投資學第三章《現(xiàn)值和內部收益率》ppt課件

1、用用 x=(x0, x1, . , xn) 表示一個現(xiàn)金流表示一個現(xiàn)金流 。我們需。我們需求找到方法來比較不同的現(xiàn)金流。求找到方法來比較不同的現(xiàn)金流。折現(xiàn)因子折現(xiàn)因子 Discount factors 和和 利率利率 Interest rates 可以在時間軸上挪動現(xiàn)金流可以在時間軸上挪動現(xiàn)金流.timex0 x1x2x3x4x5x6timePV現(xiàn)值 present value. timex0 x1x2x3x4x5x6timePV終值 final value. timex0 x1x2x3x4x5x6timeFVPresent Value (PV)Internal Rate of Return

2、(IRR)Cash Flow EvaluationDiscount factorSpot and forward rate218.75218.75218.75218.7510218.7594.2.1594.8.1595.2.1595.8.1596.2.1596.8.15201101(1006.875)(0.5)322(0.5)0.9825dd18101=2.75 d(0.5)+102.75 d(1)3221100=2.3125 d(0.5)+2.3125 d(1)32 +102.3125 d(1.5)Present Value (PV)Internal Rate of Return (IRR)

3、Cash Flow EvaluationDiscount factorSpot and forward rate( )r tAA(1+r/2)2ttime0t面值:100元小于100元121( )2()1( )tr tdt2( )112trdtAA(1+r/2)2ttime0t( )r t根據(jù)圖3.2顯示，即期利率最初具有斜率向上的期限構造，但大約到第23年后，即期利率具有斜率向下的趨勢 downward sloping52.552.5 52.5 100094.2.1594.8.1595.2.1595.8.15價錢:票息:面值:120 xPyPP120 xCyCC120 xFyFF可以組合一個

4、 zero coupon bond, 使C0=0, F0=$100這個組合要滿足3個公式:解出x和y.組合而成的零息債券的價錢 xP1+yP2Bond 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.構建的零息債券價錢:x = - 4, y = 5Coupon: x(10) + y(8) = 0Face: x(100) + y(100) = 100Create a zero:Spot Rate:201010034.57(1/2)r

5、100.11r xP1+yP2=34.57元運用即期利率，那么債券現(xiàn)值又可以表示為232 .3 1 2 52 .3 1 2 51 0 2 .3 1 2 5(0 .5 )(1)(1 .5 )111222rrr以這種方式表示，闡明現(xiàn)金流量根據(jù)發(fā)生當時的即期利率來折現(xiàn)。換句話說，投資者對于不同時期發(fā)生的現(xiàn)金流量，將賺取不同的報酬率( )r t(1.5)r r(0.5)94.2.1594.8.1595.2.1595.8.15 r(1) r(1.5)(1)r (2)r 96.2.15 r(2)12wr(0.5)= (0.5)=3.567%r 2(0.5)(1)(1)111222rrr3(0.5)(1)(

6、1.5)(1.5)1 1 112222rrrr2(0.5)(1)( )( )1 1 . 112222trrr tr t(1)(2)(3)(4)( )r t2.31252.3125102.3125(0.5)(0.5)(1)(0.5)(1)(1.5)111111222222rrrrrr232.31252.3125102.3125(0.5)(1)(1.5)111222rrrPresent Value (PV)Internal Rate of Return (IRR)Cash Flow EvaluationDiscount factorSpot and forward rate用 x=(x0, x1,

7、 . , xn) 表示現(xiàn)金流. 內部收益率指可以滿足下式的r1202.01(1)(1)nnxxxxrrr為了解出 r, 通常會設:1(1)crand solve2012.0nnxxcx cx c( , )0PV x r 對現(xiàn)金流 (-2,1,1,1)，它的內部收益率是多少?Solve:2302ccc Question: 這個多項式只需獨一的正解嗎?Answer: This can be a problem.0.81c 0.23r11cr2012.0nnxxcx cx cProof: plot2012( ).nnf cxx cx cx cf(c)cx0increasing.假設現(xiàn)金流 x=(x0

8、, x1, ., xn) 有x0 0 至少有一項為哪一項正的. 那么對以下方程式就有獨一的正解:0kx 22111222TtTtcyyP上式又可以表示為，221 (1) (1)22TTcyyPy221(1)(1)22TTcyyPyPresent Value (PV)Internal Rate of Return (IRR)Cash Flow EvaluationDiscount factorSpot and forward rate種樹為了賣木材賺錢，有兩種砍伐方案:(1) After 1 year: Cash Flow: (-1,2) (2) After 2 years: Cash Flow

9、: (-1,0,3) 假設r1 is 10% r2 is 12%,他會選哪個?(1) NPV = -1 + 2/1.1 = 0.82(2) NPV = -1 + 3/(1.12)2 = 1.392NPV says choose option (2). Choose 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=0.73510011100.080.036280.

10、036280.03628111222( )r t2342.31252.31252.3125102.3125100.37511112222yyyy4100.375(1)2y3242.3125(1)2.3125(1)2.3125(1) 102.3125222100.375(1)2yyyy這個式子的左側可以解釋為，在94.8.15獲得第一筆票息2.3125元，根據(jù)到期收益率y再投資至到期日，收益為2.3125(1+y/2)3 。以次類推，假定再投資利率為y，在到期日收取的一切金額就是100.375(1+y/2)4 ，如此那么到期收益率代表了實現(xiàn)的債券收益率。2.312594.2.1594.8.159

11、5.2.1595.8.1596.2.152.31252.3125102.3125比例變化 Scaling:IRR 表達不了變化, NPV可以 反復 RepetitionNPV可以區(qū)分現(xiàn)金流能否反復, IRR 不能NPV和IRR不一定得到一樣的結論。IRR也并不能有效區(qū)分債券的收益高低。NPV可以表達現(xiàn)金流的多種情況，所以在評價現(xiàn)金流的時候比IRR更好。 下表是各年期零息債券面值1000元的報價1、請?zhí)顚懕碇锌崭?。此處均為半年復利一次的名義利率。2、現(xiàn)金流x=(x0,x0.5,x1,x1.5)為(-1020,85,85,1085) 根據(jù)上表計算它的凈現(xiàn)值和內部收益率。1. 18%987.3102

12、018%14%987.31039.3615.4%yy23nSxxxx2341.nnxSxxxxx1(1)nxSxx11/ 2xyQ: Why does the term structure shape like this?Three Explanations1 期望實際 Expectations Theory2 流動性偏好 Liquidity Preference3 市場分割 Market Segmentation即期利率是由對利率的市場預期決議的。Spot rates are determined by the expectations of what interest rates will

13、 be.Ex: If the spot rate curve slopes upward, that is because people expect interest rates to increase in the future. Investors prefer short term securities over long term securities because long term securities “tie-up capitalthat may be needed before maturity. However, there are well developed mar

14、kets for short and long term securities. Hence investors can easily get out.More likely, long term investments are more sensitive to interest rates. To lessen risk, investors would prefershort term securities.Fixed 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. 財富/資產 收入 資本收入的市值金融實際金融實際價錢實際產權實際產權實際262.8238.911.102306.2253.06(1.10)3322.3242.15(1.10)4