The termstrc Package

Transcription

1 The termstrc Package July 9, 2007 Type Package Title Term Structure and Credit Spread Estimation Version 1.0 Date Author Maintainer Robert Ferstl Depends R (>= 2.4.0) The package offers several widely-used term structure estimation procedures, i.e. the parametric Nelson and Siegel approach, Svensson approach and cubic splines. License GPL R topics documented: aabse bond_prices bond_yields corpbonds create_cashflows_matrix create_maturities_matrix duration eurobonds gi loss_function maturity_range nelson_estim nelson_siegel plot.cubicsplines plot.nelson print.cubicsplines print.nelson

2 2 aabse print.summary.cubicsplines print.summary.nelson rmse splines_estim spotrates summary.cubicsplines summary.nelson svensson termstrc-package Index 24 aabse Average Absolute Mean Error Average absolute mean error aabse(actual, estimated) actual estimated vector, consisting of the observed values vector, consisting of the estimated values Details Calculation of the AABSE according to the formula: References AABSE = 1 n n e i e i David Bolder and David Streliski (1999): Yield Curve Modelling at the Bank of Canada.Technical Report No 84 Bank of Canada rmse i=1

3 bond_prices 3 bond_prices Bond Price Calculation Function for the calculation of bond prices according to the chosen approach (Nelson and Siegel or Svensson) based on the cashflows and maturities of the bonds. bond_prices(method = "Nelson/Siegel", beta, m, cf) method beta m cf defines the desired method, "Nelson/Siegel" for the Nelson/Siegel approach or "Svensson" for the Svensson approach parameter vector, is linked to the chosen approach maturities matrix, consists of the maturity dates which are appended to the cashflows of the bonds cashflows matrix Value Note Returns a list with: spot_rates spot rates discount_factors discount factors bond_prices bond prices References David Bolder and David Streliski (1999): Yield Curve Modelling at the Bank of Canada.Technical Report No 84 Bank of Canada svensson, nelson_siegel

4 4 bond_yields bond_yields Bond Yield Calculation Function for the calculation of bond yields. bond_yields(cashflows, m, tol = 1e-10) cashflows m tol matrix with the bonds cashflows maturity matrix desired accuracy for function uniroot Value matrix with the bond yields and the associated maturities References David Bolder and David Streliski (1999): Yield Curve Modelling at the Bank of Canada.Technical Report No 84 Bank of Canada uniroot Examples

5 corpbonds 5 corpbonds Corporate Bonds Details Note Corporate bonds data(corpbonds) The data set eurobonds consists of bonds of the rating classes AAA,AA+, AA, AA-, A+, A, A-, BBB+, BBB, BBB- If you use your own dataset, make sure that the structure is identical to the provided data sets. Use the function str() to explore the data set. Every element of the list except $RATING, $NAME is required for the estimation. The elements $COUPONRATE, $PRICE are not necessarily required. The calculation of the accrued interest is not implemented and has therefore be provided by the user. For the provided datasets the list element $ACCRUED includes the accrued interests. eurobonds Examples data(corpbonds) str(corpbonds) # You may use this to generate an empty data set, # which can be filled with bond data: ISIN <- vector() MATURITYDATE <- vector() STARTDATE <- vector() COUPONRATE <- vector() PRICE <- vector() ACCRUED <- vector() CFISIN <- vector() CF <- vector() DATE <- vector() CASHFLOWS <- list(cfisin,cf,date)

6 6 create_cashflows_matrix names(cashflows) <- c("isin","cf","date") TODAY <- vector() mycountry1 <- list(isin,maturitydate,startdate, COUPONRATE,PRICE,ACCRUED,CASHFLOWS,TODAY) mycountry2 <- list(isin,maturitydate,startdate, COUPONRATE,PRICE,ACCRUED,CASHFLOWS,TODAY) names(mycountry1) <- c("isin","maturitydate","startdate","couponrate", "PRICE","ACCRUED","CASHFLOWS","TODAY") names(mycountry2) <- c("isin","maturitydate","startdate","couponrate", "PRICE","ACCRUED","CASHFLOWS","TODAY") mybonds <- list(mycountry1,mycountry2) names(mybonds) <- c("mycountry1","mycountry2") create_cashflows_matrix Cashflows Matrix Creation Creates a matrix of cashflows for a specified group of bonds. The number of rows is the number of cashflows for the bond with the longest maturity. create_cashflows_matrix(group, include_price = FALSE) Value group group name include_price if TRUE the dirty price is included matrix create_maturities_matrix

7 create_maturities_matrix 7 create_maturities_matrix Maturity Matrix Creation Creates a matrix of maturities for a specified group of bonds. The number of rows is the number of cashflows for the bond with the longest maturity. create_maturities_matrix(group, include_price = FALSE) Value group group name include_price if TRUE the dirty price is included matrix create_cashflows_matrix duration Duration, modified Duration and Duration based Weights Duration, modified duration and duration based weights duration(cf_p, m_p, y) cf_p m_p y cashflows matrix maturity matrix yield of the bond

8 8 eurobonds Value The function returns a matrix with three columns, i.e. duration, modified duration and duration based weights. References David Bolder and David Streliski (1999): Yield Curve Modelling at the Bank of Canada.Technical Report No 84 Bank of Canada eurobonds European Government Bonds European government bonds data(eurobonds) Details The data set eurobonds consists of German, Austrian, Italian and Hungarian government bonds. Note If you use your own dataset, make sure that the structure is identical to the provided data sets. Use the function str() to explore the data set. corpbonds Examples data(corpbonds) str(corpbonds)

9 gi 9 gi Cubic Functions Calculation of the cubic functions according to an approach of McCulloch (1975) gi(t, T, i, s) t T i s maturity knot points index number of basis functions References J. Huston McCulloch (1971): Measuring the Term Structure of Interest Rates. The Journal of Business, J. Huston McCulloch (1975): The Tax-Adjusted Yield Curve. The Journal of Finance, Sanjay K. Nawalkha and Gloria M. Soto and Natalia K. Beliaeva (2005): Interest Rate Risk Modeling : The Fixed Income Valuation Course Wiley Finance,60 67 loss_function Loss Function used for the Term Structure Estimation The loss function defines the objective function used for the optimisation. loss_function(p, phat, omega, weights)

10 10 maturity_range p phat omega weights observed prices, yields estimated prices, yields weights if "none" the squared deviation of p and phat is not weighted, use "duration" for a duration weighted optimization References David Bolder and David Streliski (1999): Yield Curve Modelling at the Bank of Canada.Technical Report No 84 Bank of Canada maturity_range Maturity Range for Bond Data Set The function restricts a bond data set to a specified maturity range. maturity_range(bonddata, lower, upper) bonddata lower upper bond dataset lower bound of maturity spectrum upper bound of maturity spectrum

11 nelson_estim 11 nelson_estim Term Structure and Credit Spread Estimation with Nelson/Siegel and Svensson Method Term structure and credit spread estimation with Nelson/Siegel and Svensson method nelson_estim(group, bonddata, matrange = "all", method = "Nelson/Siegel", fit = "prices", weights = "none", startparam, control = list(eval.max = 1000)) group bonddata matrange method fit weights startparam control vector defining the group of bonds used for the estimation, e.g. c("germany","austria") a dataset of bonds in list format use "all" for no restrictions, or restrict the maturity range used for the estimation with c(lower,upper) "Nelson/Siegel" or "Svensson" use "prices" ("yields") for minimising the squared price (yield) error If a weighted minimisation of the squared price deviation is required, use "duration", otherwise "none" matrix of start parameters, for the Nelson/Siegel (Svensson) method 4 (6) parameters for each each group are required (one row per group). list of control parameters for the function nlminb Details group The first element of the vector will be used as the reference country for the credit spread estimation. The group can be either a vector of groups or a scalar. bonddata The package is tailored to the included data sets. Therefore the structure and the naming convention of other used data sets has to be identical. Use the function str() to explore the structure of the provided datasets. startparam For the estimation a matrix of start parameters must be provided. The estimation is very sensitive to the start parameters, due to multiple local minima of the objective functions. Therefore a great effort should be invested in finding a good start parameter set. control Please refer to the documentation of the function mlimnb.

12 12 nelson_estim Value The function nelson_estim returns a list with the following elements or sub-lists: group matrange method fit weights n_group zcy_curves scurves cf m duration p phat y yhat opt_result groups used from data set includes the chosen maturity range includes the chosen estimation method includes the chosen objective function type of weighting used in optimisation the number of groups used for the optimisation values for plotting the estimated zero-coupon yield curves values for plotting the spread curves cashflows matrix for all specified groups maturity matrix for all specified groups duration, weighted duration and duration based weights dirty prices estimated bond prices bond yields theoretical bond yields calculated with the estimated bond prices phat the optimal parameter vector for the specified groups according to the chosen estimation approach References Charles R. Nelson and Andrew F. Siegel (1987): Parsimonious modeling of yield curves. The Journal of Business, 60(4): Lars E.O. Svensson (1994): Estimating and interpreting forward interest rates: Sweden Technical Reports 4871, National Bureau of Economic Research. David Bolder and David Streliski (1999): Yield Curve Modelling at the Bank of Canada. Technical Report No 84 Bank of Canada for another estimation method see splines_estim Examples demo(euro01)

13 nelson_siegel 13 nelson_siegel Spot Rate Function according to Nelson/Siegel This function calculates the spot rates for certain maturity dates and a parameter vector according to Nelson/Siegel. nelson_siegel(beta, m) beta a vector of parameters (β 0, β 1, β 2, τ 1 ) m Value one maturity (or a vector of maturities) returns a vector consisting of the calculated spot rates References Charles R. Nelson and Andrew F. Siegel (1987): Parsimonious modeling of yield curves. The Journal of Business, 60(4): Examples nelson_siegel(rep(0.01,4),1:30) plot.cubicsplines S3 Plot Method for Cubic Splines S3 plot method for an object of the class cubicsplines ## S3 method for class 'cubicsplines': plot(x,matrange,pdf=false,...)

14 14 plot.nelson x matrange pdf... other arguments object of the class cubicsplines maturity range for the plot, e.g. c(2,10) if TRUE the graphics are plotted in a pdf file, which is saved in the current working directory Details A plot of the estimated zero-coupon yield curve including the calculated yields for each specified group of bonds is offered. The occurrence of dashed vertical lines in a graphic represents the knot points used for the cubic splines estimation. In addition all zero-coupon yield curves are plotted in one graphic and afterwards the spread curves are plotted. print.cubicsplines, summary.cubicsplines plot.nelson S3 Plot Method S3 plot method for an object of the class nelson ## S3 method for class 'nelson': plot(x,matrange,pdf=false,...) x matrange pdf object of the class nelson... other arguments maturity range for the plot, e.g. c(2,10) If TRUE the graphics are plotted in a pdf file, which is saved in the current working directory

15 print.cubicsplines 15 Details A plot of the estimated zero-coupon yield curve including the calculated yields for each specified group of bonds is offered. In addition all zero-coupon yield curves are plotted in one graphic and afterwards the spread curves are plotted. The occurrence of a partially dashed line in a graphic indicates that the the curve has been extrapolated. print.nelson, summary.nelson print.cubicsplines S3 Print Method for Cubicsplines S3 print method for an object of the class cubicsplines ## S3 method for class 'cubicsplines': print(x,...) x object of the class cubicsplines... other arguments Details The print method for an object of the class cubicsplines prints important input parameters of the optimisation and the results of the regression (i.e. the parameter of the cubic spline functions). plot.cubicsplines, summary.cubicsplines

16 16 print.summary.cubicsplines print.nelson S3 Print Method S3 print method for an object of the class nelson ## S3 method for class 'nelson': print(x,...) x Details object of the class nelson... other arguments The print method for an object of the class nelson prints important input parameters of the optimisation and the results of the optimisation (i.e. the optimal parameter vector). plot.nelson,summary.nelson print.summary.cubicsplines S3 Print Method S3 print method for an object of the class summary.cubicsplines ## S3 method for class 'summary.cubicsplines': print(x,...) x object of the class summary.nelson... other arguments

17 print.summary.nelson 17 print.summary.nelson print.summary.nelson S3 Print Method S3 print method for an object of the class summary.nelson ## S3 method for class 'summary.nelson': print(x,...) x object of the class summary.nelson... other arguments print.summary.cubicsplines rmse Root Mean Squared Error Root mean squared error rmse(actual, estimated)

18 18 splines_estim actual estimated vector, consisting of the observed values vector, consisting of the estimated values Details Calculation of the RMSE according to the formula: RMSE = 1 n (e i e i ) n 2 References David Bolder and David Streliski (1999): Yield Curve Modelling at the Bank of Canada.Technical Report No 84 Bank of Canada. aabse i=1 splines_estim Term Structure and Credit Spread Estimation with Cubic Splines Method Term structure and credit spread estimation with cubic splines method splines_estim(group, bonddata, matrange = "all") group bonddata matrange vector defining the group of bonds used for the estimation, e.g. c("germany","austria") a dataset of bonds in list format use "all" for no restrictions, or restrict the maturity range used for the estimation with c(lower,upper)

19 splines_estim 19 Details group The first element of the vector will be used as the reference country for the credit spread estimation. The group can be either a vector of groups or a scalar. bonddata The package is tailored to the included data sets. Therefore the structure and the naming convention of other used data sets has to be identical. Use the function str() to explore the structure of the provided datasets. Value The function splines_estim returns a list with the following elements or sub-lists: group matrange n_group zcy_curves scurves cf m duration p phat y yhat alpha groups used from data set maturity range the number of groups used for the optimisation values for plotting the estimated zero-coupon yield curves values for plotting the spread curves cashflows matrix for all specified groups maturity matrix for all specified groups duration, weighted duration and duration based weights dirty prices estimated bond prices bond yields theoretical bond yields calculated with the estimated bond prices phat OLS coefficients of cubic splines estimation References David Bolder and David Streliski (1999): Yield Curve Modelling at the Bank of Canada. Technical Report No 84 Bank of Canada J.Huston McCulloch (1971): Measuring the Term Structure of Interest Rates. The Journal of Business, J. Huston McCulloch (1975): The Tax-Adjusted Yield Curve. The Journal of Finance, Sanjay K. Nawalkha and Gloria M. Soto and Natalia K. Beliaeva (2005): Interest Rate Risk Modeling : The Fixed Income Valuation Course Wiley Finance,60 67 for another estimation method see nelson_estim

20 20 summary.cubicsplines Examples demo(euro02) spotrates Function for the Calculation of the Spot Rates The function calculates the spot rates for the chosen approach, a provided maturity vector and parameter set spotrates(method, beta, m) method beta m "Nelson/Siegel","Svensson" parameter set maturity vector summary.cubicsplines S3 Summary Method for Cubicsplines S3 summary method for objects of the class cubicsplines ## S3 method for class 'cubicsplines': summary(object,...) object object of the class cubicsplines... other arguments

21 summary.nelson 21 Details The summary method for an object of the class cubicsplines prints the solution of the goodness of fit tests (RMSE, AABSE) of the optimisation. plot.cubicsplines, print.cubicsplines, rmse, aabse Examples summary.nelson S3 Summary Method S3 summary method for objects of the class nelson ## S3 method for class 'nelson': summary(object,...) object object of the class nelson... other arguments Details The summary method for an object of the class nelson prints the solution of the goodness of fit tests (RMSE,AABSE) of the optimisation. Moreover a convergence information of the used optimiser (nlminb) is printed. nlminb, plot.nelson, print.nelson, rmse, aabse Examples

22 22 termstrc-package svensson Spot Rate Function according to Svensson This function calculates the spot-rates for certain maturity dates and a parameter vector according to Svensson. svensson(beta, m) beta a vector of parameters (β 0, β 1, β 2, τ 1, β 3, τ 2 ) m one maturity (or a vector of maturities) Value returns a vector consisting of the calculated spot rates References Lars E.O. Svensson (1994): Estimating and interpreting forward interest rates: Sweden Technical Reports 4871, National Bureau of Economic Research. Examples svensson(c(0.07,0.3,0.05,0.1,0.08,0.2),1:30) termstrc-package Term Structure and Credit Spread Estimation The package offers several widely-used term structure estimation procedures, i.e. the parametric Nelson and Siegel approach, Svensson approach and cubic splines.

23 termstrc-package 23 References Bank for International Settlements (2005). Zero-coupon yield curves: technical documentation. BIS Papers, No. 25 Robert R. Bliss (2007): Testing term structure estimation methods. Advances in Futures and Options Research, David Bolder and David Streliski (1999): Yield Curve Modelling at the Bank of Canada. Bank of Canada Technical Report, No. 84 Alois Geyer and Richard Mader (1999): Estimation of the Term Structure of Interest Rates - A Parametric Approach. OeNB Working Paper, No. 37 Michalis Ioannides (2003): A comparison of yield curve estimation techniques using UK data. Journal of Banking & Finance, J. Huston McCulloch (1971): Measuring the Term Structure of Interest Rates. The Journal of Business, J. Huston McCulloch (1975): The Tax-Adjusted Yield Curve. The Journal of Finance, Sanjay K. Nawalkha and Gloria M. Soto and Natalia K. Beliaeva (2005): Interest Rate Risk Modeling : The Fixed Income Valuation Course Wiley Finance Charles R. Nelson and Andrew F. Siegel (1987): Parsimonious modeling of yield curves. The Journal of Business, 60(4): Lars E.O. Svensson (1994): Estimating and interpreting forward interest rates: Sweden Technical Reports 4871, National Bureau of Economic Research.

24 Index Topic datasets corpbonds, 4 eurobonds, 7 Topic package termstrc-package, 22 aabse, 1, 17, 20, 21 svensson, 3, 21 termstrc (termstrc-package), 22 termstrc-package, 22 uniroot, 4 bond_prices, 2 bond_yields, 3 corpbonds, 4, 8 create_cashflows_matrix, 5, 6 create_maturities_matrix, 6, 6 duration, 7 eurobonds, 4, 7 gi, 8 loss_function, 9 maturity_range, 9 nelson_estim, 10, 19 nelson_siegel, 3, 12 nlminb, 21 plot.cubicsplines, 13, 15, 20 plot.nelson, 13, 15, 21 print.cubicsplines, 13, 14, 20 print.nelson, 14, 15, 21 print.summary.cubicsplines, 15, 16 print.summary.nelson, 16, 16 rmse, 2, 17, 20, 21 splines_estim, 11, 18 spotrates, 19 summary.cubicsplines, 13, 15, 20 summary.nelson, 14, 15, 21 24