sandbox/test_discountFactorArbitrage.R

In GARPFRM: Global Association of Risk Professionals: Financial Risk Manager

# Read in the data file
suppressMessages(library(GARPFRM))
options(digits=3)
data(bonds)


## The Cash Flows from Fixed-Rate Government Coupon Bonds
# Discount Factors and the Law of One Price
# Initialize: The Cash Flows from Fixed-Rate: treasury bonds ticking in quarters
## Example from Tuckman
cashFlow = rbind(c(100+(1+1/4)/2,0,0),c((4 +7/8)/2,100+(4+7/8)/2,0),c((4+1/2)/2,(4+1/2)/2,100+(4+1/2)/2))
# Initialize: Price of the bond
price = matrix(c(100.550, 104.513, 105.856), ncol=1)
DF = discountFactor(price, cashFlow)

## Additional Example
cashFlow = rbind(c(100, 0, 0, 0), c(2 + 7/8, 102 + 7/8, 0, 0), c(3 + 3/4, 3 + 3/4, 103 + 3/4, 0), c(3 + 3/4, 3 + 3/4, 3 + 3/4, 103 + 3/4))
# Initialize: Price of the bond
price <- matrix(c(96.8, 99.56, 100.86, 101.22), ncol=1)

# Estimate the Discount Factors (DF)
DF = discountFactor(price , cashFlow)
# To confirm solution check that price is replicable
(cashFlow%*%price)/100


## Estimate bondPrice
# Choose a 2 year bond with semiannual payments to match number of bond prices and CFs
time = seq(from=0.5, to=2, by=0.5)
# First define a bond object to be used throughout the analysis, where m is the compound frequency
bond = bondSpec(time, face=100, m=2, couponRate = 0.0475)
# Estimate price, yield, convexity and duration
price = bondPrice(bond,DF)
# Yield-to-maturity is often quoted when describing a security in terms of arates rather than price.
# Estimate yied to maturity
bondYTM(bond,DF)

# Duration measures the effect of a small parallel shift in the yield curve
# if rate goes up 10 basis points the relative P&L will change by mDuration*0.1%
mDuration = bondDuration(bond,DF)
# Duration plus convexity measure the effect of a larger parallel shift in the yield curve
# Note however, they do not measure the effect of non-parallel shifts
convexity = bondConvexity(bond,DF)

## Example with a longer compounding time sequence:
# Yields of bond with varying coupons over  Estimation and Plot
# Utilizing a discount factor trable rewrite DF 10 years semiannually
DF = rbind( 0.9615, 0.94305, 0.9246, 0.90591, 0.889, 0.87019, 0.8548, 0.8358825, 0.8219, 0.80294,
            0.7903, 0.7713, 0.7599, 0.74092, 0.7307, 0.7117325, 0.7026, 0.70059, 0.6756)
time = seq(0.5,10,0.5)
# estimate bond specs using a 4% coupon rate
bond = bondSpec(time, face=100, m=2, couponRate = 0.04)
bondYTM(bond,DF)

# Illustrating elasticity of YTM: measure 1% and 10% increase in yield on duration
DF = rbind(0.95434,0.9434,0.917232,0.89,0.85678,0.8396,0.81242,0.7921,0.7693,0.7473,0.7298,0.7050)
# Choose a 2 year bond with semiannual payments to match number of bond prices and CFs
time = seq(from=0.5, to=6, by=0.5)
# First define a bond object to be used throughout the analysis, where m is the compound frequency
bond = bondSpec(time, face=100, m=2, couponRate = 0.0475)
# Duration measures the effect of a small parallel shift in the yield curve
mDuration = bondDuration(bond,DF)
mDuration = bondDuration(bond,DF,0.01)
mDuration = bondDuration(bond,DF,0.1)


### Valuation and Risk Model Section- Yield Curve Shapes
# Vasicek Modeling to illustrate different yield curve calibrations
# Initialize Model
theta = 0.10
k = 0.8
sigma = 0.08
# Seven Yield Curves to estimate
r = seq(0, 0.15, 0.025)
length(r)
maturity = 10
# Illustration #1 for standard theta and initial r estimate yield path
yieldCurves = yieldCurveVasicek(r, k, theta, sigma, maturity)
# Plot using matplot-plot the columns of one matrix against the columns of another
maturity = seq(1,maturity,1)
matplot(maturity, yieldCurves, type="l", lty=1, main="Yield Curves")
# choose h = theta for y horizontal line
abline(h = theta, col="red", lty=2)

# Illustration #2 for high theta and low initial r estimate yield path
theta = 0.45
maturity = 10
yieldCurves = yieldCurveVasicek(r, k, theta, sigma, maturity)
# Plot using matplot-plot the columns of one matrix against the columns of another
maturity = seq(1,maturity,1)
matplot(maturity, yieldCurves, type="l", lty=1, main="Yield Curves")
# choose h = theta for y horizontal line
abline(h = theta, col="red", lty=2)


## Appliation: Idiosyncratic Pricing of US Treasury Notes and Bonds
t0 = as.Date("2013-08-15")
t1 = as.Date("2014-02-15")
tn = as.Date("2013-10-04")
currentDate = tn
# Apply a coupon rate of 4.75% bond and create a bond object
bond = bondSpec(face=100, m=2, couponRate = 0.0475)
y1 = 0.00961
bondFullPrice(bond, y1, 8, t0, t1, tn)$clean
bondFullPrice(bond, y1, 8, t0, t1, tn)$dirty
bondFullPrice(bond, y1, 8, t0, t1, tn)$accruedInterest


## Estimating the term structure: compounded rates from discount factors
# Ulitzing data in the following format: Cusip,	IssueDate,	MaturityDate,	Name,	Coupon,	Bid/Ask
head(dat)
ccRate = compoundingRate(dat, initialDate=as.Date("2000-05-15"), m=4, face=100)

years = ccRate$years
rate = ccRate$ccRate
# Plot of continuously compounded spot rates
plot(x=years, y=rate, type="l", ylab="Rate", xlab="Time to Maturity", main="Term Structure: Spot Rates")

# Spot, Forward Rates
DF = c(0.996489, 0.991306, 0.984484, 0.975616, 0.964519)
time = seq(from=0.5, to=2.5, by=0.5)
rates = spotForwardRates(time,DF)
rates