8E>48BAO @5H0BL =5 B>;L:> ;8F0<, 70=8<0NI8<AO D8=0=A>2>9 @01>B>9, => 8 =0A5;5=8N, :>340 @5H05BAO 2>?@>A > B><, :C40 2K3>4=55 2;>68BL 45=L38.  B0:8E A;CG0OE @5H5=85 A2>48BAO : >?@545;5=8N ?@>F5=B=>9 AB02:8:  INCLUDEPICTURE "http://irbis.asu.ru/mmc/econ/u_finmath/f04.gif" \* MERGEFORMATINET "0:8< >1@07><, C25;8G5=85 2:;040 70 B@8 3>40 2 B@8 @070 M:2820;5=B=> 3>4>2>9 ?@>F5=B=>9 AB02:5 2 44,3%, ?>MB><C @07<5I5=85 45=53 ?>4 46% 3>4>2KE 1C45B 1>;55 2K3>4=>. 2.3. -:2820;5=B=>ABL AB02>: 8 70<5=0 ?;0B5659   HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/2.3.1.ru.shtml" 2.3.1. -:2820;5=B=>ABL ?@>F5=B=KE AB02>:  HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/2.3.2.ru.shtml" 2.3.2. 7<5=5=85 D8=0=A>2KE CA;>289 2.3.1. -:2820;5=B=>ABL ?@>F5=B=KE AB02>:  >AB0B>G=> G0AB> 2 ?@0:B8:5 2>7=8:05B A8BC0F8O, :>340 =5>1E>48<> ?@>8725AB8 <564C A>1>9 A@02=5=85 ?> 2K3>4=>AB8 CA;>289 @07;8G=KE D8=0=A>2KE >?5@0F89 8 :><<5@G5A:8E A45;>:. #A;>28O D8=0=A>2>-:><<5@G5A:8E >?5@0F89 <>3CB 1KBL 25AL<0 @07=>>1@07=K<8 8 =0?@O<CN =5A>?>AB028<K<8. ;O A>?>AB02;5=8O 0;LB5@=0B82=KE 20@80=B>2 AB02:8, 8A?>;L7C5<K5 2 CA;>28OE :>=B@0:B>2, ?@82>4OB : 548=>>1@07=><C ?>:070B5;N. 07;8G=K5 D8=0=A>2K5 AE5<K <>6=> AG8B0BL M:2820;5=B=K<8 2 B>< A;CG05, 5A;8 >=8 ?@82>4OB : >4=><C 8 B><C 65 D8=0=A>2><C @57C;LB0BC. -:2820;5=B=0O ?@>F5=B=0O AB02:0  MB> AB02:0, :>B>@0O 4;O @0AA<0B@8205<>9 D8=0=A>2>9 >?5@0F88 40AB B>G=> B0:>9 65 45=56=K9 @57C;LB0B (=0@0I5==CN AC<<C), GB> 8 ?@8<5=O5<0O 2 MB>9 >?5@0F88 AB02:0. ;0AA8G5A:8< ?@8<5@>< M:2820;5=B=>AB8 O2;ONBAO =><8=0;L=0O 8 MDD5:B82=0O AB02:0 ?@>F5=B>2: i = (1 + j / m)m - 1. j = m[(1 + i)1 / m - 1]. -DD5:B82=0O AB02:0 87<5@O5B B>B >B=>A8B5;L=K9 4>E>4, :>B>@K9 <>65B 1KBL ?>;CG5= 2 F5;>< 70 3>4, B.5. A>25@H5==> 157@07;8G=>  ?@8<5=OBL ;8 AB02:C j ?@8 =0G8A;5=88 ?@>F5=B>2 m @07 2 3>4 8;8 3>4>2CN AB02:C i,  8 B0, 8 4@C30O AB02:8 M:2820;5=B=K 2 D8=0=A>2>< >B=>H5=88. >MB><C A>25@H5==> =5 8<55B 7=0G5=8O, :0:CN 87 ?@82545==KE AB02>: C:07K20BL 2 D8=0=A>2KE CA;>28OE, ?>A:>;L:C 8A?>;L7>20=85 8E 405B >4=C 8 BC 65 =0@0I5==CN AC<<C.  !( 2 ?@0:B8G5A:8E @0AG5B0E ?@8<5=ONB =><8=0;L=CN AB02:C, 0 2 52@>?59A:8E AB@0=0E ?@54?>G8B0NB MDD5:B82=CN AB02:C ?@>F5=B>2. A;8 425 =><8=0;L=K5 AB02:8 >?@545;ONB >4=C 8 BC 65 MDD5:B82=CN AB02:C ?@>F5=B>2, B> >=8 =07K20NBAO M:2820;5=B=K<8. @8<5@. 0:>2K 1C4CB M:2820;5=B=K5 =><8=0;L=K5 ?@>F5=B=K5 AB02:8 A ?>;C3>4>2K< =0G8A;5=85< ?@>F5=B>2 8 565<5AOG=K< =0G8A;5=85< ?@>F5=B>2, 5A;8 A>>B25BAB2CNI0O 8< MDD5:B82=0O AB02:0 4>;6=0 1KBL @02=0 25%? 5H5=85: 0E>48< =><8=0;L=CN AB02:C 4;O ?>;C3>4>2>3> =0G8A;5=8O ?@>F5=B>2: j = m[(1 + i)1 / m - 1] = 2[(1 + 0,25)1/2 - 1] = 0,23607. 0E>48< =><8=0;L=CN AB02:C 4;O 565<5AOG=>3> =0G8A;5=8O ?@>F5=B>2: j = m[(1 + i)1 / m - 1] = 4[(1 + 0,25)1/12 - 1] = 0,22523. "0:8< >1@07><, =><8=0;L=K5 AB02:8 23,61% A ?>;C3>4>2K< =0G8A;5=85< ?@>F5=B>2 8 22,52% A 565<5AOG=K< =0G8A;5=85< ?@>F5=B>2 O2;ONBAO M:2820;5=B=K<8. @8 2K2>45 @025=AB2, A2O7K20NI8E M:2820;5=B=K5 AB02:8, ?@8@02=820NBAO 4@C3 : 4@C3C <=>68B5;8 =0@0I5=8O, GB> 405B 2>7<>6=>ABL 8A?>;L7>20BL D>@<C;K M:2820;5=B=>AB8 ?@>ABKE 8 A;>6=KE AB02>:: ?@>AB0O ?@>F5=B=0O AB02:0: i = [(1 + j / m)m " n - 1] / n; A;>6=0O ?@>F5=B=0O AB02:0:  INCLUDEPICTURE "http://irbis.asu.ru/mmc/econ/u_finmath/f05.gif" \* MERGEFORMATINET . @8<5@. @54?>;0305BAO ?><5AB8BL :0?8B0; =0 4 3>40 ;81> ?>4 A;>6=CN ?@>F5=B=CN AB02:C 20% 3>4>2KE A ?>;C3>4>2K< =0G8A;5=85< ?@>F5=B>2, ;81> ?>4 ?@>ABCN ?@>F5=B=CN AB02:C 26% 3>4>2KE. 09B8 >?B8<0;L=K9 20@80=B. 5H5=85: 0E>48< 4;O A;>6=>9 ?@>F5=B=>9 AB02:8 M:2820;5=B=CN ?@>ABCN AB02:C: i = [(1 + j / m)m " n - 1] / n = [(1 + 0,2 / 2)2 " 4 - 1] / 4 = 0,2859. "0:8< >1@07><, M:2820;5=B=0O A;>6=>9 AB02:5 ?> ?5@2><C 20@80=BC ?@>AB0O ?@>F5=B=0O AB02:0 A>AB02;O5B 28,59% 3>4>2KE, GB> 2KH5 ?@54;0305<>9 ?@>AB>9 AB02:8 2 26% 3>4>2KE ?> 2B>@><C 20@80=BC, A;54>20B5;L=>, 2K3>4=55 @07<5AB8BL :0?8B0; ?> ?5@2><C 20@80=BC, B.5. ?>4 20% 3>4>2KE A ?>;C3>4>2K< =0G8A;5=85< ?@>F5=B>2. 0E>48< M:2820;5=B=CN A;>6=CN AB02:C ?@>F5=B>2 4;O ?@>AB>9 AB02:8:  INCLUDEPICTURE "http://irbis.asu.ru/mmc/econ/u_finmath/f06.gif" \* MERGEFORMATINET "0:8< >1@07><, ?@>F5=B=0O AB02:0 18,64% 3>4>2KE A ?>;C3>4>2K< =0G8A;5=85< ?@>F5=B>2 =865 20% 3>4>2KE A ?>;C3>4>2K< =0G8A;5=85< ?@>F5=B>2, B> ?5@2K9 20@80=B 2K3>4=55. 2.3.2. 7<5=5=85 D8=0=A>2KE CA;>289   ?@0:B8G5A:>9 45OB5;L=>AB8 G0AB> 2>7=8:05B =5>1E>48<>ABL 87<5=5=8O CA;>289 @0=55 70:;NG5==>3> :>=B@0:B0  >1J548=5=85 =5A:>;L:8E ?;0B5659 8;8 70<5=5 548=>2@5<5==>3> ?;0B560 @O4>< ?>A;54>20B5;L=KE ?;0B5659. AB5AB25==>, GB> 2 B0:8E CA;>28OE =8 >48= 87 CG0AB=8:>2 D8=0=A>2>9 >?5@0F88 =5 4>;65= B5@?5BL C1KB>:, 2K720==K9 87<5=5=85< D8=0=A>2KE CA;>289. 5H5=85 ?>4>1=KE 7040G A2>48BAO : ?>AB@>5=8N C@02=5=8O M:2820;5=B=>AB8, 2 :>B>@>< AC<<0 70<5=O5<KE ?;0B5659, ?@82545==0O : :0:><C-B> >4=><C <><5=BC 2@5<5=8, ?@8@02=5=0 : AC<<5 ?;0B5659 ?> =>2><C >1O70B5;LAB2C, ?@82545==><C : B><C 65 <><5=BC 2@5<5=8. ;O :@0B:>A@>G=KE :>=B@0:B>2 :>=A>;840F8O >ACI5AB2;O5BAO =0 >A=>25 ?@>ABKE AB02>:.  A;CG05 A >1J548=5=85< (:>=A>;848@>20=85<) =5A:>;L:8E ?;0B5659 2 >48= AC<<0 70<5=O5<KE ?;0B5659, ?@82545==KE : >4=>9 8 B>9 65 40B5, ?@8@02=8205BAO : =>2><C >1O70B5;LAB2C: FV0 = FVj " (1 + i " tj), 345 tj  2@5<5==>9 8=B5@20; <564C A@>:0<8, tj = n0 - nj. @8<5@ 6. 5H5=> :>=A>;848@>20BL 420 ?;0B560 A> A@>:0<8 20.04 8 10.05 8 AC<<0<8 ?;0B560 20 BKA. @C1. 8 30 BKA. @C1. !@>: :>=A>;840F88 ?;0B5659 31.05. ?@545;8BL AC<<C :>=A>;848@>20==>3> ?;0B560 ?@8 CA;>288, GB> AB02:0 @02=0 10% 3>4>2KE. 5H5=85: ?@545;8< 2@5<5==>9 8=B5@20; <564C A@>:0<8 4;O ?5@2>3> ?;0B560 8 :>=A>;848@>20==>3> ?;0B560:  HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/2.3.2.ru.shtml" \l "note8#note8" 8>>> t1= 11(0?@5;L) + 31(<09) - 1 = 41 45=L; 4;O 2B>@>3> ?;0B560 8 :>=A>;848@>20==>3> ?;0B560: t2 = 22(<09) - 1 = 21 45=L. BAN40 AC<<0 :>=A>;848@>20==>3> ?;0B560 1C45B @02=0: FVo1. = FV1 " (1 + t1 / T " i) + FV2 " (1 + t2 / T " i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n1 =04> 70<5=8BL ?;0B56>< FV>1. A> A@>:>< n>1. (n>1. > n1) ?@8 8A?>;L7>20=88 A;>6=>9 ?@>F5=B=>9 AB02:8 i, B> C@02=5=85 M:2820;5=B=>AB8 8<55B 284: FV>1. = FV1 " (1 + i)n>1. - n1 @8<5@. @54;0305BAO ?;0B56 2 45 BKA. @C1. A> A@>:>< C?;0BK G5@57 3 3>40 70<5=8BL ?;0B56>< A> A@>:>< C?;0BK G5@57 5 ;5B. 09B8 =>2CN AC<<C ?;0B560, 8AE>4O 87 ?@>F5=B=>9 AB02:8 12 % 3>4>2KE. 5H5=85: >A:>;L:C n>1. > n1, B> ?;0B56 A>AB028B: FV>1. = FV1 (1 + i)n>1. - n1 = 45'000 (1 + 0,12)5 - 3 = 56'448 @C1. "0:8< >1@07><, 2 =>2KE CA;>28OE D8=0=A>2>9 >?5@0F88 1C45B ?@54CA<>B@5= ?;0B56 56'448 @C1. "5ABK 4;O ?@>25@:8 CA2>5=8O ?@>945==>3> <0B5@80;0   7040=8OE, ?@54AB02;5==KE 2 D>@<5 B5AB0 =5>1E>48<> 2K1@0BL ?@028;L=K9 20@80=B >B25B0. =>340 ?@028;L=KE >B25B>2 <>65B 1KBL 420 8 1>;55. 0@0I5=85 MB>: A ?@>F5AA C25;8G5=8O :0?8B0;0 70 AG5B ?@8A>548=5=8O ?@>F5=B>2; B 1078A=K9 B5<? @>AB0; C >B=>H5=85 =0@0I5==>9 AC<<K : ?5@2>=0G0;L=>9 AC<<5 4>;30; D 42865=85 45=56=>3> ?>B>:0 >B =0AB>OI53> : 1C4CI5<C. $>@<C;0 ?@>ABKE ?@>F5=B>2: A FV = PV " i " n B FV = PV(1 + i)n C FV = PV(1 + ni) D FV = PV(1 + i) @>ABK5 ?@>F5=BK 8A?>;L7CNBAO 2 A;CG0OE: A @58=25AB8@>20=8O ?@>F5=B>2; B 2K?;0BK ?@>F5=B>2 ?> <5@5 8E =0G8A;5=8O; C :@0B:>A@>G=KE AAC4, A >4=>:@0B=K< =0G8A;5=85< ?@>F5=B>2; D AAC4, A 4;8B5;L=>ABLN 1>;55 >4=>3> 3>40. ">G=K9 ?@>F5=B MB>: A :0?8B0;870F8O ?@>F5=B0; B :><<5@G5A:89 ?@>F5=B; C @0AG5B ?@>F5=B>2, 8AE>4O 87 ?@>4>;68B5;L=>AB8 3>40 2 365 8;8 366 4=59; D @0AG5B ?@>F5=B>2 A B>G=K< G8A;>< 4=59 D8=0=A>2>9 >?5@0F88. ">G=>5 G8A;> 4=59 D8=0=A>2>9 >?5@0F88 <>6=> >?@545;8BL: A ?> A?5F80;L=K< B01;8F0< ?>@O4:>2KE =><5@>2 4=59 3>40; B 8A?>;L7CO ?@O<>9 AG5B D0:B8G5A:8E 4=59 <564C 40B0<8; C 8AE>4O 87 ?@>4>;68B5;L=>AB8 :064>3> F5;>3> <5AOF0 2 30 4=59; D AG8B0O 40BC 2K40G8 8 40BC ?>30H5=8O AAC4K 70 >48= 45=L. $@0=FC7A:0O ?@0:B8:0 =0G8A;5=8O ?@>F5=B>2: A >1K:=>25==K9 ?@>F5=B A ?@81;865==K< G8A;>< 4=59 D8=0=A>2>9 >?5@0F88; B >1K:=>25==K9 ?@>F5=B A B>G=K< G8A;>< 4=59 D8=0=A>2>9 >?5@0F88; C B>G=K9 ?@>F5=B A B>G=K< G8A;>< 4=59 D8=0=A>2>9 >?5@0F88; D B>G=K9 ?@>F5=B A ?@81;865==K< G8A;>< 4=59 D8=0=A>2>9 >?5@0F88. 5@<0=A:0O ?@0:B8:0 =0G8A;5=8O ?@>F5=B>2: A >1K:=>25==K9 ?@>F5=B A ?@81;865==K< G8A;>< 4=59 D8=0=A>2>9 >?5@0F88; B >1K:=>25==K9 ?@>F5=B A B>G=K< G8A;>< 4=59 D8=0=A>2>9 >?5@0F88; C B>G=K9 ?@>F5=B A B>G=K< G8A;>< 4=59 D8=0=A>2>9 >?5@0F88; D B>G=K9 ?@>F5=B A ?@81;865==K< G8A;>< 4=59 D8=0=A>2>9 >?5@0F88. =3;89A:0O ?@0:B8:0 =0G8A;5=8O ?@>F5=B>2: A >1K:=>25==K9 ?@>F5=B A ?@81;865==K< G8A;>< 4=59 D8=0=A>2>9 >?5@0F88; B >1K:=>25==K9 ?@>F5=B A B>G=K< G8A;>< 4=59 D8=0=A>2>9 >?5@0F88; C B>G=K9 ?@>F5=B A B>G=K< G8A;>< 4=59 D8=0=A>2>9 >?5@0F88; D B>G=K9 ?@>F5=B A ?@81;865==K< G8A;>< 4=59 D8=0=A>2>9 >?5@0F88. 0AG5B =0@0I5==>9 AC<<K 2 A;CG05 48A:@5B=> 87<5=ONI59AO 2> 2@5<5=8 ?@>F5=B=>9 AB02:8 ?> AE5<5 ?@>ABKE ?@>F5=B>2 8<55B A;54CNI89 284: A FV = PV (1 + n:i:) B FV = PV  (1 + n:i:) C FV = PV (1 + n1i1)(1 + n2i2) : (1 + n:i:) D FV = PV (1 + n i:) !@>: D8=0=A>2>9 >?5@0F88 ?> AE5<5 ?@>ABKE ?@>F5=B>2 >?@545;O5BAO ?> D>@<C;5: A n = I / (PV " i) B n = [(FV - PV) / (FV " t)] i C t = [(FV - PV) / (PV " i)] T D n = [(FV - PV) / (FV " t)] T A;8 2 CA;>28OE D8=0=A>2>9 >?5@0F88 >BACBAB2C5B ?@>AB0O ?@>F5=B=0O AB02:0, B>: A MB>3> =5 <>65B 1KBL; B 55 <>6=> >?@545;8BL ?> D>@<C;5 i = [(FV - PV) / (PV " t)]" T C 55 =52>7<>6=> >?@545;8BL D 55 <>6=> >?@545;8BL ?> D>@<C;5 i =  ?@>F5=B=KE G8A5; / 48287>@ $>@<C;0 A;>6=KE ?@>F5=B>2: A FV = PV(1 + ni) B FV = PV(1 + t / T " i) C FV = PV(1 + i)n D FV = PV(1 + ni)(1 + i)n 0G8A;5=85 ?> AE5<5 A;>6=KE ?@>F5=B>2 ?@54?>GB8B5;L=55: A ?@8 :@0B:>A@>G=KE D8=0=A>2KE >?5@0F8OE; B ?@8 A@>:5 D8=0=A>2>9 >?5@0F88 2 >48= 3>4; C ?@8 4>;3>A@>G=KE D8=0=A>2KE >?5@0F8OE; D 2> 2A5E 2KH5?5@5G8A;5==KE A;CG0OE. '5< 1>;LH5 ?5@8>4>2 =0G8A;5=8O ?@>F5=B>2: A B5< <54;5==55 845B ?@>F5AA =0@0I5=8O; B B5< 1KAB@55 845B ?@>F5AA =0@0I5=8O; C ?@>F5AA =0@0I5=8O =5 87<5=O5BAO; D ?@>F5AA =0@0I5=8O ?@54A:070BL =5;L7O. ><8=0;L=0O AB02:0 MB>: A 3>4>20O AB02:0 ?@>F5=B>2, 8AE>4O 87 :>B>@>9 >?@545;O5BAO 25;8G8=0 AB02:8 ?@>F5=B>2 2 :064>< ?5@8>45 =0G8A;5=8O, ?@8 =0G8A;5=88 A;>6=KE ?@>F5=B>2 =5A:>;L:> @07 2 3>4; B >B=>H5=85 AC<<K ?@>F5=B>2, 2K?;0G8205<KE 70 D8:A8@>20==K9 >B@57>: 2@5<5=8, : 25;8G8=5 AAC4K; C ?@>F5=B=0O AB02:0, ?@8<5=O5BAO 4;O 45:C@A82=KE ?@>F5=B>2; D 3>4>20O AB02:0, A C:070=85< ?5@8>40 =0G8A;5=8O ?@>F5=B>2. $>@<C;0 A;>6=KE ?@>F5=B>2 A =5>4=>:@0B=K< =0G8A;5=85< ?@>F5=B>2 2 B5G5=85 3>40: A FV = PV(1 + i) m " n B FV = PV(1 + j / m) m " n C FV = PV / m " (1 + i) n / m D FV = PV(1 + i " m) m " n -DD5:B82=0O AB02:0 ?@>F5=B>2: A =5 >B@0605B MDD5:B82=>AB8 D8=0=A>2>9 >?5@0F88; B 87<5@O5B @50;L=K9 >B=>A8B5;L=K9 4>E>4; C >B@0605B MDD5:B D8=0=A>2>9 >?5@0F88; D 7028A8B >B :>;8G5AB20 =0G8A;5=89 8 25;8G8=K ?5@2>=0G0;L=>9 AC<<K. $>@<C;0 A;>6=KE ?@>F5=B>2 A 8A?>;L7>20=85< ?5@5<5==KE ?@>F5=B=KE AB02>:: A FV = PV(1 + i1) n1 (1 + i2) n2 & (1 + ik) nk B FV = PV(1 + nkik) ! FV = PV(1 + n1i1 " n2i2 " & " nkik) nk D FV = PV(1 + in)(1 + i)  A;CG05, :>340 A@>: D8=0=A>2>9 >?5@0F88 2K@065= 4@>1=K< G8A;>< ;5B, =0G8A;5=85 ?@>F5=B>2 2>7<>6=> A 8A?>;L7>20=85<: A >1I53> <5B>40; B MDD5:B82=>9 ?@>F5=B=>9 AB02:8; C A<5H0==>3> <5B>40; D ?5@5<5==KE ?@>F5=B=KE AB02>:. !<5H0==K9 <5B>4 @0AG5B0: A FV = PV(1 + i)0 + 2 B FV = PV(1 + i)0 (1 + 2i) C FV = PV(1 + 02i)n D FV = PV(1 + i)0 (1 + i)2 5?@5@K2=>5 =0G8A;5=85 ?@>F5=B>2 MB>: A =0G8A;5=85 ?@>F5=B>2 5654=52=>; B =0G8A;5=85 ?@>F5=B>2 565G0A=>; C =0G8A;5=85 ?@>F5=B>2 565<8=CB=>; D =0G8A;5=85 ?@>F5=B>2 70 =5D8:A8@>20==K9 ?@><56CB>: 2@5<5=8. A;8 2 CA;>28OE D8=0=A>2>9 >?5@0F88 >BACBAB2C5B AB02:0 A;>6=KE ?@>F5=B>2, B>: A 55 >?@545;8BL =5;L7O; B  INCLUDEPICTURE "http://irbis.asu.ru/mmc/econ/u_finmath/f07.gif" \* MERGEFORMATINET  C i = ln(FV / PV) / ln(1 + n) D i = lim(1 + j / m)m E i = (1 + j / m)m - 1 ;020 3. ?5@0F88 48A:>=B8@>20=8O   HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/3.1.ru.shtml" 3.1. !CI=>ABL 48A:>=B8@>20=8O  HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/3.2.ru.shtml" 3.2. 0B5<0B8G5A:>5 48A:>=B8@>20=85  HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/3.3.ru.shtml" 3.3. 0=:>2A:89 CG5B 3.1. !CI=>ABL 48A:>=B8@>20=8O   D8=0=A>2>9 ?@0:B8:5 G0AB> ?@8E>48BAO @5H0BL 7040G8, >1@0B=K5 >?@545;5=8N =0@0I5==>9 AC<<K: ?> C65 8725AB=>9 =0@0I5==>9 AC<<5 (FV) A;54C5B >?@545;8BL =58725AB=CN ?5@2>=0G0;L=CN AC<<C 4>;30 (PV). "0:85 A8BC0F88 2>7=8:0NB ?@8 @07@01>B:5 CA;>289 D8=0=A>2>9 A45;:8, 8;8 :>340 ?@>F5=BK A =0@0I5==>9 AC<<K C45@6820NBAO =5?>A@54AB25==> ?@8 2K40G5 AAC4K. @>F5AA =0G8A;5=8O 8 C45@60=8O ?@>F5=B>2 2?5@54, 4> =0ABC?;5=8O A@>:0 ?>30H5=8O 4>;30, =07K20NB CG5B><, 0 A0<8 ?@>F5=BK 2 2845 @07=>AB8 =0@0I5==>9 8 ?5@2>=0G0;L=>9 AC<< 4>;30 48A:>=B>< (discount): D = FV - PV "5@<8= 48A:>=B8@>20=85 2 H8@>:>< A<KA;5 >7=0G05B >?@545;5=85 7=0G5=8O AB>8<>AB=>9 25;8G8=K =0 =5:>B>@K9 <><5=B 2@5<5=8 ?@8 CA;>288, GB> 2 1C4CI5< >=0 A>AB028B 7040==CN 25;8G8=C.  INCLUDEPICTURE "http://irbis.asu.ru/mmc/econ/u_finmath/ris06.gif" \* MERGEFORMATINET  8A. 6. >38:0 D8=0=A>2>9 >?5@0F88 48A:>=B8@>20=8O.5 @54:> B0:>9 @0AG5B =07K20NB ?@82545=85< AB>8<>AB=>3> ?>:070B5;O : 7040==><C <><5=BC 2@5<5=8, 0 25;8G8=C PV =07K20NB ?@82545==>9 (A>2@5<5==>9 8;8 B5:CI59) 25;8G8=>9 FV. "0:8< >1@07><, 48A:>=B8@>20=85  ?@82545=85 1C4CI8E 45=53 : B5:CI5<C <><5=BC 2@5<5=8, 8 ?@8 MB>< =5 8<55B 7=0G5=8O, 8<5;0 ;8 <5AB> 2 459AB28B5;L=>AB8 40==0O D8=0=A>20O >?5@0F8O 8;8 =5B, 0 B0:65 =57028A8<> >B B>3>, <>6=> ;8 AG8B0BL 48A:>=B8@C5<CN AC<<C 1C:20;L=> =0@0I5==>9. <5==> 48A:>=B8@>20=85 ?>72>;O5B CG8BK20BL 2 AB>8<>AB=KE @0AG5B0E D0:B>@ 2@5<5=8, ?>A:>;L:C 405B A53>4=OH=NN >F5=:C AC<<K, :>B>@0O 1C45B ?>;CG5=0 2 1C4CI5<. @825AB8 AB>8<>ABL 45=53 <>6=> : ;N1><C <><5=BC 2@5<5=8, 0 =5 >1O70B5;L=> : =0G0;C D8=0=A>2>9 >?5@0F88. AE>4O 87 <5B>48:8 =0G8A;5=8O ?@>F5=B>2, ?@8<5=ONB 420 2840 48A:>=B8@>20=8O: <0B5<0B8G5A:>5 48A:>=B8@>20=85 ?> ?@>F5=B=>9 AB02:5; 10=:>2A:89 CG5B ?> CG5B=>9 AB02:5. 07;8G85 2 AB02:5 ?@>F5=B>2 8 CG5B=>9 AB02:5 70:;NG05BAO 2 @07;8G88 107K 4;O =0G8A;5=89 ?@>F5=B>2: 2 ?@>F5=B=>9 AB02:5 2 :0G5AB25 107K 15@5BAO ?5@2>=0G0;L=0O AC<<0 4>;30: i = (FV - PV) / PV 2 CG5B=>9 AB02:5 70 107C ?@8=8<05BAO =0@0I5==0O AC<<0 4>;30: d = (FV - PV) / FV @>F5=BK, =0G8A;5==K5 ?> AB02:5 ?@>F5=B>2, =07K20NBAO 0=B8A8?0B82=K<8, 0 ?> CG5B=>9 AB02:5  45:C@A82=K<8. #G5B=0O AB02:0 1>;55 65AB:> >B@0605B 2@5<5==>9 D0:B>@, G5< ?@>F5=B=0O AB02:0. A;8 A@02=8BL <564C A>1>9 <0B5<0B8G5A:>5 8 10=:>2A:>5 48A:>=B8@>20=85 2 A;CG05, :>340 ?@>F5=B=0O 8 CG5B=0O AB02:0 @02=K ?> A2>59 25;8G8=5, B> 284=>, GB> ?@82545==0O 25;8G8=0 ?> ?@>F5=B=>9 AB02:5 1>;LH5 ?@82545==>9 25;8G8=K ?> CG5B=>9 AB02:5.  3.2. 0B5<0B8G5A:>5 48A:>=B8@>20=85  0B5<0B8G5A:>5 48A:>=B8@>20=85  >?@545;5=85 ?5@2>=0G0;L=>9 AC<<K 4>;30, :>B>@0O ?@8 =0G8A;5=88 ?@>F5=B>2 ?> 7040==>9 25;8G8=5 ?@>F5=B=>9 AB02:8 ( i ) ?>72>;8B : :>=FC A@>:0 ?>;CG8BL C:070==CN =0@0I5==CN AC<<C: 4;O ?@>ABKE ?@>F5=B>2 PV = FV : (1 + n " i ) = FV " 1 / (1 + n " i ) = = FV " (1 + n " i ) -1 = FV " k4, 345 k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t / T " i ) = = 310'000 " 1 / (1 + 150 / 360 " 0,08) = 300'000 @C1. PV = FV " k4 = 310'000 " 0,9677419 = 300'000 @C1. "0:8< >1@07><, ?5@2>=0G0;L=0O AC<<0 4>;30 A>AB028;0 300 BKA. @C1., 0 ?@>F5=BK 70 150 4=59  10 BKA. @C1. ;O A;>6=KE ?@>F5=B>2 PV = FV " (1 + i) -n = FV " k4, 345 k4  48A:>=B=K9 <=>68B5;L 4;O A;>6=KE ?@>F5=B>2. A;8 =0G8A;5=85 ?@>F5=B>2 ?@>872>48BAO m @07 2 3>4, B> D>@<C;0 ?@8<5B 284: PV = FV " (1 + j / m) -m " n . @8<5@. '5@57 420 3>40 D8@<5 ?>B@51C5BAO 45=L38 2 @07<5@5 30 <;= @C1., :0:CN AC<<C =5>1E>48<> A53>4=O ?><5AB8BL 2 10=:, =0G8A;ONI89 25% 3>4>2KE, GB>1K G5@57 2 3>40 ?>;CG8BL B@51C5<CN AC<<C? 5H5=85: >A:>;L:C A@>: D8=0=A>2>9 >?5@0F88 A>AB02;O5B 1>;55 3>40, GB> 8A?>;L7C5< D>@<C;C ?@82545=8O 4;O A;>6=KE ?@>F5=B>2: PV = FV " 1 / (1 + i) n = = 30'000'000 " 1 / (1 + 0,25)2 = 19'200'000 @C1. 8;8 PV = FV " k4 = 30'000'000 " 0,6400000 = 19'200'000 @C1. "0:8< >1@07><, D8@<5 A;54C5B @07<5AB8BL =0 AG5B5 19'200'000 @C1. ?>4 25% 3>4>2KE, GB>1K G5@57 420 3>40 ?>;CG8BL 65;05<K5 30'000'000 @C1. !>2@5<5==0O 25;8G8=0 8 ?@>F5=B=0O AB02:0, ?> :>B>@>9 ?@>2>48BAO 48A:>=B8@>20=85, =0E>4OBAO 2 >1@0B=>9 7028A8<>AB8: G5< 2KH5 ?@>F5=B=0O AB02:0, B5< ?@8 ?@>G8E @02=KE CA;>28OE <5=LH5 A>2@5<5==0O 25;8G8=0.  B>9 65 >1@0B=>9 7028A8<>AB8 =0E>4OBAO A>2@5<5==0O 25;8G8=0 8 A@>: D8=0=A>2>9 >?5@0F88: G5< 2KH5 A@>: D8=0=A>2>9 >?5@0F88, B5< <5=LH5 ?@8 ?@>G8E @02=KE CA;>28OE A>2@5<5==0O 25;8G8=0. 3.3. 0=:>2A:89 CG5B  0=:>2A:89 CG5B  2B>@>9 284 48A:>=B8@>20=8O, ?@8 :>B>@>< 8AE>4O 87 8725AB=>9 AC<<K 2 1C4CI5<, >?@545;ONB AC<<C 2 40==K9 <><5=B 2@5<5=8, C45@6820O 48A:>=B. ?5@0F8O CG5B0 (CG5B 25:A5;59) 70:;NG05BAO 2 B><, GB> 10=: 8;8 4@C3>5 D8=0=A>2>5 CG@5645=85 4> =0ABC?;5=8O ?;0B560 ?> 25:A5;N ?>:C?05B 53> C ?@54JO28B5;O ?> F5=5 =865 AC<<K 25:A5;O, B.5. ?@8>1@5B05B 53> A 48A:>=B><. !C<<0, :>B>@CN ?>;CG05B 25:A5;545@60B5;L ?@8 4>A@>G=>< CG5B5 25:A5;O, =07K205BAO 48A:>=B8@>20==>9 25;8G8=>9 25:A5;O. @8 MB>< 10=: C45@68205B 2 A2>N ?>;L7C ?@>F5=BK (48A:>=B) >B AC<<K 25:A5;O 70 2@5<O, >AB02H55AO 4> A@>:0 53> ?>30H5=8O. >4>1=K< >1@07>< (A 48A:>=B><) 3>AC40@AB2> ?@>405B 1>;LH8=AB2> A2>8E F5==KE 1C<03. ;O @0AG5B0 48A:>=B0 8A?>;L7C5BAO CG5B=0O AB02:0: ?@>AB0O CG5B=0O AB02:0: D = FV - PV = FV " n " d = FV " t / T " d , 345 n  ?@>4>;68B5;L=>ABL A@>:0 2 3>40E >B <><5=B0 CG5B0 4> 40BK 2K?;0BK 8725AB=>9 AC<<K 2 1C4CI5<. BAN40: PV = FV - FV " n " d = FV " (1 - n " d), 345 (1 - n " d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t = 13 (023CAB) + 30 (A5=BO1@L) + 31 (>:BO1@L) + 17 (=>O1@L) - 1 = 90 4=59. BAN40, >?@545;O5<0O AC<<0: PV = FV " (1 - t / T " d) = 5'000 " (1 - 90 / 360 " 0,08) = 4'900 @C1. ">340 48A:>=B A>AB028B: D = FV - PV = 5'000 - 4'900 = 100 @C1. 8;8 D = FV " t / T " d = 5'000 " 90 / 360 " 0,08 = 100 @C1. !;54>20B5;L=>, ?@54JO28B5;L 25:A5;O ?>;CG8B AC<<C 4'900 @C1., 0 10=: ?@8 =0ABC?;5=88 A@>:0 25:A5;O @50;87C5B 48A:>=B 2 @07<5@5 100 @C1. ?> A;>6=>9 CG5B=>9 AB02:5: PV = FV " (1 - d) n @8 8A?>;L7>20=88 A;>6=>9 CG5B=>9 AB02:8 ?@>F5AA 48A:>=B8@>20=8O ?@>8AE>48B A ?@>3@5AA8@CNI8< 70<54;5=85<, B.:. CG5B=0O AB02:0 :064K9 @07 ?@8<5=O5BAO : C<5=LH05<>9 =0 25;8G8=C 48A:>=B0 25;8G8=5. @8<5@. ?@545;8BL 25;8G8=C AC<<K, 2K40205<CN 705<I8:C, 5A;8 >= >1O7C5BAO 25@=CBL 55 G5@57 420 3>40 2 @07<5@5 55 BKA. @C1. 0=: >?@545;O5B A2>9 4>E>4 A 8A?>;L7>20=85< 3>4>2>9 CG5B=>9 AB02:8 30%. 5H5=85: A?>;L7CO D>@<C;C 48A:>=B8@>20=8O ?> A;>6=>9 CG5B=>9 AB02:5, >?@545;O5<: PV = FV " (1 - d) n = 55'000 " (1 - 0,3)2 = 26'950 @C1. 05<I8: <>65B ?>;CG8BL AAC4C 2 @07<5@5 26'950 @C1., 0 G5@57 420 3>40 25@=5B 55 BKA. @C1. 1J548=5=85 ?;0B5659 <>6=> ?@>872>48BL 8 =0 >A=>25 CG5B=>9 AB02:8, =0?@8<5@, ?@8 :>=A>;848@>20=88 25:A5;59.  MB>< A;CG05, AC<<0 :>=A>;848@>20==>3> ?;0B560 @0AAG8BK205BAO ?> A;54CNI59 D>@<C;5: FVo1 = FVj " (1 - d " tj) -1 , 345 tj  8=B5@20; 2@5<5=8 <564C A@>:0<8 25:A5;59. @8<5@. 5:A5;L =0 AC<<C 10 BKA. @C1. A> A@>:>< ?>30H5=8O 10.06, 0 B0:65 25:A5;L =0 AC<<C 20 BKA. @C1. A> A@>:>< ?>30H5=8O 01.08 70<5=ONBAO >4=8< A ?@>4;5=85< A@>:0 4> 01.10. @8 >1J548=5=88 25:A5;59 ?@8<5=O5BAO CG5B=0O AB02:0 25%. ?@545;8BL AC<<C :>=A>;848@>20==>3> 25:A5;O. 5H5=85: ;O 8A?>;L7>20=8O D>@<C;K :>=A>;848@>20==>3> ?;0B560 =5>1E>48<> >?@545;8BL A@>: ?@>;>=30F88 25:A5;59: t1 = 21 (8N=L) + 31 (8N;L) + 31 (023CAB) + 30 (A5=BO1@L) + 1 (>:BO1@L) - 1 = 113 4=59, t2 = 31 (023CAB) + 30 (A5=BO1@L) + 1(>:BO1@L) - 1 = 61 45=L. ">340, AC<<0 :>=A>;848@>20==>3> 25:A5;O: FVo = FVj " (1 - d " tj) -1 = = 10'000 " (1 - 113 / 360 " 0,25) -1 + 20'000 " (1 - 61 / 360 " 0,25) -1 = = 31'736 @C1. "0:8< >1@07><, AC<<0 :>=A>;848@>20==>3> 25:A5;O A 40B>9 ?>30H5=8O 01.10 A>AB028B 31'736 @C1.  B>< A;CG05, :>340 CG5BC ?>4;568B 4>;3>2>5 >1O70B5;LAB2>, ?> :>B>@><C ?@54CA<0B@8205BAO =0G8A;5=85 ?@>F5=B>2, ?@>8AE>48B A>2<5I5=85 =0G8A;5=8O ?@>F5=B>2 ?> ?@>F5=B=>9 AB02:5 8 48A:>=B8@>20=8O ?> CG5B=>9 AB02:5: PV2 = PV1 " (1 + n1 " i ) " (1 - n2 " d ), 345 PV1  ?5@2>=0G0;L=0O AC<<0 4>;30; PV2  AC<<0, ?>;CG05<0O ?@8 CG5B5 >1O70B5;LAB20; n1  >1I89 A@>: ?;0B56=>3> >1O70B5;LAB20; n2  A@>: >B <><5=B0 CG5B0 4> ?>30H5=8O. @8<5@. 1O70B5;LAB2> C?;0B8BL G5@57 100 4=59 AC<<C 4>;30 2 @07<5@5 50 BKA. @C1. A =0G8A;O5<K<8 =0 =55 B>G=K<8 ?@>F5=B0<8 ?> AB02:5 40%, 1K;> CGB5=> 70 25 4=59 4> A@>:0 ?>30H5=8O ?> CG5B=>9 AB02:5 25%. ?@545;8BL AC<<C, ?>;CG5==CN ?@8 CG5B5 >1O70B5;LAB20. 5H5=85: !;54C5B >1@0B8BL 2=8<0=85 =0 @07;8G85 2@5<5==KE 107, 8A?>;L7C5<KE ?@8 =0@0I5=88 8 CG5B5: PV2 = PV1 " (1 + n1 " i ) " (1 - n2 " d ) = = 50'000 " (1 + 100 / 365 " 0,4) " (1 - 25 / 360 " 0,25) = 54'516 @C1. !;54>20B5;L=>, AC<<0, ?>;CG05<0O ?@8 CG5B5 40==>3> >1O70B5;LAB20, A>AB028B 54'516 @C1. 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CABL 565<5AOG=K9 C@>25=L 8=D;OF88 2,5%. ?@545;8BL >68405<K9 C@>25=L 8=D;OF88 70 :20@B0;. 5H5=85: =45:A 8=D;OF88 70 <5AOF J = 1 +  = 1 + 0,025 = 1,025 =45:A 8=D;OF88 70 :20@B0;, B.5. 70 B@8 <5AOF0 J = (1 + )3 = 1,0253 = 1,077 #@>25=L 8=D;OF88 70 :20@B0;  = J - 1 = 1,077 - 1 = 0,077 !;54>20B5;L=>, >68405<K9 :20@B0;L=K9 C@>25=L 8=D;OF88 A>AB028B 7,7%. >:070B5;8 D8=0=A>2>9 >?5@0F88 <>3CB 1KBL ?@54AB02;5=K, :0:: =><8=0;L=K5, B.5. @0AAG8B0==K5 2 B5:CI8E F5=0E; @50;L=K5, B.5. CG8BK20NI85 2;8O=85 8=D;OF88, 8 @0AAG8B0==K5 2 A>?>AB028<KE F5=0E 1078A=>3> ?5@8>40. @8<5@. ?@545;8BL @50;L=K5 @57C;LB0BK 2:;04=>9 >?5@0F88 4;O AC<<K 5'000 @C1., @07<5I5==>9 =0 ?>;3>40 ?>4 8% 3>4>2KE, 5A;8 565<5AOG=K9 C@>25=L 8=D;OF88 A>AB02;O5B 2%. 5H5=85: 0@0I5==0O AC<<0 2:;040 FV = PV(1 + ni) = 5'000 (1 + 0,5 " 0,08) = 5'200,00 @C1. =45:A 8=D;OF88 70 A@>: E@0=5=8O 2:;040 A>AB028B J = (1 + 0,02)6 = 1,126 50;L=0O AC<<0 2:;040 FV = 5'200 / 1,126 = 4'618,11 @C1. !;54>20B5;L=>, =0@0I5==0O 25;8G8=0 ?> A2>59 ?>:C?0B5;L=>9 A?>A>1=>AB8 A CG5B>< 8=D;OF88 1C45B A>>B25BAB2>20BL AC<<5 4'618,11 @C1., B.5. <5=LH5 ?5@2>=0G0;L=>9 AC<<K. 5.2. 5B>4K CG5B0 8=D;OF88 2 D8=0=A>2KE @0AG5B0E  ;045;LFK 45=53 =5 <>3CB <8@8BLAO A 8E >15AF5=820=85< 2 @57C;LB0B5 8=D;OF88 8 ?@54?@8=8<0NB @07;8G=K5 ?>?KB:8 :><?5=A0F88 ?>B5@L >B A=865=8O 8E ?>:C?0B5;L=>9 A?>A>1=>AB8. 081>;55 @0A?@>AB@0=5==K< <5B>4>< O2;O5BAO 8=45:A0F8O AB02:8 ?@>F5=B>2, ?> :>B>@>9 ?@>872>48BAO =0@0I5=85, ?>A:>;L:C: 5A;8 C@>25=L 8=D;OF88 @025= AB02:5 =0G8A;O5<KE ?@>F5=B>2 ( = i), B> @50;L=>3> @>AB0 45=56=KE AC<< =5 1C45B, B.:. =0@0I5=85 1C45B ?>;=>ABLN ?>3;>I0BLAO 8=D;OF859; 5A;8 C@>25=L 8=D;OF88 2KH5 C@>2=O ?@>F5=B=>9 AB02:8 ( > i),B> ?@>8AE>48B "?@>540=85" :0?8B0;0, 8 @50;L=0O =0@0I5==0O AC<<0 1C45B <5=LH5 ?5@2>=0G0;L=>9 45=56=>9 AC<<K; 5A;8 C@>25=L 8=D;OF88 =865 ?@>F5=B=>9 AB02:8 ( < i), B> MB> 1C45B A>>B25BAB2>20BL @>ABC @50;L=>9 45=56=>9 AC<<K.  A2O78 A MB8< 22>48BAO ?>=OB85 =><8=0;L=0O AB02:0 ?@>F5=B0, B.5. AB02:8 A ?>?@02:>9 =0 8=D;OF8N ( i ). 1I0O D>@<C;0 4;O >?@545;5=8O ?@>AB>9 AB02:8 ?@>F5=B>2, :><?5=A8@CNI59 >68405<CN 8=D;OF8N, 8<55B A;54CNI89 284: i = [(1 + n i) " J - 1] : n 345 i  ?@>AB0O AB02:0 ?@>F5=B>2, E0@0:B5@87CNI0O B@51C5<CN @50;L=CN 4>E>4=>ABL D8=0=A>2>9 >?5@0F88 (=5BB>-AB02:0); i  ?@>F5=B=0O AB02:0 A ?>?@02:>9 =0 8=D;OF8N. @8<5@. 0=: 2K40; :;85=BC :@548B =0 >48= 3>4 2 @07<5@5 20 BKA. @C1. ?> AB02:5 6% 3>4>2KE. #@>25=L 8=D;OF88 70 3>4 A>AB028; 18%. ?@545;8BL A CG5B>< 8=D;OF88 @50;L=CN AB02:C ?@>F5=B>2 ?> :@548BC, ?>30H05<CN AC<<C 8 AC<<C ?@>F5=B>2 70 :@548B. 5H5=85: ><8=0;L=0O =0@0I5==0O AC<<0 FV = PV(1 + n i) = 20'000 (1 + 0,06) = 21'200,00 @C1. ><8=0;L=K5 =0G8A;5==K5 ?@>F5=BK I = FV - PV = 21'200 - 20'000 = 1'200,00 @C1. 50;L=0O =0@0I5==0O AC<<0 FV = FV / (1 +  ) = 21'200 / 1,18 = 17'966,10 @C1. 50;L=K5 ?@>F5=BK I = FV - PV = 17'966,10 - 20'000 = -2'033,90 @C1. "0:8< >1@07><, ?>;CG5= C1KB>: >B 40==>9 D8=0=A>2>9 >?5@0F88 2 @07<5@5 2'033,90 @C1. !B02:0 ?> :@548BC A CG5B>< 8=D;OF88 4>;6=0 1KBL @02=0 i = [(1 + n i) " I - 1] : n = (1,06 " 1,18 - 1) / 1 = 0,2508 0@0I5==0O AC<<0 FV = PV(1 + n i) = 20'000 (1 + 0,2508) = 25'016,00 @C1. >E>4 10=:0 I = FV - PV = 25'016 - 20'000 = 5'016,00 @C1. 50;L=K9 4>E>4 10=:0 I = FV - PV = 25'016 / 1,18 - 20'000 = 1'200,00 @C1. 50;L=0O 4>E>4=>ABL D8=0=A>2>9 >?5@0F88 i = I / PV = 1'200 / 20'000 = 0,06 "0:8< >1@07><, GB>1K >15A?5G8BL 4>E>4=>ABL 2 @07<5@5 6% 3>4>2KE, AB02:0 ?> :@548BC A CG5B>< 8=D;OF88 4>;6=0 A>>B25BAB2>20BL 25,1% 3>4>2K<. >4>20O AB02:0 A;>6=KE ?@>F5=B>2, >15A?5G820NI0O @50;L=CN 4>E>4=>ABL :@548B=>9 >?5@0F88, >?@545;O5BAO ?> D>@<C;5 i = i +  + i @8<5@. ?@545;8BL =><8=0;L=CN AB02:C ?@>F5=B>2 4;O D8=0=A>2>9 >?5@0F88, 5A;8 C@>25=L MDD5:B82=>AB8 4>;65= A>AB02;OBL 7% 3>4>2KE, 0 3>4>2>9 C@>25=L 8=D;OF88 22%. 5H5=85: @>F5=B=0O AB02:0 A CG5B>< 8=D;OF88 i = i +  + i = 0,07 + 0,22 + 0,07 " 0,22 = 0,3054. 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'5@57 A:>;L:> ;5B 2:;04 @07<5@>< 500 @C1. 4>AB83=5B 25;8G8=K 1000 @C1. ?@8 AB02:5 ?@>F5=B>2 13,94% A 565<5AOG=K< =0G8A;5=85< ?@>F5=B>2? 5H5=85: =>@<013,94% / 122K?;0B0-500!1000B8?=0G5=85 60,01. -B> >1I55 G8A;> @07 =0G8A;5=8O ?@>F5=B>2, 0 2 3>40E MB> 1C45B 60 / 12 = 5 ;5B. 6.2.4. ?@545;5=85 ?@>F5=B=>9 AB02:8  ;O >?@545;5=8O ?@>F5=B=>9 AB02:8 8A?>;L7C5BAO DC=:F8O  , :>B>@0O >?@545;O5B 7=0G5=85 ?@>F5=B=>9 AB02:8 70 >48= @0AG5B=K9 ?5@8>4. ;O @0AG5B0 3>4>2>9 ?@>F5=B=>9 AB02:8 ?>;CG5==>5 7=0G5=85 =5>1E>48<> C<=>68BL =0 G8A;> 40==KE ?5@8>4>2 2 3>4C. @3C<5=BK DC=:F88: :?5@; 2K?;0B0; ; !; B8?; ?@54?>;>65=85. 45AL 2A5 0@3C<5=BK, :@><5 ?>A;54=53>, C65 2AB@5G0;8AL 2 2KH5 @0AA<>B@5==KE DC=:F8O. @3C<5=B ?@54?>;>65=85 2 1>;LH8=AB25 A;CG052 =5 8A?>;L7C5BAO, ?>A:>;L:C ?> C<>;G0=8N >= AG8B05BAO @02=K< 10%, => 5A;8 DC=:F8O 2>72@0I05B 7=0G5=85 >H81:8, B> 2 MB>< A;CG05 70405BAO ?@54?>;0305<>5 7=0G5=85 ?@>F5=B=>9 AB02:8. @8<5@. 0AAG8B0BL ?@>F5=B=CN AB02:C 4;O G5BK@5E;5B=53> 709<0 @07<5@>< 7000 @C1. A 565<5AOG=K< ?>30H5=85< ?> 250 @C1. 5H5=85: ?@545;O5< ?@>F5=B=CN AB02:C 70 <5AOF: :?5@4 " 122K?;0B0-2507000!1000B8??@54?>;>65=85=0G5=85 2,46%. >4>20O ?@>F5=B=0O AB02:0 2,46 " 12 = 29,5. BAN40, 3>4>20O ?@>F5=B=0O AB02:0 4;O 40==>9 D8=0=A>2>9 >?5@0F88 A>AB02;O5B 29,5%. ;020 7. @548B=K5 @0AG5BK   HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/7.1.ru.shtml" 7.1. ;0=8@>20=85 ?>30H5=8O 4>;30 7.1. ;0=8@>20=85 ?>30H5=8O 4>;30   HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/7.1.1.ru.shtml" 7.1.1. >30H5=85 4>;30 548=>2@5<5==K< ?;0B56><  HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/7.1.2.ru.shtml" 7.1.2. >30H5=85 4>;30 2 @0AA@>G:C  HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/7.1.3.ru.shtml" 7.1.3. >B@518B5;LA:89 :@548B 7.1.1. >30H5=85 4>;30 548=>2@5<5==K< ?;0B56><  >;8G5AB25==K9 0=0;87 4>;3>A@>G=>9 704>;65==>AB8 (709<0) ?@8<5=O5BAO 4;O 4>AB865=8O A10;0=A8@>20==>AB8, B.5. 045:20B=>AB8 53> ?0@0<5B@>2 ?@8=OBK< CA;>28O< D8=0=A>2>3> A>3;0H5=8O, ?CB5< ?;0=8@>20=8O ?>30H5=8O 4>;30. ;0=8@>20=85 ?>30H5=8O 4>;30 70:;NG05BAO 2 >?@545;5=88 ?5@8>48G5A:8E @0AE>4>2, A2O70==KE A 709<><,  B0:85 @0AE>4K =07K20NBAO >1A;C6820=85< 4>;30. 07>20O AC<<0 >1A;C6820=8O 4>;30  A@>G=0O C?;0B0, 2 :>B>@CN 2E>4OB: B5:CI85 ?@>F5=B=K5 ?;0B568; A@54AB20, 4;O ?>30H5=8O (0<>@B870F88) >A=>2=>9 AC<<K 4>;30. 07<5@K A@>G=KE C?;0B 7028AOB >B CA;>289 709<0: A@>:0; =0;8G8O 8 ?@>4>;68B5;L=>AB8 ;L3>B=>3> ?5@8>40; C@>2=O ?@>F5=B=>9 AB02:8; A?>A>10 ?>30H5=8O >A=>2=>9 AC<<K 4>;30 8 2K?;0BK ?@>F5=B>2. ;O :@548B=>9 AE5<K 2 :0G5AB25 8AE>4=KE ?0@0<5B@>2 2KABC?0NB 25;8G8=0 709<0 (D), A@>: 53> ?>30H5=8O (n), ?@>F5=B ?> :@548BC (i), ?>4 :>B>@K9 2K40NBAO 45=L38, 8 ?>B>: ?;0B5659 ?> 2K?;0B5 4>;30 (Yt). 0AA<>B@8< @07;8G=K5 A?>A>1K ?>30H5=8O 704>;65==>AB8, ?>A:>;L:C >B 2K1>@0 A?>A>10 ?>30H5=8O AB>8<>ABL :@548B0 (AC<<0 2K?;0G8205<KE ?@>F5=B>2) 1C45B @07;8G=>9. 45AL 2>7<>6=K 420 20@80=B0: 0) ?>30H5=85 548=>2@5<5==K< ?;0B56><, B.5. 2>72@0B 2A59 AC<<K 2 >3>2>@5==K9 A@>:; 1) ?>30H5=85 4>;30 2 @0AA@>G:C, B.5. G0ABO<8. 7.1.1.1. >30H5=85 >A=>2=>9 AC<<K 4>;30 548=>2@5<5==K< ?;0B56>< 2 :>=F5 A@>:0 A ?>AB>O==>9 2K?;0B>9 ?@>F5=B>2. 0AA<>B@8< ?>30H5=85 548=>2@5<5==K< ?;0B56><.  ?@>AB59H5< A;CG05 :@548B ?>30H05BAO 548=K< ?;0B56>< 2 :>=F5 A@>:0: Y = D " (1 + i)n, 345 Y A@>G=0O C?;0B0; D AC<<0 4>;30. -B>B ?;0B56, :0: =0@0I5==0O AC<<0 4>;30, A>AB>8B 87 42CE G0AB59: 2>72@0B >A=>2=>9 AC<<K 4>;30 (D); 2K?;0B0 ?@>F5=B>2 ?> 4>;3C (I), 345 I = D " (1 + i )n - D.  D8=0=A>2>9 ?@0:B8:5 2AB@5G0NBAO A;CG08, :>340 C :@548B>@0 2>7=8:05B =5>1E>48<>ABL 25@=CBL G0ABL 45=53 4>A@>G=>.  B0:8E A;CG0OE 2>7=8:05B @8A: =52>72@0B0, ?>A:>;L:C B@51C5<>9 AC<<K =0 B0:>9 <><5=B 2@5<5=8 <>65B 8 =5 1KBL. @8 7=0G8B5;L=>9 AC<<5 4>;30 @07>2K9 ?;0B56 B@51C5B A>740=8O B0: =07K205<>3> D>=40 ?>30H5=8O, ?CB5< ?5@8>48G5A:8E 27=>A>2. $>=4 ?>30H5=8O 0::C<C;8@C5B 45=56=K5 A@54AB20, =0?@02;5==K5 =0 ?>30H5=85 704>;65==>AB8. 081>;55 MDD5:B82=> @07<5I5=85 D>=40 ?>30H5=8O A =0G8A;5=85< =0 27=>AK ?@>F5=B>2, =0?@8<5@, =0 A?5F80;L=>< AG5B5 2 10=:5. 5 B@C4=> 70<5B8BL, GB> B0:85 ?;0B568 ?> A2>59 ACB8 O2;ONBAO D8=0=A>2>9 @5=B>9 (0==C8B5B><), ?>MB><C 7040G0 A2>48BAO : >?@545;5=8N >4=>3> 87 ?0@0<5B@>2 D8=0=A>2>9 @5=BK  G;5=0 @5=BK. 45AL 2>7<>6=> 420 20@80=B0. 5@2K9  2K?;0B0 ?@>F5=B>2 ?> <5@5 8E =0G8A;5=8O, 0 >A=>2=0O AC<<0 45=53 2>72@0I05BAO 2 :>=F5 A@>:0 709<0.  INCLUDEPICTURE "http://irbis.asu.ru/mmc/econ/u_finmath/ris10.gif" \* MERGEFORMATINET  8A. 10. 48=>2@5<5==>5 ?>30H5=85 4>;30 A 2K?;0B>9 ?@>F5=B>2 ?> <5@5 8E =0G8A;5=8OA;8 ?@>F5=BK 2K?;0G820NBAO 5653>4=>, B>340 25;8G8=0 A@>G=>9 C?;0BK (@0AE>4>2 4>;6=8:0 ?> ?>30H5=8N 4>;30) @02=0:  INCLUDEPICTURE "http://irbis.asu.ru/mmc/econ/u_finmath/f41.gif" \* MERGEFORMATINET 345 D ?5@2>=0G0;L=0O AC<<0 4>;30; q AB02:0 ?@>F5=B>2 ?> CA;>28O< 709<0; sn; i  :>MDD8F85=B =0@0I5=8O D8=0=A>2>9 @5=BK; n A@>: 4>;30 2 3>40E; i  AB02:0 ?@>F5=B>2 ?@8 A>740=88 D>=40 ?>30H5=8O. 45AL D83C@8@CNB 425 AB02:8 ?@>F5=B>2: i  >?@545;O5B A:>@>ABL @>AB0 AC<<K D>=40 ?>30H5=8O; q  AC<<C 2K?;0G8205<KE 70 705< ?@>F5=B>2. @8<5@. >;3 100 BKA. 4>;;0@>2 2K40= ?>4 10% 3>4>2KE =0 3 3>40, A 5653>4=>9 2K?;0B>9 ?@>F5=B>2 ?> 4>;3C. ;O ?>30H5=8O AC<<K 4>;30 548=>2@5<5==K< ?;0B56>< A>7405BAO D>=4, :C40 5653>4=> 2=>AOBAO @02=K5 AC<<K, =0 :>B>@K5 =0G8A;ONBAO ?@>F5=BK ?> AB02:5 11%. 09B8 5653>4=K5 @0AE>4K 4>;6=8:0. 5H5=85: 653>4=K5 @0AE>4K 4>;6=8:0 A>AB02;ONB 25;8G8=C A@>G=>9 C?;0BK: Y = I + R, I = D " q = 100'000 " 0,1 = 10'000 4>;;0@>2,  INCLUDEPICTURE "http://irbis.asu.ru/mmc/econ/u_finmath/f42.gif" \* MERGEFORMATINET BAN40 Y = 10'000 + 29'921,31 = 39'921,31 4>;;0@>2. "0:8< >1@07><, 5653>4=K5 @0AE>4K 4>;6=8:0 ?> >1A;C6820=8N 4>;30 A>AB02OB 39'921,31 4>;;0@>2. 4=0:>, 1>;55 =03;O4=K< 8 MDD5:B82=K< A?>A>1>< ?;0=8@>20=8O 4>;30 O2;O5BAO A>AB02;5=85 B01;8F, 2 :>B>@KE >B@060NB 2A5 >A=>2=K5 E0@0:B5@8AB8:8 >1A;C6820=8O 4>;30: ;0= ?>30H5=8O 4>;30 548=>2@5<5==K< ?;0B56>< A 5653>4=>9 2K?;0B>9 ?@>F5=B>2 8 A>740=85< ?>30A8B5;L=>3> D>=40 >4>;3 (D)K?;0B0 ?@>F5=B>2 (I = D " q)7=>AK 2 ?>30A8B5;L=K9 D>=4,  INCLUDEPICTURE "http://irbis.asu.ru/mmc/econ/u_finmath/f43.gif" \* MERGEFORMATINET 5;8G8=0 A@>G=>9 C?;0BK, (Y=I+R)0:>?;5==0O AC<<0 4>;30 [FVt+1=FVt(1+i)+R]1100'00010'00029'921,3139'921,3129'921,312100'00010'00029'921,3139'921,3163'133,963100'00010'00029'921,3139'921,31100'000,00B>3>E30'00089'763,93119'763,93E "0:8< >1@07><, 87 ?@82545==>9 B01;8FK 284=>, GB> 5653>4=K5 @0AE>4K ?> >1A;C6820=8N 4>;30 A>AB02OB 39'921,31 4>;;0@>2, GB> 2 F5;>< 70 B@8 3>40 A>AB028B AC<<C 119'763,93 4>;;0@>2, ?@8G5< 2K?;0B0 ?@>F5=B>2 70 B@8 3>40 30'000 4>;;0@>2, 0 =0 ?>30H5=85 >A=>2=>3> 4>;30 2 @07<5@5 100'000 4>;;0@>2 ?@8E>48BAO 2A53> ;8HL 89'763,93 4>;;0@>2, B.5. 10'236,07 4>;;0@>2 O2;O5BAO =015602H8<8 ?@>F5=B0<8 =0 @07<5I5==K5 A@54AB20 2 D>=45 ?>30H5=8O. "0:8< >1@07><, A>740=85 D>=40 ?>30H5=8O O2;O5BAO =5>1E>48<K< M;5<5=B>< A>AB02;5=8O ?;0=0 ?>30H5=8O 4>;30, B.:. ?>72>;O5B =5 B>;L:> A=878BL @8A: =5 2>72@0B0 45=56=KE A@54AB2, => 8 A>:@0B8BL @0AE>4K ?> >1A;C6820=8N AC<<K 4>;30. 2.1.1.2. >30H5=85 >A=>2=>9 AC<<K 4>;30 8 ?@>F5=B>2 ?> =5<C 548=>2@5<5==K< ?;0B56>< 2 :>=F5 A@>:0 AAC4K. B>@>9 20@80=B ?>30H5=8O 4>;30 548=>2@5<5==K< ?;0B56>< A>AB>8B 2 2K?;0B5 ?@>F5=B>2 >4=>2@5<5==> A ?>30H5=85< 4>;30.  INCLUDEPICTURE "http://irbis.asu.ru/mmc/econ/u_finmath/ris11.gif" \* MERGEFORMATINET  8A. 11. 48=>2@5<5==>5 ?>30H5=85 4>;30 >4=>2@5<5==> A 2K?;0B>9 ?@>F5=B>2 ?> =5<C MB>< A;CG05 27=>AK 2 D>=4 ?>30H5=8O O2;ONBAO >4=>2@5<5==> 8 25;8G8=>9 A@>G=>9 C?;0BK (G;5=>< D8=0=A>2>9 @5=BK):  INCLUDEPICTURE "http://irbis.asu.ru/mmc/econ/u_finmath/f44.gif" \* MERGEFORMATINET 345 D ?5@2>=0G0;L=0O AC<<0 4>;30; q AB02:0 ?@>F5=B>2 ?> CA;>28O< 709<0; sn; i  :>MDD8F85=B =0@0I5=8O D8=0=A>2>9 @5=BK; n A@>: 4>;30 2 3>40E; i  AB02:0 ?@>F5=B>2 ?@8 A>740=88 ?>30A8B5;L=>3> D>=40. @8<5@. 0AA<>B@8< ?@54K4CI89 ?@8<5@, 87<5=82 CA;>28O: ?>30H5=85 548=>2@5<5==K< ?;0B56><, :0: AC<<K >A=>2=>3> 4>;30, B0: 8 2K?;0BK ?@>F5=B>2. 5H5=85: 5;8G8=0 A@>G=>9 C?;0BK @02=0: Y = [D " (1 + q)n] : Sn; i = 100'000 " (1 + 0,11) " 3 : 3,3421000 = 39'825,26 4>;;0@>2 "0:8< >1@07><, 25;8G8=0 5653>4=KE @0AE>4>2 ?> >1A;C6820=8N 4>;30 A>AB028B 39'825,26 4>;;0@>2, GB> =5A:>;L:> <5=LH5 0=0;>38G=>3> ?>:070B5;O 2 ?@54K4CI5< ?@8<5@5, A;54>20B5;L=>, <5=LH5 8 >1I0O AC<<0 @0AE>4>2 ?> >1A;C6820=8N 4>;30, A>AB02;ONI0O 25;8G8=C 119'475,78 4>;;0@>2. ;O 1>;55 =03;O4=>3> ?@54AB02;5=8O ?;0=0 ?>30H5=8O 4>;30 745AL B0:65 =5>1E>48<> A>AB02;5=85 B01;8FK. ;0= ?>30H5=8O 4>;30 548=>2@5<5==K< ?;0B56>< >4>;3 (Dt)7=>AK 2 ?>30A8B5;L=K9 D>=4, (Rt = Yt)0:>?;5==0O 25;8G8=0 2 ?>30A8B5;L=>< D>=45, (St)@>F5=BK ?> 4>;3C, (It)5;8G8=0 ?>30H5=8O B5:CI53> 4>;30, (St-It)1100'00039'825,2639'825,2610'00029'825,262110'00039'825,2684'031,3011'00084'020,303121'00039'825,26133'100,0012'100121'000,00B>3>133'100119'475,78E33'100E 0: 284=> 87 B01;8FK, ?@>8AE>48B 5653>4=>5 C25;8G5=85 AC<<K 4>;30 70 AG5B ?@8A>548=5=8O : =5<C ?@>F5=B>2, ?>MB><C : :>=FC A@>:0 4>;3 2>7@0AB5B 4> 133'100 4>;;0@>2, 87 :>B>@KE 2K?;0B0 ?@>F5=B>2 A>AB028B 33'100 4>;;0@>2. 4=0:> 70 AG5B C25;8G5=8O @07<5@0 27=>A>2 2 ?>30A8B5;L=K9 D>=4 >1I0O 25;8G8=0 >1A;C6820=8O 4>;30 C<5=LH05BAO. 7.1.2. >30H5=85 4>;30 2 @0AA@>G:C   ?@0:B8:5 D8=0=A>2>9 45OB5;L=>AB8 4>;3 G0AB> ?>30H05BAO 2 @0AA@>G:C, B.5. @0A?@545;5==K<8 2> 2@5<5=8 ?;0B560<8. @8 ?>30H5=88 >A=>2=>9 AC<<K 4>;30 G0ABO<8 53> B5:CI55 7=0G5=85 1C45B C<5=LH0BLAO 8, A;54>20B5;L=>, AC<<0 ?@>F5=B=KE ?;0B5659 B0:65 1C45B C<5=LH0BLAO. >30H5=85 4>;30 G0ABO<8 B0:65 <>65B >ACI5AB2;OBLAO @07;8G=K<8 A?>A>10<8.  7028A8<>AB8 >B ?@5A;54C5<KE 8=B5@5A>2 AB>@>=K <>3CB 2K18@0BL @07;8G=K5, C4>1=K5 4;O =8E @568<K 2 2845 ?>AB>O==KE 8;8 ?5@5<5==KE D8=0=A>2KE @5=B, 0 B0:65 =5@53C;O@=KE ?>B>:>2 ?;0B5659. 7.1.2.1. >30H5=85 >A=>2=>9 AC<<K 4>;30 @02=K<8 G0ABO<8. 4=8< 87 20@80=B>2 ?>30H5=8O 4>;30 2 @0AA@>G:C O2;O5BAO ?>30H5=85 >A=>2=>9 AC<<K 4>;30 @02=K<8 G0ABO<8. @8 MB>< 25;8G8=0 ?>30H5=8O 4>;30 >?@545;O5BAO A;54CNI8< >1@07><: dt = D : n = const, 345 dt  25;8G8=0 ?>30H5=8O >A=>2=>9 AC<<K 4>;30; D ?5@2>=0G0;L=0O AC<<0 4>;30; n A@>: 4>;30 2 3>40E; t  =><5@ 3>40, t = 1, 2, & , n. @>F5=BK =0G8A;ONBAO =0 C<5=LH05<CN AC<<C >A=>2=>3> 4>;30: It = Dt " q , 345 Dt  >AB0B>: 4>;30 =0 =0G0;> >G5@54=>3> 3>40; q  AB02:0 ?@>F5=B>2, =0G8A;O5<KE =0 AC<<C 4>;30. ">340 @07<5@ A@>G=>9 C?;0BK <>6=> ?@54AB028BL :0: AC<<C ?@>F5=B>2 8 AC<<C ?>30H5=8O 4>;30: Yt = It + dt , 345 Yt  A@>G=0O C?;0B0 =0 :>=5F B5:CI53> 3>40. @8<5@. !C<<0 100 BKA. 4>;;0@>2 2K40=0 ?>4 10% 3>4>2KE =0 3 3>40. ?@545;8BL 25;8G8=C A@>G=>9 C?;0BK ?@8 ?>30H5=88 >A=>2=>9 AC<<K 4>;30 @02=K<8 5653>4=K<8 G0ABO<8. 5H5=85: 5;8G8=0 AC<<K ?>30H5=8O 4>;30 @02=0: dt = D : n = 100'000 : 3 = 33'333,33 4>;;0@0. >A:>;L:C 25;8G8=0 A@>G=>9 C?;0BK ?@8 B0:>< A?>A>15 ?>30H5=8O 4>;30 <5=O5BAO 87 3>40 2 3>4, B> 2 MB>< A;CG05 157 ?>AB@>5=8O ?;0=0 ?>30H5=8O 4>;30 2 2845 B01;8FK ?@>AB> =5 >1>9B8AL. ;0= ?>30H5=8O >A=>2=>9 AC<<K 4>;30 @02=K<8 G0ABO<8 >4 (t)>;3 (D)!C<<0 ?>30H5=8O 4>;30 (dt)K?;0B0 ?@>F5=B>2 (It)5;8G8=0 A@>G=>9 C?;0BK (Yt)123451100'000,0033'333,3310'000,0043'333,33266'666,6733'333,336'666,6740'000,00333'333,3433'333,343'333,3336'666,67B>3>E100'000,0020'000,00120'000,00 "0:8< >1@07><, >1I85 @0AE>4K ?> >1A;C6820=8N 4>;30 A>AB028;8 120 BKA. 4>;;0@>2, 87 :>B>@KE 20 BKA. 4>;;0@>2 A>AB02;ONB ?@>F5=BK, 0 100 BKA. 4>;;0@>2  ?>30H5=85 >A=>2=>9 AC<<K 4>;30.  HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/7.1.2.ru.shtml" \l "note11#note11" 11>>> 7.1.2.2. >30H5=85 4>;30 8 ?@>F5=B>2 ?> =5<C @02=K<8 AC<<0<8 2 B5G5=85 A@>:0 AAC4K. >;3 B0:65 <>6=> ?>30H0BL 2 @0AA@>G:C @02=K<8 A@>G=K<8 C?;0B0<8, :>B>@K5 2:;NG0NB 2 A51O :0: ?>30H5=85 >A=>2=>9 AC<<K 4>;30, B0: 8 25;8G8=C ?@>F5=B>2 ?> =5<C: Yt = It + dt = const. @8 ?>30H5=88 4>;30 2 @0AA@>G:C 25;8G8=0 4>;30 A8AB5<0B8G5A:8 C1K205B, GB> ?@82>48B : C<5=LH5=8N ?@>F5=B>2 8, A>>B25BAB25==>, C25;8G5=8N AC<<, 84CI8E =0 ?>30H5=85 4>;30,  MB> B0: =07K205<>5 ?@>3@5AA82=>5 ?>30H5=85. >A:>;L:C A@>G=K5 C?;0BK @02=K, B> 8E ?>A;54>20B5;L=>ABL ?@54AB02;O5B A>1>9 D8=0=A>2CN @5=BC, A>2@5<5==>5 7=0G5=85 :>B>@>9 4>;6=> 1KBL @02=> AC<<5 4>;30. > D>@<C;5 4;O >?@545;5=8O @07<5@0 ?;0B560 ?>AB>O==>9 3>4>2>9 D8=0=A>2>9 @5=BK A 2K?;0B0<8 2 :>=F5 ?5@8>40, @07<5@ A@>G=>9 C?;0BK @025=:  INCLUDEPICTURE "http://irbis.asu.ru/mmc/econ/u_finmath/f45.gif" \* MERGEFORMATINET 345 Yt  25;8G8=0 A@>G=>9 C?;0BK; D ?5@2>=0G0;L=0O AC<<0 4>;30; q ?@>F5=B=0O AB02:0 =0 AC<<C 4>;30; n A@>: 4>;30 2 3>40E; t =><5@ 3>40, t = 1, 2, & , n. @8<5@. #A;>28O ?@54K4CI59 7040G8, => ?>30H5=85 4>;30 ?@54CA<0B@8205B C?;0BC @02=K<8 A@>G=K<8 2K?;0B0<8. 5H5=85: !@>G=0O C?;0B0, 2:;NG0NI0O 2 A51O ?>30H5=85 >A=>2=>9 AC<<K 4>;30 8 2K?;0BC ?@>F5=B>2 ?> 4>;3C, @02=0: Yt= 100'000,00 : 2,486851991 = 40'211,48 4>;;0@>2. BAN40 >1I85 @0AE>4K ?> ?>30H5=8N 4>;30 @02=K: Yt = 40'211,48 " 3 = 120'634,44 4>;;0@0. "0:8< >1@07><, 5653>4=K5 @0AE>4K ?> ?>30H5=8N 4>;30 1C4CB A>AB02;OBL 40'211,48 4>;;0@>2, 0 70 25AL A@>: D8=0=A>2>9 >?5@0F88  120'634,44 4>;;0@0. @8 MB>< 20@80=B5 ?>30H5=8O 4>;30 B0:65 2>7<>6=> ?>AB@>5=85 B01;8FK. ;0= ?>30H5=8O 4>;30 @02=K<8 A@>G=K<8 C?;0B0<8 >4 (t)>;3 (Dt)!@>G=0O C?;0B0 (Yt)@>F5=BK (It)!C<<0 ?>30H5=8O >A=>2=>3> 4>;30 (dt=Yt-It)123451100'000,0040'211,48100'00,0030'211,48269'788,5240'211,486'978,8533'232,63336'555,8940'211,483'655,5936'555,89B>3>E120'634,4420'634,44100'000,00 "0:8< >1@07><, >1I85 @0AE>4K ?> >1A;C6820=8N 4>;30 A>AB02;ONB 120'634,44 4>;;0@>2, 87 :>B>@KE 100 BKA. 4>;;0@>2 84CB =0 ?>30H5=85 4>;30, 0 20'634,44 4>;;0@>2  ?@>F5=BK.  B01;8F5 =03;O4=> ?@54AB02;5=> @0A?@545;5=85 AC<<K A@>G=>9 C?;0BK =0 2K?;0BC ?@>F5=B>2 8 =5?>A@54AB25==>5 ?>30H5=85 4>;30.  INCLUDEPICTURE "http://irbis.asu.ru/mmc/images/dot.gif" \* MERGEFORMATINET   HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/7.1.2.ru.shtml" \l "rev11#rev11" <<<11!@02=8B5 ?>;CG5==K5 @57C;LB0BK A 2K?;0B>9 4>;30 548=>2@5<5==K< ?;0B56><.  7.1.3. >B@518B5;LA:89 :@548B  '0AB=K< A;CG05< ?>30H5=8O 4>;30 @02=K<8 A@>G=K<8 C?;0B0<8 O2;O5BAO ?>B@518B5;LA:89 :@548B, ?@8 :>B>@>< ?@>F5=BK =0G8A;ONBAO A@07C =0 2AN AC<<C :@548B0, 0 AC<<0 704>;65==>AB8 @02=><5@=> ?>30H05BAO =0 ?@>BO65=88 2A53> A@>:0 :@548B0. @>F5=BK 2 ?>B@518B5;LA:>< :@548B5 =0G8A;ONBAO A@07C =0 2AN AC<<C 4>;30 ?> ?@>AB>9 AB02:5: I = D " n " i ">340 >1I0O AC<<0 @0AE>4>2 ?> ?>30H5=8N :@548B0 A:;04K205BAO 87 2K?;0BK ?@>F5=B>2 8 AC<<K >A=>2=>3> 4>;30: Yt = D + I !;54>20B5;L=>, @07<5@ A@>G=>9 C?;0BK >?@545;O5BAO ?> D>@<C;5: Yt = (D + I) : (n " m), 345 n A@>: :@548B0 2 3>40E; m :>;8G5AB2> 27=>A>2 2 B5G5=85 3>40. @8<5@. >B@518B5;LA:89 :@548B =0 AC<<C 5 BKA. @C1. >B:@KB =0 2 3>40 ?> AB02:5 25% 3>4>2KE. >30H5=85 :@548B0 @02=K<8 27=>A0<8 565:20@B0;L=>. ?@545;8BL AB>8<>ABL :@548B0 8 @07<5@ 565:20@B0;L=KE 27=>A>2. 5H5=85: !B>8<>ABL :@548B0 MB> ?@>F5=BK, :>B>@K5 @02=K: I = D " n " i = 5'000 " 2 " 0,25 = 2'500 @C1;59 1I0O AC<<0 @0AE>4>2 ?> >1A;C6820=8N :@548B0 @02=0: Yt = D + I = 5'000 + 2'500 = 7'500 @C1;59 65:20@B0;L=K5 27=>AK A>AB02OB 25;8G8=C: Yt = (D + I) : (n " m) = 7'500 : 2 " 4 = 937,50 @C1;59 "0:8< >1@07><, 565:20@B0;L=K5 27=>AK 2 @07<5@5 937,50 @C1;59 ?>72>;O5B 2K?;0B8BL AC<<C 4>;30 8 2K?;0B8BL ?@>F5=BK. A;8 1K 8A?>;L7>20;>AL ?@>3@5AA82=>5 ?>30H5=85, B.5. =0G8A;5=85 ?@>F5=B>2 =0 >AB0B>: 4>;30, B> MB> 1K;> 1K 70<5B=> 45H52;5 4;O 4>;6=8:0. 0AG;5=5=85 25;8G8=K A@>G=>9 C?;0BK 2 ?>B@518B5;LA:>< :@548B5 =0 ?@>F5=B=K5 ?;0B568 8 ?>30H5=85 >A=>2=>9 AC<<K 4>;30 2 <8@>2>9 ?@0:B8:5 =07K205BAO "<5B>4>< 78". -B> A2O70=> A B5<, GB> 4;O ?>B@518B5;LA:>3> :@548B0 A@>:>< 12 <5AOF52 8 565<5AOG=K< ?>30H5=85, AC<<0 ?>@O4:>2KE =><5@>2 <5AOF52 1C45B @02=0 78, GB> 8 40;> =0720=85 B0:><C <5B>4C =0G8A;5=8O ?@>F5=B>2. -B> ?@028;> <>6=> >1>1I8BL 4;O n ;5B 8 m ?;0B5659 2 3>4C: N = m " n [(m " n + 1) : 2], 345 N  AC<<0 ?>A;54>20B5;L=KE =><5@>2 2K?;0B. BAN40 >G5=L ;53:> @0AG;5=8BL A@>G=CN C?;0BC =0 ?@>F5=B=K5 ?;0B568 8 AC<<C ?>30H5=8O >A=>2=>3> 4>;30: Yt = It + dt , 345 It ?@>F5=B=K9 ?;0B56; dt  AC<<0 ?>30H5=8O >A=>2=>3> 4>;30. ">340 25;8G8=0 ?@>F5=B=>3> ?;0B560 >?@545;O5BAO A;54CNI8< >1@07><: It = I " (t / N), 0 AC<<0 ?>30H5=8O >A=>2=>3> 4>;30 :0: @07=8F0 A@>G=>9 C?;0BK 8 ?@>F5=B=KE 2K?;0B: Rt = Yt - It . 0AA<>B@8< ?@54K4CI89 ?@8<5@, @0AG;5=82 A@>G=CN C?;0BC =0 A>AB02;ONI85 M;5<5=BK, 2A5 40==K5 ?@54AB0282 2 2845 B01;8FK. ;0= ?>30H5=8O ?>B@518B5;LA:>3> :@548B0 ;0B56t>;3 (Dt=Dt-1-Rt)!@>G=0O C?;0B0 (Yt)@>F5=BK [It=I (t/N)]>30H5=85 >A=>2=>9 AC<<K 4>;30 (dt=Yt-It)123456185'000,00937,50555,56381,94274'618,06937,50486,11451,39364'166,67937,50416,67520,83453'645,84937,50347,22590,28543'055,56937,50277,78659,72632'395,84937,50208,33729,17721'666,67937,50 ;020 8. F5=:0 8=25AB8F8>==KE ?@>F5AA>2   HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/8.1.ru.shtml" 8.1. A>15==>AB8 8=25AB8F8>==KE ?@>F5AA>2 :0: >1J5:B0 D8=0=A>2>9 <0B5<0B8:8  HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/8.2.ru.shtml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http://irbis.asu.ru/mmc/econ/u_finmath/ris12.gif" \* MERGEFORMATINET  8A. 12. @0D8G5A:>5 87>1@065=85 8=25AB8F8>==>3> ?@>F5AA0@8=OB> @07;8G0BL: D8=0=A>2K5 8=25AB8F88; @50;L=K5 8=25AB8F88; 8=25AB8F88 2 =5<0B5@80;L=K5 0:B82K. $8=0=A>2K5 8=25AB8F88  2;>65=85 45=56=KE A@54AB2 2 F5==K5 1C<038; @50;L=K5 8=25AB8F88  2;>65=8O 2 >A=>2=>9 :0?8B0; 8 ?@8@>AB 70?0A>2; 2;>65=8O 2 =5<0B5@80;L=K5 0:B82K  2;>65=8O 2 @0728B85 =0CG=KE 8AA;54>20=89, ?>2KH5=85 :20;8D8:0F88 @01>B=8:>2, ?@8>1@5B5=85 ;8F5=789 8 ?@02. 50;870F8O 8=25AB8F8>==KE ?@>5:B>2 B@51C5B >B:070 >B 45=56=KE A@54AB2 A53>4=O 2 ?>;L7C ?>;CG5=8O 4>E>40 2 1C4CI5<, ?>MB><C ;N1>9 8=25AB8F8>==K9 ?@>5:B B@51C5B 0=0;870 8 >F5=:8. F5=820O MDD5:B82=>ABL 8=25AB8F8>==KE ?@>5:B>2, A;54C5B CG8BK20BL 8 AB5?5=L @8A:0,  745AL, :0: ?@028;>, 2K45;ONB 420 2840 @8A:0: ?@54?@8=8<0B5;LA:89 8 D8=0=A>2K9. @54?@8=8<0B5;LA:89 @8A:  @8A:, A2O70==K9 A 45OB5;L=>ABLN :>=:@5B=>3> 187=5A0. $8=0=A>2K9 @8A:  87<5=5=8O<8 @K=>G=>9 AB02:8 4>E>40 =0 :0?8B0;. ;O C?@>I5=8O 8AA;54>20=8O MDD5:B82=>AB8 8=25AB8F89 ?@54?>;0305BAO, GB> =5>1E>48<0O =>@<0 ?@81K;8 7040=0 8 >48=0:>20 4;O 2A5E 8=25AB8F8>==KE ?@>5:B>2 8 4;O ;N1>3> 87 @0AA<0B@8205<KE ?@>5:B>2 AB5?5=L @8A:0 >48=0:>20. 07;8G0NB ?@>ABK5 (AB0B8G5A:85) 8 CA;>6=5==K5 (48=0<8G5A:85) <5B>4K. @>ABK5 <5B>4K B@048F8>==> 8A?>;L7>20;8AL 2 A>F80;8AB8G5A:>9 M:>=><8:5 8 >B25G0;8 459AB2CNI8< B>340 CA;>28O< E>7O9AB2>20=8O.  @K=>G=KE CA;>28OE 8A?>;L7CNBAO <5B>4K, >A=>20==K5 =0 B5>@88 2@5<5==>9 AB>8<>AB8 45=53, :>B>@K5 CAB@0=8;8 =54>AB0B>: @0=55 459AB2CNI8E <5B>48:. 06=59H0O 7040G0 0=0;870 8=25AB8F8>==KE ?@>5:B>2  @0AG5B 1C4CI8E 45=56=KE ?>B>:>2, 2>7=8:0NI8E ?@8 @50;870F88 ?@>5:B0, => =5 ?@81K;8. =0;87 8=25AB8F8>==KE ?@>5:B>2 >A=>20= =0 8AA;54>20=88 4>E>4>2 8 @0AE>4>2, 2K@065==KE 2 D>@<5 45=56=KE ?>B>:>2, => =5 =0 87<5=5=8OE, 2K720==KE CA;>2=>ABO<8 1CE30;B5@A:>3> CG5B0.  40==>9 3;025 @0AA<0B@820NBAO B>;L:> <5B>4K 8 ?>:070B5;8 MDD5:B82=>AB8 8=25AB8F89, >A=>20==K5 =0 ?@8=F8?5 48A:>=B8@>20=8O. @8 0=0;875 ?>B>:>2 ?;0B5659 8A?>;L7CNBAO >1>1I0NI85 ?>:070B5;8: =0@0I5==0O AB>8<>ABL; ?@82545==0O AB>8<>ABL; =>@<0 4>E>4=>AB8. -B8 ?>:070B5;8 C65 @0AA<0B@820;8AL 2 B5>@5B8G5A:>9 G0AB8, => 4;O 8=25AB8F8>==KE ?@>F5AA>2 >=8 ?@8>1@5B0NB A2>N A?5F8D8:C.  8.2. >:070B5;8 MDD5:B0 8 MDD5:B82=>AB8 8=25AB8F8>==KE ?@>5:B>2   HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/8.2.1.ru.shtml" 8.2.1. '8ABK9 ?@82545==K9 4>E>4  HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/8.2.2.ru.shtml" 8.2.2. !@>: >:C?05<>AB8  HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/8.2.3.ru.shtml" 8.2.3. =CB@5==OO =>@<0 4>E>4=>AB8  8.2.1. '8ABK9 ?@82545==K9 4>E>4  >A:>;L:C 45=56=K5 A@54AB20 @0A?@545;5=K 2> 2@5<5=8, B> 8 745AL D0:B>@ 2@5<5=8 83@05B 206=CN @>;L. @8 >F5=:5 8=25AB8F8>==KE ?@>5:B>2 8A?>;L7C5BAO <5B>4 @0AG5B0 G8AB>3> ?@82545==>3> 4>E>40, :>B>@K9 ?@54CA<0B@8205B 48A:>=B8@>20=85 45=56=KE ?>B>:>2: 2A5 4>E>4K 8 70B@0BK ?@82>4OBAO : >4=><C <><5=BC 2@5<5=8. &5=B@0;L=K< ?>:070B5;5< 2 @0AA<0B@8205<>< <5B>45 O2;O5BAO ?>:070B5;L NPV (net present value)  B5:CI0O AB>8<>ABL 45=56=KE ?>B>:>2 70 2KG5B>< B5:CI59 AB>8<>AB8 45=56=KE >BB>:>2. -B> >1>1I5==K9 :>=5G=K9 @57C;LB0B 8=25AB8F8>==>9 45OB5;L=>AB8 2 01A>;NB=>< 87<5@5=88. @8 @07>2>9 8=25AB8F88 @0AG5B G8AB>3> ?@82545==>3> 4>E>40 <>6=> ?@54AB028BL A;54CNI8< 2K@065=85<:  INCLUDEPICTURE "http://irbis.asu.ru/mmc/econ/u_finmath/f46.gif" \* MERGEFORMATINET 345 Rk  3>4>2K5 45=56=K5 ?>ABC?;5=8O 2 B5G5=85 n ;5B, k = 1, 2, & , n; IC AB0@B>2K5 8=25AB8F88; i AB02:0 48A:>=B8@>20=8O. 06=K< <><5=B>< O2;O5BAO 2K1>@ AB02:8 48A:>=B8@>20=8O, :>B>@0O 4>;6=0 >B@060BL >68405<K9 CA@54=5==K9 C@>25=L AAC4=>3> ?@>F5=B0 =0 D8=0=A>2>< @K=:5. ;O >?@545;5=8O MDD5:B82=>AB8 8=25AB8F8>==>3> ?@>5:B0 >B45;L=>9 D8@<>9 2 :0G5AB25 AB02:8 48A:>=B8@>20=8O 8A?>;L7C5BAO A@54=52725H5==0O F5=0 :0?8B0;0, 8A?>;L7C5<>3> D8@<>9 4;O D8=0=A8@>20=8O 40==>3> 8=25AB8F8>==>3> ?@>5:B0. A;8 ?@>5:B ?@54?>;0305B =5 @07>2CN 8=25AB8F8N, 0 ?>A;54>20B5;L=>5 8=25AB8@>20=85 D8=0=A>2KE @5AC@A>2 2 B5G5=85 =5A:>;L:8E ;5B (m), B> D>@<C;0 4;O @0AG5B0 <>48D8F8@C5BAO:  INCLUDEPICTURE "http://irbis.asu.ru/mmc/econ/u_finmath/f47.gif" \* MERGEFORMATINET >:070B5;L NPV O2;O5BAO 01A>;NB=K< ?@8@>AB><, ?>A:>;L:C >F5=8205B, =0 A:>;L:> ?@82545==K9 4>E>4 ?5@5:@K205B ?@82545==K5 70B@0BK: ?@8 NPV > 0 ?@>5:B A;54C5B ?@8=OBL; ?@8 NPV < 0 ?@>5:B =5 ?@8=8<05BAO, ?@8 NPV = 0 ?@>5:B =5 8<55B =8 ?@81K;8, =8 C1KB:>2. 5>1E>48<> >B<5B8BL, GB> ?>:070B5;L NPV >B@0605B ?@>3=>7=CN >F5=:C 87<5=5=8O M:>=><8G5A:>3> ?>B5=F80;0 D8@<K 2 A;CG05 ?@8=OB8O 40==>3> ?@>5:B0. 4=> 87 206=KE A2>9AB2 40==>3> :@8B5@8O, GB> ?>:070B5;L NPV @07;8G=KE ?@>5:B>2 <>6=> AC<<8@>20BL, ?>A:>;L:C >= 0448B825= 2> 2@5<5=8. -B> ?>72>;O5B 8A?>;L7>20BL 53> ?@8 0=0;875 >?B8<0;L=>AB8 8=25AB8F8>==>3> ?>@BD5;O. @8<5@. $8@<0 @0AA<0B@8205B F5;5A>>1@07=>ABL 8=25AB8F8>==>3> ?@>5:B0, AB>8<>ABL :>B>@>3> A>AB02;O5B 210 BKA. 4>;;0@>2. > ?@>3=>70< 5653>4=K5 ?>ABC?;5=8O A>AB02OB 55 BKA. 4>;;0@>2. @>5:B @0AAG8B0= =0 5 ;5B. 5>1E>48<0O =>@<0 ?@81K;8 A>AB02;O5B 8%. !;54C5B ;8 ?@8=OBL MB>B ?@>5:B? 5H5=85: '8AB0O AB>8<>ABL ?@>5:B0 @02=0:  HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/8.2.1.ru.shtml" \l "note12#note12" 12>>> NPV = 55'000 (1,08)-1 + 55'000 (1,08)-2 + 55'000 (1,08)-3 + 55'000 (1,08)-4 + + 55'000 (1,08)-5 - 210'000 = 50'926 + 42'867 + 39'692 + 36'751 + 34'029 - 210'000 = = 204'265 - 210'000 = -5'735 4>;;0@>2. >A:>;L:C 25;8G8=0 G8AB>9 B5:CI59 AB>8<>AB8 -5'735 4>;;0@>2, B.5. NPV < 0, B> ?@>5:B =5 <>65B 1KBL ?@8=OB. 0: ?@028;>, >A=>2K20NBAO =0 B><, GB> 25;8G8=0 NPV =0E>48BAO =0 =0G0;> @50;870F88 8=25AB8F8>==>3> ?@>5:B0, >4=0:> <>6=> >?@545;OBL MBC 25;8G8=C =0 <><5=B 7025@H5=8O ?@>F5AA0 2;>65=89 8;8 =0 8=>9 <><5=B 2@5<5=8. 0?><=8<, GB> AB02:0 48A:>=B8@>20=8O  @57C;LB0B 2K1>@0, AC1J5:B82=>3> AC645=8O. @><5 B>3>, ?@8 2KA>:>< C@>2=5 AB02:8 >B40;5==K5 ?;0B568 1C4CB >:07K20BL =0 25;8G8=C NPV <0;>5 2;8O=85, ?>MB><C 20@80=BK, >B;8G0NI85AO ?> ?@>4>;68B5;L=>AB8 ?5@8>4>2 >B40G8, <>3CB >:070BLAO @02=>F5==K<8 ?> :>=5G=><C M:>=><8G5A:><C MDD5:BC.  INCLUDEPICTURE "http://irbis.asu.ru/mmc/images/dot.gif" \* MERGEFORMATINET   HYPERLINK "http://irbis.asu.ru/mmc/econ/u_finmath/8.2.1.ru.shtml" \l "rev12#rev12" <<<12040G0 @5H05BAO A 8A?>;L7>20=85< D>@<C;K A>2@5<5==>9 25;8G8=K >1KG=>9 @5=BK.  8.2.2. !@>: >:C?05<>AB8  ;O 0=0;870 8=25AB8F89 ?@8<5=ONB 8 B0:>9 ?>:070B5;L, :0: A@>: >:C?05<>AB8 (payback period method)  ?@>4>;68B5;L=>ABL 2@5<5=8, 2 B5G5=85 :>B>@>3> 48A:>=B8@>20==K5 =0 <><5=B 7025@H5=8O 8=25AB8F89 ?@>3=>78@C5<K5 45=56=K5 ?>ABC?;5=8O @02=K AC<<5 8=25AB8F89. =K<8 A;>20<8  MB> AC<<0 ;5B, =5>1E>48<KE 4;O 2>7<5I5=8O AB0@B>2KE 8=25AB8F89:  INCLUDEPICTURE "http://irbis.asu.ru/mmc/econ/u_finmath/f48.gif" \* MERGEFORMATINET B.5. NPV = 0. 5@8>4 >:C?05<>AB8 <>6=> >?@545;8BL :0: >68405<>5 G8A;> ;5B ?> C?@>I5==>9 D>@<C;5: n>: = '8A;> ;5B 4> 3>40 >:C?05<>AB8 + (5 2>7<5I5==0O AB>8<>ABL =0 =0G0;> 3>40 >:C?05<>AB8 / @8B>: =0;8G=>AB8 2 B5G5=85 3>40 >:C?05<>AB8) 0==K9 ?>:070B5;L >?@545;O5B A@>:, 2 B5G5=85 :>B>@>3> 8=25AB8F88 1C4CB "70<>@>65=K", ?>A:>;L:C @50;L=K9 4>E>4 >B 8=25AB8F8>==>3> ?@>5:B0 =0G=5B ?>ABC?0BL B>;L:> ?> 8AB5G5=88 ?5@8>40 >:C?05<>AB8. @8<5@. 0AAG8B0BL A@>: >:C?05<>AB8 ?@>5:B0, 4;O :>B>@>3> @07<5@ 8=25AB8F89 A>AB02;O5B 1 <;= @C1., 0 45=56=K5 ?>ABC?;5=8O 2 B5G5=85 5 ;5B 1C4CB A>AB02;OBL: 200; 500; 600; 800; 900 BKA. @C1. A>>B25BAB25==>. !B02:0 48A:>=B8@>20=8O 15%. 5H5=85: 0AAG8B05< 48A:>=B8@>20==K9 45=56=K9 ?>B>:: 5@8>40123455=56=K9 ?>B>:-10002005006008009008A:>=B8@>20==K9 45=56=K9 ?>B>:-10001743783944584470:>?;5==K9 48A:>=B8@>20==K9 45=56=K9 ?>B>:-1000-826-448-54404851 !@>: >:C?05<>AB8 ?@>5:B0: k>: = 3 + 54 / 458 = 3,12 "0:8< >1@07><, ?5@8>4, @50;L=> =5>1E>48<K9 4;O 2>7<5I5=8O 8=25AB8@>20==>9 AC<K, A>AB028B 3,12 3>40 8;8 3 3>40 8 44 4=O. A;8 4>E>4K <>6=> ?@54AB028BL 2 2845 0==C8B5B0, B>  INCLUDEPICTURE "http://irbis.asu.ru/mmc/econ/u_finmath/f49.gif" \* MERGEFORMATINET !@>: >:C?05<>AB8 ACI5AB2C5B, 5A;8 =5 =0@CH0NBAO >?@545;5==K5 A>>B=>H5=8O <564C ?>ABC?;5=8O<8 8 @07<5@>< 8=25AB8F89. @8 5653>4=KE ?>AB>O==KE ?>ABC?;5=8OE MB> A>>B=>H5=85 8<55B 284: Rk < IC " i, B.5. =5 2AO:89 C@>25=L 4>E>40 ?@8 ?@>G8E @02=KE CA;>28OE ?@82>48B : >:C?05<>AB8 8=25AB8F89. 8.2.3. =CB@5==OO =>@<0 4>E>4=>AB8  @8 0=0;875 MDD5:B82=>AB8 8=25AB8F8>==KE ?@>5:B>2 H8@>:> 8A?>;L7C5BAO ?>:070B5;L 2=CB@5==59 =>@<K 4>E>4=>AB8 (IRR  internal rate of return)  MB> AB02:0 48A:>=B8@>20=8O, ?@8@02=820NI0O AC<<C ?@82545==KE 4>E>4>2 >B 8=25AB8F8>==>3> ?@>5:B0 : 25;8G8=5 8=25AB8F89, B.5. 2;>65=8O >:C?0NBAO, => =5 ?@8=>AOB ?@81K;L. 5;8G8=0 MB>9 AB02:8 ?>;=>ABLN >?@545;O5BAO "2=CB@5==8<8" CA;>28O<8, E0@0:B5@87CNI8<8 8=25AB8F8>==K9 ?@>5:B. @8<5=5=85 40==>3> <5B>40 A2>48BAO : ?>A;54>20B5;L=>9 8B5@0F88 (?>2B>@5=8O) =0E>645=8O 48A:>=B8@CNI53> <=>68B5;O, ?>:0 =5 1C45B >15A?5G5=> @025=AB2> NPV = 0. K18@0NBAO 420 7=0G5=8O :>MDD8F85=B0 48A:>=B8@>20=8O, ?@8 :>B>@KE DC=:F8O NPV <5=O5B A2>9 7=0:, 8 8A?>;L7CNB D>@<C;C: IRR = i1 + NPV(i1) / [NPV(i1) - NPV(i2)] " (i2 - i1) =25AB>@ A@02=8205B ?>;CG5==>5 7=0G5=85 IRR A> AB02:>9 ?@82;5G5==KE D8=0=A>2KE @5AC@A>2 (CC  Cost of Capital): 5A;8 IRR > CC, B> ?@>5:B <>6=> ?@8=OBL; 5A;8 IRR < !!, ?@>5:B >B25@305BAO; IRR = !! ?@>5:B 8<55B =C;52CN ?@81K;L. @8<5@. 0AAG8B0BL 2=CB@5==NN AB02:C 4>E>4=>AB8 ?> ?@>5:BC, 345 70B@0BK A>AB02;ONB 1200 BKA. @C1., 0 4>E>4K 50; 200; 450; 500 8 600 BKA. @C1. 5H5=85: 0AG5B ?> AB02:5 5%: NPV = 47619 + 181406 + 388767 + 411351 + 470116 - 1200000 = 299259. >A:>;L:C NPV > 0, B> =>20O AB02:0 48A:>=B8@>20=8O 4>;6=0 1KBL 1>;LH5 5%. 0AG5B ?> AB02:5 15%: NPV = 43478 + 151229 + 295882 + 285877 + 298306 - 1200000 = -125228. KG8A;O5< 2=CB@5==NN AB02:C 4>E>4=>AB8: IRR = 5 + [299259 / [299259 - (-125228)]] " (15 - 5) = 12,05. =CB@5==OO =>@<0 4>E>4=>AB8 ?@>5:B0 @02=0 12,05%. ">G=>ABL 2KG8A;5=8O >1@0B=0 25;8G8=5 8=B5@20;0 <564C 2K18@05<K<8 ?@>F5=B=K<8 AB02:0<8, ?>MB><C 4;O CB>G=5=8O 25;8G8=K ?@>F5=B=>9 AB02:8 4;8=0 8=B5@20;0 ?@8=8<05BAO 70 1%. @8<5@. #B>G=8BL 25;8G8=C AB02:8 4;O ?@54K4CI53> ?@8<5@0. 5H5=85: ;O ?@>F5=B=>9 AB02:8 11%: NPV = 45045 + 162324 + 329036 + 329365 + 356071 - 1200000 = 21841. ;O ?@>F5=B=>9 AB02:8 12%: NPV = 44643 + 159439 + 320301 + 317759 + 340456 - 1200000 = -17402. #B>G=5==0O 25;8G8=0: IRR = 11 + [21841 / [21841 - (-17402)]] " (12 - 11) = 11,56. !B02:0 11,56 % O2;O5BAO 25@E=8< ?@545;>< ?@>F5=B=>9 AB02:8, ?> :>B>@>9 D8@<0 <>65B >:C?8BL :@548B 4;O D8=0=A8@>20=8O 8=25AB8F8>==>3> ?@>5:B0. @8;>65=85 1  >@O4:>2K5 =><5@0 4=59 2 =5 28A>:>A=>< 3>4C 5=L5AOF/=20@L$52@0;L0@B?@5;L09N=LN;L23CAB!5=BO1@L:BO1@L>O1@L5:01@L11326091121152182213244274305335223361921221531832142452753063363334629312315418421524627630733744356394124155185216247277308338553664951251561862172482783093396637659612615718721724927931034077386697127158188219250280311341883967981281591892202512813123429940689912916019022125228231334310104169100130161191222253283314344111142701011311621922232542843153451212437110213216319322425528531634613134472103133164194213256286317347141445731041341651952262572873183481515467410513516619622725828831934916164775106136167197228259289320350171748761071371681982292602903213511818497710813816919923026129132235219195078109139170200231262292323353202051791101401712012322632933243542121528011114117220223326429432535522225381112142173203234265295326356232354821131431742042352662963273572424558311414417520523626729732835825255684115145176206237268298329359262657851161461772072382692993303602727588611714717820823927030033136128285987118148179209240271301332362292988119149180210241272302333363303089120150181211242273303334364313190151212243304365 @8;>65=85 2  =>68B5;8 =0@0I5=8O ?> A;>6=K< ?@>F5=B0< '8A;> ?5@8>4>2!B02:0 ?@>F5=B>2 70 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34pւTHkd$$IfT" X i:qt!$06444434ab4pւT$$If!v h555x5M555l5i5 5 5 u5 5 `#v#v#vx#vM#v#v#vl#vi#v #v #v u#v #v `:V 06, 5 / 34pւTHkd$$IfT" X i:qt!$06444434ab4pւT$$If!v h555x5M555l5i5 5 5 u5 5 `#v#v#vx#vM#v#v#vl#vi#v #v #v u#v #v `:V 06, 5 / 34pւTHkd#$$IfT" X i:qt!$06444434ab4pւT$$If!v h555x5M555l5i5 5 5 u5 5 `#v#v#vx#vM#v#v#vl#vi#v #v #v u#v #v `:V 06, 5 / 34pւTHkd($$IfT" X i:qt!$06444434ab4pւT$$If!v h555x5M555l5i5 5 5 u5 5 `#v#v#vx#vM#v#v#vl#vi#v #v #v u#v #v `:V 06, 5 / 34pւTHkd-$$IfT" X i:qt!$06444434ab4pւT$$If!v h555x5M555l5i5 5 5 u5 5 `#v#v#vx#vM#v#v#vl#vi#v #v #v u#v #v `:V 06, 5 / 34pւTHkd2$$IfT" X i:qt!$06444434ab4pւT$$If!v h555x5M555l5i5 5 5 u5 5 `#v#v#vx#vM#v#v#vl#vi#v #v #v u#v #v `:V 06, 5 / 34pւTHkd8$$IfT" X i:qt!$06444434ab4pւTDd$V  3 C" $88Dd$V  3 C" $88$$If!vh55#v#v:V 06,5/ 34pT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd5?$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkdB$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd/F$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkdI$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd)M$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkdP$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd#T$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkdW$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd[$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd^$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkdb$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkde$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkdi$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkdl$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd p$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkds$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkdw$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkdz$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd}$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd|$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkdv$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkdp$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkdj$$IfTִ" j B06    34ab pPTDd$V  3 C" $88$$If!vh55#v#v:V 06,5/ 34pT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkdN$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd˞$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkdH$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkdť$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkdB$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd<$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd6$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd0$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd*$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd$$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd $$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd$$IfTִ" j B06    34ab pPTDd$V  3 C" $88Dd$V  3 C" $88$$If!vh55#v#v:V 06,5/ 34pT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd $$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkdz $$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd $$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkdt$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkdn$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkdh$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd!$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkdb%$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd($$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd\,$$IfTִ" j B06    34ab pPT5$$If!vh55H5H5H5H5H5H5W#v#vH#vW:V 06,5/ 34pPTDkd/$$IfTִ" j B06    34ab 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