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1、第一章:利息理论基础第一节:利息的度量一、利息的定义利息产生在资金的所有者和使用者不统一的场合,它的实质是资金的使用者付给资金所有者的租金,用以补偿所有者在资金租借期内不能支配该笔资金而蒙受的损失。二、利息的度量利息可以按照不同的标准来度量,主要的度量方式有1、 按照计息时刻划分:期末计息:利率期初计息:贴现率2、 按照积累方式划分:(1)线性积累:单利计息单贴现计息(2)指数积累:复利计息复贴现计息(3)单复利/贴现计息之间的相关关系 单利的实质利率逐期递减,复利的实质利率保持恒定。单贴现的实质利率逐期递增,复贴现的实质利率保持恒定。 时,相同单复利场合,复利计息比单利计息产生更大的积累值。
2、所以长期业务一般复利计息。时,相同单复利场合,单利计息比复利计息产生更大的积累值。所以短期业务一般单利计息。3、按照利息转换频率划分:(1)一年转换一次:实质利率 (实质贴现率 )(2)一年转换 次:名义利率 (名义贴现率 )(3)连续计息(一年转换无穷次):利息效力 特别,恒定利息效力场合有三、变利息1、 什么是变利息2、 常见的变利息情况(1)连续变化场合(2)离散变化场合第二节:利息问题求解原则一、利息问题求解四要素1、 原始投资本金2、投资时期的长度3、利率及计息方式4、本金在投资期末的积累值二、利息问题求解的原则1、本质任何一个有关利息问题的求解本质都是对四要素知三求一的问题。2、工
3、具现金流图:一维坐标图,记录资金按时间顺序投入或抽出的示意图。3、方法建立现金流分析方程(求值方程)4、原则在任意时间参照点,求值方程等号两边现时值相等。第三节:年金一、 年金的定义与分类1、 年金的定义:按一定的时间间隔支付的一系列付款称为年金。原始含义是限于一年支付一次的付款,现已推广到任意间隔长度的系列付款。2、 年金的分类:(1) 基本年金约束条件:等时间间隔付款付款频率与利息转换频率一致每次付款金额恒定(2) 一般年金 不满足基本年金三个约束条件的年金即为一般年金。(3) 二、基本年金1、 分类(1)付款时刻不同:初付年金/延付年金(2)付款期限不同:有限年金/永久年金2、 基本年金
4、公式推导3、 变利率年金问题(1) 时期变利率(第 个时期利率为 )(2) 付款变利率(第 次付款的年金始终以利率 计息)三、一般年金 1、分类(1)支付频率不同于计息频率(2)变额年金2、支付频率不同于计息频率年金(1)支付频率小于计息频率的年金分析方法一:利率转换方法二:年金的代数分析(2)支付频率大于计息频率的年金分析方法一:利率转换方法二:年金的代数分析(3) 连续年金特别,在常数利息效力场合3、变额年金(1) 等差年金 初始投资P元,等差Q元的年金的一般公式:现时值: 积累值: 特别地,递增年金:P=Q=1现时值: 积累值: 递减年金:P=n,Q=-1现时值: 积累值: (2) 等比
5、年金(下一期年金值为前一期年金值的( )倍)现时值: 积累值: 第四节:收益率一、收益率的概念 1、贴现资金流与现金流动表2、收益率的定义:使得投资返回净现时值等于零时的利率称为收益率。也称为“内返回率”二、 收益率的唯一性判别 1、 由于收益率是高次方程的解,所以它的解很可能不唯一。2、 Descartes符号判别定理:收益率的最大重数小于等于资金流的符号改变次数。3、 收益率唯一性判别定理二:整个投资期间未动用投资余额始终为正,收益率唯一。三、再投资率1、 本金的再投资率2、 利息的再投资率四、基金的利息度量1、 币值加权方法2、 时间加权方法第五节:分期偿还表和偿债基金一、分期偿还和偿债
6、基金的概念1、 分期偿还:借款人按一定的周期用分期付款的办法偿还贷款,这种还贷方法称为分期偿还。2、 偿债基金:借款人在贷款期末用一次的集中付款来偿还贷款人。利息则在此期间分期付款,并假设借款人周期性地付款给一个“基金”,该“基金”在贷款期末的积累值正好可以偿还贷款本金。二、分期偿还表时期付款金额支付利息偿还本金未偿还贷款余额0-11110总计 三、偿债基金时期付款金额支付利息存入偿债基金偿债基金积累值未偿还贷款余额0-110总计 对偿债基金而言,第 次付款的实际支付利息为:第 次付款的实际偿还本金为:1、 At a certain interest rate the present valu
7、es of the following two payment patterns are equal: (1) 200 at the end of 5 years plus 500 at the end of 10 years;(2) 400.94 at the end if 5 years.At the same interest rate, 100 invested now plus 120 invested at the end of 5 years will accumulate to P at the end of 10 years. Calculate P. 2、 An inves
8、tment if 1 will double in 27.72 year at a force if interest . An investment if 1 will increase to 7.04 in n years at a nominal rate of interest numerically equal to and convertible once every two years. Calculate n. 3、 Fund A accumulates at a rate of 12% convertible monthly. Fund B accumulates with
9、a force of interest , for all t .At time t0, 1 is deposited in each fund. T is the time that the two funds are equal, T0. Determine T (in years).4、 A 100,000 loan is to be repaid by 30 equal payments at the end of each year. The outstanding balance is amortized at 4%. In addition to the annual payme
10、nts, the borrower must pay an origination fee at the time the loan is made. This fee is 2% if the loan but does not reduce the loan balance. When the second payment is due, the borrower pays the remaining loan balance. Determine the yield to the lender considering the origination fee and the early p
11、ay-off if the loan. 5、 A perpetuity pay 1 at the end of every year plus an additional 1 at the end of every second year. The present value if the perpetuity if K for i0. Determine K.6、 An annuity pays 1 at the end of each 4-year period for 40 years. Given , find the present value of the annuity.7、 A
12、 35-year loan is to be repaid in equal annual installments. The amount of interest paid in the installment is 135.The amount of interest paid in the installment is 108.calculate the amount of interest paid in the installment.8、 Determine the present value of 1 payable at the end of years 7,11,15,19,
13、23,27.A、 C、 B、 D、 9、 You are given two series of payments. Series A is a perpetuity with payments of 1 at the end of each of the first 2 years, 2 at the end of each of the next 2 years,3 at the end of each if the next of the next 2 years, and so on. Series B is a perpetuity with payments if K at the
14、 end of each of the first 3 years, 2K at the end of each of the next 3 years, 3K at the end of each of the next 3 years, and so on. The present values of the two series of payments are equal. Calculate K. 10、Mary purchases an increasing annuity-immediate for 50,000 that makes twenty annual payments as follows:(1) P, 2p,10p in years 1 through 10(2) 10(1.05)p , 10(1.052)p, to 10(1.0510)p in years 11 through 20The annual effective interest rate is 7% for the first 10 years and 5