收益率 Yield Rate 大学教学课件

1、1,收益率 Yield Rate,孟生旺 中国人民大学统计学院 ,2,主要内容,现金流分析 净现值 收益率 再投资收益率 基金的收益率 币值加权收益率 时间加权收益率 收益分配 投资组合法 投资年度法,3,问题：已知一个项目的现金流，如何评价其收益的好坏？ 方法一：净现值法（net present value，NPV） 净现值大于零，表示项目有利可图。 净现值越大，表示项目的收益越好。,1. 现金流分析,4,方法二：收益率法 当资金流入的现值与资金流出的现值相等时，所对应的利率 i 称为收益率（yield rate）或 内部报酬率（internal rate of return，IRR） 资金

2、流入的现值与资金流出的现值之差就是净现值，所以收益率也是使得净现值为零的利率： 收益率越大，表示项目的收益越高。,5,例: 如果期初投资20万元，可以在今后的5年内每年末获得5万元的收入。假设投资者A所要求的年收益率为7%，投资者B所要求的年收益率为8%，试通过净现值法和收益率法分别分析投资者A和投资者B的投资决策。 解：（1）该项目的净现值为 按A所要求的收益率7%计算，净现值为0.5万元，大于零，可投资。 按B所要求的收益率8%计算，净现值为 0.036万元，小于零，不可投资。,6,（2）如果令净现值等于零，即 可以计算出该项目的收益率为7.93%， 大于A所要求的收益率（7%），可行 小

3、于B所要求的收益率（8%），不可行,7,Project P requires an investment of 4000 at time 0. The investment pays 2000 at time 1 and 4000 at time 2. Project Q requires an investment of x at time 2. The investment pays 2000 at time 0 and 4000 at time 1. The net present values of the two projects are equal at an interest r

4、ate of 10%. Calculate x.,Example,8,Solution:,Equate net present values:,9,例（不存在）： ，求收益率。 解: 净现值为 由于 ，因此该方程无实数解，不存在收益率。,收益率的唯一性,10,Example（不唯一）：Consider a transaction in which a person makes payments of $100 immediately and $132 at the end of two years in exchange for a payment in return of $230 at t

5、he end of one year. 解：价值方程为,100,132,230,11,多重收益率情况下的净现值（续前例）,100,132,230,多重收益率：,净现值：,（下图）,12,要求10%以下的收益率，净现值小于零，项目不可行。 要求10%20%的收益率，净现值大于零，项目又是可行的！ 要求20%以上的收益率，净现值小于零，项目又不可行。,13,收益率唯一性的条件,准则一：计算资金净流入，如果资金净流入只改变过一次符号，收益率将是唯一的。 准则二：用收益率计算资金净流入的累积值 ，如果始终为负，直至在最后一年末才为零，那么这个收益率就是唯一的。 （见下页例，收益率为9.98396% ）

6、,14,投资项目的资金流出和资金流入,15,2. 再投资收益率,再投资：投资收入按新的利率进行投资。 例：考虑两种可选的投资项目 A）投资5年，每年的利率为9 B）投资10年，每年的利率为8 如果两种投资在10年期间的收益无差异，计算项目A在5年后的再投资收益率应为多少？,16,例：有一笔1000万元的贷款，期限为10年，年实际利率为 9%，有下面三种还款方式： 本金和利息在第10年末一次还清； 每年末偿还当年的利息，本金在第10年末归还。 在10年内每年末偿还相同的金额。 假设偿还给银行的款项可按7的利率再投资，试比较在这三种还款方式下银行的年收益率。,17,解: （1）无需再投资，收益率为

7、 i9 （2）所有付款在第10年末的累积值为 价值方程：,18,（3）所有付款在第10年末的累积值为 价值方程： i=7.97%,19,Exercise,Mary invests 1000 at the end of each year for 5 years at an annual effective rate of 9%, and reinvests the interest at an annual effective rate of 9%. At the end of 5 years, her investment has value of X. John invests 1000

8、at the beginning of each year for 5 years at an annual effective rate of 10% and reinvests the interest at an annual effective rate of 8%.At the end of 5 years, his investment has value Y . Calculate Y X.,20,Solution:,The future value at time 5 of Marys investments is,First，we find the balance in Jo

9、hns account:,Hence, the interest payments John gets are:,The accumulation of these interest payments at time 5 is:,The total accumulation of Johns investments is 6669.91.,Hence, the difference is 6669.91 5984.71 = 685.2.,21,Exercise:,Payments of 1000 are invested at the end of each year for 5 years.

10、 The payments earn interest at an annual effective rate of 10%. The interest can be reinvested at an annual effective rate of 6% in the first 4 years and at an annual effective rate of k there after. The amount in the fund at the end of 5 years is 6090. Calculate k.,22,Solution :,k = 10.51%,Hence, t

11、he interest payments are:,23,Exercise:,Eric deposits 12 into a fund at time 0 and an additional 12 into the same fund at time 10. The fund credits interest at an annual effective rate of i. Interest is payable annually and reinvested at an annual effective rate of 0.75i. At time 20, the accumulated

12、amount of the reinvested interest payments is equal to 64. Calculate i, i 0.,24,The future value at time 20 of this cash flow is,i = 9.57%,25,Exercise:,Victor invests 300 into a bank account at the beginning of each year for 20 years. The account pays out interest at the end of every year at an annu

13、al effective interest rate of i . The interest is reinvested at an annual effective rate of (i/2). The yield rate on the entire investment over the 20 year period is 8% annual effective. Determine i .,26,Solution:,Since the yield rate on the entire investment over the 20 year period is 8% annual eff

14、ective, the accumulated value at the end of 20 years is,The cash flow of the interests is,So, the future value at time 20 of this cahsflow is,27,Exercise:,At time t = 0, Sebastian invests 2000 in a fund earning 8% convertible quarterly, but payable annually. He reinvests each interest payment in ind

15、ividual separate funds each earning 9% convertible quarterly, but payable annually. The interest payments from the separate funds are accumulated in a side fund that guarantees an annual effective rate of 7%. Determine the total value of all funds at t = 10.,28,Solution:,A nominal rate of 8% convert

16、ible quarterly is equivalent to an effective annual rate of 8.2432%. The interest earned each year is (2000)(0.082432) = 164.86. A nominal rate of 9% convertible quarterly is equivalent to an effective annual rate of 9.3083%. The interest obtained from 164.86 invested at an effective annual rate is

17、9.3083% is (164.86)(0.093083) =15.3457. So, the cash flow going into the account paying an annual effective rate of 7% is,29,The future value of this cash flow is,The total value of all funds is 2000 + (10)(164.84) + 836.66 = 4485.06.,30,Exercise:,Susan invests Z at the end of each year for seven ye

18、ars at an annual effective interest rate of 5%. The interest credited at the end of each year is reinvested at an annual effective rate of 6%. The accumulated value at the end of seven years is X . Lori invests Z at the end of each year for 14 years at an annual effective interest rate of 2.5%. The

19、interest credited at the end of each year is reinvested at an annual effective rate of 3%. The accumulated value at the end of 14 years is Y . Calculate Y/X .,31,Solution:,We may assume that Z = 1. Susans accumulated value consists of her $7 invested plus the cashflow for her interest. The cashflow

20、of her interest payments are,The accumulated value of this cash flow is,So, Susans accumulated value is X = 8.1653.,32,The accumulated value of this cash flow is,So, Susans accumulated value is Y = 16.5719. We have that Y/X = 2.0296.,Loris accumulated value consists of her $14 invested plus the cash

21、 flow for her interest. The cash flow of her interest payments are,33,Exercise :,At time t = 0, John invests 5 in a fund earning i annually. He reinvests each interest payment in individual separate funds each earning j annually. The interest payments from the separate funds are accumulated in a sid

22、e fund that guarantees an annual effective rate of h. Determine the total value of all funds at t = 10.,34,基金的收益率：币值加权收益率(dollar-weighted yield rate ),基金的利息度量(Interest measurement of a fund) ： 币值加权收益率(dollar-weighted yield rate )：投资者 时间加权收益率(time-weighted yield rate )：经理人,35,币值加权收益率：简单近似,假设：本金在当期的变化

23、是平稳的。 符号（仅考虑一个时期） 期初的本金余额：A0 期末累积值：A1 当期产生的利息： I 期末的本金余额：(A1 I) 当期的平均本金余额：(A0 + A1 I )/2 当期的收益率可近似表示为,36,Example,Find the effective rate of interest earned during a calendar year by an insurance company with the following data: Asset, beginning of year 10,000（初始本金） Premium income 1,000（新增投资） Gross i

24、nvestment income 530 Policy benefits 420（负的新增投资） Investment expenses 20 Other expense 180（负的新增投资）,37,解：假设保费收入和各项支出在年内均匀分布 年初本金 A010,000 年末累积值 A110,0001,00053042020180 10,910 当年利息 I = 53020 = 510 或 I = A1 A0 ( 1000420 180) = 510,38,币值加权收益率：精确计算,假设：期初的本金为A0，在时刻t的新增投资为Ct，投资收益率为i，在期末的累积值可表示为（仅考虑一个时期）,注：

25、时刻t的新增投资额Ct 产生收益的时间长度为(1 t),用A1表示期末的累积值，则有,注：Ct 0表示增加投资； Ct 0表示减少投资。,解此方程即可求得收益率 i,39,币值加权收益率：近似计算,对于不足一个时期的新增投资，用单利近似复利，即令,则期末累积值可表示为,40,对近似公式的解释：,（1）分母是加权平均本金余额，以本金产生利息的时间长度为权数。,（2）如果进一步假设新增投资发生在期中，即t = 0.5，则,41,Example:,At the beginning of the year, an investment fund was established with an init

26、ial deposit of 1000. A new deposit of 1000 was made at the end of 4 months. Withdrawals of 200 and 500 were made at the end of 6 months and 8 months, respectively. The amount in the fund at the end of the year is 1560. Calculate the dollar-weighted yield rate earned by the fund during the year.,42,解

27、：,年初本金A0： 1000 新增投资 Ct： 1000，200，500 新增合计：C300 新增投资时间：4月末，6月末，8月末 年末累积值 A1：1560 当年利息： I=A1A0C == 260,注：分母可以看作是平均本金余额。,43,投资期限超过1年的情形,已知投资期是从0到n，第n期末的累积值为An 。第j期的新增投资为Cj ，在复利方式下，第n期末的累积值为 通过数值方法可以解出币值加权收益率 i。 注：当 时，不能使用单利来近似计算。,44,如果假设新增投资是连续的，即Ct 是 t 的连续函数，表示时刻t的支付率。用At表示时刻t的累积值，则 解释：在时

28、刻t的累积值等于初始本金积累t个时期，加上新增投资Cs积累到时刻t。,45,4. 基金的收益率： 时间加权收益率（time-weighted rates of interest）,币值加权收益率： 受本金增减变化的影响。而本金的增减变化由投资者决定。 可以衡量投资者的收益，但不能衡量经理人的经营业绩。 时间加权收益率：扣除了本金增减变化的影响后所计算的一种收益率。 衡量经理人的业绩。,46,时间加权收益率（i）的一般公式,期初的本金为A(0)。在第 k 个时间区间末的累积值为Ak，新增投资为Ck（k = 1, 2, , n）。在期末T 的累积值为A(T),47,Example:,On Janu

29、ary 1 an investment account is worth $100,000. On May 1 the value has increased to $112,000 and $30,000 of new principal is deposited. On November 1 the value has declined to $125,000 and $42,000 is withdrawn. On January 1 of the following year the investment account is again worth $100,000. Compute

30、 the yield rate by: (1) the dollar-weighted method; (2) the time-weighted method.,48,解：（1）币值加权收益率 由公式 A1A0CI 有 100,000100,000+(30,000-42,000)+I I =12,000 故,49,（2）时间加权收益率 分析：为什么两种收益率差距这么大？在效益差的时期投入了新的资本，在效益好的时期提取了部分本金。,(1.12),(0.88),(1.20),1.12,0.88,1.20,50,Example:,1,000 is deposited into a fund on

31、January 1, 1999. Another deposit is made into the fund on July 1, 1999. On January 1, 2000, the balance in the fund is 2,000. The time-weighted yield rate is 10% and the dollar-weighted yield rate is 9%. Calculate the annual effective interest rate earned on the fund during the first six month of 19

32、99.,1000,A0.5 C0.5,2000,？,51,解：设1999年7月1日的投入额为 ，基金余额为 。由币值加权收益率公式有,由时间加权收益率公式有,故,1000,A0.5 C0.5,2000,52,5. 收益分配(Allocating investment income),问题： 基金包括不同时期的投资。 如何把基金的投资收入分配给不同时期的投资？ 分配方法： 投资组合法(portfolio method) 投资年度法(investment method),53,投资组合法(portfolio methods),适用情况：基金的收益率水平一直保持恒定 方法：按照基金的平均收益率分配投

33、资收入。 例：如果基金的收益率一直保持在6%的水平，某个投资者的投资额是10000元，存入基金的时间是9个月，那么应该分配给他的利息收入应该是 复利法：10000(1 + 0.06)0.75 10000 = 446.71 单利法：100006%9/12=450,54,存在的问题：在短期内，简单易行。如果投资期限较长，收益率波动较大，可能不公平。 例： 假设：基金在最近3年的平均年收益率为8， 当年的收益率达到了10， 如果对新投资者仍然以8分配收益， 结果：可能会使其放弃对该基金的投资，或者不能吸引更多的新投资者。 如何解决？用投资年度方法分配收益。,55,投资年度法(investment y

34、ear method，new year method),适用情况：收益率波动较大。 方法：新存入的资金在若干年内按投资年度利率分配利息，而超过一定年数以后，再按组合利率分配利息。 投资年度利率：考虑投资发生日期的利率，投资发生的日期不同，投资年度利率是不同的。 组合利率：不同年度投资的平均利率。,56,投资年度利率：例,57,Example:,The following table shows the annual effective interest rates being credited by an investment account, by calendar year of inve

35、stment. The investment year method is applicable for the first 3 years, after which a portfolio rate is used: An investment of 100 is made at the beginning of years 1990, 1991, and 1992. The total amount of investment interest credited by the fund during the year 1993 is equal to 28.40 Find r.,58,An

36、 investment of 100 is made at the beginning of years 1990, 1991, and 1992. The total amount of investment interest credited by the fund during the year 1993 is equal to 28.40. Find r.,59,解： 90年初的投资到93年初的积累值为：100(1.10)(1.10)(1+r)=121(1+r) 故在93年的利息收入： 121(1+r)(0.08) 91年初的投资到93年初的积累值为100(1.12)(1.05)117

37、.6 故在93年的利息收入为 117.6(0.1)=11.76 92年初的投资到93年初的积累值为：100(1.08)108 故在93年的利息收入为 108(r-0.02) 总利息收入：121 (1+r) (0.08) + 11.76 + 108(r-0.02)=28.4 故 r = 7.75%,60,Exercise：,You are given the following table of interest rates. A person deposits 1000 on January 1, 1997. Let the following be the accumulated value of the 1000