03现值和内部收益率

1、02-02-1 1章章 贤贤 军军02-02-2 2Present Value (PV)Internal Rate of Return (IRR)Cash Flow EvaluationDiscount factorSpot and forward rate02-02-3 3用用 x=(x0, x1, . , xn) 表示一个现金流表示一个现金流timex0 x1x2x3x4x5x6timePV02-02-4 4现值 present value. timex0 x1x2x3x4x5x6timePV终值 final value. timex0 x1x2x3x4x5x6timeFV02-02-5

2、5Present Value (PV)Internal Rate of Return (IRR)Cash Flow EvaluationDiscount factorSpot and forward rate02-02-6 602-02-7 702-02-8 8218.75218.75218.75218.7510218.7594.2.1594.8.1595.2.1595.8.1596.2.1596.8.1502-02-9 902-02-1010201101(1006.875)(0.5)322(0.5)0.9825dd18101=2.75 d(0.5)+102.75 d(1)3202-02-11

3、1121100=2.3125 d(0.5)+2.3125 d(1)32 +102.3125 d(1.5)02-02-121202-02-131302-02-1414Present Value (PV)Internal Rate of Return (IRR)Cash Flow EvaluationDiscount factorSpot and forward rate02-02-151502-02-1616( )r tAA(1+r/2)2ttime0t面值:100元小于100元02-02-1717121( )2()1( )tr tdt已知折现因子d(t)表示t年之后的1元的现值，因此有：2(

4、)112trdtAA(1+r/2)2ttime0t02-02-1818( )r t02-02-191902-02-2020根据图3.2显示，即期利率最初具有斜率向上的期限结构，但大约到第23年后，即期利率具有斜率向下的趋势 downward sloping02-02-212102-02-222252.552.5 52.5 100094.2.1594.8.1595.2.1595.8.1502-02-2323价格:票息:面值:120 xPyPP120 xCyCC120 xFyFF可以组合一个 zero coupon bond, 使C0=0, F0=$100这个组合要满足3个公式:解出x和y.组合而

5、成的零息债券的价格 xP1+yP202-02-2424Bond 1: 10 year, 10% coupon, P1=98.72元Bond 2: 10 year, 8% coupon, P2=85.89元 面值100元Portfolio: x Bond 1, y Bond 2.构建的零息债券价格:x = - 4, y = 5Coupon: x(10) + y(8) = 0Face: x(100) + y(100) = 100Create a zero:Spot Rate:201010034.57(1/2)r100.11r xP1+yP2=34.57元02-02-2525使用即期利率，则债券现值

6、又可以表示为232 .3 1 2 52 .3 1 2 51 0 2 .3 1 2 5(0 .5 )(1)(1 .5 )111222rrr02-02-2626以这种方式表示，说明现金流量根据发生当时的即期利率来折现。换句话说，投资者对于不同时期发生的现金流量，将赚取不同的报酬率02-02-2727( )r t02-02-2828 A(1+ ) A(1+ )（二）（二） 一年期债券一年期债券 收益率收益率 A(1+ )2 两年期债券两年期债券 第一年第一年的收益率的收益率 A(1+ )202-02-292902-02-3030(1.5)r r(0.5)94.2.1594.8.1595.2.1595

7、.8.15 r(1) r(1.5)(1)r (2)r 96.2.15 r(2)02-02-3131r(0.5)= (0.5)=3.567%r 2(0.5)(1)(1)111222rrr3(0.5)(1)(1.5)(1.5)1 1 112222rrrr2(0.5)(1)( )( )1 1 . 112222trrr tr t(1)(2)(3)(4)02-02-3232( )r t02-02-333302-02-34342.31252.3125102.3125(0.5)(0.5)(1)(0.5)(1)(1.5)111111222222rrrrrr232.31252.3125102.3125(0.5)

8、(1)(1.5)111222rrr02-02-3535Present Value (PV)Internal Rate of Return (IRR)Cash Flow EvaluationDiscount factorSpot and forward rate02-02-3636用 x=(x0, x1, . , xn) 表示现金流. 内部收益率指能够满足下式的r1202.01(1)(1)nnxxxxrrr为了解出 r, 通常会设:1(1)crand solve2012.0nnxxcx cx c( , )0PV x r 02-02-3737对现金流 (-2,1,1,1)，它的内部收益率是多少?S

9、olve:2302ccc Question: 这个多项式只有惟一的正解吗?Answer: This can be a problem.0.81c 0.23r11cr02-02-38382012.0nnxxcx cx cProof: plot2012( ).nnf cxx cx cx cf(c)cx0increasing.如果现金流 x=(x0, x1, ., xn) 有x0 0 至少有一项是正的. 那么对以下方程式就有惟一的正解:0kx 02-02-393902-02-404022111222TtTtcyyP上式又可以表示为，221 (1) (1)22TTcyyPy02-02-4141221(

10、1)(1)22TTcyyPy02-02-4242Present Value (PV)Internal Rate of Return (IRR)Cash Flow EvaluationDiscount factorSpot and forward rate02-02-4343When to cut a tree？种树为了卖木材赚钱，有两种砍伐方案:(1) After 1 year: Cash Flow: (-1,2) (2) After 2 years: Cash Flow: (-1,0,3) 如果r1 is 10% r2 is 12%,你会选哪个?02-02-4444(1) NPV = -1

11、+ 2/1.1 = 0.82(2) NPV = -1 + 3/(1.12)2 = 1.392NPV says choose option (2). 02-02-4545Choose the one with the greatest internal rate of return. Solution:IRR says choose option (1)(1) IRR: -1+2c=0 c=0.5 r=1.0(2) IRR: -1+3c2=0 c=0.577 r=0.702-02-464602-02-47473510011100.080.036280.036280.0362811122202-0

12、2-484802-02-494902-02-50502342.31252.31252.3125102.3125100.37511112222yyyy4100.375(1)2y02-02-51513242.3125(1)2.3125(1)2.3125(1) 102.3125222100.375(1)2yyyy这个式子的左侧可以解释为，在94.8.15获得第一笔票息2.3125元，根据到期收益率y再投资至到期日，收益为2.3125(1+y/2)3 。以次类推，假定再投资利率为y，在到期日收取的所有金额就是100.375(1+y/2)4 ，如此则到期收益率代表了实现的债券收益率。02-02-5252

13、2.312594.2.1594.8.1595.2.1595.8.1596.2.152.31252.3125102.312502-02-5353比例变化 Scaling:IRR 体现不了变化, NPV可以 02-02-5454重复 RepetitionNPV可以区分现金流是否重复, IRR 不能02-02-5555NPV和IRR不一定得到相同的结论。IRR也并不能有效区分债券的收益高低。NPV可以体现现金流的多种情况，所以在评价现金流的时候比IRR更好。 02-02-565602-02-5757下表是各年期零息债券（面值1000元）的报价1、请填写表中空格。此处均为半年复利一次的名义利率。2、现

14、金流x=(x0,x0.5,x1,x1.5)为(-1020,85,85,1085) 根据上表计算它的净现值和内部收益率。02-02-58581. 02-02-595918%987.3102018%14%987.31039.3615.4%yy02-02-606002-02-616123nSxxxx2341.nnxSxxxxx1(1)nxSxx11/ 2xy02-02-626202-02-6363Q: Why does the term structure shape like this?Three Explanations1 期望理论 Expectations Theory2 流动性偏好 Liqu

15、idity Preference3 市场分割 Market Segmentation02-02-6464即期利率是由对利率的市场预期决定的。Spot rates are determined by the expectations of what interest rates will be.Ex: If the spot rate curve slopes upward, that is because people expect interest rates to increase in the future. 02-02-6565Investors prefer short term s

16、ecurities over long term securities because long term securities “tie-up” capitalthat may be needed before maturity. However, there are well developed markets for short and long term securities. Hence investors can easily get out.More likely, long term investments are more sensitive to interest rate

17、s. To lessen risk, investors would prefershort term securities.02-02-6666Fixed income market is segmented by maturity dates. The rate for each maturity date is determined by the group of investors interested in that maturity date. Hence, simple supply and demand among those investors determines the interest rate. 02-02-676702-02-6868财富/资产 收入 资本收入的市值金融理论金融理论价格理论产权理论产权理论02-02-696902-02-707002-02-717102-02-727202-02-737302-02-747402-02-7575262.8238.911.102306.2253.06(1.10)3322.3242.15(1