BondValuation: An R Package for Fixed Coupon Bond Analysis

The purpose of this paper is to introduce the R package BondValuation for the analysis of large datasets of fixed coupon bonds. The conceptual heterogeneity of fixed coupon bonds traded in the global markets imposes a high degree of complexity on their comparative analysis. Contrary to baseline fixed income theory, in practice, most bonds feature coupon period irregularities. In addition, there are a multitude of day count methods that determine the interest accrual, the cash flows and the discount factors used in bond valuation. Several R packages, e.g., fBonds, RQuantLib, and YieldCurve, provide tools for fixed income analysis. Nevertheless, none of them is capable of evaluating bonds featuring irregular first and/or final coupon periods, and neither provides adequate coverage of day count conventions currently used in the global bond markets. The R package BondValuation closes this gap using the generalized valuation methodology presented in Djatschenko (2019).

Wadim Djatschenko (European University Viadrina)
2020-01-06

1 Introduction

Although bond valuation using the traditional present value approach is fundamental in financial theory and practice, the R community lacks applications that comprehensively handle the peculiarities of real-world fixed coupon bonds. A possible reason for the slow development of adequate computation tools concerns the matter’s theoretical intricacy, characterized by a complex interaction of day count conventions (DCC) and irregularities in the temporal structure of the fixed income instruments.

A day count convention is an instrument-specific set of rules that prescribes the way in which calendar dates are converted to numerical values. Thus, given a schedule of a bond’s anniversary dates (i.e., issue date, coupon payment dates, maturity date), a day count convention is used, e.g., to determine the fraction of regular coupon periods between two calendar dates within the bond’s life. Irregular first and final coupon periods occur irrespective of the stipulated day count convention. The lengths of the first and final coupon periods are measured in fractions of regular coupon periods and calculated according to the rules of the specified day count convention. A fist or final coupon period is irregular, if its length differs from \(1\), which is the length of a regular coupon period.1

The R package RQuantLib (Eddelbuettel et al. 2018) provides access to parts of QuantLib (QuantLib Team 2018), which is the leading open-source software library for quantitative finance. Currently, QuantLib incorporates methods for the analysis and valuation of a wide variety of financial instruments, such as options, swaps, various financial derivatives, and several types of bonds, including fixed rate bonds. Nevertheless, QuantLib does not implement methods for handling irregular coupon periods, and the coverage of DCCs is not exhaustive with nine different conventions.

A closer examination of bond market data reveals the importance of this problem. According to the Thomson Reuters EIKON database, \(99.66\%\) of the plain vanilla fixed coupon bonds that were issued worldwide in \(2017\) are spread over \(15\) different DCCs, and \(67\%\) of them feature irregular first and/or final coupon periods. Given the enormous size of the global bond market, neglecting irregular coupon periods potentially leads to cash flow miscalculations in the tens of billions of US dollars, as Djatschenko (2019) points out.

Essentially, DCCs influence bond valuation in three places. First, the amounts of interest payable at the end of any irregular coupon period are computed according to the respective convention. Second, the powers of the discount factors used in present value calculations depend on the stipulated DCC. Finally, in contrast to stocks, the full prices of bonds are usually not directly observable but need to be calculated as the sum of the quoted clean price and accrued interest, which is paid by the buyer to the seller if the transaction is conducted between two coupon payment dates. Accrued interest is computed conformal to the stipulated bond- and market-specific DCCs.

Djatschenko (2019) addresses these three aspects and proposes a generalized valuation methodology for fixed coupon bonds that allows for irregular first and final coupon periods and is compatible with any conceivable DCC. In summary, the methodology can be described as follows. In a first step, Djatschenko (2019) introduces a standardized bond-specific temporal structure, which is determined by the stipulated DCC. Based on this time structure, a valuation formula is derived that allows for first and final coupon periods of any lengths. The novelty of this proposed evaluation formula lies in the isolation of each DCC-dependent parameter, resulting in a modular structure that can easily integrate any conceivable DCC. In addition, Djatschenko (2019) presents closed-form solutions for the valuation formula’s first and second derivatives, which are useful in the Newton-Raphson based determination of the bond’s yield as well as in calculation of duration and convexity. The approach outlined in Djatschenko (2019) relies exclusively on information that is typically provided by financial data vendors and is seamlessly implemented in the R package BondValuation.

The remainder of this paper consists of the two main sections, “2” and “3”. The section entitled “2” provides an overview of the functions implemented in the R package BondValuation and briefly illustrates the underlying theoretical concepts. The subsection entitled “2.1” introduces the DCCs covered by BondValuation and demonstrates their impact on interest accrual using the function AccrInt(). Subsequently, the 2.2 and its implementation within the function AnnivDates() are illustrated. In the following subsection, the calculation of 2.3 is demonstrated using the functions AnnivDates() and DP(). Next, the functions BondVal.Yield() and BondVal.Price() are used to compute 2.4. The section entitled “3” demonstrates how the R package BondValuation can be used for the analysis of large data frames of fixed coupon bonds. The paper ends with a short “4”.

2 The BondValuation package

The R package BondValuation consists of five functions, AccrInt(), AnnivDates(), BondVal.Price(), BondVal.Yield(), and DP(), and four data frames, List.DCC, NonBusDays.Brazil, PanelSomeBonds2016, SomeBonds2016.

The workhorse function of the package, AnnivDates(), performs a variety of sanity checks on the input data and, if possible, automatically corrects corrupted entries. It determines the bond-specific temporal structure and cash flows. The output of AnnivDates() is used in the downstream processes of the functions BondVal.Price(), BondVal.Yield(), and DP(). While AnnivDates(), BondVal.Price(), BondVal.Yield(), and DP() require bond data as input, the function AccrInt() simply computes the amount of interst accruing from some start date to some end date.

The data frames PanelSomeBonds2016 and SomeBonds2016 provide simulated data of 100 plain vanilla fixed coupon corporate bonds issued in 2016. List.DCC provides an overview of the DCCs implemented in the R package BondValuation. NonBusDays.Brazil is used with the BusDay/252 (Brazilian) convention and contains all non-business days in Brazil from 1946-01-01 to 2299-12-31 based on the Brazilian national holiday calendar.

Day count conventions

All DCCs that are identified by Thomson Reuters EIKON for plain vanilla fixed coupon bonds in 2017, and, additionally, the 30E/360 (ISDA) method, are covered by the R package BondValuation2:

> # example 1
> library(BondValuation)
> print(List.DCC, row.names = FALSE)
DCC                DCC.Name                     DCC.Reference \\
  1          ACT/ACT (ISDA)                [@ISDA1998]; [@ISDA2006] section 4.16 (b)\\
  2          ACT/ACT (ICMA)                [@ICMA2010]; [@ISDA2006] section 4.16 (c)\\
  3           ACT/ACT (AFB)                     [@ISDA1998]; [@EBF2004]; [@SWX2003]\\
  4                ACT/365L                               [@ICMA2010]; [@SWX2003]\\
  5                  30/360     [@ISDA2006] section 4.16 (f); [@MSRB2017] Rule G-33\\
  6                 30E/360   [@ICMA2010]; [@ISDA2006] section 4.16 (g); [@SWX2003]\\
  7          30E/360 (ISDA)                            [@ISDA2006] section 4.16 (h)\\
  8         30/360 (German)            [@EBF2004]; [@SWX2003]\\
  9               30/360 US                                [@Mayle1993]; [@SWX2003]\\
 10         ACT/365 (Fixed)                [@ISDA2006] section 4.16 (d); [@SWX2003]\\
 11             ACT(NL)/365                     [@Krgin2002]; Thomson Reuters EIKON\\
 12                 ACT/360                [@ISDA2006] section 4.16 (e); [@SWX2003]\\
 13                  30/365                     [@Krgin2002]; Thomson Reuters EIKON\\
 14 ACT/365 (Canadian Bond)                      [@IIAC2018]; Thomson Reuters EIKON\\
 15                 ACT/364                                   Thomson Reuters EIKON\\
 16  BusDay/252 (Brazilian)   `        [@CaputoSilva2010], [@Itau2017]\\

DCC.Reference
(International Swaps and Derivatives Association, Inc. 1998); (International Swaps and Derivatives Association, Inc. 2006) section 4.16 (b)
(International Capital Market Association 2010); (International Swaps and Derivatives Association, Inc. 2006) section 4.16 (c)
(International Swaps and Derivatives Association, Inc. 1998); (Banking Federation of the European Union 2004); (Christie and SWX Swiss Exchange 2003) (International Capital Market Association 2010); (Christie and SWX Swiss Exchange 2003)
(International Swaps and Derivatives Association, Inc. 2006) section 4.16 (f); (Municipal Securities Rulemaking Board 2017) Rule G-33
(International Capital Market Association 2010); (International Swaps and Derivatives Association, Inc. 2006) section 4.16 (g); (Christie and SWX Swiss Exchange 2003)
(International Swaps and Derivatives Association, Inc. 2006) section 4.16 (h)
(Banking Federation of the European Union 2004); (Christie and SWX Swiss Exchange 2003)
(Mayle 1993); (Christie and SWX Swiss Exchange 2003)
(International Swaps and Derivatives Association, Inc. 2006) section 4.16 (d); (Christie and SWX Swiss Exchange 2003)
(Krgin 2002); Thomson Reuters EIKON (International Swaps and Derivatives Association, Inc. 2006) section 4.16 (e); (Christie and SWX Swiss Exchange 2003)
(Krgin 2002); Thomson Reuters EIKON
(Investment Industry Association of Canada (IIAC) 2018); Thomson Reuters EIKON
Thomson Reuters EIKON
(Caputo Silva et al. 2010), (Itaú Unibanco 2017)

The function AccrInt() can be used to compare the differences in interest accrual between the day count methods. As an example, the code below returns the number of days and the amount of interest (in percent of the bond’s par value) accrued from Start = 2011-08-31 to End = 2012-02-29 with different DCCs, ceteris paribus. In this example, we assume that CpY = 2 coupons are paid per year, the nominal interest rate p.a. is Coup = 5.25%, the bond is redeemed at RV = 100% of its par value, and payments follow the End-of-Month rule3, i.e., EOM = 1. In addition, some of the DCCs require specification of the next coupon payment’s year figure, YearNCP = 2012, and the maturity date, Mat = 2021-08-31.

> # example 2
> library(BondValuation)
> DCC_Comparison<-data.frame(Start = rep(as.Date("2011-08-31"), 16),
+                            End = rep(as.Date("2012-02-29"), 16),
+                            Coup = rep(5.25, 16),
+                            DCC = seq(1, 16),
+                            DCC.Name = List.DCC[, 2],
+                            RV = rep(100, 16),
+                            CpY = rep(2, 16),
+                            Mat = rep(as.Date("2021-08-31"), 16),
+                            YearNCP = rep(2012, 16),
+                            EOM = rep(1, 16))
> AccrIntOutput <- suppressWarnings(
+   apply(
+     DCC_Comparison[, c('Start', 'End', 'Coup', 'DCC', 'RV', 'CpY', 'Mat',
+                        'YearNCP', 'EOM')], 1,
+     function(y) AccrInt(y[1], y[2], y[3], y[4], y[5], y[6], y[7], y[8], y[9])
+   )
+ )
> Accrued_Interest <- do.call(rbind, lapply(AccrIntOutput, function(x) x[[1]]))
> Days_Accrued <- do.call(rbind, lapply(AccrIntOutput, function(x) x[[2]]))
> DCC_Comparison <- cbind(DCC_Comparison, Accrued_Interest, Days_Accrued)
> print(DCC_Comparison[, c('DCC.Name', 'Start', 'End', 'Days_Accrued', 
+                          'Accrued_Interest')], row.names = FALSE)
               DCC.Name        Start          End   Days_Accrued   Accrued_Interest
         ACT/ACT (ISDA)   2011-08-31   2012-02-29            182           2.615490
         ACT/ACT (ICMA)   2011-08-31   2012-02-29            182           2.625000
          ACT/ACT (AFB)   2011-08-31   2012-02-29            182           2.617808
               ACT/365L   2011-08-31   2012-02-29            182           2.610656
                 30/360   2011-08-31   2012-02-29            179           2.610417
                30E/360   2011-08-31   2012-02-29            179           2.610417
         30E/360 (ISDA)   2011-08-31   2012-02-29            180           2.625000
        30/360 (German)   2011-08-31   2012-02-29            180           2.625000
              30/360 US   2011-08-31   2012-02-29            179           2.610417
        ACT/365 (Fixed)   2011-08-31   2012-02-29            182           2.617808
            ACT(NL)/365   2011-08-31   2012-02-29            182           2.617808
                ACT/360   2011-08-31   2012-02-29            182           2.654167
                 30/365   2011-08-31   2012-02-29            179           2.574658
ACT/365 (Canadian Bond)   2011-08-31   2012-02-29            182           2.625000
                ACT/364   2011-08-31   2012-02-29            182           2.625000
 BusDay/252 (Brazilian)   2011-08-31   2012-02-29            124           2.549769

Bond-specific temporal structure

The function AnnivDates() evaluates bond-specific information and returns the bond’s time-invariant characteristics in the data frame Traits, the bond’s temporal structure in the data frame DateVectors and, if the nominal interest rate is passed, the bond’s cash flows in the data frame PaySched.4 The classes and formats of input data are checked and adjusted, if possible. Moreover, AnnivDates() performs several plausibility tests, e.g., whether the provided calendar dates are in a correct chronological order and whether there are inconsistencies among the provided parameters. The results of these sanity checks are reported in the data frame Warnings.

The minimum accepted input for AnnivDates() are two calendar dates, of which the first is interpreted as the bond’s issue date and the second as its maturity date. If, as illustrated below, only two dates are passed to AnnivDates(), several parameters take on default values, which are reported in warning messages.

> # example 3
> library(BondValuation)
> AnnivDates(as.Date("2019-05-31"), "2021-07-31")
\$`Warnings`
Em_FIAD_differ    EmMatMissing    CpYOverride   RV_set100percent     NegLifeFlag
             0               0              1                  1               0
      ZeroFlag   Em_Mat_SameMY ChronErrorFlag    FIPD_LIPD_equal IPD_CpY_Corrupt
             0               0              0                  0               0
 EOM_Deviation     EOMOverride    DCCOverride            NoCoups
             0               1              1                  0
\$Traits
DateOrigin           CpY           FIAD           Em      Em_Orig          FIPD
1970-01-01             2           <NA>   2019-05-31   2019-05-31          <NA>
 FIPD_Orig      est_FIPD           LIPD    LIPD_Orig     est_LIPD           Mat
      <NA>    2019-07-31           <NA>         <NA>   2021-01-31    2021-07-31
     Refer       FCPType      FCPLength      LCPType    LCPLength           Par
2021-07-31         short      0.3370166      regular            1           100
CouponInPercent.p.a  DayCountConvention     EOM_Orig      est_EOM      EOM_used
                 NA                   2           NA            1             1

\$DateVectors
 RealDates   RD_indexes    CoupDates   CD_indexes   AnnivDates   AD_indexes
2019-05-31    0.6629834   2019-07-31            1   2019-01-31            0
2019-07-31    1.0000000   2020-01-31            2   2019-07-31            1
2020-01-31    2.0000000   2020-07-31            3   2020-01-31            2
2020-07-31    3.0000000   2021-01-31            4   2020-07-31            3
2021-01-31    4.0000000   2021-07-31            5   2021-01-31            4
2021-07-31    5.0000000         <NA>           NA   2021-07-31            5

Warning messages:
1: In InputFormatCheck(Em = Em, Mat = Mat, CpY = CpY, FIPD = FIPD,  :
   The maturity date (Mat) is supplied as a string of class "character" in the
   format "yyyy-mm-dd". It is converted to class "Date" using the command
   "as.Date(Mat,"%Y-%m-%d")" and processed as Mat = 2021-07-31 .
2: In AnnivDates(as.Date("2019-05-31"), "2021-07-31") :
   Number of interest payments per year (CpY) is missing or NA. CpY is set 2!
3: In AnnivDates(as.Date("2019-05-31"), "2021-07-31") :
   Redemption value (RV) is missing or NA. RV is set 100!
4: In AnnivDates(as.Date("2019-05-31"), "2021-07-31") :
   EOM was not provided or NA! EOM is set 1 .
   Note: The available calandar dates suggest that EOM = 1 .
5: In AnnivDates(as.Date("2019-05-31"), "2021-07-31") :
   The day count indentifier (DCC) is missing or NA. DCC is set 2 (Act/Act (ICMA))!

Since neither the first nor the penultimate coupon payment date is passed to AnnivDates(), the calendar dates in the data frame DateVectors are constructed backwards starting from the maturity date 2021-07-31. This results in a bond with a short first coupon period having a length of $Traits$FCPLength = 0.3370166 regular coupon periods.

The data frame DateVectors contains three vectors of calendar dates, RealDates, CoupDates, and AnnivDates, and their corresponding indexes, RD_indexes, CD_indexes, and AD_indexes. The vector RealDates comprises the bond’s issue date, maturity date, and all coupon payment dates in between, while CoupDates contains only the coupon payment dates. The vector AnnivDates consists of the bond’s so-called anniversary dates, i.e., scheduled coupon dates and notional coupon dates located before the first and after the penultimate coupon payment dates. The lengths of the first ($Traits$FCPLength) and final coupon periods ($Traits$LCPLength) are calculated as differences between the coresponding values of the vectors AD_indexes and RD_indexes. RD_indexes are used in the functions BondVal.Price(), BondVal.Yield(), and DP() to determine the powers of discount factors in pricing formulas.

As warning message 4 in the example above reports, the function AnnivDates() analyzes the provided calendar dates to ascertain whether the bond follows the End-of-Month rule (EOM). An automated replacement of the provided value of EOM by the value determined by the function AnnivDates() can be activated by setting option FindEOM = TRUE, which defaults to FALSE.

The following example illustrates the effect of EOM on DateVectors. In addition to issue date (Em) and maturity date (Mat), the first coupon payment date (FIPD), the penultimate coupon payment date (LIPD), the number of coupon payments p.a. (CpY), and EOM are passed to the function AnnivDates():

> # example 4
> library(BondValuation)
> # example 4a: computing DateVectors for EOM = 1
> EOM.input <- 1
> AnnivDates(Em = as.Date("2019-05-31"),
+            Mat = as.Date("2021-07-31"),
+            CpY = 2,
+            FIPD = as.Date("2020-02-29"),
+            LIPD = as.Date("2021-02-28"),
+            EOM = EOM.input)\$DateVectors
   RealDates   RD_indexes    CoupDates   CD_indexes   AnnivDates   AD_indexes
1 2019-05-31    -0.500000   2020-02-29     1.000000   2019-02-28           -1
2 2020-02-29     1.000000   2020-08-31     2.000000   2019-08-31            0
3 2020-08-31     2.000000   2021-02-28     3.000000   2020-02-29            1
4 2021-02-28     3.000000   2021-07-31     3.831522   2020-08-31            2
5 2021-07-31     3.831522         <NA>           NA   2021-02-28            3
6       <NA>           NA         <NA>           NA   2021-08-31            4
Warning messages:
1: In AnnivDates(Em = as.Date("2019-05-31"), Mat = as.Date("2021-07-31"),  :
   Redemption value (RV) is missing or NA. RV is set 100!
2: In AnnivDates(Em = as.Date("2019-05-31"), Mat = as.Date("2021-07-31"),  :
   The day count indentifier (DCC) is missing or NA. DCC is set 2 (Act/Act (ICMA))!
>
> # example 4b: computing DateVectors for EOM = 0
> EOM.input <- 0
> AnnivDates(Em = as.Date("2019-05-31"),
+            Mat = as.Date("2021-07-31"),
+            CpY = 2,
+            FIPD = as.Date("2020-02-29"),
+            LIPD = as.Date("2021-02-28"),
+            EOM = EOM.input)\$DateVectors
   RealDates   RD_indexes    CoupDates   CD_indexes   AnnivDates   AD_indexes
1 2019-05-31   -0.4945055   2020-02-29     1.000000   2019-02-28           -1
2 2020-02-29    1.0000000   2020-08-29     2.000000   2019-08-29            0
3 2020-08-29    2.0000000   2021-02-28     3.000000   2020-02-29            1
4 2021-02-28    3.0000000   2021-07-31     3.840659   2020-08-29            2
5 2021-07-31    3.8406593         <NA>           NA   2021-02-28            3
6       <NA>           NA         <NA>           NA   2021-08-29            4
Warning messages:
1: In AnnivDates(Em = as.Date("2019-05-31"), Mat = as.Date("2021-07-31"),  :
   Redemption value (RV) is missing or NA. RV is set 100!
2: In AnnivDates(Em = as.Date("2019-05-31"), Mat = as.Date("2021-07-31"),  :
   The available calandar dates suggest that EOM = 1 .
   Option FindEOM = FALSE is active. Provided EOM is not overridden and remains
   EOM = 0 .
3: In AnnivDates(Em = as.Date("2019-05-31"), Mat = as.Date("2021-07-31"),  :
   The day count indentifier (DCC) is missing or NA. DCC is set 2 (Act/Act (ICMA))!

In contrast to example 3, the bond in example 4 features a long first and a short final coupon period. The function AnnivDates() has checked whether the provided dates FIPD and LIPD are on each other’s annivesary dates and constructed the calendar dates in DateVectors backwards and forwards from LIPD. The values of RD_indexes and AD_indexes are illustrated in 1, where \(E\) corresponds to the first element of RD_indexes and \(M\) is the final element of RD_indexes.

As shown in example 4, all else equal, the value of EOM affects the DCC-conformal temporal locations of the issue date \(E\) and the maturity date \(M\) and, hence, the lengths of the first and final coupon periods, which, in turn, determine the amounts of interest paid on the first and final coupon payment dates. So far, no value of DCC was passed to the function AnnivDates(). As reported in the warning messages in example 4, the parameter DCC defaults to the Act/Act (ICMA) convention. The following, example 5, illustrates how RD_indexes, FCPLength and LCPLength vary across DCCs for EOM = 0.

graphic without alt text
Figure 1: Timeline illustration of the bonds in examples 4 and 5.
> # example 5
> library(BondValuation)
> TempStruct.by.DCC <- data.frame(Em = rep(as.Date("2019-05-31"), 16),
+                                 Mat = rep(as.Date("2021-07-31"), 16),
+                                 CpY = rep(2, 16),
+                                 FIPD = rep(as.Date("2020-02-29"), 16),
+                                 LIPD = rep(as.Date("2021-02-28"), 16),
+                                 FIAD = rep(as.Date("2019-05-31"), 16),
+                                 DCC = seq(1, 16),
+                                 EOM = rep(0, 16),
+                                 DCC.Name = List.DCC[, 2])
> # Applying AnnivDates() to the data frame TempStruct.by.DCC for EOM = 0
> suppressWarnings(
+   FullAnalysis.EOM0 <- apply(
+     TempStruct.by.DCC[, c('Em','Mat','CpY','FIPD','LIPD','FIAD','DCC','EOM')],
+     1, function(y) AnnivDates(
+       y[1], y[2], y[3], y[4], y[5], y[6], , , y[7], y[8])
+   )
+ )
> FCPLength.EOM0 <- lapply(lapply(lapply(FullAnalysis.EOM0, `[[`, 2), `[[`, 15)
+                          , na.omit)
> FCPLength.EOM0 <- as.data.frame(do.call(rbind, lapply(FCPLength.EOM0, round, 4)))
> LCPLength.EOM0 <- lapply(lapply(lapply(FullAnalysis.EOM0, `[[`, 2), `[[`, 17)
+                          , na.omit)
> LCPLength.EOM0 <- as.data.frame(do.call(rbind, lapply(LCPLength.EOM0, round, 4)))
> TempStruct.EOM0 <- lapply(lapply(lapply(FullAnalysis.EOM0, `[[`, 3), `[[`, 2)
+                           , na.omit)
> TempStruct.EOM0 <- lapply(TempStruct.EOM0, `length<-`, 
+                           max(lengths(TempStruct.EOM0)))
> TempStruct.EOM0 <- as.data.frame(do.call(rbind, lapply(TempStruct.EOM0, round, 4)))
> TempStruct.by.DCC.EOM0 <- cbind(TempStruct.by.DCC, TempStruct.EOM0,
+                                 FCPLength.EOM0, LCPLength.EOM0)
> names(TempStruct.by.DCC.EOM0)[c(10:16)] <- c("E", "01", "02", "03", "M",
+                                              "FCPLength", "LCPLength")
> print(TempStruct.by.DCC.EOM0[, c(9:ncol(TempStruct.by.DCC.EOM0))],
+       row.names = FALSE)
               DCC.Name         E   01   02   03        M  FCPLength  LCPLength
         ACT/ACT (ISDA)   -0.4945    1    2    3   3.8407     1.4945     0.8407
         ACT/ACT (ICMA)   -0.4945    1    2    3   3.8407     1.4945     0.8407
          ACT/ACT (AFB)   -0.4945    1    2    3   3.8407     1.4945     0.8407
               ACT/365L   -0.4945    1    2    3   3.8407     1.4945     0.8407
                 30/360   -0.4862    1    2    3   3.8453     1.4862     0.8453
                30E/360   -0.4917    1    2    3   3.8398     1.4917     0.8398
         30E/360 (ISDA)   -0.4972    1    2    3   3.8380     1.4972     0.8380
        30/360 (German)   -0.4972    1    2    3   3.8380     1.4972     0.8380
              30/360 US   -0.4862    1    2    3   3.8453     1.4862     0.8453
        ACT/365 (Fixed)   -0.4945    1    2    3   3.8407     1.4945     0.8407
            ACT(NL)/365   -0.4945    1    2    3   3.8407     1.4945     0.8407
                ACT/360   -0.4945    1    2    3   3.8407     1.4945     0.8407
                 30/365   -0.4862    1    2    3   3.8453     1.4862     0.8453
ACT/365 (Canadian Bond)   -0.4945    1    2    3   3.8407     1.4945     0.8407
                ACT/364   -0.4945    1    2    3   3.8407     1.4945     0.8407
 BusDay/252 (Brazilian)   -0.5040    1    2    3   3.8425     1.5040     0.8425

The output of example 5 shows that for the specified bond, there is little variation in temporal structure across the DCCs. Specifically, with DCC \(\in \{1,\ 2,\ 3,\ 4,\ 10,\ 11,\ 12,\ 14,\ 15\}\), the values of RD_indexes are \(\{-0.4945,\ 1,\ 2,\ 3,\ 3.8407\}\); with DCC \(\in \{5,\ 9,\ 13\}\), it holds RD_indexes \(= \{-0.4862,\ 1,\ 2,\ 3,\ 3.8453\}\), and with DCC \(\in \{7,\ 8\}\), we get RD_indexes \(= \{-0.4972,\ 1,\ 2,\ 3,\ 3.8380\}\). Only the DCCs \(6\) and \(16\) produce a unique temporal structure for this bond. Nevertheless, it would be wrong to infer from this that the temporal structure always has so little variation across the DCCs, as example 6 illustrates.

> # example 6
> library(BondValuation)
> TempStruct.by.DCC <- data.frame(Em = rep(as.Date("2019-10-31"), 16),
+                                 Mat = rep(as.Date("2024-02-29"), 16),
+                                 CpY = rep(2, 16),
+                                 FIPD = rep(as.Date("2020-03-30"), 16),
+                                 LIPD = rep(as.Date("2023-03-30"), 16),
+                                 FIAD = rep(as.Date("2019-10-31"), 16),
+                                 DCC = seq(1, 16),
+                                 EOM = rep(0, 16),
+                                 DCC.Name = List.DCC[, 2])
> 
> # Applying AnnivDates() to the data frame TempStruct.by.DCC for EOM = 0
> suppressWarnings(
+   FullAnalysis.EOM0 <- apply(
+     TempStruct.by.DCC[, c('Em','Mat','CpY','FIPD','LIPD','FIAD','DCC','EOM')],
+     1, function(y) AnnivDates(
+       y[1], y[2], y[3], y[4], y[5], y[6], , , y[7], y[8])
+   )
+ )
> FCPLength.EOM0 <- lapply(lapply(lapply(FullAnalysis.EOM0, `[[`, 2), `[[`, 15)
+                          , na.omit)
> FCPLength.EOM0 <- as.data.frame(do.call(rbind, lapply(FCPLength.EOM0, round, 4)))
> LCPLength.EOM0 <- lapply(lapply(lapply(FullAnalysis.EOM0, `[[`, 2), `[[`, 17)
+                          , na.omit)
> LCPLength.EOM0 <- as.data.frame(do.call(rbind, lapply(LCPLength.EOM0, round, 4)))
> TempStruct.EOM0 <- lapply(lapply(lapply(FullAnalysis.EOM0, `[[`, 3), `[[`, 2)
+                           , na.omit)
> TempStruct.EOM0 <- lapply(TempStruct.EOM0, `length<-`, 
+                           max(lengths(TempStruct.EOM0)))
> TempStruct.EOM0 <- as.data.frame(do.call(rbind, lapply(TempStruct.EOM0, round, 4)))
> TempStruct.by.DCC.EOM0 <- cbind(TempStruct.by.DCC, TempStruct.EOM0,
+                                 FCPLength.EOM0, LCPLength.EOM0)
> names(TempStruct.by.DCC.EOM0)[c(10:20)] <- c("E","01","02","03","04","05","06",
+                                              "07","M","FCPLength","LCPLength")
> print(TempStruct.by.DCC.EOM0[, c(9:ncol(TempStruct.by.DCC.EOM0))],
+       row.names = FALSE)
               DCC.Name       E 01 02 03 04 05 06 07      M FCPLength LCPLength
         ACT/ACT (ISDA)  0.1706  1  2  3  4  5  6  7 8.8354    0.8294    1.8354
         ACT/ACT (ICMA)  0.1703  1  2  3  4  5  6  7 8.8352    0.8297    1.8352
          ACT/ACT (AFB)  0.1708  1  2  3  4  5  6  7 8.8375    0.8292    1.8375
               ACT/365L  0.1703  1  2  3  4  5  6  7 8.8352    0.8297    1.8352
                 30/360  0.1667  1  2  3  4  5  6  7 8.8278    0.8333    1.8278
                30E/360  0.1667  1  2  3  4  5  6  7 8.8278    0.8333    1.8278
         30E/360 (ISDA)  0.1667  1  2  3  4  5  6  7 8.8278    0.8333    1.8278
        30/360 (German)  0.1667  1  2  3  4  5  6  7 8.8333    0.8333    1.8333
              30/360 US  0.1667  1  2  3  4  5  6  7 8.8278    0.8333    1.8278
        ACT/365 (Fixed)  0.1703  1  2  3  4  5  6  7 8.8352    0.8297    1.8352
            ACT(NL)/365  0.1713  1  2  3  4  5  6  7 8.8398    0.8287    1.8398
                ACT/360  0.1703  1  2  3  4  5  6  7 8.8352    0.8297    1.8352
                 30/365  0.1667  1  2  3  4  5  6  7 8.8278    0.8333    1.8278
ACT/365 (Canadian Bond)  0.1703  1  2  3  4  5  6  7 8.8352    0.8297    1.8352
                ACT/364  0.1703  1  2  3  4  5  6  7 8.8352    0.8297    1.8352
 BusDay/252 (Brazilian)  0.1840  1  2  3  4  5  6  7 8.8279    0.8160    1.8279

The output of example 6 reveals that, all else equal, the specified bond can feature \(7\) different temporal structures, depending on the stipulated DCC. While in example 5 the “ACT/ACT” family of DCCs produced the same temporal structure, in example 6 most of the “30/360” DCCs result in the same day count.

Cash flows, accrued interest, and dirty price

In the preceding examples of AnnivDates(), no information on nominal interest rate (Coup) and redemption value (RV) was passed to the function. If this information, however, is provided, then AnnivDates() generates the data frame PaySched, consisting of the scheduled coupon dates and the corresponding cash flows. As Djatschenko (2019) points out, precise application of the respective DCC’s mathematical rule results in varying interest payments. While this is intended for calculation of the cash flows paid at the ends of irregular first and final coupon periods, most issuers design their bonds to pay the same cash flow at the end of each regular period. With default RegCF.equal = 0 the function AnnivDates() calculates all cash flows according to the mathematical rule of the respective DCC. Passing any other value to RegCF.equal forces all regular cash flows to be equal sized. The following, example 7, uses the same input as example 6 supplemented by information on nominal interest rate p.a. (Coup \(= 10 \%\)) and redemption value (RV \(= 100\%\)) and illustrates the differences in cash flows (in percent of the bond’s par value) by DCC between the two modes of RegCF.equal.

> # example 7
> library(BondValuation)
> CashFlows.by.DCC <- data.frame(Em = rep(as.Date("2019-10-31"), 16),
+                                Mat = rep(as.Date("2024-02-29"), 16),
+                                CpY = rep(2, 16),
+                                FIPD = rep(as.Date("2020-03-30"), 16),
+                                LIPD = rep(as.Date("2023-03-30"), 16),
+                                FIAD = rep(as.Date("2019-10-31"), 16),
+                                RV = rep(100, 16),
+                                Coup = rep(10, 16),
+                                DCC = seq(1, 16),
+                                EOM = rep(0, 16),
+                                DCC.Name = List.DCC[, 2])
> 
> # Applying AnnivDates() to the data frame CashFlows.by.DCC for EOM = 0
> # with option RegCF.equal = 0 and RegCF.equal = 1
> Suffix <- c("RegCFvary","RegCFequal")
> for (i in c(0,1)) {
+   suppressWarnings(
+     FullAnalysis <- apply(
+       CashFlows.by.DCC[, c('Em','Mat','CpY','FIPD','LIPD','FIAD','RV',
+                            'Coup','DCC','EOM')],
+       1, function(y) AnnivDates(
+         y[1],y[2],y[3],y[4],y[5],y[6],y[7],y[8],y[9],y[10], RegCF.equal = i)
+     )
+   )
+   CashFlows <- lapply(lapply(lapply(FullAnalysis, `[[`, 4), `[[`, 2)
+                       , na.omit)
+   CashFlows <- as.data.frame(do.call(rbind, lapply(CashFlows, round, 4)))
+   CashFlows <- cbind(CashFlows.by.DCC, CashFlows)
+   names(CashFlows)[c(12:19)] <- c(
+     "CF.1","CF.2","CF.3","CF.4","CF.5","CF.6","CF.7","CF.M")
+   assign(paste0("CashFlows.by.DCC.",Suffix[i+1]),CashFlows)
+   rm(FullAnalysis,CashFlows)
+ }
> 
> # RegCF.equal = 0, \textit{i.e.}, regular cash flows may vary
> print(CashFlows.by.DCC.RegCFvary[, c(11:ncol(CashFlows.by.DCC.RegCFvary))],
+       row.names = FALSE)
               DCC.Name   CF.1   CF.2   CF.3   CF.4   CF.5   CF.6   CF.7   CF.M
         ACT/ACT (ISDA) 4.1303 5.0273 4.9519 5.0411 4.9589 5.0411 4.9589 9.2011
         ACT/ACT (ICMA) 4.1484 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 9.1758
          ACT/ACT (AFB) 4.1257 5.0411 4.9589 5.0411 4.9589 5.0411 4.9589 9.2055
               ACT/365L 4.1257 5.0273 4.9589 5.0411 4.9589 5.0411 4.9589 9.1803
                 30/360 4.1667 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 9.1389
                30E/360 4.1667 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 9.1389
         30E/360 (ISDA) 4.1667 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 9.1389
        30/360 (German) 4.1667 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 9.1667
              30/360 US 4.1667 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 9.1389
        ACT/365 (Fixed) 4.1370 5.0411 4.9589 5.0411 4.9589 5.0411 4.9589 9.2055
            ACT(NL)/365 4.1096 5.0411 4.9589 5.0411 4.9589 5.0411 4.9589 9.2055
                ACT/360 4.1944 5.1111 5.0278 5.1111 5.0278 5.1111 5.0278 9.3333
                 30/365 4.1096 4.9315 4.9315 4.9315 4.9315 4.9315 4.9315 9.0137
ACT/365 (Canadian Bond) 4.1370 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 9.1644
                ACT/364 4.1484 5.0549 4.9725 5.0549 4.9725 5.0549 4.9725 9.2308
 BusDay/252 (Brazilian) 3.9332 4.8809 4.8809 4.8809 4.8809 4.8809 4.8809 9.0060
> 
> # RegCF.equal = 1, \textit{i.e.}, regular cash flows forced to be equal
> print(CashFlows.by.DCC.RegCFequal[, c(11:ncol(CashFlows.by.DCC.RegCFequal))],
+       row.names = FALSE)
               DCC.Name   CF.1   CF.2   CF.3   CF.4   CF.5   CF.6   CF.7   CF.M
         ACT/ACT (ISDA) 4.1303 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 9.2011
         ACT/ACT (ICMA) 4.1484 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 9.1758
          ACT/ACT (AFB) 4.1257 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 9.2055
               ACT/365L 4.1257 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 9.1803
                 30/360 4.1667 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 9.1389
                30E/360 4.1667 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 9.1389
         30E/360 (ISDA) 4.1667 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 9.1389
        30/360 (German) 4.1667 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 9.1667
              30/360 US 4.1667 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 9.1389
        ACT/365 (Fixed) 4.1370 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 9.2055
            ACT(NL)/365 4.1096 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 9.2055
                ACT/360 4.1944 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 9.3333
                 30/365 4.1096 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 9.0137
ACT/365 (Canadian Bond) 4.1370 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 9.1644
                ACT/364 4.1484 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 9.2308
 BusDay/252 (Brazilian) 3.9332 4.8809 4.8809 4.8809 4.8809 4.8809 4.8809 9.0060

Please note that, irrespective of the value of RegCF.equal passed to AnnivDates(), the cash flows at the ends of all regular coupon periods are equal sized with the conventions ACT/ACT (ICMA), ACT/365 (Canadian Bond), and BusDay/252 (Brazilian). This is due to the DCC-specific rules described in Djatschenko (2019). While with the majority of DCCs, the cash flows are computed based upon the ratio of the nominal interest rate p.a. and the number of interest payments per year, which yields regular cash flows of \(5 \%\), BusDay/252 (Brazilian) determines them exponentially, resulting in regular cash flows of \(4.8809 \%\).

The vast majority of bonds are quoted clean, i.e., their observable prices do not contain accrued interest. The actual price that a bond buyer pays to the seller is called full or dirty price and computed as the sum of the quoted clean price and accrued interest, which is calculated according to the respective DCC. Accrued interest and the dirty price of a specific bond can be calculated using the function DP(). In addition to the input parameters required by AnnivDates(), the clean price (CP) and the settlement date (SETT) need to be passed to the function DP(). The following, example 8, returns the accrued interest and dirty price by DCC for the same bond as used in example 7, assuming that on the settlement dates SETT1 = 2020-09-28, SETT2 = 2023-03-30, and SETT3 = 2024-01-15, the quoted clean price is \(105\%\) of the bond’s par value.

> # example 8
> library(BondValuation)
> AccrIntDP.by.DCC <- data.frame(CP = 105,
+                                SETT1 = rep(as.Date("2020-09-28"), 16),
+                                SETT2 = rep(as.Date("2023-03-30"), 16),
+                                SETT3 = rep(as.Date("2024-01-15"), 16),
+                                Em = rep(as.Date("2019-10-31"), 16),
+                                Mat = rep(as.Date("2024-02-29"), 16),
+                                CpY = rep(2, 16),
+                                FIPD = rep(as.Date("2020-03-30"), 16),
+                                LIPD = rep(as.Date("2023-03-30"), 16),
+                                FIAD = rep(as.Date("2019-10-31"), 16),
+                                RV = rep(100, 16),
+                                Coup = rep(10, 16),
+                                DCC = seq(1, 16),
+                                EOM = rep(0, 16),
+                                DCC.Name = List.DCC[, 2])
> 
> Suffix <- c("SETT1","SETT2","SETT3")
> for (i in c(1:3)) {
+   DP.Output<-suppressWarnings(
+     apply(AccrIntDP.by.DCC[,c('CP',paste0('SETT',i),'Em','Mat','CpY','FIPD',
+                               'LIPD','FIAD','RV','Coup','DCC')],
+           1,function(y) DP(y[1],y[2],y[3],y[4],y[5],y[6],y[7],
+                            y[8],y[9],y[10],y[11])))
+   AI<-do.call(rbind,lapply(lapply(lapply(DP.Output, `[[`, 2), `[[`, 3), round, 4))
+   DP<-do.call(rbind,lapply(lapply(lapply(DP.Output, `[[`, 2), `[[`, 1), round, 4))
+   AccrIntDP.by.DCC<-cbind(AccrIntDP.by.DCC,AI,DP)
+   names(AccrIntDP.by.DCC)[
+     c((ncol(AccrIntDP.by.DCC) - 1) : ncol(AccrIntDP.by.DCC))] <- c(
+     paste0("AI.", Suffix[i]), paste0("DP.", Suffix[i]))
+   rm(DP.Output,AI,DP)
+ }
> print(AccrIntDP.by.DCC[,c(15:ncol(AccrIntDP.by.DCC))], row.names = FALSE)
               DCC.Name AI.SETT1 DP.SETT1 AI.SETT2 DP.SETT2 AI.SETT3 DP.SETT3
         ACT/ACT (ISDA)   4.9727 109.9727        0      105   7.9716 112.9716
         ACT/ACT (ICMA)   4.9457 109.9457        0      105   7.9396 112.9396
          ACT/ACT (AFB)   4.9863 109.9863        0      105   7.9726 112.9726
               ACT/365L   4.9727 109.9727        0      105   7.9508 112.9508
                 30/360   4.9444 109.9444        0      105   7.9167 112.9167
                30E/360   4.9444 109.9444        0      105   7.9167 112.9167
         30E/360 (ISDA)   4.9444 109.9444        0      105   7.9167 112.9167
        30/360 (German)   4.9444 109.9444        0      105   7.9167 112.9167
              30/360 US   4.9444 109.9444        0      105   7.9167 112.9167
        ACT/365 (Fixed)   4.9863 109.9863        0      105   7.9726 112.9726
            ACT(NL)/365   4.9863 109.9863        0      105   7.9726 112.9726
                ACT/360   5.0556 110.0556        0      105   8.0833 113.0833
                 30/365   4.8767 109.8767        0      105   7.8082 112.8082
ACT/365 (Canadian Bond)   4.9863 109.9863        0      105   7.9315 112.9315
                ACT/364   5.0000 110.0000        0      105   7.9945 112.9945
 BusDay/252 (Brazilian)   4.8412 109.8412        0      105   7.7354 112.7354

Yield to maturity, duration, and convexity

The yield to maturity p.a. is determined as the value \(y\) that fulfills equation ((1)).

\[\label{eq:MidPeriodTVM_odd} DP_{\tau} = CP_{\tau} + AC(t_{\tau}) = \dfrac{CN(t_{\tau})}{\left( 1+\dfrac{y}{h} \right)^{w}} + \sum_{i=1}^{\eta} \dfrac{CF_{i+k}}{\left( 1+\dfrac{y}{h} \right)^{w+i}} + \dfrac{CF_{M} + RV}{\left( 1+\dfrac{y}{h} \right)^{w+\eta+z}}. \tag{1}\]

In equation ((1)), \(DP_{\tau}\) denotes the dirty price, consisting of the quoted clean price \(CP_{\tau}\) and accrued interest \(AC(t_{\tau})\). Conformal with the notation in Djatschenko (2019), \(t_{\tau}\) is the settlement date and \(\tau\) its index in the temporal structure established by the function AnnivDates(). On the right side of equation ((1)), \(CN(t_{\tau})\) denotes the next coupon payment after the settlement date \(t_{\tau}\), and \(w\) is the fraction of a regular coupon period left until this payment. The set \(CF_{i+k}\) with \(i \in \left\{ x \in \mathbb{N} \mid x \in [1,\eta] \right\}\) contains all interest payments after \(t_k\), excluding the final coupon payment, \(CF_M\), where \(k\) is the index of the next coupon date after \(t_{\tau}\), \(\eta\) is the number of interest payment dates between \(t_{\tau}\) and the penultimate coupon date, and \(M\) is the index corresponding to the bond’s maturity date. \(RV\) denotes the redemption payment, \(z\) represents the length of the final coupon period, and \(h\) represents the number of regular interest payments per year.

The dirty price \(DP_{\tau}\) and the accrued interest \(AC(\tau)\) are computed as illustrated in example 8. The cash flows \(CN(t_{\tau})\), \(CF_{i+k}\), and \(CF_M\) are calculated as demonstrated in example 7. The powers in the denominators in equation ((1)) are found based on the temporal structure established by the function AnnivDates(), as shown in example 6.

Essentially, the same DCC is used for computation of cash flows, accrued interest and the indexes of the temporal structure. Nevertheless, the option Calc.Method in the functions BondVal.Price() and BondVal.Yield() allows for switching the calculation method for the temporal structure to DCC = 2, i.e., ACT/ACT (ICMA), while keeping the DCC passed to the function for determination of cash flows and accrued interest.

The function BondVal.Price() can be used to compute a bond’s clean price, (\(CP_{\tau}\)), given its yield to maturity p.a. (\(y\)), while the function BondVal.Yield() returns \(y\) given \(CP_{\tau}\). Besides accrued interest (\(AC(t_{\tau})\)) and dirty price (\(DP_{\tau}\)), both functions return \(\tau\), MacAulay duration, modified duration, and convexity of the specified bond.5 The following, example 9, demonstrates the use of the function BondVal.Yield() for the bond analyzed in example 8. In the output of example 9, YtM denotes the bond’s yield to maturity p.a. in percent, DUR is the bond’s modified duration in years, and Conv is the bond’s convexity in years. The suffixes .S1, .S2, and .S3 correspond to the three analyzed settlement dates, SETT1 = 2020-09-28, SETT2 = 2023-03-30, and SETT3 = 2024-01-15. For space reasons, the first column of the displayed data frame contains the DCC-codes instead of their names.

> # example 9
> library(BondValuation)
> YtM.by.DCC <- data.frame(CP = 105,
+                          SETT1 = rep(as.Date("2020-09-28"), 16),
+                          SETT2 = rep(as.Date("2023-03-30"), 16),
+                          SETT3 = rep(as.Date("2024-01-15"), 16),
+                          Em = rep(as.Date("2019-10-31"), 16),
+                          Mat = rep(as.Date("2024-02-29"), 16),
+                          CpY = rep(2, 16),
+                          FIPD = rep(as.Date("2020-03-30"), 16),
+                          LIPD = rep(as.Date("2023-03-30"), 16),
+                          FIAD = rep(as.Date("2019-10-31"), 16),
+                          RV = rep(100, 16),
+                          Coup = rep(10, 16),
+                          DCC = seq(1, 16),
+                          EOM = rep(0, 16))
> 
> Suffix <- c("S1","S2","S3")
> i<-1
> for (i in c(1:3)) {
+   BondValYield.Output<-suppressWarnings(
+     apply(YtM.by.DCC[,c('CP',paste0('SETT',i),'Em','Mat','CpY','FIPD',
+                         'LIPD','FIAD','RV','Coup','DCC')],
+           1,function(y) BondVal.Yield(y[1],y[2],y[3],y[4],y[5],y[6],y[7],
+                                       y[8],y[9],y[10],y[11])))
+   
+   YtM<-do.call(rbind,lapply(lapply(BondValYield.Output, `[[`, 4), round, 3))
+   ModDUR<-do.call(rbind,lapply(lapply(BondValYield.Output, `[[`, 5), round, 4))
+   Conv<-do.call(rbind,lapply(lapply(BondValYield.Output, `[[`, 7), round, 4))
+   YtM.by.DCC<-cbind(YtM.by.DCC,YtM,ModDUR,Conv)
+   names(YtM.by.DCC)[
+     c((ncol(YtM.by.DCC) - 2) : ncol(YtM.by.DCC))] <- c(
+       paste0("YtM.", Suffix[i]), paste0("DUR.", Suffix[i]),
+       paste0("Conv.", Suffix[i]))
+   rm(BondValYield.Output,YtM,ModDUR,Conv)
+ }
> print(YtM.by.DCC[,c(13,15:ncol(YtM.by.DCC))], row.names = FALSE)
 DCC  YtM.S1  DUR.S1  Conv.S1  YtM.S2  DUR.S2  Conv.S2   YtM.S3  DUR.S3  Conv.S3
   1   8.244  2.7593   4.9777   4.360  0.8824   0.7786  -27.035  0.1277   0.0163
   2   8.252  2.7590   4.9766   4.334  0.8825   0.7788  -26.956  0.1279   0.0164
   3   8.245  2.7595   4.9793   4.360  0.8833   0.7803  -26.899  0.1282   0.0164
   4   8.238  2.7604   4.9813   4.339  0.8824   0.7787  -27.002  0.1279   0.0164
   5   8.251  2.7564   4.9676   4.313  0.8792   0.7730  -27.373  0.1265   0.0160
   6   8.251  2.7564   4.9676   4.313  0.8792   0.7730  -27.373  0.1265   0.0160
   7   8.251  2.7564   4.9676   4.313  0.8792   0.7730  -27.373  0.1265   0.0160
   8   8.252  2.7584   4.9746   4.329  0.8817   0.7774  -26.568  0.1293   0.0167
   9   8.251  2.7564   4.9676   4.313  0.8792   0.7730  -27.373  0.1265   0.0160
  10   8.248  2.7587   4.9763   4.365  0.8822   0.7784  -26.973  0.1279   0.0164
  11   8.243  2.7604   4.9824   4.354  0.8845   0.7823  -26.825  0.1286   0.0165
  12   8.381  2.7505   4.9554   4.498  0.8812   0.7765  -26.824  0.1279   0.0163
  13   8.120  2.7645   4.9882   4.183  0.8802   0.7748  -27.521  0.1265   0.0160
  14   8.236  2.7593   4.9776   4.322  0.8826   0.7789  -26.983  0.1279   0.0164
  15   8.274  2.7570   4.9722   4.391  0.8820   0.7780  -26.943  0.1279   0.0164
  16   8.032  2.7729   5.0104   4.175  0.8803   0.7750  -26.038  0.1314   0.0173

3 Application of the package BondValuation

This section demonstrates how the R package BondValuation can be applied for the analysis of large data frames. For this purpose, the two sample data frames, SomeBonds2016 and PanelSomeBonds2016, are used. SomeBonds2016 contains time-invariant information of \(100\) hypothetical bonds. PanelSomeBonds2016 provides daily clean prices and yields of the same bonds in long format.

Checking the data with AnnivDates()

Since erroneous entries in the data are often an issue, the function AnnivDates() performs several plausibility checks. Example 10 provides a summary of SomeBonds2016 and illustrates a strategy for error identification in this data frame.

> # example 10
> library(BondValuation)
> summary(SomeBonds2016)
    ID.No               Coup.Type              Issue.Date           FIAD.Input        
Min.   :  1.00        Length:100            Min.   :2016-01-01    Min.   :2016-01-01  
1st Qu.: 25.75        Class :character      1st Qu.:2016-04-25    1st Qu.:2016-04-25  
Median : 50.50        Mode  :character      Median :2016-06-12    Median :2016-06-12  
Mean   : 50.50                              Mean   :2016-06-19    Mean   :2016-06-19  
3rd Qu.: 75.25                              3rd Qu.:2016-08-23    3rd Qu.:2016-08-23  
Max.   :100.00                              Max.   :2016-10-14    Max.   :2016-10-28  
  FIPD.Input            LIPD.Input             Mat.Date             CpY.Input   
Min.   :2016-04-24    Min.   :2016-08-23    Min.   :2017-01-24    Min.   : 1.0  
1st Qu.:2016-09-30    1st Qu.:2019-02-14    1st Qu.:2019-07-24    1st Qu.: 2.0  
Median :2016-12-15    Median :2020-06-08    Median :2020-11-20    Median : 2.5  
Mean   :2016-12-15    Mean   :2021-11-11    Mean   :2022-05-18    Mean   : 4.3  
3rd Qu.:2017-03-02    3rd Qu.:2022-11-04    3rd Qu.:2023-05-05    3rd Qu.: 6.0  
Max.   :2017-08-23    Max.   :2056-05-20    Max.   :2056-08-31    Max.   :12.0  
  Coup.Input            RV.Input               DCC.Input           EOM.Input   
Min.   : 0.010        Min.   :100           Min.   : 1.00         Min.   :0.00  
1st Qu.: 0.800        1st Qu.:100           1st Qu.: 5.00         1st Qu.:1.00  
Median : 1.410        Median :100           Median :10.00         Median :1.00  
Mean   : 2.270        Mean   :100           Mean   : 8.95         Mean   :0.79  
3rd Qu.: 2.869        3rd Qu.:100           3rd Qu.:13.00         3rd Qu.:1.00  
Max.   :24.020        Max.   :100           Max.   :16.00         Max.   :1.00

The summary information above reveals that all bonds in the data frame were issued (Issue.Date) and started to accrue interest (FIAD.Input) in \(2016\). The terms to maturity (Mat.Date) span from about \(1\) to approximately \(40\) years. The summary of variable CpY.Input shows that there are no zero coupon bonds in the dataset and the number of interest payments per year varies from \(1\) to \(12\). Nominal interest rates (Coup.Input) average \(2.27\%\), varying from \(0.01\%\) to \(24.02\%\). All bonds are redeemed (RV.Input) at \(100\%\) of their respective par values and \(79\%\) of them follow the End-of-Month rule (EOM.Input). Now AnnivDates() is used to analyze the data for plausibility.

> # example 10: continued (I)
>
> # Applying AnnivDates() to the data frame SomeBonds2016.
> FullAnalysis<-suppressWarnings(
+   apply(
+     SomeBonds2016[,c('Issue.Date','Mat.Date','CpY.Input','FIPD.Input',
+                      'LIPD.Input','FIAD.Input','RV.Input','Coup.Input',
+                      'DCC.Input','EOM.Input')], 1,
+     function(y) AnnivDates(y[1], y[2], y[3], y[4], y[5], y[6], y[7],
+                            y[8], y[9], y[10])
+   )
+ )
> # Extracting the data frame Warnings and binding the Warnings to the bonds
> BondsWithWarnings<-cbind(
+   SomeBonds2016, do.call(
+     rbind, lapply(FullAnalysis, `[[`, 1)
+   )
+ )
> summary(BondsWithWarnings[,c((ncol(SomeBonds2016)+1):ncol(BondsWithWarnings))])
Em_FIAD_differ    EmMatMissing    CpYOverride  RV_set100percent    NegLifeFlag
  Min.   :0.00       Min.   :0      Min.   :0         Min.   :0      Min.   :0   
  1st Qu.:0.00       1st Qu.:0      1st Qu.:0         1st Qu.:0      1st Qu.:0   
  Median :0.00       Median :0      Median :0         Median :0      Median :0   
  Mean   :0.04       Mean   :0      Mean   :0         Mean   :0      Mean   :0   
  3rd Qu.:0.00       3rd Qu.:0      3rd Qu.:0         3rd Qu.:0      3rd Qu.:0   
  Max.   :1.00       Max.   :0      Max.   :0         Max.   :0      Max.   :0   
      ZeroFlag   Em_Mat_SameMY ChronErrorFlag   FIPD_LIPD_equal IPD_CpY_Corrupt
     Min.   :0       Min.   :0   Min.   :0.00      Min.   :0.00    Min.   :0.00   
     1st Qu.:0       1st Qu.:0   1st Qu.:0.00      1st Qu.:0.00    1st Qu.:0.00   
     Median :0       Median :0   Median :0.00      Median :0.00    Median :0.00   
     Mean   :0       Mean   :0   Mean   :0.01      Mean   :0.02    Mean   :0.09   
     3rd Qu.:0       3rd Qu.:0   3rd Qu.:0.00      3rd Qu.:0.00    3rd Qu.:0.00   
     Max.   :0       Max.   :0   Max.   :1.00      Max.   :1.00    Max.   :1.00   
 EOM_Deviation     EOMOverride    DCCOverride           NoCoups    
  Min.   :0.00    Min.   :0.00      Min.   :0      Min.   :0.00  
  1st Qu.:0.00    1st Qu.:0.00      1st Qu.:0      1st Qu.:0.00  
  Median :1.00    Median :1.00      Median :0      Median :0.00  
  Mean   :0.69    Mean   :0.68      Mean   :0      Mean   :0.01  
  3rd Qu.:1.00    3rd Qu.:1.00      3rd Qu.:0      3rd Qu.:0.00  
  Max.   :1.00    Max.   :1.00      Max.   :0      Max.   :1.00

The summary information in example 10: continued (I) reveals that \(1\%\) of the bonds suffer from a chronological error (ChronErrorFlag) and \(9\%\) feature inconsistencies between the coupon payment dates and the number of interest payment dates per year CpY (IPD_CpY_Corrupt). To illustrate the rationale behind the plausibility analysis, a manual inspection of the affected bonds is performed below.6

> # example 10: continued (II)
>
> # manual examination of the rows where ChronErrorFlag = 1
> print(BondsWithWarnings[
+   which(BondsWithWarnings\$ChronErrorFlag == 1),
+   c('ID.No', 'Issue.Date', 'FIAD.Input', 'FIPD.Input', 'LIPD.Input', 'Mat.Date')],
+   row.names = FALSE)
ID.No   Issue.Date   FIAD.Input   FIPD.Input   LIPD.Input     Mat.Date
   17   2016-08-23   2016-08-23   2017-08-23   2016-08-23   2017-08-23
> 
> # manual examination of the rows where IPD_CpY_Corrupt = 1
> print(BondsWithWarnings[
+   which(BondsWithWarnings\$IPD_CpY_Corrupt == 1),
+   c('ID.No', 'Issue.Date', 'FIAD.Input', 'FIPD.Input', 'LIPD.Input', 'Mat.Date',
+     'CpY.Input')], row.names = FALSE)
ID.No   Issue.Date   FIAD.Input   FIPD.Input   LIPD.Input     Mat.Date   CpY.Input
    2   2016-06-23   2016-06-23   2016-07-15   2019-05-15   2019-06-15           4
    4   2016-05-24   2016-05-24   2016-05-31   2017-04-30   2017-05-31           2
   19   2016-09-28   2016-09-28   2017-02-28   2021-08-31   2021-09-28           1
   56   2016-07-26   2016-07-26   2017-01-26   2020-07-26   2020-10-26           1
   64   2016-04-13   2016-04-13   2016-04-24   2017-03-24   2017-04-24           6
   65   2016-09-30   2016-09-30   2016-10-31   2018-02-28   2018-03-29           1
   70   2016-08-26   2016-08-26   2016-11-20   2056-05-20   2056-08-31           1
   82   2016-06-30   2016-06-30   2016-07-15   2028-09-15   2028-12-15           2
   84   2016-07-20   2016-07-20   2016-07-24   2016-09-24   2017-01-24           2

The chronological error occurred because the provided penultimate coupon date (LIPD.Input) is located prior to the supplied first interest payment date (FIPD.Input). Since the authenticity of FIPD.Input and LIPD.Input is unclear in this case, both are automatically dropped by AnnivDates(), and the calculation continues based upon the provided values of Issue.Date and Mat.Date.

As can be seen in the manual examination of the rows where IPD_CpY_Corrupt = 1, for all of them, there are inconsistencies between the value of CpY.Input and the interval between FIPD.Input and LIPD.Input. In the first row, for example, CpY.Input indicates that coupons are paid quarterly. If the value of FIPD.Input is correct, coupon payments should occur on July 15th, October 15th, January 15th, and April 15th. If the value of LIPD.Input is genuine, however, interest should be paid on May 15th, August 15th, November 15th, and February 15th. Finally, in this case, FIPD.Input and LIPD.Input can both be correct, lying on each other’s anniversary dates for a value of CpY.Input = 6. Since it is not clear which of the three values is correct, AnnivDates() cannot automatically revise the input but only helps the user to identify the inconsistency. If the data are passed to AnnivDates() as they are, CpY.Input is assumed to be genuine, FIPD.Input and LIPD.Input are dropped, and the execution continues as if they were not provided in the first place, resulting in the temporal structure and cash flows provided below in example 10: continued (III).

> # example 10: continued (III)
> # Printing data frame DateVectors for bond with ID.No = 2
> print(
+   as.data.frame(do.call(rbind, lapply(FullAnalysis, `[[`, 3)[2])),
+   row.names = FALSE)
 RealDates    RD_indexes    CoupDates   CD_indexes   AnnivDates   AD_indexes
2016-06-23    0.08888889   2016-09-15            1   2016-06-15            0
2016-09-15    1.00000000   2016-12-15            2   2016-09-15            1
2016-12-15    2.00000000   2017-03-15            3   2016-12-15            2
2017-03-15    3.00000000   2017-06-15            4   2017-03-15            3
2017-06-15    4.00000000   2017-09-15            5   2017-06-15            4
2017-09-15    5.00000000   2017-12-15            6   2017-09-15            5
2017-12-15    6.00000000   2018-03-15            7   2017-12-15            6
2018-03-15    7.00000000   2018-06-15            8   2018-03-15            7
2018-06-15    8.00000000   2018-09-15            9   2018-06-15            8
2018-09-15    9.00000000   2018-12-15           10   2018-09-15            9
2018-12-15   10.00000000   2019-03-15           11   2018-12-15           10
2019-03-15   11.00000000   2019-06-15           12   2019-03-15           11
2019-06-15   12.00000000         <NA>           NA   2019-06-15           12
> 
> # Printing data frame PaySched for bond with ID.No = 2
> print(
+   as.data.frame(do.call(rbind, lapply(FullAnalysis, `[[`, 4)[2])),
+   row.names = FALSE)
 CoupDates   CoupPayments
2016-09-15      0.7368611
2016-12-15      0.8087500
2017-03-15      0.8087500
2017-06-15      0.8087500
2017-09-15      0.8087500
2017-12-15      0.8087500
2018-03-15      0.8087500
2018-06-15      0.8087500
2018-09-15      0.8087500
2018-12-15      0.8087500
2019-03-15      0.8087500
2019-06-15      0.8087500

The consequences of the plausibility-check-induced automated data revision by AnnivDates() are stored in the data frame Traits. Alongside the values that were initially provided and that are actually used in the subsequent calculations, Traits contains information on the types and lengths of the first and final coupon periods. Example 10: continued (IV) demonstrates how the data frame Traits can be extracted from the output of AnnivDates() and provides summary information on the lengths and types of the first and final coupon periods in the data frame SomeBonds2016. Of the \(100\) bonds in SomeBonds2016, only \(20\) have regular first coupon periods and \(28\) feature final coupon periods of regular length. The lengths of the first coupon periods vary from \(1.37\%\) to \(1,200\%\) of the bond-specific regular coupon period length, while the final coupon periods average \(238\%\) and span from \(2.78\%\) to \(1,200\%\) of the respective bond’s regular coupon period length.

> # example 10: continued (IV)
> # Extracting the data frame Warnings and binding the Warnings to the bonds
> BondsWithTraits<-cbind(
+   SomeBonds2016, do.call(
+     rbind, lapply(FullAnalysis, `[[`, 2)
+   )
+ )
> summary(BondsWithTraits[, c('FCPType', 'LCPType', 'FCPLength', 'LCPLength')])
   FCPType       LCPType         FCPLength            LCPLength       
long   :49    long   :51     Min.   : 0.01366     Min.   : 0.02778  
short  :31    regular:28     1st Qu.: 0.92150     1st Qu.: 1.00000  
regular:20    short  :21     Median : 1.00000     Median : 1.25140  
                             Mean   : 2.19291     Mean   : 2.38417  
                             3rd Qu.: 3.00000     3rd Qu.: 3.00000  
                             Max.   :12.00000     Max.   :12.00000

Applying BondVal.Yield() to long format data

In addition to the time-invariant information in SomeBonds2016, the data frame PanelSomeBonds2016 provides daily clean prices (CP.Input) and yields to maturity (YtM.Input) that correspond to the trade dates, TradeDate, and settlement dates, SETT. TradeDate is the calendar date on which the transaction is initiated and the quoted clean price is observed; SETT is the actual calendar date on which the transfer of cash and assets is completed. The settlement date is used for the following computation. Example 11 below shows that PanelSomeBonds2016 has \(12,718\) rows and \(16\) columns and provides summary information regarding the time-variant variables. The clean prices span from \(90.38\%\) to \(224.16\%\), while the yields to maturity average \(-0.01593\%\), varying from \(-1.725\%\) to \(2\%\).

> # example 11
> library(BondValuation)
> dim(PanelSomeBonds2016)
[1] 12718    16
> summary(PanelSomeBonds2016[, c(13:16)])
         TradeDate                  SETT          CP.Input           YtM.Input       
Min.   :2016-01-29    Min.   :2016-02-02    Min.   : 90.38    Min.   :-1.72500  
1st Qu.:2016-08-03    1st Qu.:2016-08-05    1st Qu.:102.73    1st Qu.:-0.35000  
Median :2016-10-03    Median :2016-10-05    Median :105.79    Median :-0.02500  
Mean   :2016-09-20    Mean   :2016-09-23    Mean   :112.53    Mean   :-0.01593  
3rd Qu.:2016-11-17    3rd Qu.:2016-11-21    3rd Qu.:113.21    3rd Qu.: 0.27500  
Max.   :2016-12-30    Max.   :2017-01-03    Max.   :224.16    Max.   : 2.00000

In the following, example 12, the function BondVal.Yield() is used to determine \(\tau\), accrued interest, dirty price, yield to maturity, modified duration, MacAulay duration, and convexity for each bond and settlement date in PanelSomeBonds2016. In alternative 1, the function BondVal.Yield() is applied to every row of the data frame PanelSomeBonds. Alternative 2 demonstrates a significantly faster approach, where AnnivDates() is applied to every bond’s time-invariant characteristics before its output is passed to BondVal.Yield() for every settlement date. Alternative 2 takes less than half the time of alternative 1.7

> # example 12
> # analysis of PanelSomeBonds2016 with BondValuation
> library(BondValuation)
> Panel <- PanelSomeBonds2016
> Vars <- c("tau","AccrInt","DP","YtM","ModDUR","MacDUR","Conv")
> Panel[, c((ncol(Panel) + 1) : (ncol(Panel) + length(Vars)))] <- as.numeric(NA)
> names(Panel)[(ncol(Panel) - length(Vars) + 1) : ncol(Panel)] <- Vars
> 
> # Alternative 1: loop through the data frame 
> #                applying BondVal.Yield to each row
> Time.Alt01 <- system.time(
+   for (i in c(1:nrow(Panel))) \{
+     BondVal.Out <- suppressWarnings(
+       BondVal.Yield(CP = Panel\$CP.Input[i],
+                     SETT = Panel\$SETT[i],
+                     Em = Panel\$Issue.Date[i],
+                     Mat = Panel\$Mat.Date[i],
+                     CpY = Panel\$CpY.Input[i],
+                     FIPD = Panel\$FIPD.Input[i],
+                     LIPD = Panel\$LIPD.Input[i],
+                     FIAD = Panel\$FIAD.Input[i],
+                     RV = Panel\$RV.Input[i],
+                     Coup = Panel\$Coup.Input[i],
+                     DCC = Panel\$DCC.Input[i],
+                     EOM = Panel\$EOM.Input[i],
+                     Precision = .Machine\$double.eps^0.5
+       )
+     )
+     Panel[i, c((ncol(Panel) - length(Vars) + 1) : ncol(Panel))] <-
+       round(as.numeric(BondVal.Out[c(11, 2 : 7)]), 4)
+   \}, gcFirst = TRUE
+ )
> 
> 
> # Alternative 2: Run AnnivDates() once per Bond-ID and pass its output
> #                to BondVal.Yield() for every row with the same Bond-ID
> NonDuplID <- c(which(!duplicated(Panel\$ID.No)), (nrow(Panel)+1))
> Time.Alt02 <- system.time(
+   for (i in c(1 : (length(NonDuplID) - 1))) \{
+     BondCount <- NonDuplID[i]
+     AnnivDates.Out <- suppressWarnings(
+       AnnivDates(Em = Panel\$Issue.Date[BondCount],
+                  Mat = Panel\$Mat.Date[BondCount],
+                  CpY = Panel\$CpY.Input[BondCount],
+                  FIPD = Panel\$FIPD.Input[BondCount],
+                  LIPD = Panel\$LIPD.Input[BondCount],
+                  FIAD = Panel\$FIAD.Input[BondCount],
+                  RV = Panel\$RV.Input[BondCount],
+                  Coup = Panel\$Coup.Input[BondCount],
+                  DCC = Panel\$DCC.Input[BondCount],
+                  EOM = Panel\$EOM.Input[BondCount]
+       )
+     )
+     for (j in c(NonDuplID[i] : (NonDuplID[i + 1] - 1))) \{
+       BondVal.Out <- suppressWarnings(
+         BondVal.Yield(CP = Panel\$CP.Input[j],
+                       SETT = Panel\$SETT[j],
+                       Em = Panel\$Issue.Date[j],
+                       Mat = Panel\$Mat.Date[j],
+                       CpY = Panel\$CpY.Input[j],
+                       FIPD = Panel\$FIPD.Input[j],
+                       LIPD = Panel\$LIPD.Input[j],
+                       FIAD = Panel\$FIAD.Input[j],
+                       RV = Panel\$RV.Input[j],
+                       Coup = Panel\$Coup.Input[j],
+                       DCC = Panel\$DCC.Input[j],
+                       EOM = Panel\$EOM.Input[j],
+                       InputCheck = 0,
+                       Precision = .Machine\$double.eps^0.5,
+                       AnnivDatesOutput = AnnivDates.Out
+         )
+       )
+       Panel[j, c((ncol(Panel) - length(Vars) + 1) : ncol(Panel))] <-
+         round(as.numeric(BondVal.Out[c(11, 2 : 7)]), 4)
+     \}
+   \}, gcFirst = TRUE
+ )
>
> round(Time.Alt02[[3]] / Time.Alt01[[3]], 2)
[1] 0.48

4 Conclusion

This article introduces the R package BondValuation and provides guidance on its application for analysis of large data frames of fixed coupon bonds. The theoretical foundation of the package is the generalized valuation methodology developed by Djatschenko (2019). Its seamless implementation in BondValuation is framed by a set of routines that assist the user in data quality evaluation and automatically correct corrupted entries.

BondValuation is the first R package that properly handles irregular first and final coupon periods of fixed coupon bonds and provides a comprehensive coverage of the day count conventions (DCC) used in the global bond markets. Currently, \(16\) different DCCs are implemented, which account for the vast majority of the methods used in the global fixed income markets. Within its scope, the R package BondValuation performs correctly and efficiently. Nevertheless, the current version of the software remains open for further development and refinement. Essentially, the calculations are performed under the assumption that interest accrual and temporal structure follow the same DCC. The option CalcMethod in the functions BondVal.Price() and BondVal.Yield() can be used to force the temporal structure to follow the ACT/ACT (ICMA) method, while the DCC passed to the respective function is used to compute accrued interest. In future versions of the package, I intend to implement an explicit assignment of DCC to both interest accrual and temporal structure, which will increase the flexibility of the package.

A further limitation is that the calendar dates of the temporal structure are currently returned, regardless of whether or not they are business days. Although this is the common approach in theoretical bond valuation, including the possibility of business day adjustments for cash flows would be particularly appealing to practitioners. Along with business day adjustments, future versions of BondValuation can be extended by methods for bond portfolio analysis.

The current version of BondValuation is designed for processing non-callable, option-free, non-sinkable fixed coupon bonds and zero bonds. With the implemented methods, callable bonds can be analyzed through appropriate adjustment of the maturity date to the next call date, returning the so-called yield-to-worst and the corresponding duration and convexity measures. Based on the implemented functions, the R package BondValuation can be extended to incorporate methods for explicit treatment of callable, sinkable and convertible fixed and floating rate bonds.

The R package BondValuation provides the computational foundation for the exploration of a variety of interesting research questions related to the analysis of fixed income securities across markets. Even considering the limitations described above, the software is also useful to practitioners. I intend to continuously extend and improve the package, and I highly appreciate feedback from the users.

5 Acknowledgments

I would like to thank Ingo Geishecker for our frequent discussions and his profound advice. I am also grateful to Karl Ludwig Keiber and Philipp Otto for their helpful comments and suggestions, and to Inna Keil for her excellent research assistance. All remaining errors are my own responsibility.

CRAN packages used

BondValuation, fBonds, RQuantLib, YieldCurve

CRAN Task Views implied by cited packages

Finance

Note

This article is converted from a Legacy LaTeX article using the texor package. The pdf version is the official version. To report a problem with the html, refer to CONTRIBUTE on the R Journal homepage.

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Itaú Unibanco. Brazilian Sovereign Fixed Income and Foreign Exchange Markets. 1st ed Itaú Unibanco, 2017.
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References

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Citation

For attribution, please cite this work as

Djatschenko, "BondValuation: An R Package for Fixed Coupon Bond Analysis", The R Journal, 2020

BibTeX citation

@article{RJ-2019-055,
  author = {Djatschenko, Wadim},
  title = {BondValuation: An R Package for Fixed Coupon Bond Analysis},
  journal = {The R Journal},
  year = {2020},
  note = {https://rjournal.github.io/},
  volume = {11},
  issue = {2},
  issn = {2073-4859},
  pages = {124-141}
}

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