利息基本理论和利息度量

1、1The Theory of Interest 利息理论利息理论5789项目（存款）项目（存款）年利率（年利率（%）城乡居民及单位存款（一）活期0.35（二）定期1.整存整取三个月2.856个月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）利率的表现形式:例、甲乙双方商定贷款事宜，因业务发展需要，甲方向乙方借款1000万元，期限2年,到期后，甲方一次性归还乙方1150万元。则甲方贷款的

2、年利率是：1150-100010002=7.5%风险太大,可能无法归还本息16Aicntt11PiAiInntt11117债券价格债券价格与收益率与收益率反相关反相关PA,折价发行折价发行P=A,平价发行平价发行PA,溢价发行溢价发行1819PPAi20价格与收益价格与收益率反向变化率反向变化23242526利率i=6.56%pi 债券需求（对可贷资金的供给） 债券供给（对可贷资金的需求）债券数量/可贷资金量282930323334 nniiiii 1111132135ngggggn 3213637通货膨胀率通货膨胀率名义利率实际利率通货膨胀率名义利率实际利率11114041424344454

3、6)(ta1) 0(a)()(,)(2121tatattta有是增函数，即47)()()0()()()0(),0(taKtaAtAtAtAA时刻的累积值为为常数，则在内的利息额为时期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时，当时，当）（为增函数时当)()()(0)2)()(2)()(2)()(,2121212121212122212121tatatatttttttttttttttatatt51) 1() 1() 1()(1naKIAInAnAnAinnnn实际利率)0( ,1)(,tittaitt为利率为任意值为度量时期设niPIniininiinnananainisatastasitiistnn、的实际利率时刻开始的一个时期内为时点、设、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,试计算i5540,)1 ()(titatititititititiiiinananaistsatastatttnnnn11,11111113111) 1() 1()(20, 0),()()(111时当时，当时，、当、551 . 01 . 0,) 1 . 01 ()() 1 . 01 (100)()(.,) 1 . 1 (100)(105105iiiitataAtAiitAttt确定如果5657 3 , 2 , 1,) 1%(51

6、%5) 1(1nnniiinn123456in5%4.76%4.55%4.35%4.17%4%58n123456单利1.051.101.151.21.251.3复利1.051.10251.15761.21551.27631.3401复利超过单利的%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为贴现因子iviviid11,1dvviiid111111ttdvta1)(1ttdvta1)(这两数之积两数之差 d1idiididiivdd) 1n(1dd) 1n(1nd11nd)(1d) 1n(1)nd1 (a(n)1)-a(na(n)d1 -11n67dtdtta10,1)(1对1t,dt1d-131t,dt1d-121t0,dt1d-111,d0ttt如果）（如果）（如果）（有：设680,1)(1tdvtatt对280012. 0336p12. 0i ,12. 13

8、00336i1697011111)()()(mmmmmimimiiimi为：则年实际利率次利息的年名义利率，表示一年结转设m171nnnnnnvndndndddnd11)()()(111111年名义贴现率为：为：则年实际贴现率时期收取一次贴现值，表示年名义贴现率，每设n172diidnnmmndmi)()(1111)(nnndimmmid)(11时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年度实际利率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个时期内，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：，根据复利累计函数有为为年利率，则季度利率设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利息总量RRa11363.8063972.1210000s10000D,10000sD10000sD005. 0 ,12005. 0 ,12ni 114nniniias1nniniias1 isanini11dsanini 11115)1 ( iaann )1 (issnn 11nnaa 11nnss 116假定：计算的日期离开每次付款日期为整数个时期假定：计算的日期离开每次付款日期为整数个时期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（扣减付款）：在第15年末的求值方程是：支付在时刻 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 整数若若n不是整数，应该如何处理？不是整数，应该如何处理？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 总收益为14570.191886342.15300000300000RaR,024695. 0105. 1,2042Rjj则有：设每次的偿还额为146147结转周期结转周期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；设不足额部分为次足额领取次数为：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是名义贴现率是名义贴现率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终值为：170171 dnvaIaiaInnnn 1 dnsdnsIsis Innnn111 2iQiPIa172 iidiinvaIaIannnnn1111limlim 22111limiddnvaa Innn 173 42005. 01105. 0111