| DRF.5F.tVar {DeRezende.Ferreira} | R Documentation |
Estimation of the De Rezende-Ferreira 5 Factor model's parameters with variable \ \tau
Description
The command estimates the parameters of the De Rezende-Ferreira 5 Factor model
using variable \ \tau_{1} and \ \tau_{2}
Usage
DRF.5F.tVar(rate, maturity)
Arguments
rate |
Vector or matrix of class "zoo", which contains interest rates |
maturity |
Vector of class "numeric", wich contains the maturities |
Details
The De Rezende-Ferreira model used to fit the forward rates is:
f_{t}\left(m\right) = \beta_{0t} +
\beta_{1t} e^{-\frac{m}{\tau_{1t}}} + \beta_{2t} e^{-\frac{m}{\tau_{2t}}} +
\beta_{3t} \left ( {\frac{m}{\tau_{1t}}} e^{-\frac{m}{\tau_{1t}}} \right ) +
\beta_{4t} \left ( {\frac{m}{\tau_{2t}}} e^{-\frac{m}{\tau_{2t}}} \right )
The spot rates, derived from the forward rates f_{t}\left(m\right), are given by:
y_{t}\left ( m \right ) = \beta_{0t} +
\beta_{1t} \left (\frac{1 - e^{-\frac{m}{\tau_{1t}}}}{\frac{m}{\tau_{1t}}}\right) +
\beta_{2t} \left (\frac{1 - e^{-\frac{m}{\tau_{2t}}}}{\frac{m}{\tau_{2t}}}\right) +
\beta_{3t} \left (\frac{1 - e^{-\frac{m}{\tau_{1t}}}}{\frac{m}{\tau_{1t}}} - e^{-\frac{m}{\tau_{1t}}}\right ) +
\beta_{4t} \left (\frac{1 - e^{-\frac{m}{\tau_{2t}}}}{\frac{m}{\tau_{2t}}} - e^{-\frac{m}{\tau_{2t}}}\right )
The set of optimal parameters will be chosen according to the lowest RMSE value:
\left (\widehat{\tau}_{1t},\widehat{\tau}_{2t} \right) = argmin\left \{\frac{1}{N}\sum_{t=1}^{N}
\sqrt{\frac{1}{T}\sum_{t=1}^{T}\left [ y_{t}\left (t_{n} \right ) - \widehat{y}_{t}\left (t_{n},\tau_{1t},
\tau_{2t},\widehat{\beta_{t}}\right )\right ]^{2} } \right \}
Value
An object of class "zoo", that contains
\ \left (\beta_{0t},\ \beta_{1t},\ \beta_{2t},\ \beta_{3t},\ \beta_{4t},\ \tau_{1t},\ \tau_{2t},\ SSR_{t},\ R^{2}_{t} \right)
Examples
#
# De Rezende-Ferreira 5F model on the Brazilian Data-Set
#
data(ZC_Brazil)
real.rate = ZC_Brazil
ZC_Brazil[["Date"]] = NULL
rate = zoo(ZC_Brazil)
index(rate) = as.POSIXct(paste(real.rate[["Date"]]))
maturity <- c(0.5, 0.75, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10)
RF.5F.Parameters <- DRF.5F.tVar(rate, maturity)
par(mfrow=c(3,2))
plot(RF.5F.Parameters[,"beta0"],xlab="Date",ylab="BETA0",ylim=c(9.5,12.0),col="blue",lwd=1)
grid(nx=12, ny=12)
plot(RF.5F.Parameters[,"beta1"],xlab="Date",ylab="BETA1",ylim=c(-18.0,2.3),col= "blue",lwd=1)
grid(nx=12, ny=12)
plot(RF.5F.Parameters[,"beta2"],xlab="Date",ylab="BETA2",ylim=c(-6.0,13.0),col= "blue",lwd=1)
grid(nx=12, ny=12)
plot(RF.5F.Parameters[,"beta3"],xlab="Date",ylab="BETA3",ylim=c(-10.0,0.0),col= "blue",lwd=1)
grid(nx=12, ny=12)
plot(RF.5F.Parameters[,"beta4"],xlab="Date",ylab="BETA4",ylim=c(-5.0,5.0),col="blue",lwd=1)
grid(nx=12, ny=12)
par(mfrow=c(1,1))
par(mfrow=c(2,1))
plot(RF.5F.Parameters[,"tau1"],xlab="Date",ylab="TAU1",ylim=c(0.2,1.3),col="blue",lwd=1)
grid(nx=12, ny=12)
plot(RF.5F.Parameters[,"tau2"],xlab="Date",ylab="TAU2",ylim=c(2.5,5.5),col="blue",lwd=1)
grid(nx=12, ny=12)
par(mfrow=c(1,1))
#
# De Rezende-Ferreira 5F on the Russian Data-Set
#
data(ZC_Russia)
real.rate = ZC_Russia
ZC_Russia[["Date"]] = NULL
rate = zoo(ZC_Russia)
index(rate) = as.POSIXct(paste(real.rate[["Date"]]))
maturity <- c(0.25, 0.5, 0.75, 1,2,3,5,7,10,15,20,30)
RF.5F.Parameters <- DRF.5F.tVar(rate, maturity)
par(mfrow=c(3,2))
plot(RF.5F.Parameters[,"beta0"],xlab="",ylab="BETA0",ylim=c(10.5,12.5),col="blue",lwd=1)
grid(nx=12, ny=12)
plot(RF.5F.Parameters[,"beta1"],xlab="Date",ylab="BETA1",ylim=c(-1.5,0.5),col="blue",lwd=1)
grid(nx=12, ny=12)
plot(RF.5F.Parameters[,"beta2"],xlab="Date",ylab="BETA2",ylim=c(-7.0,-3.5),col="blue",lwd=1)
grid(nx=12, ny=12)
plot(RF.5F.Parameters[,"beta3"],xlab="Date",ylab="BETA3",ylim=c(-1.5,3.5),col="blue",lwd=1)
grid(nx=12, ny=12)
plot(RF.5F.Parameters[,"beta4"],xlab="Date",ylab="BETA4",ylim=c(-5.5,-0.1),col="blue",lwd=1)
grid(nx=12, ny=12)
par(mfrow=c(1,1))
par(mfrow=c(2,1))
plot(RF.5F.Parameters[,"tau1"],xlab="Date",ylab="TAU1",ylim=c(0.1,1.9),col="blue",lwd=1)
grid(nx=12, ny=12)
plot(RF.5F.Parameters[,"tau2"],xlab="Date",ylab="TAU2",ylim=c(7.5,16.8),col="blue",lwd=1)
grid(nx=12, ny=12)
par(mfrow=c(1,1))