利息基本理論和利息度量

1、1The Theory of Interest 利息理論利息理論5789項(xiàng)目（存款）項(xiàng)目（存款）年利率（年利率（%）城鄉(xiāng)居民及單位存款（一）活期0.35（二）定期1.整存整取三個(gè)月2.856個(gè)月3.05一年3.25二年3.75三年4.25五年4.75期限期限利率（利率（%）（）（20140219）O/N2.40301W3.76202W4.67801M5.29423M5.60006M4.99989M5.00001Y5.000112利率=利息本金（2）利率的表現(xiàn)形式:例、甲乙雙方商定貸款事宜，因業(yè)務(wù)發(fā)展需要，甲方向乙方借款1000萬元，期限2年,到期后，甲方一次性歸還乙方1150萬元。則甲方貸款的

2、年利率是：1150-100010002=7.5%風(fēng)險(xiǎn)太大,可能無法歸還本息16Aicntt11PiAiInntt11117債券價(jià)格債券價(jià)格與收益率與收益率反相關(guān)反相關(guān)PA,折價(jià)發(fā)行折價(jià)發(fā)行P=A,平價(jià)發(fā)行平價(jià)發(fā)行PA,溢價(jià)發(fā)行溢價(jià)發(fā)行1819PPAi20價(jià)格與收益價(jià)格與收益率反向變化率反向變化23242526利率i=6.56%pi 債券需求（對(duì)可貸資金的供給） 債券供給（對(duì)可貸資金的需求）債券數(shù)量/可貸資金量282930323334 nniiiii 1111132135ngggggn 3213637通貨膨脹率通貨膨脹率名義利率實(shí)際利率通貨膨脹率名義利率實(shí)際利率11114041424344454

3、6)(ta1) 0(a)()(,)(2121tatattta有是增函數(shù)，即47)()()0()()()0(),0(taKtaAtAtAtAA時(shí)刻的累積值為為常數(shù)，則在內(nèi)的利息額為時(shí)期tAtAtI)0()()()(tA)(tA)(tA)(tA)1()() 1()(nanaKnAnAIn49)(300) 1508. 0(100)5(108. 0)(08. 0172)9(100172)3(100)3(110)0()(222元AttaabaaAbbaabattabatta2)(50iatattttatttataAtAAA12312131) 1 (11302031)0(03231)(32)(3)()(3

4、33020022222時(shí)，當(dāng)時(shí)，當(dāng)）（為增函數(shù)時(shí)當(dāng))()()(0)2)()(2)()(2)()(,2121212121212122212121tatatatttttttttttttttatatt51) 1() 1() 1()(1naKIAInAnAnAinnnn實(shí)際利率)0( ,1)(,tittaitt為利率為任意值為度量時(shí)期設(shè)niPIniininiinnananainisatastasitiistnn、的實(shí)際利率時(shí)刻開始的一個(gè)時(shí)期內(nèi)為時(shí)點(diǎn)、設(shè)、4) 1(1) 1(1) 1(11) 1() 1()(131)()()(2)(152530425. 020. 105. 0405. 0105. 045

5、 . 0 . 01405. 01505. 01)4()4()5(05. 01)()05. 01 (100)()(5aaaittattaAtA如果A（t）=100+5t,試計(jì)算i5540,)1 ()(titatititititititiiiinananaistsatastatttnnnn11,11111113111) 1() 1()(20, 0),()()(111時(shí)當(dāng)時(shí)，當(dāng)時(shí)，、當(dāng)、551 . 01 . 0,) 1 . 01 ()() 1 . 01 (100)()(.,) 1 . 1 (100)(105105iiiitataAtAiitAttt確定如果5657 3 , 2 , 1,) 1%(51

6、%5) 1(1nnniiinn123456in5%4.76%4.55%4.35%4.17%4%58n123456單利1.051.101.151.21.251.3復(fù)利1.051.10251.15761.21551.27631.3401復(fù)利超過單利的%00.2270.6631.292.13.1591)0()()(1)0(t1AtAitAiAt60 sststtiii111%03. 81%51%611%51%61233 , 23 , 223ii6263itta11)(111111,11)(iivvitattiiiiinanananAnAnAnAnIdnnnn1111)() 1()()() 1()()

7、()(16465niiniinnianadn11)1(11a(n)1)-(n)(151%1051%1015niid66為貼現(xiàn)因子iviviid11,1dvviiid111111ttdvta1)(1ttdvta1)(這兩數(shù)之積兩數(shù)之差 d1idiididiivdd) 1n(1dd) 1n(1nd11nd)(1d) 1n(1)nd1 (a(n)1)-a(na(n)d1 -11n67dtdtta10,1)(1對(duì)1t,dt1d-131t,dt1d-121t0,dt1d-111,d0ttt如果）（如果）（如果）（有：設(shè)680,1)(1tdvtatt對(duì)280012. 0336p12. 0i ,12. 13

8、00336i1697011111)()()(mmmmmimimiiimi為：則年實(shí)際利率次利息的年名義利率，表示一年結(jié)轉(zhuǎn)設(shè)m171nnnnnnvndndndddnd11)()()(111111年名義貼現(xiàn)率為：為：則年實(shí)際貼現(xiàn)率時(shí)期收取一次貼現(xiàn)值，表示年名義貼現(xiàn)率，每設(shè)n172diidnnmmndmi)()(1111)(nnndimmmid)(11時(shí)nmmdmimm1)()(11mdmimdmimmmm)()()()(732054)02. 1 (500)408. 01 (500)20(A12121)97. 0(1000206. 011000)12(A%36. 8i0836. 1)12%81 (1

9、2d1i11212)12(74（元）（）270180.9550000%51 (50000)2d1 (50000)d1 (50000)6(a)6(APV)6(aPV)6(A121262)2(617576%41004d%74.17104167. 01%167. 496496961004年度實(shí)際利率i77)()()()(tatatAtAtdstseta0)(, 2t0 ,01t. 0t元）（元）(2 .15)ee (1000) 1 (A1000e) 1 (A)2(AI10051000e1000e1000e) 1 (1000a) 1 (A201. 001. 0201tdt. 02201. 00.01t

10、dtdt201010t 78 7911)1()()1()1()()()(210s nnttenananananaieeenaneetands個(gè)時(shí)期內(nèi)，有：，在id%7 . 708.ln1) i1 (ln（元）04.808500e500e)8(A48. 088081itiitittatat111)()(iiiiiitatattttt1ln11ln111)()(82tttata)()(11t83tiAtA1)0()(8401234n-1nx1x2y1y2y385tiAtA1)0()(861)0()(1)0()(1ttAtAiiAtA%91. 7019776. 04019776. 011000160

11、041600411000424124iiiii：，根據(jù)復(fù)利累計(jì)函數(shù)有為為年利率，則季度利率設(shè)87)()()()()()()()(0)(, 0)(,0)(1)0()(1211210121012102121ifififiiiiififififiiiiififiitAiAift)(if)(if8850001200011000,21420)2(jjij0500012000)1 (1000)(1420jjjf89032178. 0052815. 0)032177. 0(,025919. 0)032178. 0(60420. 0)03217. 0(,18346. 0)03218. 0(759. 1)032

12、2. 0(,114. 6)0321. 0(%42. 60642. 00321. 020321. 071.16817.22771.16800050. 00300. 017.227)0350. 0(71.1680300. 0)2(jffffffijff9076.186%)41 (600%)41 (%)41 (200100%42%8%,8162010)2(xxii91)1ln()0(ln)(ln)1ln()0(ln)(ln1)0()(iAtAtitAtAiAtAt年859. 6,717.1321500)2%61 (10002nnn922)1 (ti)1ln(2lnitiiiit72. 0)1ln(2

13、ln1P31，表1.2.193nsss 21nnnntntttntnttsssstststtvsssvsvsvstvsssvsvsvsnn 21221121212121lnln2121ntttnt 21194年6500200100850032002100t年832. 575236. 0500200100800832tvvvvvtt95969798%510095100d%988. 425. 5125. 51iid%26. 5%95%51ddi99101102104v2v1nvinnavivvvvvannnni 112110521 i21niiisnn1110680.92004016.185005

14、0040220%5 . 42%9%,9045. 0,40)2(aAanjinj解：10756.363555.2710001000%24%8%,8i02. 0,40)4(RRaj解：108v2v1nvdvvvvannni 1112 10910921 i21nini)1( disnn11 110iivaannnin11limlimddvaannnn11limlim 11121.591000100007. 0,1107. 0,11SRSR 11236.1367100062367.336.2367)09. 01 (1000)10() 110利息A2 .558100082.1551082.1554167

15、58. 61000,100009. 0,10利息總量RRa11363.8063972.1210000s10000D,10000sD10000sD005. 0 ,12005. 0 ,12ni 114nniniias1nniniias1 isanini11dsanini 11115)1 ( iaann )1 (issnn 11nnaa 11nnss 116假定：計(jì)算的日期離開每次付款日期為整數(shù)個(gè)時(shí)期假定：計(jì)算的日期離開每次付款日期為整數(shù)個(gè)時(shí)期117mivm118mnmnmnmnassvia )1(mnmas 119mnmmnssis12）期初付年金）期初付年金mnmmnssis )1 (120元4

16、91650236. 77000700007. 0,10a元24993)0236. 75940.10(700070001020aa元25842)5940.1007. 01(7000)(700020aa07.21s100)05. 1 (1000 x)05. 1 (1000 xs1001415215214 方案二（扣減付款）：在第15年末的求值方程是：支付在時(shí)刻 14.20672067.0k2067.1405.ln1ln2k142)05.1 (10iv11k0 ,10a1000100ak14k14k14k1427.2005. 01)05. 1 (100 x2067. 03125%79.12i1279

17、. 1i1) i1 () i1 (5001200) i1 () i1 (s222 126127128)1()(2!2)1(! 3)2)(1(!2)1(!2)0()0()0(22 nnknikinnnainnninnnififfakannn129) 1()(2)211 (11)121211 (1122 nkkniinnkininnan130131kiikiikiiivanttnnnn1111111) 11、132111111111niikiiiitnttnttt120nkkni%28. 4042775. 0i042775. 0i042775. 0i ,0427482. 0i04545. 0) 1

18、10(8)810(2i3210133134iAiniiAn1ln1ln11isin1ln1ln n可能不是 整數(shù)若若n不是整數(shù)，應(yīng)該如何處理？不是整數(shù)，應(yīng)該如何處理？135136nnniiia 11122111122111111 nnniiia 1122221)1 (1111 nnnnniiiis nnnniiis1221111 137 11211121111111111 nniiiiiia 1112111211111111111 nniiiiiia 1382111111)1 (11iiiiiisnnnnnn 111111111iiiiiisnnnnnn 1391401411421431442

19、4.32461716. 91008170.20200100200:030301. 0101. 1%111, 8,163124jjssjj 總收益為14570.191886342.15300000300000RaR,024695. 0105. 1,2042Rjj則有：設(shè)每次的償還額為146147結(jié)轉(zhuǎn)周期結(jié)轉(zhuǎn)周期148knknkknkknkkksaivvvvvvv 11112knnknssisa11491884.20,29.2035.20,35.2010000010000,10000010000035. 0,38035. 0,36035. 0, 22035. 0,2035. 0, 2035. 02

20、2035. 0, 2035. 0,2naaaasasannnn可知由且且，150元97.100845027. 3035. 229.201000010000010000010000%5 . 31035. 0,2035. 0,3636RsaR151knknknkkkaavvvvv 11112knnknasiaa1152kkkkkkisivvvv111112 kkkkiavvv11112 （元）有；設(shè)不足額部分為次足額領(lǐng)取次數(shù)為：5 .483x)005. 01 (10000ss500 xx,2367.2337172n7110000sa500693,0.00569,0.005005. 0, 3005.

21、 0,n153（元）07.36941940985. 2973456.2666967.811000ass100001. 0, 301. 0,2401. 0,60154155111m次1/m1/m1/m156nmmnmmnmnmnmmmnaiiivvvvmvvvvma )(1111211111 nmmnmnnmnsiiiiais)(1)1 (1157 11m1mmiimi158)12(, 3)12(,13)12(,103)12(,2)12(,12)12(,102480014800)248004800) 1iiiiiisssiaaav nmmnmnmnmmmnadddvvvmvvvma )(1121

22、111111159 nmmnmnnmnsdddiais )(1)1 (1 md 11 m1mmdd是名義貼現(xiàn)率是名義貼現(xiàn)率161mmnnmiaa1lim mmnnmdaa1lim 16221. 011 . 011212 . 0, 51,10222)2()2()2()2(,iiiiai163%46.232346. 019 . 019 . 0119 . 01111025 . 05 . 05 . 05 . 15 . 0 iiivvvvv164nnneva11111nnneisnnaiannsis16510203ss%93. 60693. 0102ln2, 010120231311010101010201020eeeeeeee是增根166167168169 invanvvvvIannnn 3232 insinsIaiIsnnnnn) 1(1)(1 invaQPavQ) 1n(P2Q)vP(v)QP(Pvnnnn32 insQPs) i1)(invaQ(Pannnnnn終值為：170171 dnvaIaiaInnnn 1 dnsdnsIsis Innnn111 2iQiPIa172 iidiinvaIaIannnnn1111limlim 22111limiddnvaa Innn 173 42005. 01105. 0111