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xcel.Sheet.8  8A 3. >45;L D0:B>@=>3> 0=0;870 :>=:C@5=B>A?>A>1=>AB8 ?@54?@8OB8O. "01;8F0 2 - !8AB5<0 ?>:070B5;59 D0:B>@=>9 >F5=:8 :>=:C@5=B>A?>A>1=>AB8 ?@><KH;5==>3> ?@54?@8OB8O. 08<5=>20=85 ?>:070B5;O5B>4 @0AG5B0%0@0:B5@8AB8:0 ?>:070B5;OI @C??0. >:070B5;8 >F5=:8 D8=0=A>2>3> D0:B>@0 1. =B53@0;L=K9 :>MDD8F85=B ;8:284=>AB8 (I;8:2). I;8:2 =  EMBED Equation.DSMT4 , 345:  (:>MDD8F85=B 01A>;NB=>9 ;8:284=>AB8) = (45=. A@54AB20 + :@0B:>A@. D8=. 2;>65=8O) / B5:CI85 ?0AA82K  (:>MDD8F85=B 1KAB@>9 ;8:284=>AB8) = (45=. A@54AB20 + :@0B:>A@. D8=. 2;>65=8O + 4518B. 704>;65==>ABL) / B5:CI85 ?0AA82K " (:>MDD8F85=B B5:CI59 ;8:284=>AB8) = B5:CI85 0:B82K/ B5:CI85 ?0AA82KB@0605B 4>AB0B>G=>ABL C ?@54?@8OB8O 45=56=KE 8 4@C38E A@54AB2, 4;O >?;0BK >1O70B5;LAB2 2 B5:CI89 <><5=B.2. =B53@0;L=K9 :>MDD8F85=B AB@C:BC@K :0?8B0;0 (IA:)IA: =  EMBED Equation.DSMT4 , 345:  (:>MDD8F85=B 02B>=><88) = A>1AB25==K9 :0?8B0; />1I0O AC<<0 2A5E A@54AB2 D8=0=A8@>20=8O $ (:>MDD8F85=B D8=0=A>2>3> ;525@8460) = 4>;3>A@>G=K5 >1O70B5;LAB20 + B5:CI85 >1O70B5;LAB20) / A>1AB25==K9 :0?8B0;  (:>MDD8F85=B 7028A8<>AB8 >B 2=5H=8E 709<>2) = (4>;3>A@>G=K5 >1O70B5;LAB20 + B5:CI85 >1O70B5;LAB20) / >1I0O AC<<0 2A5E A@54AB2 D8=0=A8@>20=8O%0@0:B5@87CNB AB5?5=L D8=0=A>2>9 =57028A8<>AB8 :><?0=88 >B :@548B>@>2.3. =B53@0;L=K9 :>MDD8F85=B @5=B015;L=>AB8 ?@54?@8OB8O (I@5=B) I@5=B =  EMBED Equation.DSMT4 , 345: ! (@5=B015;L=>ABL A>1AB25==>3> :0?8B0;0) = G8AB0O ?@81K;L/ A>1AB25==K9 :0?8B0; ! (@5=B015;L=>ABL >1>@>B=KE A@54AB2) = G8AB0O ?@81K;L / >1>@>B=K5 A@54AB20  (@5=B015;L=>ABL 8=25AB8F8>==>3> :0?8B0;0) = G8AB0O ?@81K;L/ (A>1AB25==K9 :0?8B0; + 4>;3>A@>G=K5 >1O70B5;LAB20)B@0605B ?@81K;L=>ABL 45OB5;L=>AB8 :><?0=88.  A;CG05, :>340 548=8G=K5 ?>:070B5;8 @5=B015;L=>AB8 8<5NB >B@8F0B5;L=>5 7=0G5=85  8=B53@0;L=K9 ?>:070B5;L B0: 65 1C45B 8<5BL >B@8F0B5;L=>5 7=0G5=85, GB> A2845B5;LAB2C5B >1 C1KB>G=>9 @01>B5 ?@54?@8OB8O.4. =B53@0;L=K9 ?>:070B5;L 45;>2>9 0:B82=>AB8 (I40) I40 =  EMBED Equation.DSMT4 , 345:  (:>MDD8F85=B >1>@0G8205<>AB8 4518B>@A:>9 704>;65==>AB8) = 2K@CG:0 >B @50;870F88/ A@. 4518B>@A:0O 704>;65==>ABL  (:>MDD8F85=B >1>@0G8205<>AB8 70?0A>2) = A515AB>8<>ABL ?@>4C:F88 / AB>8<>ABL 70?0A>2  (:>MDD8F85=B >1>@0G8205<>AB8 0:B82>2) = 2K@CG:0 >B @50;870F88 / A@. AB>8<>ABL 0:B82>2 >72>;O5B >F5=8BL :>=:C@5=B>A?>A>1=>ABL ?@54?@8OB8O 2 7028A8<>AB8 >B MDD5:B82=>AB8 8A?>;L7>20=8O A>1AB25==KE A@54AB2.II @C??0. >:070B5;8 >F5=:8 ?@>872>4AB25==>3> D0:B>@0 1. $>=4>>B40G0 ($>) $> = /!@, 345:  - >1J5< 2K?CA:0 ?@>4C:F88, BKA. @C1. !@ - A@54=53>4>20O AB>8<>ABL >A=>2=KE ?@>872>4AB25==KE D>=4>2, BKA. @C1.1>1I0NI89 ?>:070B5;L, E0@0:B5@87CNI89 C@>25=L MDD5:B82=>AB8 8A?>;L7>20=8O >A=>2=KE ?@>872>4AB25==KE D>=4>2.2. @>872>48B5;L=>ABL B@C40 (B) B =  /!!', 345:  - >1J5< 2K?CA:0 ?@>4C:F88, BKA.@C1. !!' - A@54=5A?8A>G=0O G8A;5==>ABL @01>B=8:>2, G5;.B@0605B MDD5:B82=>ABL 8A?>;L7>20=8O :04@>2 ?@54?@8OB8O.3. !515AB>8<>ABL 548=8FK ?@>4C:F88 (!54) !54 = ! / , 345: !  ?>;=0O A515AB>8<>ABL, BKA. @C1.  - >1J5< 2K?CA:0 ?@>4C:F88, "# %0@0:B5@87C5B B5E=8G5A:89 8 >@30=870F8>==K9 C@>25=L @01>BK ?@54?@8OB89, ?>:07K205B MDD5:B82=>ABL 8A?>;L7>20=8O 8745@65: ?@>872>4AB20 2 873>B>2;5=88 A>>B25BAB2CNI8E B>20@>2 8 CA;C34. >MDD8F85=B =5@02=><5@=>AB8 ?>ABC?;5=89 =0 A:;04 (=) == Qmax/QA@, 345: Qmax <0:A8<0;L=>5 ?>ABC?;5=85 3@C70 70 <5AOF, B>== QA@ - A@54=55 ?>ABC?;5=85 %0@0:B5@87C5B 8=B5=A82=>ABL @01>BK A:;040 8 ?>:07K205B MDD5:B82=>ABL @01>BK ;>38AB8G5A:>9 A8AB5<K ?@54?@8OB8O.III @C??0. >:070B5;8 >F5=:8 <0@:5B8=3>2>3> D0:B>@01.>MDD8F85=B MDD5:B82=>AB8 @5:;0<=>9 45OB5;L=>AB8 (@5:) @5: = (* )/  , 345:  (:>MDD8F85=B 87<5=5=8O >1J5<0 ?@>406) = >1J5< ?@>406 =0 :>=5F >BG5B=>3> ?5@8>40/ >1J5< ?@>406 =0 =0G0;> >BG5B=>3> ?5@8>40.    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F5=0, CAB0=>2;5==0O ?@54?@8OB85<, @C1>:07K205B @>AB 8;8 A=865=85 :>=:C@5=B>A?>A>1=>AB8 ?@54?@8OB8O 70 AG5B 48=0<8:8 F5= =0 ?@>4C:B.3. =B53@0;L=K9 ?>:070B5;L :>=:C@5=B>A?>A>1=>AB8 ?@54?@8OB8O ?> :0G5AB2C (I:=) I:= = ";8= * "@07, 345: ";8=- :>MDD8F85=B :>=:C@5=B>A?>A>1=>AB8 ?> ?0@0<5B@C ;8=59=0O ?;>B=>ABL ";8= = min / B, 345: B - 25;8G8=0 ?0@0<5B@0 ;8=59=0O ?;>B=>ABL 0=0;878@C5<>3> B>20@0 min- <8=8<0;L=>5 7=0G5=85 ?0@0<5B@0 ;8=59=0O ?;>B=>ABL A@548 2A59 0=0;878@C5<>9 3@C??K B>20@>2. "@07- :>MDD8F85=B :>=:C@5=B>A?>A>1=>AB8 ?> ?0@0<5B@C @07@K2=0O =03@C7:0 "@07 = B / max, 345: B- 25;8G8=0 ?0@0<5B@0 @07@K2=0O =03@C7:0 0=0;878@C5<>3> B>20@0 max - <0:A8<0;L=>5 7=0G5=85 ?0@0<5B@0 @07@K2=0O =03@C7:0 A@548 2A59 0=0;878@C5<>9 3@C??K B>20@>2. %0@0:B5@87C5B 7028A8<>ABL :>=:C@5=B>A?>A>1=>AB8 B>20@0 >B 53> :0G5AB25==KE ?0@0<5B@>2.! 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"0:65 1K;> ?@>2545=> @0=68@>20=8e >A=>2=KE ?>:070B5;59 >F5=:8 :064>3> D0:B>@0. ;O ?@>25@:8 AB5?5=8 4>AB>25@=>AB8 M:A?5@B=>9 >F5=:8 ?> :064>9 3@C??5 ?>:070B5;59 1K; @0AAG8B0= :>MDD8F85=B :>=:>@40F88 W, :>B>@K9 ?>:07K205B, =0A:>;L:> <=5=8O M:A?5@B>2 A>3;0A>20=K, B> 5ABL ?@8=04;560B : >4=>9 8 B>9 65 35=5@0;L=>9 A>2>:C?=>AB8 >F5=>:. >MDD8F85=B :>=:>@40F88 @0AAG8B0= ?> D>@<C;5:  EMBED Equation.DSMT4  345: 12 - ?>AB>O==0O 25;8G8=0 2 D>@<C;5 @0AG5B0 :>MDD8F85=B0 :>=:>@40F88, ?@54;>65==>9 5=40;;><; N  :>;8G5AB2> M:A?5@B>2; n  :>;8G5AB2> :@8B5@852 >F5=:8; ri- AC<<0 @0=3>2 i- 3> ?>:070B5;O; r- A@54=OO AC<<0 10;;>2 2A5E ?>:070B5;59. 5;8G8=0 :>MDD8F85=B0 :>=:>@40F88 <>65B <5=OBLAO 2 480?07>=5 >B 0 4> 1, 345 53> @025=AB2> 548=8F5 >7=0G05B ?>;=CN A>3;0A>20==>ABL <=5=89 M:A?5@B>2, 0 @025=AB2> =C;N A2845B5;LAB2C5B > B><, GB> A2O78 <564C >F5=:0<8 =5 ACI5AB2C5B.  A;CG05, :>340 0,2 d" W d" 0,4 8<55B <5AB> A;010O A>3;0A>20==>ABL <=5=89 M:A?5@B>2, 0 ?@8 W e" 0,6 <>6=> 3>2>@8BL > ACI5AB2>20=88 A8;L=>9 A>3;0A>20==>AB8 <=5=89 M:A?5@B>2. @><5 B>3> 4;O >?@545;5=8O :>MDD8F85=B>2 25A><>AB8 :064>3> D0:B>@0 <K 2>A?>;L7>20;8AL D>@<C;>9 $8H1>@=0: ai = 2 * (n-ri+1) / n*(n+1), (4;O 2A5E i >B 1 4> n) 345: ai  :>MDD8F85=B 25A><>AB8 i  3> :@8B5@8O, n - :>;8G5AB2> :@8B5@852 >F5=:8 ri @0=3, ?@8A2>5==K9 i-<C ?>:070B5;N 57C;LB0BK M:A?5@B=>9 >F5=:8 25A><>AB8 D0:B>@>2 :>=:C@5=B>A?>A>1=>AB8 8 ?@>25@:8 A>3;0A>20==>AB8 <=5=89 M:A?5@B>2 ?@54AB02;5=K 2 B01;.3. "01;8F0 3 - -:A?5@B=0O >F5=:0 25A><>AB8 D0:B>@>2 :>=:C@5=B>A?>A>1=>AB8 ?@54?@8OB8O @8B5@88 >F5=:8 M:A?5@B=0O >F5=:01234567891011121314151617181920$8=0=A>2K9 D0:B>@55445455555545555554@>872>4AB25==K9 D0:B>@423333233232223222330@:5B8=3>2K9 D0:B>@43424444333333434344>=:C@5=B>A?>A>1=>ABL B>20@054555555444454545455B>3>@>4>;65=85 B01;8FK 3      @8B5@88 >F5=:8!C<<0 10;;>2B:;>=5=85 >B A@54=53>204@0B >B:;>=5=8O 0=3>MDD8F85=B 25A><>AB8$8=0=A>2K9 D0:B>@951832410,40@>872>4AB25==K9 D0:B>@52-2562540,100@:5B8=3>2K9 D0:B>@69-86430,20>=:C@5=B>A?>A>1=>ABL B>20@0921522520,30B>3>30801238 EMBED Equation.DSMT4 = 0,62 !B5?5=L A>3;0A>20==>AB8 M:A?5@B=KE >F5=>: <>6=> AG8B0BL ?@85<;5<>9, B0: :0: W = 0,62 > 0,60 081>;55 7=0G8<K< A>3;0A=> <=5=8N M:A?5@B>2, O2;O5BAO D8=0=A>2K9 D0:B>@, 0 =08<5=55  ?@>872>4AB25==K9. @><5 B>3>, 2 48AA5@B0F8>==>9 @01>B5 CAB0=>2;5=K 2708<>A2O78 <564C ?>:070B5;O<8, E0@0:B5@87CNI8<8 :>=:C@5=B>A?>A>1=>ABL ?@54?@8OB8O. >AB865=85 :>=:C@5=B>A?>A>1=>AB8 ?@>872>48B5;5< 1C45B A>>B25BAB2>20BL 4>AB865=8N B0:>3> ?>;>65=8O, ?@8 :>B>@>< >= 8<55B >?@545;5==K5 2>7<>6=>AB8 @0728B8O, ?@>872>4AB25==K5 8 @K=>G=K5 ?@58<CI5AB20. 4=8< 87 8A?>;L7C5<KE ?>:070B5;59 045:20B=>AB8 C@02=5=8O O2;O5BAO :>MDD8F85=B 45B5@<8=0F88, =86=59 3@0=8F59 7=0G5=89 :>B>@>3> <>65B 1KBL ?@8=OB0 25;8G8=0 0,7. >;55 4>AB>25@=K9 @57C;LB0B ?@>3=>70 <>65B 1KBL 4>AB83=CB ?@8 1>;LH8E 7=0G5=8OE R (0,8-0,9). 0AG5B ?@>2>48;AO A ?><>ILN DC=:F89 MS Excel 87 :0B53>@88 !B0B8AB8G5A:85:  (@0AG5B ;8=59=>9 @53@5AA88). 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