北大版金融数学引论第二章答案
(完整版)北大版金融数学引论第二章答案
版权所有，翻版必究第二章习题答案1．某家庭从子女出生时开始累积大学教育费用5万元。
如果它们前十年每年底存 款1000元，后十年每年底存款1000+X 元，年利率7%。
计算X 。
解：S = 1000s 20¬p7%+Xs 10¬p7%X =50000 − 1000s 20¬p7% s 10¬p7%= 651.722．价值10,000元的新车。
购买者计划分期付款方式：每月底还250元，期限4年。
月结算名利率18%。
计算首次付款金额。
解： 设首次付款为X ，则有10000 = X + 250a 48¬p1.5%解得X = 1489.363．设有n 年期期末年金，其中年金金额为n ，实利率i =1。
试计算该年金的现值。
解：P V = na¬n pi1 − v nn= n 1n=(n + 1)nn 2− n n+2 (n + 1)n4．已知：a¬n p= X ，a 2¬n p= Y 。
试用X 和Y 表示d 。
解： a 2¬n p= a¬n p+ a¬np (1 − d)n则Y − X1d = 1 − ( X ) n5．已知：a¬7p= 5.58238, a 11¬p= 7.88687, a 18¬p= 10.82760。
计算i 。
解：a 18¬p = a¬7p + a 11¬p v 7解得6.证明： 11−v 10=s 10¬p +a ∞¬p。
s 10¬pi = 6.0%北京大学数学科学学院金融数学系第 1 页版权所有，翻版必究证明：s 10¬p + a ∞¬p(1+i)10−1+11 s 10¬p=i(1+i)10−1ii= 1 − v 107．已知：半年结算名利率6%，计算下面10年期末年金的现值：开始4年每半 年200元，然后减为每次100元。
专题2 金融数学的实际应用含答案
专题2 金融数学的实际应用含答案金融数学是金融领域中的重要学科，它通过运用数学方法和模型来解决各种金融问题。
本文将探讨金融数学的实际应用，并提供相关问题的答案。
1. 期权定价期权是金融市场中常见的金融衍生产品之一。
金融数学可以应用于期权定价，通过使用著名的Black-Scholes模型，可以计算出期权的合理价格。
该模型考虑了标的资产价格、行权价格、无风险利率、资产波动率等因素，从而帮助投资者确定期权的合理价格。
2. 简单利率与复利率的计算金融数学还可以用于计算利率。
在金融市场中，人们经常需要计算简单利率和复利率。
简单利率是指每期利息基于本金计算，而复利率是指每期利息基于本金和已累积利息计算。
通过使用金融数学中的利率计算公式，可以准确计算出简单利率和复利率。
3. 货币时间价值货币时间价值是指货币随着时间推移而产生的价值变化。
金融数学可以帮助人们计算货币时间价值，确定现金流的现值和未来价值。
通过将未来现金流的价值折算到现在，金融数学可以帮助人们做出更准确的投资决策。
4. 统计分析金融数学对于统计分析也有重要应用。
通过应用统计学方法，金融数学可以帮助分析市场数据，预测未来的趋势和变化。
例如，通过使用回归分析，金融数学可以帮助评估不同因素对证券价格的影响，并建立预测模型。
5. 投资组合优化金融数学还可以应用于投资组合优化。
投资组合优化是指通过合理配置不同资产，以达到最佳的风险和收益平衡。
金融数学可以帮助分析资产间的相关性和波动性，从而帮助投资者构建理想的投资组合。
以上是金融数学在实际应用中的一些例子。
通过应用金融数学的方法和模型，人们可以更好地理解和解决各种金融问题，提高投资决策的准确性和效率。
以上是针对专题2 "金融数学的实际应用" 的简要介绍和答案。
希望对您有所帮助！。
金融数学引论答案第二版
金融数学引论答案第二版【篇一：北大版金融数学引论第二章答案】>第二章习题答案1．某家庭从子女出生时开始累积大学教育费用5万元。
如果它们前十年每年底存款1000元，后十年每年底存款1000+x 元，年利率7%。
计算x 。
解：s = 1000s?7%+xs?7%20p10p20px = 50000 ? 1000s?7% = 651.72s?p7%102．价值10,000元的新车。
购买者计划分期付款方式：每月底还250元，期限4年。
月结算名利率18%。
计算首次付款金额。
解：设首次付款为x ，则有10000 = x + 250a?p1.5%48解得x = 1489.3613．设有n年期期末年金，其中年金金额为n，实利率i =n解：p v = na?npi= 1nn+2 =(n + 1)nn2n4．已知：a?pn= x，a?p2n= y 。
试用x和y 表示d 。
解： a?p2n= a?pn+ a?p (1 ? d)则nny ? xd = 1 ? ( x ) n5．已知：a?p7= 5.58238, a?= 7.88687, a?= 10.82760。
计算i。
11p18p解：a?p = a?p + a?p v718711解得=i = 6.0%10?p +a∞?p6.证明： 11?v10s。
s10?p北京大学数学科学学院金融数学系第 1 页版权所有，翻版必究证明：s?p + a∞?p=s?10p10+101 = 107．已知：半年结算名利率6%，计算下面10年期末年金的现值：开始4年每半年200元，然后减为每次100元。
解：p v = 100a?+ 100a20?8p3% p3% = 2189.7168．某人现年40岁，现在开始每年初在退休金帐号上存入1000元，共计25年。
然后，从65岁开始每年初领取一定的退休金，共计15年。
设前25年的年利率为8%，后15年的年利率7%。
计算每年的退休金。
解：设每年退休金为x，选择65岁年初为比较日=解得x = 8101.658。
最新金融数学习题部分答案
第一章
注：（2）：P=1/4即可。
第二章
注：i=4.5%,以半年为换算周期
注：本题是期初年金，故年金额组成等比数列，取其于网络如有侵权请联系管理员删除收集于网络如有侵权请联系管理员删除收集于网络如有侵权请联系管理员删除收集于网络如有侵权请联系管理员删除收集于网络如有侵权请联系管理员删除收集于网络如有侵权请联系管理员删除收集于网络如有侵权请联系管理员删除收集于网络如有侵权请联系管理员删除收集于网络如有侵权请联系管理员删除
金融数学课后答案
金融数学课后答案【篇一：金融数学(利息理论)复习题练习题】购买一张3年期，面值为1000元的国库券，每年末按息票率为8%支付利息，第三年末除支付80元利息外同时偿付1000元的债券面值，如果该债券发行价为900元，请问他做这项投资是否合适？ 2.已知：1） 1?i2） 1?由于(1?m)?(1?n)?1?i 由于(1?)?(1?)?1?d3. 假设银行的年贷款利率12%，某人从银行借得期限为1年，金额为100元的贷款。
银行对借款人的还款方式有两种方案：一、要求借款人在年末还本付息；二、要求借款人每季度末支付一次利息年末还本。
试分析两种还款方式有何区别？哪一种方案对借款人有利？4. 设m?1，按从小到大的顺序排列i,i(m)(m)(m)(m)m?(1?i5)(1?i6)?1 求m?? ?(1?d(5)d(6)?1)(1?6) 求m?? 5(5)(6)d(m)mm(n)nm(n)n,d,d(m),?解：由i?d?i?d? i?dd(m?1)?d(m) ? d?d(m) i(m)?d(n) ? d(m)?i(m) i(m?1)?i(m)?i(m)?ii(m)?limd(m)?? 1?i?e??1?? ， limm??m???d?d(m)???i(m)?i5. 两项基金x,y以相同的金额开始，且有：（1）基金x以利息强度5%计息；（2）基金y以每半年计息一次的名义利率j计算；（3）第8年末，基金x中的金额是基金y中的金额的1.5倍。
求j.6. 已知年实际利率为8%，乙向银行贷款10,000元,期限为5年，计算下列三种还款方式中利息所占的额度：1）贷款的本金及利息积累值在第五年末一次还清； 2）每年末支付贷款利息，第五年末归还本金； 3）贷款每年年末均衡偿还（即次用年金方式偿还）。
三种还款方式乙方支付的利息相同吗? 请你说明原因？7.某人在前两年中，每半年初在银行存款1000元，后3年中，每季初在银行存款2000元，每月计息一次的年名义利率为12% 计算5年末代储户的存款积累值。
高等代数(北大版)第2章习题参考答案
第二章 行 列 式1. 求以下9级排列的逆序数,从而决定它们的奇偶性1) 1 3 4 7 8 2 6 9 5; 2) 2 1 7 9 8 6 3 5 4; 3)9 8 7 6 5 4 3 2 1;解:1) 所求排列的逆序数为:()1011033110134782695=+++++++=τ， 所以此排列为偶排列。
2) 所求排列的逆序数为:()1810345401217986354=+++++++=τ， 所以此排列为偶排列。
3) 所求排列的逆序数为:()()36219912345678987654321=-=+++++++=τ， 所以此排列为偶排列。
2.选择i 与k 使1) 1274i 56k 9成偶排列； 2) 1i 25k 4897成奇排列。
解: 1) 当3,8==k i 时, 所求排列的逆序数为:()()10011314001274856399561274=+++++++==ττk i ，故当3,8==k i 时的排列为偶排列.。
2)当6,3==k i 时, 所求排列的逆序数为:()()5110110101325648974897251=+++++++==ττk i ，故当6,3==k i 时的排列为奇排列。
3.写出把排列12345变成排列25341的那些对换。
解: 12345()()()2534125431214354,35,22,1−−→−−−→−−−→−。
4.决定排列()211 -n n 的逆序数,并讨论它的奇偶性。
解: 因为1与其它数构成1-n 个逆序,2与其它数构成2-n 个逆序,……n n 与1-构成1个逆序,所以排列()211 -n n 的逆序数为()[]()()()时排列为奇排列。
当时，排列为偶排列；故当34,2414,4211221211++=+=-=+++-+-=-k k n k k n n n n n n n τ5.如果排列n n x x x x 121- 的逆序数为k ,排列121x x x x n n -的逆序数是多 少?解: 因为比i x 大的数有i x n -个,所以在121x x x x n n -与n n x x x x 121- 这两个排列中,由i x 与比它的 各数构成的逆序数的和为i x n -.因而,由i x 构成的逆序总数 恰为 ()()21121-=-+++n n n 。
《金融数学引论第二版》复习提纲
《金融数学引论》复习提纲第一章 利息的基本计算 第一节 利息基本函数一. 累积函数a(t)与总量函数A(t)某一度量期的实际利率是指该度量期内得到的利息金额与此度量期开始时投资的本金金额之比，通常用字母i 来表示。
利息金额I n =A(n)-A(n-1)对于实际利率保持不变的情形，i=I 1/A(0)； 对于实际利率变动的情形，则i n =I n /A(n-1)； 二．单利和复利考虑投资一单位本金，（1） 如果其在t 时刻的积累函数为 a(t)=1+i*t ，则称这样产生的利息为单利；实际利率 )()()()(1111-+=---=n i in a n a n a i n（2） 如果其在t 时刻的积累函数为a(t)=(1+i)t ，则称这样产生的利息为复利。
实际利率 i i n =三.. 贴现函数一个度量期的实际贴现率为该度量期内取得的利息金额与期末的投资可回收金额之比，通常用字母d 来表示实际贴现率。
等价的利率i 、贴现率d 和贴现因子（折现因子）v 之间关系如下：,(1),1111,,,1d ii d i i d d iv d d iv v i d idi=+==-+=-==-=+四．名利率与名贴现率用()m i 表示每一度量期支付m 次利息的名义利率，这里的m 可以不是整数也可以小于1。
所谓名义利率，是指每1/m 个度量期支付利息一次，而在每1/m 个度量期的实际利率为()/m i m 。
与()m i 等价的实际利率i 之间的关系：()1(1/)m m i i m +=+。
名义贴现率()m d ，()1(1/)m m d d m -=-。
名义利率与名义贴现率之间的关系：()()()()m m m m i d i d m m m m-=⋅。
五．连续利息计算定义利息强度（利息力）为()()()()t A t a t A t a t δ''==， 0()ts ds a t e δ⎰=一个常用的关系式如下：()()11[1]1(1)[1]m p m p i d i v d e m pδ---+=+==-=-=要求：δ,,,,)()(p m d i d i ，之间的计算。
徐景峰《金融数学》1-4章习题解答
《利息理论》习题详解 第一章 利息的基本概念1.解：（1）)()0()(t a A t A =又()25A t t =+(0)5()2()1(0)55A A t a t t A ∴===++ （2）3(3)(2)11(92 2.318I A A =-=== （3）4(4)(3)0.178(3)A A i A -===2.解：15545(4)(3)(1)100(10.04)0.05 5.2nn n I i A I A i A i i -=∴==+=+⨯=3.证明： （1）123(1)()(2)(1)(3)(2)()(1)m m m m k I A m A m I A m A m I A m A m I A m k A m k ++++=+-=+-+=+-+=+-+-123123()()()()()m m m m k m m m n I I I I A m k A m n m k A n A m I I I I m n +++++++∴++++=+-=+-=++++<令有（2）()(1)()1(1)(1)n A n A n A n i A n A n --==---()1(1)()(1)(1)n n A n i A n A n i A n ∴+=-∴=+-4.证明： （1）112123123(1)(0)(0)(2)(0)(0)(0)(3)(0)(0)(0)(0)()(0)(0)(0)(0)(0)k nk i a a a i a a a i a i a a a i a i a i a n a a i a i a i a i ∴=+=++=+++=+++++第期的单利利率是又(0)1a =123123()1()(0)()1nna n i i i i a n a a n i i i i ∴=+++++∴-=-=++++（2）由于第5题结论成立，当取0m =时有12()(0)n A n A I I I -=+++5.解：（1）以单利积累计算1205003i =⨯1200.085003i ∴==⨯800(10.085)1120∴+⨯=（2）以复利积累计算3120500500(1)i +=+0.074337i ∴=5800(10.074337)1144.97∴+=6.解：设原始金额为(0)A 有(0)(10.1)(10.08)(10.06)1000A +++=解得 (0)794.1A =7.证明：设利率是i ，则n 个时期前的1元钱的当前值为(1)ni +，n 个时期后的1元钱的当前值为1(1)ni +又22211[(1)](1)20(1)(1)n n n ni i i i +-=++-≥++，当且仅当221(1)(1)1(1)n n n i i i +=⇒+=+，0i =即或者n=0时等号成立。
数理金融学导论补充练习及参考答案
6．如果一家银行所提供的利息是以名义利率5%连续地计算利息，那么每年的有效利率应该是多少？
解：有效利率应为：
。
即有效利率是每年 。
7．一家公司在未来的5年中需要一种特定型号的机器。这家公司当前有1台这种机器，价值6000美元，未来3年内每年折旧2000美元，在第三年年末报废。该机器开始使用后第一年运转费用在该年年初值为9000美元，之后在此基础上每年增加2000美元。在每年的年初可以按固定价格22000美元购买1台新机器。1台新机器的寿命是6年，在最初使用的两年中每年折旧3000美元，这之后每年折旧4000美元。新机器在第一年的运转成本是6000美元，在随后的每年中将增加1000美元。如果利率为10%，公司应在何时购买新机器？
其中，
因此股票的价格为：
股票价格的现值
第三步，将从步骤1和步骤2得到的这两个现值相加得到普通股的价值。
为了验证上面所列现金流的正确性，假设公司将在第三年的年初购买新机器，则公司在第一年的成本为旧机器9000美元的运转成本；在第二年的成本为旧机器11000的运转成本；在第三年的成本为新机器22000的购买成本，加上6000美元的运转成本，再减去从替换机器中得到的2000美元；在第四年的成本是7000美元的运转成本；在第五年的成本是8000美元的运转成本；在第六年的成本是-12000美元，它是已经使用了三年的机器价值的负值。其他的三个现金流序列可以通过相似的方法推得。
解：这家公司可以在第1、2、3、4年的年初购买新机器，其对应的六年现金流如下（以1000美元为单位）：
在第一年的年初购买新机器：22，7，8，9，10，-4；
在第二年的年初购买新机器：9，24，7，8，9，-8；
在第三年的年初购买新机器：9，11，26，7，8，-12；
金融数学引论答案第一章__北京大学出版[1]
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第一章习题答案1. 设总量函数为A(t) = t 2 + 2t + 3 。
试计算累积函数a(t) 和第n 个时段的利息I n 。
解: 把t = 0 代入得A(0) = 3 于是:a(t) =A(t)/A(0)=（t 2 + 2t + 3）/3I n = A(n) − A(n − 1)= (n 2 + 2n + 3) − ((n − 1)2+ 2(n − 1) + 3))= 2n + 12. 对以下两种情况计算从t 时刻到n(t < n) 时刻的利息: (1)I r (0 < r <n); (2)I r = 2r (0 < r < n).解: ()n n-1t 11I A(n)A(t)I I I n(n 1)/2t(t 1)/2+=-=+++=+-+・・・ (2)1t 11I A(n)A(t) 22n n k k t I ++=+=-==-∑3. 已知累积函数的形式为: 2a(t) at b =+。
若0 时刻投入的100 元累积到3 时刻为172 元，试计算：5 时刻投入的100 元在10 时刻的终值。
解: 由题意得a(0) = 1, a(3) =A(3)/A(0)= 1.72⇒ a = 0.08, b = 1∴ A(5) = 100A(10) = A(0) ・ a(10) = A(5) ・ a(10)/a(5)= 100 × 3 = 300.4. 分别对以下两种总量函数计算i 5 和i 10 :(1) A(t) = 100 + 5t; (2)t A(t) 100(1 0.1)=+.解：(1)i 5 =(A(5) − A(4))/A(4)=5120≈ 4.17%i 10 =(A(10) − A(9))/A(9)=5145≈ 3.45%(2)i 5 =(A(5) − A(4))/A(4)()()()544109109100(1 0.1)100(1 0.1) 10%100(1 0.1)100(1 0.1)100(1 0.1)i (A 10A 9)/A 9 10%100(1 0.1)+-+==++-+=-==+5.设()n A 4 1000, i 0.01n ==. 试计算A(7) 。
金融数学引论答案.docx
⾦融数学引论答案.docx第⼀章习题答案1.设总量函数为A(t) = t2 + 2/ + 3 o试计算累积函数a(t)和第n个吋段的利息【仇°解：把t =()代⼊得4(()) = 3于是：4(t) t? + 2t + 3啲=丽=3In = 4(北)⼀A(n⼀1)=(n2 + 2n + 3) — ((n — I)2 + 2(n — 1) + 3))= 2n+l2.对以下两种情况计算从t时刻到冗(￡ < n)时刻的利息：(1)⼛(0 < r < n);(2)/r =2r(0解：(1)I = A(n) - A(t)—In + in-1+ ? ? ? + A+l n(n + 1) t(t + 1)=2 2I = A(n) - A(t)n n=乞h = ⼟hk=t+l A：=t+13.已知累积函数的形式为：Q(t) = at2 +⼏若0时刻投⼊的100元累积到3吋刻为172元,试计算：5时刻投⼊的10()元在10时刻的终值。
解：由题意得。
(0) = 1, ?(3) = = L72=> a = 0.0& 6=14(5) = 100>1(10) = 4(0) ? ?(10) = 4⑸? W = 100 x 3 = 300.a(5)4.分别对以下两种总量函数计算订和讪:(1) A(t) = 100 + 5t; (2) A(t) = 100(1 + 0.1⼫?解：(1)_ 4(5) - 4(4)5 _ 4(4)5⼆⾯-.17% . 4(10)-4(9)210 =—4(9)—5=—^ 3.45%145⑵_ 4(5) - 4(4)5 - 4⑷_ 100(1 + 0.1)5 - 100(1 + 0.1)4 = 100(1+ 0.1)4=10%. 4(10) —4(9)⽫=_ 100(1+ O.1)10-100(1+ 0.1)9 = 100(1 + 0.1)9=10%5?设4(4) = 1000, i n = O.Oln.试计算4(7)。
北大版金融数学引论第二章答案
北大版金融数学引论第二章答案版权所有，翻版必究第二章习题答案1．某家庭从子女出生时开始累积大学教育费用5万元。
如果它们前十年每年底存 款1000元，后十年每年底存款1000+X 元，年利率7%。
计算X 。
解：S = 1000s 20¬p7%+Xs 10¬p7%X =50000 −1000s 20¬p 7%s 10¬p7%= 651.722．价值10,000元的新车。
购买者计划分期付款方式：每月底还250元，期限4年。
月结算名利率18%。
计算首次付款金额。
解： 设首次付款为X ，则有10000 = X + 250a 48¬p1.5%解得X = 1489.363．设有n 年期期末年金，其中年金金额为n ，实利率i =1。
试计算该年金的现值。
解：P V = na¬n pi1 − v nn= n 1n=(n + 1)nn 2− n n+2 (n + 1)n4．已知：a¬n p= X ，a 2¬n p= Y。
试用X 和Y 表示d 。
解： a 2¬n p= a¬n p+ a¬np (1 − d )n则Y − X1d = 1 − ( X ) n5．已知：a¬7p= 5.58238, a 11¬p= 7.88687, a 18¬p= 10.82760。
计算i 。
解：a 18¬p = a¬7p + a 11¬p v 7解得6.证明： 11−v10=i = 6.0%s。
10¬p +a∞¬ps10¬p北京大学数学科学学院金融数学系第 1 页版权所有，翻版必究证明：s 10¬p + a ∞¬p(1+i)10−1+11 s 10¬p=i(1+i)10−1ii= 1 − v 107．已知：半年结算名利率6%，计算下面10年期末年金的现值：开始4年每半 年200元，然后减为每次100元。
国际金融专题答案北大
国际金融专题答案北大《国际金融专题》复习题答案（供参考）――李妍婷（整理）第一章第二章导论 ...................................................... ......................................................... ..................... 1 国际收支基础 ...................................................... ......................................................... . (1)1. 国际收支平衡表的记账原则和记帐方法。
（必考）..................................................... .... 1 2. 国际收支平衡表的局部差额分析。
.................................................... ................................ 4 3. 金本位下、固定汇率纸币本位下、浮动汇率下国际收支的自我调节机制（重点）. 6 4. 概念： ............................................................................................................. . (7)（1）（2）（3）（4）（5）（6）第三章国际收支平衡表 ...................................................... ............................................ 7 自主性交易和补偿性交易 ...................................................... ............................ 7 赤字和盈余 ...................................................... .. (7)全面性控制与选择性控制 ...................................................... ............................ 8 冲销政策 ...................................................... .........................................................8 外汇缓冲政策 ...................................................... (9)国际收支理论 ...................................................... ......................................................... . (9)1. M－L条件的推导及其含义（重点）..................................................... ............................ 9 2. 弹性论对货币贬值三个阶段和J曲线的论述（重点） (10)3. 货币论中不同汇率制度下收入、价格和货币扩张对国际收支的影响（重点）....... 11 4. 用货币论的基本模型比较固定汇率制度和浮动汇率制度的优劣（重点）............... 15 5. 名词解释： .................................................... ......................................................... (15)（1）贸易条件 ...................................................... . (15)经济学院07金融 I（2）（3）（4）（5）第四章贬值的价格效应与数量效应 ...................................................... ..................... 16 贬值的消费者和生产者反应时滞（重点） (16)外汇市场操作 ...................................................... .............................................. 17 冲销性外汇操作 ...................................................... (17)外汇与外汇基础 ...................................................... .......................................................171. 外汇的基本特征 ...................................................... ......................................................... .... 17 2. 汇率的标价方法和套算汇率的方法（重点）..................................................... ............ 18 3. 通过掉期率来表示远期汇率的方法（重点）..................................................... ............ 20 4. 金本位下通过黄金输送点保持固定汇率的作用机制（重点）.................................... 21 5. Walras拍卖者和造市者的不同，外汇市场均衡价格的产生机制 .............................. 22 6. 几个概念： .................................................... ......................................................... (23)（1）（2）（3）（4）（5）逆风行事原则 ...................................................... .............................................. 23 货币替换 ...................................................... .......................................................23 联合干预 ...................................................... .......................................................24 肮脏浮动 ...................................................... .......................................................24 单向投机的自我实现 ...................................................... . (24)7. 描述固定汇率制的运行机制 ...................................................... ........................................ 24 第五章汇率理论 ...................................................... (26)1. 一价定率和商品套购的过程与结果 ...................................................... ............................ 26 2. 绝对购买力平价和相对购买力平价的含义和计算（重点）........................................ 27 3. 抛补套利和非抛补套利的含义 ...................................................... .................................... 27 4. 利率平价公式的推导过程和含义（重点）..................................................... . (28)经济学院07金融 II5. 用预期收益率曲线分析利率变动对均衡汇率的影响 .....................................................30 6. 比较汇率的弹性价格货币模型与粘性价格模型的异同。
《金融学》第二章答案 金融系统
CHAPTER 2THE FINANCIAL S YS TEMObjectives∙To provide a conceptual framework for understanding how the financial system works and how it changes over time.∙To understand the meaning and determinants of rates of return on different classes of assets.Outline2.1 What Is the Financial System?2.2 The Flow of Funds2.3 The Functional Perspective2.4 Financial Innovation and the “Invisible Hand”2.5 Financial Markets2.6 Financial Market Rates2.7 Financial Intermediaries2.8 Financial Infrastructure and Regulation2.9 Governmental and Quasi-Governmental OrganizationsSummary∙The financial system is the set of markets and intermediaries used by households, firms, and governments to implement their financial decisions. It includes the markets for stocks, bonds, and other securities, as well as financial intermediaries such as banks and insurance companies.∙Funds flow through the financial system from entities that have a surplus of funds to those that have a deficit.Often these fund flows take place through a financial intermediary.∙There are six core functions performed by the financial system:1.To provide ways to transfer economic resources through time, across borders, and among industries.2.To provide ways of managing risk.3.To provide ways of clearing and settling payments to facilitate trade.4.To provide a mechanism for the pooling of resources and for the subdividing of shares in variousenterprises.5.To provide price information to help coordinate decentralized decision-making in various sectors of theeconomy.6.To provide ways of dealing with the incentive problems created when one party to a transaction hasinformation that the other party does not or when one party acts as agent for another.∙The fundamental economic force behind financial innovation is competition, which generally leads to improvements in the way financial functions are performed. The basic types of financial assets traded in markets are debt, equity, and derivatives.∙Debt instruments are issued by anyone who borrows money—firms, governments, and households.∙Equity is the claim of the owners of a firm. Equity securities issued by corporations are called common stocks.∙Derivatives are financial instruments such as options and futures contracts that derive their value from the prices of one or more other assets.∙An interest rate is a promised rate of return, and there are as many different interest rates as there are distinct kinds of borrowing and lending. Interest rates vary depending on the unit of account, the maturity, and the default risk of the credit instrument. The nominal interest rate is the promised amount of money you receive per unit you lend.∙The real rate of return is defined as the nominal interest rate you earn corrected for the change in the purchasing power of money. For example, if you earn a nominal interest rate of 8% per year and the rate of price inflation is also 8% per year, then the real rate of return is zero.∙There are four main factors that determine rates of return in a market economy:∙the productivity of capital goods—expected rates of return on mines, dams, roads, bridges, factories, machinery, and inventories,∙the degree of uncertainty regarding the productivity of capital goods,∙time preferences of people—the preference of people for consumption now versus consumption in the future, and∙risk aversion—the amount people are willing to give up in order to reduce their exposure to risk.∙Indexing is an investment strategy that seeks to match the returns of a specified stock market index.∙Financial intermediaries are firms whose primary business is to provide customers with financial products that cannot be obtained more efficiently by transacting directly in securities markets. A mong the main types of intermediaries are banks, investment companies, and insurance companies. Their products include checking accounts, loans, mortgages, mutual funds, and a wide range of insurance contracts.Solutions to Problems at End of Chapter1. Do you agree with Adam Smith’s view that society can rely more on the “invisible hand” than on government to promote economic pros perity?Student answers will vary of course.SAMPLE ANSWER:The communist system is the exact opposite of Adam Smith’s invisible hand. And of course we have recently seen the downfall of many of the communist countries around the world. In the communist world, it was believed that government could make better decisions promoting economic prosperity than individuals could. Clearly this system failed to promote economic prosperity. It seems that Adam Smith’s view was that competitive market systems as a whole (rather than government) could best allocate resources to promote economic prosperity. However, a completely unfettered capitalist society such as in the late 1800s in the Western world may n ot have been the perfect system either as the invisible hand helped the “rich get richer” while the poor and needy had no formal assistance. This outraged the moral fabric of society and government programs were eventually set up to formally address thisi ssue of general welfare and “fairness”.2. How does the financial system contribute to economic security and prosperity in a capitalist society?In a capitalist society, it is the price system which helps make capital resource decisions. Capital flows to those operations which can employ it to earn the highest rate of return. This therefore allocates capital to its most productive use, thereby enhancing society’s economic prosperity. In addition, the financial system has markets and intermediaries which transfer risks from those who are least willing to bear it to those who are most willing to bear it. This benefits society as a whole without costing it anything. In addition, by allowing individuals to reduce or eliminate risks, it fosters an atmosphere of undertaking business ventures which also benefits society.3. Give an example of how each of the six functions of the financial system are performed more efficiently today than they were in the time of Adam Smith (1776).Clearing and settling payments:In Adam Smith’s day, just as today there was paper and coin currency. However, due to technological innovations (primarily the computer) today there are many additional forms of payment settlement such as personal checks, credit cards, debit cards and electronic transfer of funds. In addition, certain credit cards and traveler’s checks are accepted everywhere in the world making currency exchange a relic of the past. Pooling resources and subdividing shares:In Adam Smith’s day, most businesses were s mall and were financed by sole proprietorships. Therefore the need to pool resources to finance large investments was not as prevalent or as important as it is today. Again, the technological revolution of computers and telephones allow for global capital marke ts to efficiently finance today’s much larger businesses. Today these companies can access huge pools of money around the world and find the cheapest source of financing for large scale projects.Transfer economic resources: Today there is a worldwide financial system which facilitates the transfer of resources and risk from one individual to another and from one point in time to another. In Adam Smith’s day, although there were financial markets which played a limited role, they were localized, small and much less efficient and innovative than they are today.Managing risk: Of course during Adam Smith’s day individuals and businesses faced many of the same risks they do today (risk of property damage, risk of financial loss, risk of crop failure etc.) Ho wever, there were limited means to offset this risk. There were some insurance companies in place at that time, however, they concentrated on managing business risk rather than personal risk and certainly there was not the same type of insurance. A good ex ample is that in Adam Smith’s day, there was no unemployment insurance. In Adam Smith’s day, there was very little a farmer could do about reducing his risk of crop failure or lower crop prices. Today there are a vast number of markets and securities which can be used to offset individual and business risk as well as a huge network of insurance companies whose role is to transfer risk from those who want to reduce risk to those who want to take on more risk.Price information: During Adam Smith’s day, info rmation traveled slowly. Of course, there were no phones, televisions or radios. News traveled by newspaper and by the mail. Today, information travels around the worldinstantaneously. Due primarily to the growth and innovation in computer and telephone t echnology, information about security prices and performance is known at virtually the same time everywhere around the world.Incentive problems: As discussed above, today’s financial system is large, innovative and global. In Adam Smith’s day, while there were problems of moral hazard and adverse selection (but less of a principal-agent problem) there was not the same financial system and sophistication to deal with these problems as there is today.4. How does a competitive stock market accomplish the result that Adam Smith describes? Should the stock market be regulated? How and why?Student answers will vary.SAMPLE ANSWER:Adam Smith talked about free and competitive markets as a system which allocates capital to its most productive use and greatest value. In a competitive stock market, prices are set through supply and demand. Those companies returning the highest return will be rewarded with the highest prices (or cheapest source of financing). Those companies which are under performing will not be allocated as much capital because they are not as productive. Because the universe of possible investments is huge and because it is at times difficult for investors to discern which companies are the most productive employers of capital, regulation shou ld be required to make sure relevant and standardized information is disseminated to potential investors. This would include regulation on disclosure and also insider trading and stock manipulation. However other forms of market regulation are perhaps not so important from a market efficiency point of view and may even impede society’s overall financial welfare.5. Would you be able to get a student loan without someone else offering to guarantee it?Since most students do not have any earning power (yet) or source of savings or other capital, it is doubtful any intermediary would take that credit risk at any reasonable interest rate.6. Give an example of a new business that would not be able to get financing if insurance against risk were not available.EXAMPLES:∙Chemical company∙Child safety products company∙Airline∙Bank∙Hospital∙Environmental consulting∙Hazardous waste disposal7. Suppose you invest in a real-estate development deal. The total investment is $100,000. You invest $20,000 of your own money and borrow the other $80,000 from the bank. Who bears the risk of this venture and why?The $20,000 of my own money is considered the equity capital and the $80,000 is debt financing. In general it is the equity investors who absorb the primary risk of business failure. This is because if the business goes bankrupt, I will unlikely get any or my money back as the debt holders get paid back before I do. However, the debt holder also faces some risk that it will not even get back all its principal and interest. So lenders do share some of the business risk along with the equity investors.8. You are living in the United States and are thinking of traveling to Germany 6 months from now. You can purchase an option to buy marks now at a fixed rate of $0.75 per mark 6 months from now. How is the option like an insurance policy?An option means you have a choice. In this example you can choose to buy the marks at $0.75 in 6 months but you do not have to. You will only buy the marks at this price if it is cheaper for you to do so (if the spot market at that time is higher). Therefore, like an insurance policy you are protected against a potential loss. You know that the maximum price you will have to pay is $0.75 per mark and that you are protected against any higher price. Presumably you will have to pay something for the price of that option and that can be equated to an insurance premium.9. Give an example of how the problem of moral hazard might prevent you from getting financing for something you want to do. Can you think of a way of overcoming this problem?SAMPLE ANSWER:Suppose I want to start a biotechnology business and I need a lot of financing. The trouble is, I do not want to disclose my technology secrets to potential equity and debt investors. I will have great difficulty raising financing. But I could do the following: At a minimum, I could require all potential lenders and investors to sign agreements saying they will not disclose any of my secrets. Secondly, I could share some of my equity with potential lenders (equity-kickers) and investors (stock and stock options). At least that way they will not be motivated to disclose my secrets to others. Finally, if I decided I did not want to share secrets, I could give collateral in my new plant to the debt lenders and that might make them more comfortable with the issue of moral hazard.10. Give an example of how the problem of adverse selection might prevent you from getting financing for something you want to do. Can you think of a way of overcoming this problem?SAMPLE ANSWER:Suppose I want to start a car leasing business. Initially my plan was to purchase several automobiles and lease them out at attractive annual rates. However, potential lenders were worried that my business would attract individuals who drive great distances each year. Rather than buy their own car and lose significant value, they would lease my cars and take a new one each year. I would not be able to obtain financing for this business until I instituted annual mileage restrictions. This alteration in the business plan was enough to make the lenders comfortable with the potential problem of adverse selection.11. Give an example of how the principal-agent problem might prevent you from getting financing for something you want to do. Can you think of a way of overcoming this problem?SAMPLE ANSWER:Suppose you want to start a personal care products company. However, you have the idea for the business, but you do not want to actually run the business. To do that you have hired an executive from a competitor. He will own no equity in the business but will be paid a salary of $100,000 to start up the business.Trouble with this example is that the executive you have hired has little incentive to make the business really work other than his salary (which presumably he could earn at many different companies). What if this executive is really a spy? It may be difficult to get financing for this venture. The way to solve the problem is if you the owner decide to run the business (you certainly are motivated for it do well) or at a minimum, grant your new employee stock or stock options in the business.12. Why is it that a country’s postage stamps are not as good a medium of exchange as its paper currency? Postage stamps would be much easier to copy (to counterfeit) than paper currency which has intricate designs and is made of special fibers (not easily duplicated). Secondly, postage stamps would not be as durable as paper currency and because of their other use, could easily stick to other items! Finally, because postage stamps are used for another purpose, one might run out of them and have to make a special trip to the post office to get more. Of course, the post office is not as convenient as an ATM machine for getting a new supply of currency.13. Who is hurt if I issue counterfeit U.S. dollars and use them to purchase valuable goods and services?If this were done in great size, everyone would be hurt through the inflation that would result in the increased money supply. However, if done in a s mall amount, the individuals accepting the currency are taking on the risk (without knowing it) that the dollars will not be accepted by others as a medium of exchange.14. Some say the only criterion to use in predicting what will serve as money in the future is the real resource cost of producing it, including the transaction costs of verifying its authenticity. According to this criterion what do you think will be the money of the future?SAMPLE ANSWER:Payments via electronic transfer may become the medium of choice. It is a very cheap way to create currency. The biggest challenge will be to create security systems that do not allow for tampering and fraud. Once this is done and once most individuals and retail establishments have access to the system (through bank accounts and linking computer systems) then this should become the “currency” of choice.15. Should all governments issue debt that is indexed to their domestic price level? Is there a moral hazard problem that citizens face with regard to their public officials when government debt is fixed in units of the domestic currency?The answer is that all governments should issue debt that is indexed to their domestic price level. This is due to the fact that if debts are not indexed to the domestic price level, governments have the incentive to print money to repay those debts, thereby increasing domestic inflation which negatively impacts all of society.16. Describe your country’s system for financing higher education. Wh at are the roles played by households, voluntary non-profit organizations, businesses and government?SAMPLE ANSWER:In the United States, the vast majority of higher education is paid for by individuals through savings. These sums can be supplemented in whole or in part by government-guaranteed loans and through student loans and scholarships provided by universities themselves as well as by private foundations such as those provided by the Fulbright scholarship.17. Describe your country’s system for fin ancing residential housing. What are the roles played by households, businesses and government?SAMPLE ANSWER:In the United States individuals and individual borrowings from savings and loans, commercial banks and mortgage lending companies finance the vast majority of residential housing through individual equity savings. The government guarantees a certain amount of low income mortgages and local governments finance some low-income housing. Businesses play a role through the lending business as well as through the financial markets which provide liquidity for portfolios of certain standardized mortgages.18. Describe your country’s system for financing new enterprises. What are the roles played by households, businesses and government?SAMPLE ANSWER:In the United States, the vast majority of new enterprises is financed through individual savings and through initial public offerings made to the general public. These sources of financing are augmented by established firms which spend research and development (R&D) dollars developing new products and businesses and by venture capital institutions which also provide start-up financing.19. Describe your country’s system for financing medical research. What are the roles played by voluntary non-profit organizations, businesses and government?SAMPLE ANSWER:In the United States, medical research is financed both by non-profit organizations (such as universities and medical facilities as well as organizations such as the American Heart Association) as well as by businesses such as Merck, Johnson & Johnson and Genentech. The government is involved in research grants, primarily to universities.20. Assume there are only two stocks traded in the stock market, and you are trying to construct an index to show what has happened to stock prices. Let us say that in the base year the prices were $20 per share for stock 1 with 100 million shares outstanding and $10 for stock 2 with 50 million shares outstanding. A year later, the prices are $30 per share for stock 1 and $2 per share for stock 2. Using the two different methods explained in the chapter, compute stock indexes showing what has happened to the overall stock market. Which of the two methods do you prefer and why? (See appendix that follows.)DJI-Type Index = Average of Current Prices/Average of Base Prices * 100 = 106.67S&P-Type Index = (Weight of Stock 1 * Current Price of Stock 1 / Base Price of Stock 1 + Weight of Stock 2 * Current Price of Stock 2/Base Price of Stock 2) * 100 = 124The S&P-Type Index accurately reflects what has happened to the total market value of all stocks.。
金融数学引论答案第5—7章北京大学出版
版权所有，翻版必究第五章习题答案1. 已知某10年期零息票债券兑现值为1000，试对收益率为10%和9%分别计算当前价格。
并说明如果收益率下调10%，债券价格上涨的百分比。
解：(1)记P为买价，则有价值方程:P1(1 + 10%)10 = 1000P2(1 + 9%)10 = 1000解得:P1 = 385.54元P2 = 422.41元(2)收益率下降后P1(1 + 10% ×90%)10 = 1000P2(1 + 9% ×90%)10 = 1000解得:P1 = 422.41元，上涨百分比:9.56%; P2 = 458.93元，上涨百分比:8.65%。
2. 已知26周的短期国债的发行价格为9600元，到期兑现10,000元。
1〕按短期国债计算天数的典型方法计算贴现率；2〕假定投资期恰为半年，计算年收益率。
解：(1)由短期国债的定价公式10000(1 −Y dt360) = 9600解得：Y d = 7.91%(2)由定义设年换算收益率为i，则：9600(1 + i)12 = 10000解得：i = 8.51%3. 短期国债的贴现率均为8%，计算52 周国债与13 周短期国债的年利率之比。
52周实际天数已经超过360，如何处理；年利率之比是指等价年利率之比还是贴现率的比。
4. 某10年期面值为100元的债券半年名息率10%，到期兑现105元，如果收益率为半年换算8%，计算债券的买价。
北京大学数学科学学院金融数学系第1 页版权所有，翻版必究解：由基本公式：P = Fra n p i + Cv n = 100 ×5% ×13.5903 + 105 ×1.04¡20 = 115.875. 由债券价格计算公式，给出以下导数的计算公式，并解释其含义。
1) ∂P∂i , ∂P∂n和∂P∂g2) ∂n∂P和∂n∂P解：(1.1)由基本公式对i求导：∂P∂i= Fr(Da)n p i −nP(n + 1, i) < 0解释：债券的买价随着年限的增加而递减。
金融数学课后复习题答案 2
第一章习题答案1. 设总量函数为A(t) = t2 + 2t + 3 。
试计算累积函数a(t) 和第n 个时段的利息In 。
解: 把t = 0 代入得A(0) = 3 于是:a(t) =A(t)A(0)=t2 + 2t + 33In = A(n) − A(n −1)= (n2 + 2n + 3) −((n −1)2 + 2(n −1) + 3))= 2n + 12. 对以下两种情况计算从t 时刻到n(t < n) 时刻的利息: (1)Ir(0 < r <n); (2)Ir = 2r(0 < r < n).解:(1)I = A(n) − A(t)= In + In¡ 1 + ・・・+ It+1=n(n + 1)2− t(t + 1)2(2)I = A(n) − A(t)=Σnk=t+1Ik =Σnk=t+1Ik= 2n+1 −2t+13. 已知累积函数的形式为: a(t) = at2 + b 。
若0 时刻投入的100 元累积到3 时刻为172 元，试计算：5 时刻投入的100 元在10 时刻的终值。
第1 页解: 由题意得a(0) = 1, a(3) =A(3)A(0)= 1.72⇒ a = 0.08, b = 1∴A(5) = 100A(10) = A(0) ・ a(10) = A(5) ・ a(10)a(5)= 100 ×3 = 300.4. 分别对以下两种总量函数计算i5 和i10 :(1) A(t) = 100 + 5t; (2) A(t) = 100(1 + 0.1)t.解：(1)i5 =A(5) − A(4)A(4)=5120≈4.17%i10 =A(10) − A(9)A(9)=5145≈3.45%(2)i5 =A(5) − A(4)A(4)=100(1 + 0.1)5 −100(1 + 0.1)4100(1 + 0.1)4= 10%i10 =A(10) − A(9)A(9)=100(1 + 0.1)10 −100(1 + 0.1)9100(1 + 0.1)9= 10%第2 页5.设A(4) = 1000, in = 0.01n. 试计算A(7) 。
金新学案 北师大高中数学选修检测：第二章 概率 § 含答案
第二章§6一、选择题1．把一个正态曲线a沿着横轴方向向右移动2个单位，得到新的一条曲线b.下列说法中不正确的是()A．曲线b仍然是正态曲线B．曲线a和曲线b的最高点的纵坐标相等C．以曲线b为概率密度曲线的总体的期望比以曲线a为概率密度曲线的总体的期望大2 D．以曲线b为概率密度曲线的总体的方差比以曲线a为概率密度曲线的总体的方差大2 答案： D2．已知随机变量X服从正态分布N(2，σ2)，P(X<4)＝0.84，则P(X≤0)＝()A．0.16B．0.32C．0.68 D．0.84解析：由X～N(2，σ2)，可知其正态曲线如图所示，对称轴为x＝2，则P(X≤0)＝P(X≥4)＝1－P(X<4)＝1－0.84＝0.16.故选A．答案： A3．如图是当σ取三个不同值σ1、σ2、σ3的三种正态曲线N(0，σ2)的图像，那么σ1、σ2、σ3的大小关系是()A．σ1＞1＞σ2＞σ3＞0 B．0＜σ1＜σ2＜1＜σ3C．σ1＞σ2＞1＞σ3＞0 D．0＜σ1＜σ2＝1＜σ3解析：当μ＝0，σ＝1时，正态曲线f(x)＝12πe－x22在x＝0处取最大值12π，故σ2＝1.由正态曲线的性质，当μ一定时，曲线的形状由σ确定，当σ越小，曲线越“瘦高”，反之越“矮胖”，故选D．答案： D4．已知随机变量X ～N (0,1)，则X 在区间(－3，＋∞)内取值的概率等于( ) A ．0.887 4 B ．0.002 6 C ．0.001 3D ．0.998 7解析： 由X ～N (0,1)可知，μ＝0，σ＝1. 结合密度函数的图像可知 P (X ≥－3)＝12P (－3≤X ≤3)＋12＝0.998 7. 答案： D 二、填空题5．在某项测量中，测量结果X 服从正态分布N (1，σ2)(σ>0)，若X 在(0,1)内取值的概率为0.4，则X 在(0,2)内取值的概率为____________.解析： 由X ～N (1，σ2)(σ>0)，知正态曲线的对称轴为x ＝1，从而由图像可知P (0<X <1)＝P (1<X <2)，所以P (0<X <2)＝2P (0<X <1)＝2×0.4＝0.8. 答案： 0.86．已知正态总体的数据落在区间(－3，－1)里的概率和落在区间(3,5)里的概率相等，那么这个正态总体的数学期望为____________.解析： 区间(－3，－1)和区间(3,5)关于x ＝1对称(－1关于x ＝1的对称点是3，－3关于x ＝1的对称点是5)，所以正态分布的数学期望就是1.答案： 1 三、解答题7．一台机床生产一种尺寸为10 mm 的零件，现在从中抽测10个，它们的尺寸分别如下(单位：mm)：10.2,10.1,10,9.8,9.9,10.3，9.7,10,9.9,10.1.如果机床生产零件的尺寸Y 服从正态分布，求正态分布的概率密度函数式．解析： μ＝ 110(10.2＋10.1＋10＋9.8＋9.9＋10.3＋9.7＋10＋9.9＋10.1)＝10， σ2＝110[(10.2－10)2＋(10.1－10)2＋(10－10)2＋(9.8－10)2＋(9.9－10)2＋(10.3－10)2＋(9.7－10)2＋(10－10)2＋(9.9－10)2＋(10.1－10)2]＝0.03.即μ＝10，σ2＝0.03.所以Y 的概率密度函数为f (x )＝ 106πe － 50(x －10)23.8．设X ～N (6,1)，求P (4<x <5)．解析： 由已知μ＝6，σ＝1， ∵P (5<x <7)＝P (μ－σ<x <μ＋σ)＝0.683， P (4<x <8)＝P (μ－2σ<x <μ＋2σ)＝0.954，P (4<x <5)＋P (7<x <8)＝P (4<x <8)－P (5<x <7)＝0.271， 如图：由正态曲线的对称性知 P (4<x <5)＝P (7<x <8)，∴P (4<x <5)＝12[P (4<x <8)－P (5<x <7)]尖子生题库☆☆☆9．工厂制造的某机械零件尺寸X 服从正态分布N ⎝⎛⎭⎫4，19，问在一次正常的试验中，取1 000个零件时，不属于区间(3,5)这个尺寸范围内的零件大约有多少个？解析： 不属于区间(3,5)的概率为P (X ≤3)＋P (X ≥5) ＝1－P (3<X <5)＝1－P (|X －4|<1)． 因为X ～N ⎝⎛⎭⎫4，19，所以，μ＝4，σ＝13， 所以1－P (|X －4|<1)＝1－P (|X －μ|<3σ)＝1－0.9974 ＝0.0026≈0.003. 1000×0.003＝3(个)．即不属于(3,5)这个尺寸范围内的零件大约有3个．。
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版权所有，翻版必究~第二章习题答案1．某家庭从子女出生时开始累积大学教育费用5万元。
如果它们前十年每年底存 款1000元，后十年每年底存款1000+X 元，年利率7%。
计算X 。
解：S = 1000s 20¬p7%+Xs 10¬p7%X =50000 − 1000s 20¬p7% s 10¬p7%=2．价值10,000元的新车。
购买者计划分期付款方式：每月底还250元，期限4年。
月结算名利率18%。
计算首次付款金额。
解： 设首次付款为X ，则有10000 = X + 250a 48¬%解得X =3．设有n 年期期末年金，其中年金金额为n ，实利率i =1。
试计算该年金的现值。
解：P V = na¬n pi1 − v nn= n 1n=(n + 1)nn 2− n n+2 (n + 1)n4．已知：a¬n p= X ，a 2¬np = Y。
试用X 和Y 表示d 。
解： a 2¬n p= a¬np + a¬n p(1 − d)n则Y − Xd = 1 − ( X )5．已知：a¬7p= , a 11¬p= , a 18¬p= 。
计算i 。
解：a 18¬p = a¬7p + a 11¬p v 7解得6.证明：1 1−v=si = %¬+a¬。
s¬北京大学数学科学学院金融数学系第 1 页版权所有，翻版必究证明：s 10¬p + a ∞¬p(1+i)−1+11 s 10¬p=i (1+i)−1ii= 1 − v 107．已知：半年结算名利率6%，计算下面10年期末年金的现值：开始4年每半 年200元，然后减为每次100元。
解：P V = 100a¬8p3% + 100a 20¬p 3% =8．某人现年40岁，现在开始每年初在退休金帐号上存入1000元，共计25年。
然后，从65岁开始每年初领取一定的退休金，共计15年。
设前25年的年利率为8%， 后15年的年利率7%。
计算每年的退休金。
解： 设每年退休金为X ，选择65岁年初为比较日1000¨25¬p8%=X¨15¬p7%解得9．已知贴现率为10%，计算¨¬8p。
X =解： d = 10%，则 i =110.求证： (1) ¨¬np = a¬np + 1 −v n；1−d− 1 =19¨¬= (1 + i)1 − v 8i= (2) ¨¬np = s¬ −np 1 + (1 + i)n并给出两等式的实际解释。
证明： (1)¨¬np =1−dv =1−v =1−vi+ 1− v n所以(2)¨¬np =(1+i)−1¨¬np = a¬np + 1 − v n(1+i )−1=(1+i)−1n− 1=di + (1 + i)所以¨¬= s¬ −n p 1 + (1 + i)nn p版权所有，翻版必究12.从1980年6月7日开始，每季度年金100元，直至1991年12月7日，季结算名利 率6%，计算：1）该年金在1979年9月7日的现值；2）该年金在1992年6月7日的终 值。
解：P V = 100a 49¬%− 100a¬% = AV = 100s 49¬%− 100s¬%=13.现有价值相等的两种期末年金A 和B 。
年金A 在第1－10年和第21－30年中每年1元，在第11－20年中每年2元；年金B 在第1－10年和第21－30年中每年付款金 额为Y ，在第11－20年中没有。
已知：v 10=1，计算Y 。
解： 因两种年金价值相等，则有2a 30¬pi +a 10¬pi v 10=Y a 30¬ −pi Y a 10¬pi v 10所以 Y =3−v−2v1+v−2v=14.已知年金满足：2元的2n 期期末年金与3元的n 期期末年金的现值之和为36；另 外，递延n 年的2元n 期期末年金的现值为6。
计算i 。
解： 由题意知，2a 2¬npi + 3a¬npi = 362a¬npi v n = 6解得a¬7pa¬3p + s X¬pi = %15.已知a 11¬p = a Y¬p + s Z¬p 。
求X ，Y 和Z 。
解： 由题意得解得 1 − v 71 − v 11=(1 + i)X− v 3(1 + i)Z − v Y16.化简a 15¬p (1 + v 15+ v 30)。
解：X = 4, Y = 7, Z = 4a 15¬p (1 + v 15+ v 30) = a 45¬p北京大学数学科学学院金融数学系第 3 页版权所有，翻版必究17.计算下面年金在年初的现值：首次在下一年的4月1日，然后每半年一次2000元，半年结算名利率9%。
解：年金在4月1日的价值为P=1+%%×2000 = ，则P V =P (1 + i)2+=18.某递延永久年金的买价为P ，实利率i，写出递延时间的表达式。
解：设递延时间为t，有1解得t = −ln(1+ln iP i)P =i v t19.从现在开始每年初存入1000元，一直进行20年。
从第三十年底开始每年领取一定的金额X，直至永远。
计算X。
解：设年实利率为i，由两年金的现值相等，有X1000¨ 20¬pi=i v29解得X = 1000((1 + i)30− (1 + i)10)20.某人将遗产以永久年金的方式留给后代A、B、C、和D：前n年，A、B和C三人平分每年的年金，n年后所有年金由D一人继承。
如果四人的遗产份额的现值相同。
计算(1 + i)n。
解：设遗产为１，则永久年金每年的年金为i，那么A,B,C得到的遗产的现值为i3a¬n pi，而D得到遗产的现值为v n。
由题意得所以1 − v n3(1 + i)= 4= v n21.永久期末年金有A、B、C、和D四人分摊，A接受第一个n年，B接受第二个n年，C接受第三个n 年，D接受所有剩余的。
已知：C与A的份额之比为，求B与D的份额之比。
版权所有，翻版必究解：由题意知那么P V CP V AP V B==a¬n pv2na¬n pa¬n pv n13n== P V i v元年利率%的贷款从第五年底开始每年还贷100元，直至还清，如果最后一次的还款大于100元。
计算最后一次还款的数量和时间。
100a¬%v4<1000解：100a n+1¬%v4>1000解得n = 17列价值方程解得100a16¬%+Xv21 = 1000 X =年的期末年金每次4元，另有18年的期末年金每次5元；两者现值相等。
如果以同样的年利率计算货币的价值在n年内将增加一倍，计算n。
解：两年金现值相等，则 4 × a36¬p i= 5× 18，可知v18=由题意，(1 + i)n= 2 解得n = 924.某借款人可以选择以下两种还贷方式：每月底还100元，5年还清；k个月后一次还6000元。
已知月结算名利率为12%，计算k。
解：由题意可得方程100a60¬p1% = 6000(1 + i)−k解得25.已知a¬2pi= ，求i。
解：由题意得解得k = 291 − v2=i = %26.某人得到一万元人寿保险赔付。
如果购买10年期末年金可以每年得到1538元，20年的期末年金为每年1072元。
计算年利率。
解：版权所有，翻版必究27.某人在银行中存入一万元10年定期存款，年利率4%，如果前5年半内提前支取，银行将扣留提款的5% 作为惩罚。
已知：在第4、5、6和7年底分别取出K元，且第十年底的余额为一万元，计算K 。
解：由题意可得价值方程10000 = 105Ka¬2p4%v3+Ka¬2p4% + 10000v10则K = 10000−10000v105a¬v+a¬v=28.贷款P 从第六个月开始分十年逐年还清。
第一次的还款额为后面还款的一半，前四年半的年利率为i，后面的利率为j。
计算首次付款金额X的表达式。
解：选取第一次还款日为比较日，有价值方程P (1 + i)= X + 2Xa¬4pi+ 2Xa¬5pj (1 + i)−4所以P (1 + i)X =1 + 2a¬4pi+ 2a¬5pj (1 + i)−429.已知半年名利率为7%，计算下面年金在首次付款8年后的终值：每两年付款2000元，共计8次。
解：30.计算下面十年年金的现值：前5年每季度初支付400元，然后增为600元。
已知年利率为12%。
（缺命令）解：P V = 4 × 400 + 4 × 600v5=31.已知半年结算的名贴现率为9%，计算每半年付款600元的十年期初年金的现值表达式。
解：32.给出下面年金的现值：在第7、11、15、19、23和27年底支付一个货币单位。
解：P V =1s¬4pia24¬p i v3=(1 +i)24− 1(1 + i)27[(1 + i)4 − 1]=a28¬ −p a¬4ps¬3p + s¬1p北京大学数学科学学院金融数学系第 6 页版权所有，翻版必究元的永久年金和每20年付款750元的永久年金可以用每次R元的30年期末年金代替，半年换算名利率4%，求R的表达式。
解：设年实利率为i，则(1 + 2%)2= 1 + i。
有题意得750 i +750s20¬pi i=Ra30¬pi解得R =34.已知每三年付款一元的永久期末年金的现值为125/91，计算年利率。
解：由题意知解得i = 20%1is¬3pi=1259135.已知：1元永久期初年金的现值为20，它等价于每两年付款R元的永久期初年金，计算R。
解：由题意得解得R = 20 =1d=Ra¬2pi i36.已知每半年付款500元的递延期初年金价格为10000元。