Microsoft PowerPoint - 4Term Stucture of Interest Rate(1st revised).ppt [相容模式]

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1 Term Structure of Interest Rates Fnancal Engneerng and Computatons Da, Tan-Shyr

2 此章內容 Fnancal Engneerng & Computaton 教課書的第五章 Term Structure of Interest Rates C 財務程式設計的第四章 4-1,

3 Outlne Introducton Spot Rates Extractng Spot Rates from Yeld Curves Spot Rate Curve, Forward drate Curve, Yeld ldcurve Forward Rates Lockng n the Forward Rates Term Structure Theory

4 Term Structure of Interest Rates The nterest rates vary wth maturty.

5 Term Structure of Interest Rates Concerned wth how nterest rates change wth maturty. The set of yelds to maturty for bonds forms the term structure. The bonds must be of equal qualty. Credt spread. They dffer solely n ther terms to maturty.

6 Yeld Curve

7 Four Shapes A normal yeld curve s upward slopng. An nverted yeld curve s downward slopng. A flat yeld curve s flat. A humped yeld curve s upward slopng at frst but then turns downward slopng. Normal curve Inverted curve

8 Bond yeld 和 Zero Rate Bond prce 的計算 cf c x n x= c y c y c F y n 不同的時間現金流用相同的 y折現 y 為市場交易的價格不同的現金流結構無法用來折現使用 zero rate 替代 n year zero rate zero coupon rate: 可視為 n n-year zero rate zero coupon rate: 可視為 n 年後到期的零息債券的 yeld rate

9 Spot Rate Zero rate It s defned as yeld of zero coupon bond A spot rate curve zero-coupon yeld curve s a plot of spot rates aganst maturty. Bond prce can be expressed : CF C n P P = C 1 S1 1 S: zero rate for th perod C 1 y: bond yeld S C F 1 S n P = C y C y C y F y 1 2 n n n

10 市場上無法直接觀察長天期 Spot rate 市場上交易的債券中短天期的債券多為 zero coupon bond 長天期的皆為 coupon bearng bond 無法觀察長天期的 Spot rate 可是市場折現利率應和 maturty 相關, 求長天期的 Spot rate 實屬必須藉由 coupon bearng bond 的拆解可得

11 方法簡介考慮一個二期的模型, 假定有兩個債券 B1,B2 到期日分別為第一和第二期. Bond yeld 分別為 y1,y2, y2 債券價格如下 P 1 c F = 1 y 1 P c c F = 1 y 2 1 y 考慮在第一期到期的零息債券, 其 zero rate S1=y1 考慮在第二期到期的零息債券, 其 zero rate S2 計算如下 : B2 可拆解成兩個零息債券, 一個在第一期到期, 面值為 c => 用 S1 折現另一個在第二期到期, 面值為 cf => 用 S2 折現可得以下算式 : c c F P2 = 1 S1 1 S2 2 因 S1 已知, 可求出 S2

12 n 期的 Zero Rate 計算假定 S1, S2, S3, Sn-1 皆已知 Sn 滿足以下式子 : Pn c c c F = S1 1 S2 1 S n n 移項可得 n 1 Sn = c F c c c Pn... 1 S 1 1 S 2 1 S n 1 n 2 n 1 Sn c F = 1 n c c c Pn... 2 n1 1 S1 1 S2 1 S n1 The procedure s called bootstrappng.

13 float ZeroRate[5]; 程式宣告 float Yeld[5]; float C; 輸入殖利率 scanf"%f",&c; fornt =0;<5;=1 { prntf" 輸入 Yeld rate %d:",1; scanf"%f",&yeld[]; } ZeroRate[0]=Yeld[0]; 第一期 Zero rate=yeld for=1;<=4; 計算第 1 期 zero rate { float BondValue=0; fornt =0;<=;=1 { 計算債券價格 P1 float Dscount=1; fornt k=0;k<=;k { Dscount=Dscount/1Yeld[]; for=0;<;=1 Bn... 2 n1 1 Z1 1 Z 2 1 Zn 1 { float PV=C; fornt k=0;k<=;k { PV=PV/1ZeroRate[]; } BondValue=BondValue-PV; } ZeroRate[]=powC100/BondValue,1.0/1-1 } for=0;<=4; { prntf" 第 %d 期 zero rate=%f\n",,zerorate[]; } } 列印 zero rate c c c } BondValue=BondValueDscount*C; f== { BondValue=BondValueDscount*100; dvl B dvl 見第四章 ZeroCurve proect } }

14 Compare Yeld Curve and dspot rate curve Spot rate curve s zero coupon yeld curve or zero curve. Spot rate curve s conssted of zero rate. zero coupon bond Yeld curve s conssted of bond yeld. coupon bearng bond zero coupon bond

15 Example Suppose the 1-year T-bll has yeld of 8%. Because ths securty s a zero-coupon bond, the 1-year spot rate s 8%. When the 2-year 10% T-note s tradng at 90, the 2-year spot rate satsfes = S 2 = % S2 or 2

16 Spot Rate Curve and Yeld Curve If the yeld curve s flat, the spot rate curve concdes wth the yeld curve. y k : yeld to maturty for the k-perod coupon bond. Sk y k, f y 1 < y 2 < yeld curve s normal. Sk y k, f y 1 >y 2 > yeld curve s nverted. Sk y k, f S1 < S2 < spot rate curve s normal. Sk y k, f S1 > S2 > spot rate curve s nverted.

17 Fgure 5.6: Shapes of Curves Forward rate curves wll be dscussed later.

18 Shapes The spot rate curve often has the same shape as the yeld curve. -If the spot rate curve s nverted, then the yeld curve s nverted, and vce versa. -However, a normal yeld curve does not guarantee a normal spot rate curve. When the fnal prncpal payment s relatvely nsgnfcant, the spot rate curve and the yeld curve do share the same shape. Bonds of hgh coupon rates and long maturtes.

19 Shapes Consder a 3-perod coupon bond that pays $1 per perod and repays the prncpal of $100 at maturty. Assume spot rates S1 = 0.1, S2 = 0.9, and S3 = Yelds to maturty are y 1 = 0.1, y 2 = , and y 3 = , not strctly ncreasng!

20 Yeld Spread Yeld spread s the dfference between the IRR of the rsky bond and that of a rsk-free bond wth comparable same maturty. P P rsk free rskey = n = 1 C1 y F 1 y n n n = c 1 y y ' F 1 y y' = 1 Where y' s the yeld spread.

21 Statc Spread The statc spread s the amount s by whch the spot rate curve has to shft n parallel n order to prce the rsky bond correctly, P rsky n Ct = t= 1 1 s S t Unlke the yeld spread, the statc spread ncorporates nformaton from the zero rate structure. The amount of statc spread can be consdered as the constant credt spread to the Treasury spot rate curve that reflects the rsk premum of a corporate bond. t

22 #Homework 5 假定 A 公司發行一個 n 期的債券, 每一期所要償還的票息為 C, 到期日時還需償還票面價值 100 元市場上 n 期債券的報酬率爲 R, 該債券的信用風險可用 yeld spread S 來表示, 所以該債券價格可示如下債券價格 = n = 1 C R S 1 R S 假定市場上的第期的零息利率可用 n 表示, 該公司的 statc spread 用 s 表示, 則債券價格為 Z

23 #Homework 5 n C 100 債券價格 = 1 1 Z s 1 Z s = n 請使用上述關係式, 撰寫程式只輸入每一期的 yeld rate 期數債息跟 yeld spread 差等訊息來計算 statc spread n 可參考 C 財務設式設計等章節

24 Forward Rate The forward rate reflect nformaton regardng future nterest rates mpled by the market. If we nvest $1 from now to th perod. 1 S 1 S = 1 S 1 f, f, = 1 1 S Tme 0 f,

25 Forward Rates By defnton, f0, = S. f, s called the mpled forward rates. -More precsely, the --perod forward rate perods from now.

26 Example: Spot and dforward drate In ths example, f $1 s nvest n 5-perod zerocoupon bond maturty strategy, t wll grow to be $ 1 1 S 5 5 An alternatve strategy s to nvest $1 n one-perod zero-coupon bonds and then renvest at the oneperod forward rates rollover strategy. The result s exactly the same as expected. $ 1 f 0,1 f 1,2 f 2,3 f 3,4 f 4,5 Forward rate

27 Forward Rate and Future Zero Rate We dd not assume any a pror relaton between f, and future spot rate S,. -Ths s the subect of the term structure theores. Term structure theores have dfferent explanaton. -Unbased expectaton theory. f,=es, -Lqudty preference theory. f,>es,

28 Unbased Expectatons Theory Forward rate equals the average future spot rate, fa, b = E[ Sa, b ] Imples that a normal spot rate curve s due to the fact that the market expects the future spot rate to rse. - f 1>S, 1 f and only f S 1>S > S. Therefore, E[ S, 1 ] > S 1 > > S1 f and only f S 1> > >S1 S1. Conversely, the spot rate s expected to fall f and only f the spot rate curve s nverted.

29 Lqudty Preference Theory The lqudty preference holds that nvestors demand a rsk premum for holdng long-term bonds. Ths mples that fa,b >E Sa,b. Even f people expect the nterest rate to declne and rse equally, the theory asserts that the curve s upward slopng more often.

30 Spot and Forward Rate under p Contnuous Compoundng The formula for the forward rate: e e e f S S =, S S f f S S e e e = = =,, The spot rate s an arthmetc average of forward rates. f f f S S e e e e e f f f S S 1, 2,3... 1, , 2,3 1,2 1 = = f f f f S f f f 1, 2,3... 1,2 0,1,,, =

31 Spot and Forward Rate under Contnuous Compoundng The one-perod forward rate: f, 1 = 1 S 1 S 5.10 Under contnuous tme nstead of fdscrete tme, the nstantaneous forward rate at T tme equals f T, T ΔT f T lm = S T f T, T ΔT ΔT S T ΔT S T = S T T S ΔT 0 T T T Δ 5.11 Note that ft >ST f and only f ƏS/ƏT >0

32 Example: Spot and Forward Rate Compute the one-perod forward rates from ths spot rate curve: S1:2.0%, S2:2.5%, S3:3.0%, S4:3.5%, S5:4.0%. Answer: 2 f 1,2 = 2.5 f 1,2 = 3% f 2,3 = 3 f 2,3 = 4% f 3,4 = 3.5 f 3,4 = 5% f 4,5 = 4 f 4,5 = 6% 5

33 殖利率曲線, 零息利率曲線, 遠期利率曲線關係 Normal curve Inverted curve b a ba 1 Z b = 1 Za 1 f a, b 當 Zb>Za=> fa,b>zb>za

34 Spot Rate and Forward Rate When the spot rate curve s normal, the forward rate domnates the spot rates, f, > S > > S. When the spot rate curve s nverted, the forward rate s domnated dby the spot rates, f, < S < < S.

35 Lockng n the Forward Rates Forward rates may not be realzes n the future f, S,, but we can lock n any forward rate f,. Now we can make followng strateges. -Buy B 1 unt -year zero-coupon bond. -Sell 1 S 1 S unts -year zero-coupon bonds. No net ntal nvestment, because 1 S 1 S 1 1 S S = 0

36 Lockng n the Forward Rates At tme there wll be a cash nflow of $1. At tme there wll be a cash outflow of $ 1 S 1S The cash flow stream mples the rate f, between tmes and. 1 S 1 f, = 1 1 S $ 1S 1 S $0 f, $1

37 C: 遠期利率的鎖定時間時可拿到 $1 時間時需支付 $ 利率計算 => 1 Z 1 Z 1 Z 1 f, 1 Z = 1 計算購買一單位的時間到期的零息債券, 須賣出多少單位時間到期的零息債券, 才能鎖住遠期利率 f,, 其中 0<<<=5, 並將結果存在二維陣列 LockStrategy[,] 中

38 LockStratrgy 陣列行編號列編號 X X X X X X 1 X X 2 X X X 1 Z 1 3 X X X X 1 Z Z1 1 Z1 1 Z1 2 Z Z3 1 Z4 1 Z5 1 Z 2 1 Z 2 1 Z Z 3 1 Z 4 1 Z Z3 1 Z Z4 1 Z5 4 1 Z4 5 1 Z5 4 X X X X X 5 X X X X X X

39 #nclude <math.h> #nclude <stdo.h> 完整程式碼 vod man { double unt_short[6][6]; double ZeroRate[6]; nt,; // 輸入零息利率 for=1;<=6; { } prntf" 請輸入第 %d 期的零息利率 : ",; scanf"%lf",&zerorate[-1]; } // 計算鎖定 f, 需放空零息債券的單位數 for =0;<6; for=;<6; unt_short[][] = pow1zerorate[],/pow1zerorate[],; // 輸出結果 for =0;<5; for=1;<6; prntf 鎖定 f%d,%d -- 放空 %lf 單位第 %d 期零息債券, 買入一單位第 %d 期的零息債券 \n",1,1,unt_short[][],1,1;

40 Homework 6 Exercse The fact that forward rate can be locked n today means that future spot rates must equal today s forward rates, or Sa,b b=fa,b, n a certan economy. Why? How about an uncertan economy? Hnt: 可舉一個簡單實例, 用套利的觀念來說明

東吳大學 104 學年度碩士班研究生招生考試試題第 2 頁, 共 7 頁 5. Consider a project with the following cash flows. Year Cash Flow 0 -$16, , ,000 What s the IRR o

東吳大學 104 學年度碩士班研究生招生考試試題第 2 頁, 共 7 頁 5. Consider a project with the following cash flows. Year Cash Flow 0 -$16, , ,000 What s the IRR o 東吳大學 104 學年度碩士班研究生招生考試試題第 1 頁, 共 7 頁一選擇題 60 分 ( 單選每題 3 分 ) 1. Which of the following items can be found on an income statement? a. Accounts receivable b. Long-term debt c. Sales d. Inventory 2. A 15-year,