2

-

. . . .

, . , , . , (.. ) .

() . ( .. . ) . . , .

, . , .

, : ) ) . , . . . . . 100 5 . , 5 . 5 . , . , . . ) ) ) .

. . .

. . , . . .

, . , . (.. ). . (opportunity cost) . , . () () . , .

, . . . , , , . .

100 , 100. . , . 100 () 5 , 5/100=0,05 5% .

-, . . .

P , i N . P i

I=PNi

: 1000 5% 3 . ,

I=1000*0,05*3 =150

.

:

1100.005.00105.00

2105.005.25110.25

3110.255.51115.76

15.75 15. . .

i =

N =

P =

. (present worth)

F =

.

(future worth)

A =

.

:

1. .

2. ( ) ( ).

F=?

i=5%

1 2 3 4 5 6 7

P=100

:

P = F= . i= ()

1 2 -1

P () i% P F=P+Pi = P(1+i). :

F= P(1+i) +P(1+i)i =P(1+i)2

F= P(1+i)-1 +P(1+i)-1i +.. =P(1+i)

(1+i) (F/P, i%, N) , :

F=P(F/P, i%, N) .. EXCEL (F/P, i%, N) i .

(F/P, i%,N) ( )

1%2%3%4%5%6%7%8%9%10%11,00%12,00%13,00%14,00%15,00%16,00%

11,01001,02001,03001,04001,05001,06001,07001,08001,09001,10001,11001,12001,13001,14001,15001,1600

21,02011,04041,06091,08161,10251,12361,14491,16641,18811,21001,23211,25441,27691,29961,32251,3456

31,03031,06121,09271,12491,15761,19101,22501,25971,29501,33101,36761,40491,44291,48151,52091,5609

41,04061,08241,12551,16991,21551,26251,31081,36051,41161,46411,51811,57351,63051,68901,74901,8106

51,05101,10411,15931,21671,27631,33821,40261,46931,53861,61051,68511,76231,84241,92542,01142,1003

61,06151,12621,19411,26531,34011,41851,50071,58691,67711,77161,87041,97382,08202,19502,31312,4364

71,07211,14871,22991,31591,40711,50361,60581,71381,82801,94872,07622,21072,35262,50232,66002,8262

81,08291,17171,26681,36861,47751,59381,71821,85091,99262,14362,30452,47602,65842,85263,05903,2784

91,09371,19511,30481,42331,55131,68951,83851,99902,17192,35792,55802,77313,00403,25193,51793,8030

101,10461,21901,34391,48021,62891,79081,96722,15892,36742,59372,83943,10583,39463,70724,04564,4114

111,11571,24341,38421,53951,71031,89832,10492,33162,58042,85313,15183,47853,83594,22624,65245,1173

121,12681,26821,42581,60101,79592,01222,25222,51822,81273,13843,49853,89604,33454,81795,35035,9360

131,13811,29361,46851,66511,88562,13292,40982,71963,06583,45233,88334,36354,89805,49246,15286,8858

141,14951,31951,51261,73171,97992,26092,57852,93723,34173,79754,31044,88715,53486,26137,07577,9875

151,16101,34591,55801,80092,07892,39662,75903,17223,64254,17724,78465,47366,25437,13798,13719,2655

161,17261,37281,60471,87302,18292,54042,95223,42593,97034,59505,31096,13047,06738,13729,357610,7480

171,18431,40021,65281,94792,29202,69283,15883,70004,32765,05455,89516,86607,98619,276510,761312,4677

181,19611,42821,70242,02582,40662,85433,37993,99604,71715,55996,54367,69009,024310,575212,375514,4625

191,20811,45681,75352,10682,52703,02563,61654,31575,14176,11597,26338,612810,197412,055714,231816,7765

201,22021,48591,80612,19112,65333,20713,86974,66105,60446,72758,06239,646311,523113,743516,366519,4608

211,23241,51571,86032,27882,78603,39964,14065,03386,10887,40028,949210,803813,021115,667618,821522,5745

221,24471,54601,91612,36992,92533,60354,43045,43656,65868,14039,933612,100314,713817,861021,644726,1864

231,25721,57691,97362,46473,07153,81974,74055,87157,25798,954311,026313,552316,626620,361624,891530,3762

241,26971,60842,03282,56333,22514,04895,07246,34127,91119,849712,239215,178618,788123,212228,625235,2364

251,28241,64062,09382,66583,38644,29195,42746,84858,623110,834713,585517,000121,230526,461932,919040,8742

261,29531,67342,15662,77253,55574,54945,80747,39649,399211,918215,079919,040123,990530,166637,856847,4141

271,30821,70692,22132,88343,73354,82236,21397,988110,245113,110016,738621,324927,109334,389943,535355,0004

281,32131,74102,28792,99873,92015,11176,64888,627111,167114,421018,579923,883930,633539,204550,065663,8004

291,33451,77582,35663,11874,11615,41847,11439,317312,172215,863120,623726,749934,615844,693157,575574,0085

301,34781,81142,42733,24344,32195,74357,612310,062713,267717,449422,892329,959939,115950,950266,211885,8499

F= P(1+i) P

(1+i)- (P/F, i%, N) , :

P=F(P/F, i%, N)

.. EXCEL (P/F, i%, N) i .

(P/F, i%,N) ( )1%2%3%4%5%6%7%8%9%10%11%12%13%14%15%16%

10,99010,98040,97090,96150,95240,94340,93460,92590,91740,90910,90090,89290,88500,87720,86960,8621

20,98030,96120,94260,92460,90700,89000,87340,85730,84170,82640,81160,79720,78310,76950,75610,7432

30,97060,94230,91510,88900,86380,83960,81630,79380,77220,75130,73120,71180,69310,67500,65750,6407

40,96100,92380,88850,85480,82270,79210,76290,73500,70840,68300,65870,63550,61330,59210,57180,5523

50,95150,90570,86260,82190,78350,74730,71300,68060,64990,62090,59350,56740,54280,51940,49720,4761

60,94200,88800,83750,79030,74620,70500,66630,63020,59630,56450,53460,50660,48030,45560,43230,4104

70,93270,87060,81310,75990,71070,66510,62270,58350,54700,51320,48170,45230,42510,39960,37590,3538

80,92350,85350,78940,73070,67680,62740,58200,54030,50190,46650,43390,40390,37620,35060,32690,3050

90,91430,83680,76640,70260,64460,59190,54390,50020,46040,42410,39090,36060,33290,30750,28430,2630

100,90530,82030,74410,67560,61390,55840,50830,46320,42240,38550,35220,32200,29460,26970,24720,2267

110,89630,80430,72240,64960,58470,52680,47510,42890,38750,35050,31730,28750,26070,23660,21490,1954

120,88740,78850,70140,62460,55680,49700,44400,39710,35550,31860,28580,25670,23070,20760,18690,1685

130,87870,77300,68100,60060,53030,46880,41500,36770,32620,28970,25750,22920,20420,18210,16250,1452

140,87000,75790,66110,57750,50510,44230,38780,34050,29920,26330,23200,20460,18070,15970,14130,1252

150,86130,74300,64190,55530,48100,41730,36240,31520,27450,23940,20900,18270,15990,14010,12290,1079

160,85280,72840,62320,53390,45810,39360,33870,29190,25190,21760,18830,16310,14150,12290,10690,0930

170,84440,71420,60500,51340,43630,37140,31660,27030,23110,19780,16960,14560,12520,10780,09290,0802

180,83600,70020,58740,49360,41550,35030,29590,25020,21200,17990,15280,13000,11080,09460,08080,0691

190,82770,68640,57030,47460,39570,33050,27650,23170,19450,16350,13770,11610,09810,08290,07030,0596

200,81950,67300,55370,45640,37690,31180,25840,21450,17840,14860,12400,10370,08680,07280,06110,0514

210,81140,65980,53750,43880,35890,29420,24150,19870,16370,13510,11170,09260,07680,06380,05310,0443

220,80340,64680,52190,42200,34180,27750,22570,18390,15020,12280,10070,08260,06800,05600,04620,0382

230,79540,63420,50670,40570,32560,26180,21090,17030,13780,11170,09070,07380,06010,04910,04020,0329

240,78760,62170,49190,39010,31010,24700,19710,15770,12640,10150,08170,06590,05320,04310,03490,0284

250,77980,60950,47760,37510,29530,23300,18420,14600,11600,09230,07360,05880,04710,03780,03040,0245

260,77200,59760,46370,36070,28120,21980,17220,13520,10640,08390,06630,05250,04170,03310,02640,0211

270,76440,58590,45020,34680,26780,20740,16090,12520,09760,07630,05970,04690,03690,02910,02300,0182

280,75680,57440,43710,33350,25510,19560,15040,11590,08950,06930,05380,04190,03260,02550,02000,0157

290,74930,56310,42430,32070,24290,18460,14060,10730,08220,06300,04850,03740,02890,02240,01740,0135

300,74190,55210,41200,30830,23140,17410,13140,09940,07540,05730,04370,03340,02560,01960,01510,0116

1 2 -1

P F

P=

F=

i=

=

P ( .

F A, i

i%. F .

:

(1+i)-1 , :

(1+i) :

(F/, i%, N) , :

F=(F/, i%, N)

.. EXCEL (F/, i%, N) i .

(F/A, i%,N) ( )

1%2%3%4%5%6%7%8%9%10%

11,00001,00001,00001,00001,00001,00001,00001,00001,00001,0000

22,01002,02002,03002,04002,05002,06002,07002,08002,09002,1000

33,03013,06043,09093,12163,15253,18363,21493,24643,27813,3100

44,06044,12164,18364,24654,31014,37464,43994,50614,57314,6410

55,10105,20405,30915,41635,52565,63715,75075,86665,98476,1051

66,15206,30816,46846,63306,80196,97537,15337,33597,52337,7156

77,21357,43437,66257,89838,14208,39388,65408,92289,20049,4872

88,28578,58308,89239,21429,54919,897510,259810,636611,028511,4359

99,36859,754610,159110,582811,026611,491311,978012,487613,021013,5795

1010,462210,949711,463912,006112,577913,180813,816414,486615,192915,9374

1111,566812,168712,807813,486414,206814,971615,783616,645517,560318,5312

1212,682513,412114,192015,025815,917116,869917,888518,977120,140721,3843

1313,809314,680315,617816,626817,713018,882120,140621,495322,953424,5227

1414,947415,973917,086318,291919,598621,015122,550524,214926,019227,9750

1516,096917,293418,598920,023621,578623,276025,129027,152129,360931,7725

1617,257918,639320,156921,824523,657525,672527,888130,324333,003435,9497

1718,430420,012121,761623,697525,840428,212930,840233,750236,973740,5447

1819,614721,412323,414425,645428,132430,905733,999037,450241,301345,5992

1920,810922,840625,116927,671230,539033,760037,379041,446346,018551,1591

2022,019024,297426,870429,778133,066036,785640,995545,762051,160157,2750

2123,239225,783328,676531,969235,719339,992744,865250,422956,764564,0025

2224,471627,299030,536834,248038,505243,392349,005755,456862,873371,4027

2325,716328,845032,452936,617941,430546,995853,436160,893369,531979,5430

2426,973530,421934,426539,082644,502050,815658,176766,764876,789888,4973

2528,243232,030336,459341,645947,727154,864563,249073,105984,700998,3471

2629,525633,670938,553044,311751,113559,156468,676579,954493,3240109,1818

2730,820935,344340,709647,084254,669163,705874,483887,3508102,7231121,0999

2832,129137,051242,930949,967658,402668,528180,697795,3388112,9682134,2099

2933,450438,792245,218952,966362,322773,639887,3465103,9659124,1354148,6309

3034,784940,568147,575456,084966,438879,058294,4608113,2832136,3075164,4940

(F/A, i%,N) ( )

12%15%18%20%25%

11,00001,00001,00001,00001,0000

22,12002,15002,18002,20002,2500

33,37443,47253,57243,64003,8125

44,77934,99345,21545,36805,7656

56,35286,74247,15427,44168,2070

68,11528,75379,44209,929911,2588

710,089011,066812,141512,915915,0735

812,299713,726815,327016,499119,8419

914,775716,785819,085920,798925,8023

1017,548720,303723,521325,958733,2529

1120,654624,349328,755132,150442,5661

1224,133129,001734,931139,580554,2077

1328,029134,351942,218748,496668,7596

1432,392640,504750,818059,195986,9495

1537,279747,580460,965372,0351109,6868

1642,753355,717572,939087,4421138,1085

1748,883765,075187,0680105,9306173,6357

1855,749775,8364103,7403128,1167218,0446

1963,439788,2118123,4135154,7400273,5558

2072,0524102,4436146,6280186,6880342,9447

2181,6987118,8101174,0210225,0256429,6809

2292,5026137,6316206,3448271,0307538,1011

23104,6029159,2764244,4868326,2369673,6264

24118,1552184,1678289,4945392,4842843,0329

25133,3339212,7930342,6035471,98111054,7912

26150,3339245,7120405,2721567,37731319,4890

27169,3740283,5688479,2211681,85281650,3612

28190,6989327,1041566,4809819,22332063,9515

29214,5828377,1697669,4475984,06802580,9394

30241,3327434,7451790,94801181,88163227,1743

P A F= P(1+i) F

(P/, i%, N) , :

P=(P/, i%, N) .. EXCEL (P/, i%, N) i .

(P/A, i%,N) ( )

1%2%3%4%5%6%7%8%9%

10,99010,98040,97090,96150,95240,94340,93460,92590,9174

21,97041,94161,91351,88611,85941,83341,80801,78331,7591

32,94102,88392,82862,77512,72322,67302,62432,57712,5313

43,90203,80773,71713,62993,54603,46513,38723,31213,2397

54,85344,71354,57974,45184,32954,21244,10023,99273,8897

65,79555,60145,41725,24215,07574,91734,76654,62294,4859

76,72826,47206,23036,00215,78645,58245,38935,20645,0330

87,65177,32557,01976,73276,46326,20985,97135,74665,5348

98,56608,16227,78617,43537,10786,80176,51526,24695,9952

109,47138,98268,53028,11097,72177,36017,02366,71016,4177

1110,36769,78689,25268,76058,30647,88697,49877,13906,8052

1211,255110,57539,95409,38518,86338,38387,94277,53617,1607

1312,133711,348410,63509,98569,39368,85278,35777,90387,4869

1413,003712,106211,296110,56319,89869,29508,74558,24427,7862

1513,865112,849311,937911,118410,37979,71229,10798,55958,0607

1614,717913,577712,561111,652310,837810,10599,44668,85148,3126

1715,562314,291913,166112,165711,274110,47739,76329,12168,5436

1816,398314,992013,753512,659311,689610,827610,05919,37198,7556

1917,226015,678514,323813,133912,085311,158110,33569,60368,9501

2018,045616,351414,877513,590312,462211,469910,59409,81819,1285

2118,857017,011215,415014,029212,821211,764110,835510,01689,2922

2219,660417,658015,936914,451113,163012,041611,061210,20079,4424

2320,455818,292216,443614,856813,488612,303411,272210,37119,5802

2421,243418,913916,935515,247013,798612,550411,469310,52889,7066

2522,023219,523517,413115,622114,093912,783411,653610,67489,8226

2622,795220,121017,876815,982814,375213,003211,825810,81009,9290

2723,559620,706918,327016,329614,643013,210511,986710,935210,0266

2824,316421,281318,764116,663114,898113,406212,137111,051110,1161

2925,065821,844419,188516,983715,141113,590712,277711,158410,1983

3025,807722,396519,600417,292015,372513,764812,409011,257810,2737

(P/A, i%,N) ( )

10%12%15%18%20%25%

10,90910,89290,86960,84750,83330,8000

21,73551,69011,62571,56561,52781,4400

32,48692,40182,28322,17432,10651,9520

43,16993,03732,85502,69012,58872,3616

53,79083,60483,35223,12722,99062,6893

64,35534,11143,78453,49763,32552,9514

74,86844,56384,16043,81153,60463,1611

85,33494,96764,48734,07763,83723,3289

95,75905,32824,77164,30304,03103,4631

106,14465,65025,01884,49414,19253,5705

116,49515,93775,23374,65604,32713,6564

126,81376,19445,42064,79324,43923,7251

137,10346,42355,58314,90954,53273,7801

147,36676,62825,72455,00814,61063,8241

157,60616,81095,84745,09164,67553,8593

167,82376,97405,95425,16244,72963,8874

178,02167,11966,04725,22234,77463,9099

188,20147,24976,12805,27324,81223,9279

198,36497,36586,19825,31624,84353,9424

208,51367,46946,25935,35274,86963,9539

218,64877,56206,31255,38374,89133,9631

228,77157,64466,35875,40994,90943,9705

238,88327,71846,39885,43214,92453,9764

248,98477,78436,43385,45094,93713,9811

259,07707,84316,46415,46694,94763,9849

269,16097,89576,49065,48044,95633,9879

279,23727,94266,51355,49194,96363,9903

289,30667,98446,53355,50164,96973,9923

299,36968,02186,55095,50984,97473,9938

309,42698,05526,56605,51684,97893,9950

F

(A/F, i%, N) , , :

=F(A/F, i%, N) .. EXCEL (A/F, i%, N) i .

(A/F, i%,N) ( )

1%2%3%4%5%6%7%8%9%10%12%15%

11,00001,00001,00001,00001,00001,00001,00001,00001,00001,00001,00001,0000

20,49750,49500,49260,49020,48780,48540,48310,48080,47850,47620,47170,4651

30,33000,32680,32350,32030,31720,31410,31110,30800,30510,30210,29630,2880

40,24630,24260,23900,23550,23200,22860,22520,22190,21870,21550,20920,2003

50,19600,19220,18840,18460,18100,17740,17390,17050,16710,16380,15740,1483

60,16250,15850,15460,15080,14700,14340,13980,13630,13290,12960,12320,1142

70,13860,13450,13050,12660,12280,11910,11560,11210,10870,10540,09910,0904

80,12070,11650,11250,10850,10470,10100,09750,09400,09070,08740,08130,0729

90,10670,10250,09840,09450,09070,08700,08350,08010,07680,07360,06770,0596

100,09560,09130,08720,08330,07950,07590,07240,06900,06580,06270,05700,0493

110,08650,08220,07810,07410,07040,06680,06340,06010,05690,05400,04840,0411

120,07880,07460,07050,06660,06280,05930,05590,05270,04970,04680,04140,0345

130,07240,06810,06400,06010,05650,05300,04970,04650,04360,04080,03570,0291

140,06690,06260,05850,05470,05100,04760,04430,04130,03840,03570,03090,0247

150,06210,05780,05380,04990,04630,04300,03980,03680,03410,03150,02680,0210

160,05790,05370,04960,04580,04230,03900,03590,03300,03030,02780,02340,0179

170,05430,05000,04600,04220,03870,03540,03240,02960,02700,02470,02050,0154

180,05100,04670,04270,03900,03550,03240,02940,02670,02420,02190,01790,0132

190,04810,04380,03980,03610,03270,02960,02680,02410,02170,01950,01580,0113

200,04540,04120,03720,03360,03020,02720,02440,02190,01950,01750,01390,0098

210,04300,03880,03490,03130,02800,02500,02230,01980,01760,01560,01220,0084

220,04090,03660,03270,02920,02600,02300,02040,01800,01590,01400,01080,0073

230,03890,03470,03080,02730,02410,02130,01870,01640,01440,01260,00960,0063

240,03710,03290,02900,02560,02250,01970,01720,01500,01300,01130,00850,0054

250,03540,03120,02740,02400,02100,01820,01580,01370,01180,01020,00750,0047

260,03390,02970,02590,02260,01960,01690,01460,01250,01070,00920,00670,0041

270,03240,02830,02460,02120,01830,01570,01340,01140,00970,00830,00590,0035

280,03110,02700,02330,02000,01710,01460,01240,01050,00890,00750,00520,0031

290,02990,02580,02210,01890,01600,01360,01140,00960,00810,00670,00470,0027

300,02870,02460,02100,01780,01510,01260,01060,00880,00730,00610,00410,0023

P

(A/P, i%, N) , , :

=P(A/P, i%, N) .. EXCEL (A/P, i%, N) i . (A/P, i%,N) ( )

1%2%3%4%5%6%7%8%9%10%12%15%18%20%25%

11,01001,02001,03001,04001,05001,06001,07001,08001,09001,10001,12001,15001,18001,20001,2500

20,50750,51500,52260,53020,53780,54540,55310,56080,56850,57620,59170,61510,63870,65450,6944

30,34000,34680,35350,36030,36720,37410,38110,38800,39510,40210,41630,43800,45990,47470,5123

40,25630,26260,26900,27550,28200,28860,29520,30190,30870,31550,32920,35030,37170,38630,4234

50,20600,21220,21840,22460,23100,23740,24390,25050,25710,26380,27740,29830,31980,33440,3718

60,17250,17850,18460,19080,19700,20340,20980,21630,22290,22960,24320,26420,28590,30070,3388

70,14860,15450,16050,16660,17280,17910,18560,19210,19870,20540,21910,24040,26240,27740,3163

80,13070,13650,14250,14850,15470,16100,16750,17400,18070,18740,20130,22290,24520,26060,3004

90,11670,12250,12840,13450,14070,14700,15350,16010,16680,17360,18770,20960,23240,24810,2888

100,10560,11130,11720,12330,12950,13590,14240,14900,15580,16270,17700,19930,22250,23850,2801

110,09650,10220,10810,11410,12040,12680,13340,14010,14690,15400,16840,19110,21480,23110,2735

120,08880,09460,10050,10660,11280,11930,12590,13270,13970,14680,16140,18450,20860,22530,2684

130,08240,08810,09400,10010,10650,11300,11970,12650,13360,14080,15570,17910,20370,22060,2645

140,07690,08260,08850,09470,10100,10760,11430,12130,12840,13570,15090,17470,19970,21690,2615

150,07210,07780,08380,08990,09630,10300,10980,11680,12410,13150,14680,17100,19640,21390,2591

160,06790,07370,07960,08580,09230,09900,10590,11300,12030,12780,14340,16790,19370,21140,2572

170,06430,07000,07600,08220,08870,09540,10240,10960,11700,12470,14050,16540,19150,20940,2558

180,06100,06670,07270,07900,08550,09240,09940,10670,11420,12190,13790,16320,18960,20780,2546

190,05810,06380,06980,07610,08270,08960,09680,10410,11170,11950,13580,16130,18810,20650,2537

200,05540,06120,06720,07360,08020,08720,09440,10190,10950,11750,13390,15980,18680,20540,2529

210,05300,05880,06490,07130,07800,08500,09230,09980,10760,11560,13220,15840,18570,20440,2523

220,05090,05660,06270,06920,07600,08300,09040,09800,10590,11400,13080,15730,18480,20370,2519

230,04890,05470,06080,06730,07410,08130,08870,09640,10440,11260,12960,15630,18410,20310,2515

240,04710,05290,05900,06560,07250,07970,08720,09500,10300,11130,12850,15540,18350,20250,2512

250,04540,05120,05740,06400,07100,07820,08580,09370,10180,11020,12750,15470,18290,20210,2509

260,04390,04970,05590,06260,06960,07690,08460,09250,10070,10920,12670,15410,18250,20180,2508

270,04240,04830,05460,06120,06830,07570,08340,09140,09970,10830,12590,15350,18210,20150,2506

280,04110,04700,05330,06000,06710,07460,08240,09050,09890,10750,12520,15310,18180,20120,2505

290,03990,04580,05210,05890,06600,07360,08140,08960,09810,10670,12470,15270,18150,20100,2504

300,03870,04460,05100,05780,06510,07260,08060,08880,09730,10610,12410,15230,18130,20080,2503

(deferred annuities)

J . .

:

0 1 J-1 J 1 2 3 -1

F

i=%

=

, , 5% 5000 18, 19, 20 21 .

17 .. 17 .

=5000(3.5460)=17730

4 5000 18, 19, 20 21 17 17730. .

7.735.

. ( ) . (P) (F)

=100

-1 0 1 2 3 4 5

P-1 P0 F 5

i=10%

=

5 100. ( 0) . 10% .

.

P0 P-1 , P0 F0 ( P-1)

P-1=100(3.7908)=379.08

P0=F0=379.08(1.100)=416.99

F5 P0.

F5 P0 F4 (P4) () .

F4=100(6.1051)=610.51

F5=610.51(1.10)=671.56

(nominal) (effective)

. . . . 100 6% 6 . 103. 6 ( ) 106.9. 100 106.9 6.09%. .

r, i M :

i = (F/P, r/M, M)-1

10 1000 . 10 12% 3 .

12/3=3%. 1000 4 3%

A=A(F/A, 3%, 4) = 1000(0.2390)=239.

. 10 1000 40 239 . 10

F=A(F/A, 3%, 40)=239(74.40)=18.021

S , i ,

A=S(A/F,i%,k)

i , S ,

A=S(A/P, i%, k)

.

(perpetuity) . P, i% , ,

A=Pxi

A=Pxi

P

P = A/i P .

(.. ).

( ) S i% , S

S=I(F/A, i%, k) = Xi(F/A, i%, k)

: i% S ; ( S )

(F/P, i%, k)=S+X .

. .

6 :

() . . , . , , , . . . .

. r , ,

[1+(r/M)]M. M/r=k

(1+1/k)k k e, . er , r% er

(F/P, r%, N)= er er (1+i)N er (1+i). i :

i = er -1

(P/F), (F/A) (P/A) i er 1. :

EXCEL .

1

500 1 2000 1 2002 6% , ;

I=P*i*N=500*0,06*2,5=75

2 5% 100000 15 ;

P=F(P/F, 5%, 15)=100000*0,4810=48100

EXCEL. EXCEL . PV (Present Worth- ) (FV future value)

48.101,7 . ( ). 3 15 100000. 5% ;

P=F(P/F, 5%, 15)=100000*0,4810=48.100

excel. PV FV.

4 1 1 2000 8% , 1 2100;

F=1(F/P, 5%,100)=1*131,501=131,501

EXCEL.

5 6% ;

F=P(F/P, 6%, n)=2P (F/P, 6%, N)=2

(F/P, 6%, 12)=2,0122

12 .

excel NPER. ( PV -1 FV 2). 11, 89 12 . 6 5000 10 4% .

F=P(F/P, 2%, 20)=5000*1,486=7,430

7 15000 . 4% . 5% .

()

=F(A/F, 4%, 6)= 15000*0,1508=2,262

F=P(F/P, 5%, 6)=15000*1,3401=20.101,50

A=20101,50(A/F, 4%,6)=20101,50*0,1508=3.031,31

8 500. . 6% , .

500(F/A, 6%,10)(F/P, 6%,21)+500(F/A,6%,15)=

500(13.181)(3.400)+500(23.276)=34.00.70 9 1000 8% 2000. 5 ;

1000 2000.

: 1000(F/P, 4%,4)+2000(F/P,4%,2) =1000*1,1699+2000*1,0816 = 1169,90+2163,20=3.333,10

2 . 5 .

18 =3333,10*(P/F,4%,1)=3333,10*0,9615=3.204,78A=P(A/P, 4%,5)=3.204,78*0.2246=719,79

10 500 6% 6 . 20 ;

6 500.

(A/F, 3%, 2) A= 500(a/f, 3%, 2)=500*0,49261=246,31

F=246,31(F/A, 3%, 40)=246,31*75,401=18,572.02

11

12% , 10000 ) ) .

A=P(A/P, 1%, 24)=1000*0,0471=471

F=P(F/P, 1%,24)=1000*1,2697=1.269,70

12

6% ;

(1+r/M)M 1 = (1+0,005)12 -1 =1,0616-1=0,0616 6,17%

13

10000 15 . 4%. 10 . ;

=P(A/P, 4%,15)=10000*0,0899=899

P=A(P/A,4%,5)=899*4,4518=4.022,17

14

:

2 12000

3 6000

5 20000

8 13000

8 8%;

12000*(P/F, 8%,2)+6000(P/F,8%,3)+2000(P/F,8%,5)+ 13000(P/F,8%, 8)=

12000*0,8573 + 6000*0,7938 + 2000*0,6806 + 13000*0,5403=35.686,30

A=P(A/P, 8%,8)=35686,30*0,1740=6209.42

15

5 25000. 5000 20 6%. 5 . . 3% . , 3%, 6%;

A=P(A/P,6%,20)=20000*0,0872=1744

5 =

P=A*(P/A, 6%,15)=1744*9,7122 = 16938,08

16938,08*0,03 = 508,14

F=P(F/P, 6%,5)=5000(1,3382)= 6691,00

F=A(F/A, 6%,5)=1744*5,6351= 9831.10 33.968,22

16

5000000 . 1200000 100000 . 6% ;

5

P= 100000/i (A/F,6%,5)=(100000/0,06)*0,1774=295.666,60 =100000(A/F,6%,5)=100.000*0,1774=17740 P=17740/0,06=295.666,60

=5000000-1200000-295666,60=3.504.333,40 3.504.333,40*0,06=210.240

17 1000 8% 10 . 1000 8% . ;

=P(A/P, 8%, 10=1000*0,149=149

4 . 4 6 .

P=A(P/A, 8%, 4)=149*3,3121=493,5029

1000 1493,5029. ( P ).

F=P(F/P,8%,1)=1493,5029*1,08=1612,9 10

A=P(A/P,8%,10)=1612,9*0,1490=240,13 18

60 . 3000 8% . , (59) . 5 , 5 ;

F= A(F/A, 20, 8%)=3000*45,7620=137286

59 . . .

F=A(F/A,8%,2 )*(F/P, 8%, 8)=3000*2,0800*1,8509=11.549,62 137286-11549,62=125.689.38

5

=P(A/P,5,8%)=125689,38*0,2505=31496,96

19

10000 8% . 4, 6, 8 2000.

) 4 10 , . (2 )

) 3 10 , . (2 )

9

10000(F/P, 8%,9) + 2000(F/P,8%, 6)+ 2000(F/P,8%, 4)+ 2000(F/P,8%, 2)=10000*1,1999+2000*1,5869+2000*1,3605+2000*1,1664=28217,57a) A=28217,57*(A/P,8%, 4)=28217,57*0,3019=8518,90b) 6

14 . 14 .

F=10000(F/P, 8%,14) + 2000(F/P,8%, 11)+ 2000(F/P,8%, 9)+ 2000(F/P,8%, 7)=

10000*2,9372+2000*2,3316+2000*1,999+2000*1,7138=41.460,80=41460,80*(/F,8%,6)=41460,80*0,1363=5651,107 5651,107*(F/A,8%,2)=5651,107*2,0800=11754,303: 8 P 6 .F=10000(F/P, 8%,8) + 2000(F/P,8%, 5)+ 2000(F/P,8%, 3)+ 2000(F/P,8%, 1)=

10000*1,8509+2000*1,4693+2000*1,2597+ 2000*1,08 =26127=41460,80*(/P,8%,6)=26127*0,2163=5651,107 5651,107*(F/A,8%,2)=5651,107*2,0800=11754,303

EXCEL 9 4

110000-19.990,05

42000-3.173,75

62000-2.720,98

82000-2.332,80

-28.217,57 -8.519,47

20 excel (P/F, i%, n), (F/P, i%, n), (P/A, i%, n), (A/P, i%, n), (A/F, i%, n), (F/A, i%, n) 0,25% 0,25% 20% n=1 1 250.

EMBED Equation.3

PAGE 33

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