YIELD CURVE FITTING 2.0 Constructing Bond and Money Market Yield Curves using Cubic B-Spline and Natural Cubic Spline Methodology.

Transcription

1 YIELD CURVE FITTING 2.0 Constructng Bond and Money Market Yeld Curves usng Cubc B-Splne and Natural Cubc Splne Methodology Users Manual

2 YIELD CURVE FITTING 2.0 Users Manual Authors: Zhuosh Lu, Moorad Choudhry YeldCurve.Com Copyrght Zhuosh Lu Verson 2.0 March 2005

3 Table of Contents INTRODUCTION 1 YCF 2.0 USER INTERFACE 2 MAIN PAGE 2 BOND MARKET 3 A. INPUT INITIAL PARAMETERS SECTION 3 B. INPUT COUPON BOND MARKET DATA SECTION 3 C. YIELDS CURVE OUTPUT SECTION 4 MONEY MARKET AND SWAPS 7 A. INPUT INITIAL PARAMETERS SECTION 7 B. INPUT MONEY MARKET RATES SECTION 8 C. INPUT SWAP RATES SECTION 8 D. YIELDS CURVE OUTPUT SECTION 10 YIELD CURVE ESTIMATING AND FITTING METHODS 13 BASIC FIXED INCOME FORMULAE 13 A. DISCOUNT FUNCTIONS 13 B. ZERO RATES 13 C. FORWARD RATES 14 D. DAY COUNT FRACTION 14 CURVE FITTING METHODS 14 A. NATURAL CUBIC SPLINE 14 B. CUBIC B-SPLINE 15 DISCOUNT FACTORS ESTIMATION 16 A. CUBIC B-SPLINE METHOD 16 B. BOOTSTRAPPING METHOD 17 YIELD CURVE CONSTRUCTION 20 A. WITH COUPON BOND DATA 20 B. WITH MONEY MARKET DATA 20 C. WITH MONEY MARKET AND SWAPS DATA 20 TROUBLESHOOTING AND Q&A 22 TROUBLESHOOTING 22 FREQUENTLY ASKED QUESTIONS 23 REFERENCES: 25

4 Introducton Welcome to the Yeld Curve Fttng (YCF) verson 2.0 manual. YCF s an Excel-based program that calculates dscount factors, zero-coupn nterest rates and forward rates usng bond market, money market and nterest rate swap nput data. YCF has three sheets, whch are ManPage, BondMarket, and MoneyMarketAndSwaps respectvely. In the BondMarket sheet, users calculate varous yeld curves usng bond market data by means of the Cubc B-Splne Fttng Methodology. In the MoneyMarketAndSwaps sheet, users calculate varous yeld curves ether usng pure money market data through Natural Cubc Splne Interpolaton or combnng wth nterest rate swap data by the Bootstrap method. Users can access BondMarket or MoneyMarketAndSwaps va the ManPage. System requrements: Wndows NT/2000/XP Mcrosoft Excel 2000 or hgher Acrobat Reader 5.0 or hgher 1

5 Chapter 1 YCF 2.0 User Interface Man Page The Man Page s shown as follows. Clck help n the left upper corner to vew the help manual. Clck Bond Market lnk to access BondMarket page. Clck Money Market & Swaps lnk to access MoneyMarketAndSwaps page. Fg 1 2

6 Bond Market The BondMarket page has 3 parts. These are Input Intal Parameters Secton (shown n Fg 2), Input Coupon Bond Market Data Secton (shown n Fg 2), and Yelds Curve Output Secton (shown n Fg 3). A. Input Intal Parameters Secton Users nput Settlement Date, Results/Forward Rates Frequency, Input Day- Count Bass and Output Day-Count Bass n ths secton. Settlement Date: The value date on whch bonds yelds are quoted. The nput date format s: dd-mmm-yyyy, e.g. 23-Mar Results/Forward Rates Frequency: Ths s the frequency for both forward rates and results dsplay. For example, f you choose Monthly : The calculated yeld data wll be dsplayed n a monthly frequency and the forward rates are calculated usng monthly dscount factors or zero rates. Input Day-Count Bass: Day Count-Bass used n calculatng dscount factors from coupon bond data. Output Day-Count Bass: Day Count-Bass used n calculatng zerocoupon rates and forward rates from the dscount factors. B. Input Coupon Bond Market Data Secton Users nput Coupon bond data n ths secton, whch ncludes: Maturty Date: The date on whch the bond wll mature. The nput date format s: dd-mmm-yyyy, e.g. 23-Mar Coupon Rates: Indcatng the annual percentage rate used to determne the coupons payable on a bond. For a 5% coupon rate, please nput 5, nstead of say, Coupon Frequency: Coupon payment frequency. Users have three optons to choose from, Quarterly, Sem-annual and Annual coupon payment. Yelds: Bond yeld to maturty on a sem-annual bass. For a 5% bond yeld, please nput 5, nstead of say,

7 When users fnsh nputtng coupon bond data, they can proceed by clckng the button below. Calculate Curve: Clck t, and you wll get the calculated results for the dscount factors, zero-coupon rates, and forward rates. If you want to clear the old coupon bond data from the table, please clck the button below. Clear Table: Clck t, and you wll clear all the data n the table except Coupon Frequency whch are now reset to sem-annual. Notes for bond data nput 1. Please nput bond yelds n ther maturty date order. 2. Clck the calendar button to choose the maturty date. 3. Please nput 0 f the bond has no coupon. 4. You cannot use the Calculate Curve button whle the cursor stll focuses n the cell. Fg 2 C. Yelds Curve Output Secton 4

8 After clck Calculate Curve button, the spreadsheet automatcally moves the focus to ths secton, and dsplays the followng yeld curve output: Dscount factor: The rato of the present value of a fnancal asset to ts future value. Zero Rate: The annual nterest rates that would be earned on a bond that provdes no coupons. Forward Rate: The annual nterest rate for a future perod of tme that are mpled by the dscount factors that preval today. YTM: Annual yeld to maturty of bonds. The frequency of the yelds (and dscount factors) shown above s determned by the choce of Results/Forward Rates Frequency n the Input Intal Parameters Secton. You can plot the varous curves by clckng followng buttons. Plot Dscount Factors: Clck t, and you wll get the dscount curve graph. Plot Other Curves: After you clck t, a popup wndow wll appear and ask you whch yeld curve you want plot. You can choose to plot any yeld curve n the same graph by checkng them. The yeld curves you can plot are bond yeld to maturty curve, zero-coupon curve, and forward rates yeld curve. You can calculate curve data for any selected value date. To do ths, frst select value date, and then choose the type of the curve, fnally clck the Calculate button to get the result. Value date can be any dates from the earlest maturty date to the latest maturty date that are dsplayed n the output table. Its format s dd-mmmyyyy Clck Clear Contents button to clear all the output data n the table. Ths s not an oblgatory step. Notes for the yeld curve output Secton 1. Please make sure that you have curve data n the output table before plottng curves or calculatng the curve data for any selected value date. 5

9 Fg 3 Tps for the BondMarket Spreadsheet There are convenent lnks on the rght hand sde of each orange band. By clckng I, II or III, you can go to Secton I, Secton II or Secton III. Due to the large sze of the spreadsheet, ths feature s very helpful. 6

10 Money Market and Swaps MoneyMarekt&Swaps page ncludes 4 parts. They are Input Intal Parameters Secton (shown n Fg 4), Input Money Market Rates Secton (shown n Fg 4), Input Swap Rates Secton (shown n Fg 5), and Yelds Curve Output Secton (shown n Fg 6). A. Input Intal Parameters Secton Users nput Settlement Date, Results/Forward Rates Frequency, and Output Day-Count Bass n ths secton. Settlement Date: The value date on whch bonds yelds are quoted. The nput date format s: dd-mmm-yyyy, e.g. 23-Mar Results/Forward Rates Frequency: Ths s the frequency for both forward rates and results dsplay. For example, f you choose Monthly, the calculated yeld data wll be dsplayed n a monthly frequency and the forward rates are calculated usng monthly dscount factors or zero rates. Output Day-Count Bass: Day Count-Bass used n calculatng zerocoupon rates and forward rates from the dscount factors. Fg 4 7

11 B. Input Money Market Rates Secton Users nput money market data and related parameters n ths secton, whch ncludng: Maturty Date: The date on whch the money market nstruments wll mature. The nput date format s: dd-mmm-yyyy, e.g. 23-Mar Rates: Indcatng the annual percentage return of the money market nstruments. For a 5% rate, please nput 5, nstead of Money Market Day-Count Bass: Day Count-Bass used n calculatng the dscount factors from the money market rates. Users can calculate yeld curves wth money market data only. In ths case, users should clck the button below to proceed. Calculate Curve: Clck t, and you wll get the estmated results for the dscount factors, zero rates, and forward rates. If you want to estmate yeld curves wth money market rates and swap rates, smply move on to the next secton to nput the swap rates. If you want to clear the old money market data from the table, please clck the button below. Clear Table: Clck t, and you ll clear all the data n both money market rates table and swap rates table. Notes for money market data nput 1. Please nput money market rates n ther maturty date order. 2. Clck the calendar button to choose the maturty date. 3. You cannot use the Calculate Curve button whle the cursor stll focuses n the cell. C. Input Swap Rates Secton Users nput nterest rate swaps data and related parameters n ths secton, whch ncludes: 8

12 Maturty Date: The date on whch the nterest rate swaps wll mature. The nput date format s: dd-mmm-yyyy, e.g. 23-Mar Rates: The annual swap rates. For a 5% rate, please nput 5, nstead of Swap Frequency: Ths s the frequency of the cash exchange between fxed nterest payment and floatng nterest payment for a swap agreement. Swap Day-Count Bass: Day Count-Bass used n calculatng the dscount factors from the swap rates. After users nput both money market rates and swap rates, they can proceed by clckng the button below. Calculate Curve: Clck t, and you wll get the estmated dscount factors, zero rates, and forward rates. If you want to clear the old swap data from the table, please clck the button below. Clear Table: Clck t, and you ll clear all the data n both money market rates table and swap rates table. Notes for swap rates nput 1. Please nput swap rates n ther maturty date order. 2. Please make sure you have also nput money market data before you calculate yeld curves. 3. The frst swap maturty date must be earler than or equal to the last money market rate maturty. 4. You cannot use the Calculate Curve button whle the cursor stll focuses n the cell. 9

13 Fg 5 D. Yelds Curve Output Secton After clck Calculate Curve button, the spreadsheet automatcally moves the focus to ths secton, and dsplays the followng yeld curve output: Dscount factor: The rato of the present value of a fnancal asset to ts future value. Zero Rate: The annual nterest rates that would be earned on a bond that provdes no coupons. Forward Rate: The annual nterest rate for a future perod of tme; these are mpled by the dscount factors that preval today. The frequency of the yelds (and dscount factors) shown above s determned by the choce of Results/Forward Rates Frequency n the Input Intal Parameters Secton. You can plot the varous curves by clckng followng buttons. Plot Dscount Factors: Clck t, and you wll plot the dscount curve. Plot Other Curves: After clck t, a popup wndow wll appear and ask you whch yeld curve you want plot. You can choose to plot any yeld curve n the same graph by checkng them. The yeld curves you can plot are zero-coupon curve and forward rates yeld curve. 10

14 You can calculate curve data for any selected value date. To do ths, frst select value date, and then choose the type of the curve, fnally Clck Calculate button to get the result. Value date can be any dates from the earlest maturty date to the latest maturty date that are dsplayed n the output table. Its format has the form of dd-mmm-yyyy Clck Clear Contents button to clear all the output data n the table. It s not an oblgatory step. Notes for the yeld curve output Secton Please make sure that you have curve data n the output table before plottng curves or calculatng the curve data for any selected value date. Tps for the MoneyMarket&Swaps Spreadsheet There are convenent lnks on the rght hand sde of each orange band. By clckng I, II, III or IV, you can go to Secton I, Secton II, Secton III or Secton IV. Due to the large sze of the spreadsheet, ths feature s very helpful. Fg 6 11

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16 Chapter 2 Yeld Curve Estmatng and Fttng Methods Basc Fxed Income Formulae A. Dscount Functons If we have annual nterest rate (spot or zero-coupon rate) r, the dscount functon wth annual compoundng can be calculated by df = 1 (1 + r) B Where: r s the zero rate; df s the dscount factor; B s the Day Count Fracton, dfferent day count bass wll lead to dfferent value of B and also dfferent df. B. Zero Rates If we have the dscount factor, the zero rate r wth annual compoundng can be calculated by r 1 df = 1/ B where: 1 r s the zero rate; df s the dscount factor; B s the Day Count Fracton, dfferent day count bass wll lead to dfferent value of B and also dfferent df. 13

17 C. Forward Rates A set of forward rate perods are determned from value date to the last date of the curve usng the frequency set n the Input Intal Parameter Secton of the spreadsheet. The forward rate s then calculated as follows: F df = df E 1 B S 1 where: df S s the dscount factor for the start of the forward rate perod; df E s the dscount factor for the end of the forward rate perod; B s the Day Count Fracton for the relevant forward perod. D. Day Count Fracton Day Count Fracton B s determned by the followng formula B = d DaysOfYear where: d s the number of days between the value date and the contract maturty,.e. the perod of the contract n days. For 30/360 day count bass, however, d does not equal exactly the actual number of days n the contract perod snce t s assumed each month has 30 days. DaysOfYear s number of the days n a year. For Actual/Actual day count bass, t s the number of actual days n that year; for Actual/365, t s 365; for Actual /360 and 30/360, t s 360. Therefore, dfferent choce on day count bass wll affect the value of day count fracton. Curve Fttng Methods A. Natural Cubc Splne A Natural cubc splne s a smooth splne constructed of pecewse thrd-order polynomals whch pass through a set of m control ponts. Natural cubc splne s an nterpolaton curve and can be used to connect the control ponts wth smooth curve. 14

18 Consder a splne for a set of n+1 control ponts ( t t,..., ) Bartels et al. (1998), the th pece of the splne can be represented by 2 3 Y ( t) = a + b ( t t ) + c( t t ) + d( t t ), = 0,1,..., n 1 ; and the splne curve as a whole can be expressed as Y0 ( t) t [ t0, t1] Y1 ( t) t [ t1, t2 ] Y ( t) = Y2 ( t) t [ t2, t3]... Yn 1( t) t [ tn 1, tn ] 0, 1 t n. Followng So there s a total of 4n coeffcents, { a, b, c, d} = 0,... n 1 to be determned and we need at least 4n functons to constran those coeffcents. At each nteror pont, we have four condtons: curve passng through the pont must have a contnuous frst and second order dervatve, and the curve must pass through the pont. Ths gves us four tmes the n-1 nteror ponts = 4n-4 equatons. We get two extra equatons from the curve touchng the exteror ponts, yeldng 4n-2. The last two we choose artfcally, such as settng the second dervatve of the curve at the exteror ponts as 0. B. Cubc B-Splne The B-Splne s approxmatng splne curve, so t need not necessarly pass through every control pont. It has advantages - such as the degree of the polynomal s ndependent from the number of ponts. Also, the shape s controlled locally, so adjustng a sngle pont does not requre total recomputaton of the curve. It s a common practce to use B-splnes to construct cubc splnes for ts favourable propertes. The Cubc B-splne model dscussed later uses B-splnes to construct the smooth "dscount functon". A B-splne s a generalzaton of the Bezer curve. Let a vector known as the knot vector be defned T = { t0, t1,..., tm }, where T s a non-decreasng sequence. Defne the B-splne bass functons of degree p as B k,0 1 f ( t) = 0 t k t < t otherwse k + 1 and tk < t k + 1 B t t t 1 t k k + p+ ( t) = Bk, p 1 ( t) + Bk + 1, p 1 ( ) t t t t k, p t k + p k k + p+ 1 k

19 Q Gven S control ponts ( ),..., 1 S 1 = Y ( t) Q k = 0 k B k, p ( t) = 0 S wth respect to the knot sequence, a B-splne curve of degree p s defned as t t1 t S + p 1 < t S + p The cubc B-splne curve s B-splne curve of degree 3 (.e. p=3) and can be constructed usng the equaton above. Dscount Factors Estmaton YCF uses both Cubc B-Splne and Bootstrappng methods to estmate dscount factors. The nputs for the former method are coupon bond data and the nputs for the latter are nterest-rate swap rates data. A. Cubc B-Splne Method Ths method assumes the dscount functon as a smooth curve and constructs t usng the Cubc B-Splne functon wth the followng form, S 1 = df ( x) Q k = 0 k B k, 3 ( x) Where: df (x) : s the dscount factor w.r.t. date x; Q : s the S-dmensonal control vector whose components ( Q ),..., 1 are the = 0 S unknown control ponts of the Cubc B-splne curve. B k, 3 ( x) : s the cubc B-splne bass functon explaned ncurve fttng method chapter. The theoretcal value of a coupon bond can be expressed as follows: P ˆ = n j C df ( x j j ) = n j C S 1 j k = 0 Q k B k, 3( x j ) where: ( j ) j= 1,2,... n 1 C s the coupon payment before the maturty date, and C n s the sum of the coupon and prncpal payment on the maturty date for the bond ; x s the date at whch C s expected to be pad. j j 16

20 The unknown coeffcents of the dscount functon (vector Q) can be determned through mnmzng the followng functon: F( Q) = = N 2 T ( P Pˆ ) + λ [ df ''( x) ] T 2 P C j Qk Bk,3( x ) + λ j = 1 where: N = 1 n j S 1 k = dx 0 S 1 k = 0 Q k B k,3 ( x) / x P : s the observed market value(drty prce) of bond number ; λ : s a parameter that determnes the relatve mportance of goodness versus smoothness of ft to the observed data. The frst term measures the gap between a set of bonds' theoretcal prces and ther observed market prces, whle the second term measures the smoothness of the dscount functon. By solvng the mnmzaton problem above, we can fnd the values for vector Q, through whch one can construct a smooth and accurate dscount functon. The knot ponts { t 0, t1,..., tm} of the Cubc B-Splne curve are connected to the maturty of the bonds used n the calculaton. B. Bootstrappng Method Ths method calculates the dscount factors n a bootstrappng fashon from successve swap rates combned wth money market data. The quoted swap rates can be consdered as par yelds for bonds n the same class. So for a swap, ts coupon equals ts rates. In the dagram below we show Dscount Factors C3, C4, C5 that have been calculated from the money market rates. These, wth some form of nterpolaton, are used to calculate an approprate df (dscount factor) for coupon ponts 1, 2, and dx 17

21 The basc functon to calculate the unknown df,.e. the df for pont S1 s: df = N (1 r t 1 = 1 ( df B ) 1+ (1 r B ) t ) where: N s Notonal (or 1); r s the quoted swap rate df Dscount factor for each coupon from 1 to t-1 (.e. all excludng last coupon); B Day Count Fracton for each coupon from 1 to t-1 (.e. all excludng last coupon); and t s the last coupon date. In other words we can calculate the dscount factor, df, for the perod of the swap f we can extract the df for each and every coupon payment. However, a small dffculty arses when the swap has a coupon, other than the last coupon, whch falls beyond the prevously calculated df. In the dagram below coupon 3 falls after the last calculated df at pont C5. We don t know the value of df for 3. 18

22 In order to solve ths problem we create a fcttous swap. Ths fcttous nstrument (SF) matches S1 n ts coupon payment dates up to and ncludng coupon 3 but excludes the last coupon. In other words the prncpal s repad at pont 3. A dscount factor for SF can now be calculated usng the formula set out above. However, we do not have a swap rate to use. A rate for SF s manufactured by lnearly nterpolatng between the prevous swap rate (S0) and the next swap rate (S1), that s, S1 S0 SF = S0 + ( T T 0). T1 T 0 where: T: tme to maturty for SF; T0: tme to maturty for S0; T1: tme to maturty for S1. For ths reason t s an mportant requrement that the yeld curve market quotes for the cash overlap the frst swap quote. Each swap df can now be calculated n turn. S1 once calculated can be fed nto the calculaton of S2 and S2 nto S3 and so on. Ths s what s termed "bootstrappng". It s derved from the dea of dong up your bootstraps. You must start from the bottom and work your way up to the top. The next step s dependent on the prevous step (hence an error early on wll be magnfed all the way up each subsequent step). 19

23 Yeld Curve Constructon A. Wth Coupon Bond Data If you want to construct a yeld curve (e.g. zero-coupon yeld curve, forward rate yeld curve, etc.) usng coupon bond data, please go to the Bond Market Page. In the bond market page, the program frst calculates dscount factors usng the Cubc B-Splne method. Then the zero-coupon rates and forward rates are calculated from the estmated dscount factors. Further, the yelds to maturty of a bond are nterpolated usng natural cubc splne nterpolaton. To reduce the oscllatons n the forward rate curve, we mpose further smoothness restrctons on the calculated forward rates and get a smoother forward curve. The forward rate of the last perod s assumed to be the same as that of prevous perod snce there s no further dscount data to calculate t. B. Wth Money Market Data If you want to construct a yeld curve (e.g. zero-coupon yeld curve, forward rate yeld curve, etc.) usng Money market data only, please go to the Money Market& Swaps Page. In ths page, please nput money market rates and clck the calculate curve button to get the results. The program frst uses natural cubc splne to nterpolate the money market rates, and then calculates dscount factors from the zero rates. Forward rates are calculated from the estmated dscount factors. The forward rate of the last perod s assumed to be the same as that of the prevous perod snce there s no further dscount data to calculate t. You are recommended to choose weekly or monthly frequency for the Results/forward rates frequency nput, followng market practce. C. Wth Money Market and Swaps Data If you want to construct a yeld curve (e.g. zero coupon yeld curve, forward rate yeld curve, etc.) usng Money market data and swaps, please go to the Money Market& Swaps Page. In ths page, please nput money market rates and swap rates, then clck the calculate curve button to get the results. The program frst uses natural cubc splne to nterpolate the money market rates, and then calculates dscount factors from the zero rates. The dscount factors wth maturtes longer than the last money market rate are derved usng bootstrappng method wth swap data. Fnally, forward rates are calculated from the estmated dscount factors. 20

24 The forward rate of the last perod s assumed to be the same as that of prevous perod snce there s no further dscounts data to calculate t. Remember to have an overlap n maturty date between the last money market rate and the frst swap rate. 21

25 Chapter 3 Troubleshootng and Q&A Troubleshootng Ths secton provdes a table showng errors you may encounter usng YCF 2.0, probable causes for these errors, and suggested solutons. Table 2: YCF 2.0 Errors and Suggested Solutons Message Probable Cause Suggested Soluton Error n YeldCurveFttng.YeldCu rvefttng.2_0: Error gettng data converson flags. Error n YeldCurveFttng.YeldCu rvefttng.2_0: Undefned functon or varables Error n VBAProject: Automaton error The specfed module could not be found. Error n VBAProject: ActveX component can't create object. Usually caused by mwcomutl.dll not beng regstered. Some requred data have not been nput n the spreadsheet. Ths usually occurs f MATLAB runtme lb s not on the system path. 1. Project DLL s not regstered. 2. An ncompatble MATLAB DLL exsts somewhere on the system path. Fnd the EXE fle YeldCurveFttng.exe n the followng drectory: C:\WINDOWS\system32 or C:\WINNT\system32, and execute t. Input all the data requred. Place %SystemRoot%\system3 2\bn\wn32 on your system path. Fnd the EXE fle YeldCurveFttng.exe n the followng drectory: C:\WINDOWS\system32 or C:\WINNT\system32, and execute t. 22

26 Table 2: Excel Errors and Suggested Solutons Message Probable Cause Suggested Soluton Macros n ths project are dsabled. Please refer to the onlne help or documentaton of the host applcaton to determne how to enable macros. The Maturty dates n the result table are 4 years earler than the true date The macro securty for Excel s set to Hgh. Excel s usng 1904 date system nstead of default 1900 date system. Set Excel macro securty to Medum on the Securty Level tab (Tools > Macro > Securty). To change to the 1990 date system, clck Optons on the Tools menu, clck the Calculaton tab, and then unselect the 1904 date system check box. Frequently asked questons How to nstall YCF 2.0 Double clck the Setup fle n the nstallaton CD, and nstall t accordng to the screen nstructons. You must restart your computer before usng ths program. Durng the nstallaton, a command lne wndow wll pop up and ask you where to nstall the MATLAB runtme lbrares. You should choose the default drectory (just press ENTER to choose t),.e. C:\Wndows\System32 or C:\WINNT\System32, to nstall them. The Yeld Fttng nstallaton wzard wll automatcally add the nstalled drectory to your PATH envronment varable, and you don t have to do t manually. If MATLAB runtme lbrares have been prevously nstalled n the other drectory rather than C:\Wndows\System32 or C:\WINNT\System32, you must un-nstall them before you contnue ths Installaton Wzard. How to access YCF 2.0 You can start YCF 2.0 by clckng the YCF2.0 con n the desktop or from Start Menu/Programs/YeldCurveFttng2.0. How to remove YCF 2.0 To remove YCF 2.0, please go to Settngs/Control Panel/Add &Remove programs, Select t and clck the Remove button. 23

27 Can I save YCF2.0 spreadsheet fle YCF 2.0 s a Read-Only spreadsheet fle, and you cannot save the modfed YCF2.0 wth ts orgnal name. However, you can always save t usng a dfferent name. Though you can construct the yeld curve wth your saved new spreadsheet, errors may occur due to the changes of the structure n the spreadsheet. 24

28 References: Anderson, N. and J. Sleath (2001). New estmates of the UK real and nomnal yeld curves. The Bank of England's workng paper. Anderson, N. and J. Sleath (2001). New estmates of the UK real and nomnal yeld curves. The Bank of England's workng paper. Bartels, R. H. and J. C. Beatty (1998). An Introducton to Splnes for Use n Computer Graphcs and Geometrc Modellng. San Francsco, Morgan Kaufmann. Bolder, D. J. and S. Gusba (2002). Exponentals, Polynomals, and Fourer Seres: More Yeld Curve Modelng at the Bank of Canada. Bank of Canada Workng Paper. Choudhry, M., D. Joannas, et al. (2002). Captal Market Instrument Analyss and Valuaton, Fnancal Tmes Prentce Hall. Fabozz, F. J. (2002). Interest Rate, Term Structure, and Valuaton Modelng, Wley. Jeffrey, A. and O. L. a. T. Nguyen (2000). Flexble Term Structure Estmaton: Whch Method s Preferred? Yale SOM Workng Paper. McCulloch, J. H. (1971). " Measurng the Term Structure of Interest Rates." The Journal of Busness 44. McCulloch, J. H. (1975). "The Tax-Adjusted Yeld Curve." The Journal of Fnance