Bond Caculate and Term Structure-债券计算及期限结构

R Markdown

This is an R Markdown document. \(\color{#8D0012}{Markdown}\) is a simple formatting syntax for authoring HTML, PDF, and MS Word documents. For more details on using R Markdown see http://rmarkdown.rstudio.com.

Bond price,ytm,duration and convexity

Description

Basic yield, pricing, duration and convexity calculations. These functions perform simple present value calculations assuming that all periods between payments are the same length. Unlike bond functions in Excel, for example, settlement and maturity dates are not used. By default, duration is Macaulay duration.

Usage

  • bondpv(coupon, mat, yield, principal, freq)
  • bondyield(price, coupon, mat, principal, freq)
  • duration(price, coupon, mat, principal, freq, modified)
  • convexity(price, coupon, mat, principal, freq)

Arguments

  1. coupon: annual coupon
  2. mat: maturity in years
  3. yield: annual yield to maturity. If freq > 1, the yield is freq times the per period yield.
  4. principal: maturity payment of the bond, in addition to the final coupon. Default value is $1,000. If the instrument is an annuity, set principal to zero.
  5. freq: number of payments per year.
  6. price: price of the bond
  7. modified: If true, compute modified duration, otherwise compute Macaulay duration. FALSE by default.

Value

Return price, yield, duration and convexity. The bond price mathematical formula is: \[P = \sum_{t=1}^{t} \frac{C_{t-1}} {(1+ytm)^{t-1}} + \frac{C_{t}+P_{0}} {(1+ytm)^{t}}\]

The bond duartion formula is: \[MacD = \sum_{t=1}^{t}t*\frac{CF_{t}} {(1+y)^{t}}\frac{1}{P}\]

The modified duration formula is: \[ModD = -\frac{1}{P} \frac{\mathrm{d} P}{\mathrm{d} y} = \frac{1}{P} \frac{1}{1+y} \sum_{n=1}^{t} \frac{t*CF_{t}} {(1+y)^{t}}= \frac{MacD}{1+y}\]

The convexity formula is: \[Conv = \frac{1}{P} \frac{\mathrm{d}^2 P}{\mathrm{d} y^2} = \frac{1}{P}\frac{1}{(1+y)^2}\sum_{n=1}^{t} \frac{t(t+1)CF_{t}} {(1+y)^{t}}\]

The change of bond price can be caculated by the formula with ModD and Conv approximately: \[\frac{\Delta P}{P} \approx -ModD*{\Delta y} + \frac{1}{2} Conv*{\Delta y^2}\]

In fixed income practice, effective duration and effective convexity are used to describe the interest risk more often: \[EffD = \frac{1}{P} \frac{P_--P_+}{y_+-y_-} = \frac{P_--P_+}{2\Delta y P}\], \[EffConv = \frac{D_--D_+}{2 \Delta y} = \frac{P_--P_+-2P}{2 (\Delta y)^2 P}\]

Examples of bond caculate

library(derivmkts)

coupon <- 6; mat <- 3.65; freq <- 2; principal <- 100; yield <- 0.045;

price <- bondpv(coupon, mat, yield, principal, freq); price
## [1] 104.8077
bondyield(coupon, mat, price=price, principal, freq) 
## [1] 0.045
duration(price, coupon, mat, principal, freq, modified=FALSE) 
## [1] 3.216609
duration(price, coupon, mat, principal, freq, modified=TRUE) 
## [1] 3.145828
convexity(price, coupon, mat, principal, freq) 
## [1] 11.95165

In this example, the bond price is 104.81, and the modified duration is 3.15.

Term Structure Modeling

The functions in fBonds package are:

  • NelsonSiegel Nelson-Siegel Term Structure,
  • Svensson Nelson-Siegel-Svensson Term Structure.

Usage

  • NelsonSiegel(rate, maturity, doplot = TRUE)
  • Svensson(rate, maturity, doplot = TRUE)

Arguments

  1. doplot: a logical. Should a plot be displayed?
  2. maturity: a numeric vector of maturities on an annual scale.
  3. rate: a numeric vector of forward rates.

Value

a list object with entries returned from the optimization function nlminb.

Examples of term structure

Table 1: Bond yield and maturity
Maturity Yield
0.083 0.04984
0.167 0.05283
0.250 0.05549
0.333 0.05777
0.417 0.05961
0.500 0.06102
0.583 0.06216
0.667 0.06314
0.750 0.06403
0.833 0.06488 
par(mar=c(5,4,2,2))
plot(dt,pch=16,col=sample(rainbow(6),nrow(dt),replace = T))
grid()
Figure 1: Dot plot of bond data

Figure 1: Dot plot of bond data 

par(mar=c(5,4,2,2))
NelsonSiegel(Yield, Maturity,doplot=T)
Figure 2: Plot of NelsonSiegel Term Structure

Figure 2: Plot of NelsonSiegel Term Structure

Figure 2: Plot of NelsonSiegel Term Structure

Figure 2: Plot of NelsonSiegel Term Structure 

## $par
##         beta0         beta1         beta2          tau1 
##  8.885514e-02 -3.671098e-02 -1.737138e-09  1.025116e+00 
## 
## $objective
## [1] 0.0001602499
## 
## $convergence
## [1] 0
## 
## $iterations
## [1] 25
## 
## $evaluations
## function gradient 
##       37      125 
## 
## $message
## [1] "relative convergence (4)"
par(mar=c(5,4,2,2))
Svensson(Yield, Maturity,doplot=T)
Figure 3: Plot of Nelson-Siegel-Svensson Term Structure

Figure 3: Plot of Nelson-Siegel-Svensson Term Structure

Figure 3: Plot of Nelson-Siegel-Svensson Term Structure

Figure 3: Plot of Nelson-Siegel-Svensson Term Structure

Figure 3: Plot of Nelson-Siegel-Svensson Term Structure

Figure 3: Plot of Nelson-Siegel-Svensson Term Structure 

## $par
##         beta0         beta1         beta2         beta3          tau1 
##  0.0677180818 -0.0202951750  0.0003743188  0.0568608140  0.5829797701 
##          tau2 
## 21.0000767906 
## 
## $objective
## [1] 0.0290922
## 
## $convergence
## [1] 1
## 
## $iterations
## [1] 61
## 
## $evaluations
## function gradient 
##      132      503 
## 
## $message
## [1] "false convergence (8)"