YieldCurve (version 5.1)

Svensson: Estimation of the Svensson parameters

Description

Returns the estimated coefficients of the Svensson's model.

Usage

Svensson(rate, maturity )

Value

Returns a data frame with the estimated coefficients: \(\beta_{0}\), \(\beta_{1}\), \(\beta_{2}\),\(\beta_{3}\), \(\lambda_1\) and \(\lambda_2\).

Arguments

rate

vector or matrix which contains the interest rates.

maturity

vector which contains the maturity (in months) of the rate. The vector's length must be the same of the number of columns of the rate.

Author

Sergio Salvino Guirreri

Details

The Svensson's model to describe the forward rate is: $$y_t(\tau) = \beta_{0} + \beta_{1} \exp\left( -\frac{\tau}{\lambda_1} \right) + \beta_2 \frac{\tau}{\lambda_1} \exp \left( -\frac{\tau}{\lambda_1} \right) + \beta_3 \frac{\tau}{\lambda_2} \exp \left( -\frac{\tau}{\lambda_2} \right) $$

The spot rate can be derived from forward rate and it is given by: $$ y_t(\tau) = \beta_0 + \beta_1 \frac{ 1- \exp( -\frac{\tau}{\lambda_1}) }{\frac{\tau}{\lambda_1} } + \beta_2 \left[\frac{ 1- \exp( -\frac{\tau}{\lambda_1}) }{\frac{\tau}{\lambda_1} } - \exp( -\frac{\tau}{\lambda_1}) \right] + \beta_3 \left[\frac{ 1- \exp(-\frac{\tau}{\lambda_2}) }{\frac{\tau}{\lambda_2} } - \exp( -\frac{\tau}{\lambda_2}) \right]$$

References

Svensson, L.E. (1994), Estimating and Interpreting Forward Interest Rates: Sweden 1992-1994, IMF Working Paper, WP/94/114.

Nelson, C.R., and A.F. Siegel (1987), Parsimonious Modeling of Yield Curve, The Journal of Business, 60, 473-489.

Examples

data(ECBYieldCurve)
maturity.ECB <- c(0.25,0.5,seq(1,30,by=1))
A <- Svensson(ECBYieldCurve[1:10,], maturity.ECB )
Svensson.rate <- Srates( A, maturity.ECB, "Spot" )
plot(maturity.ECB, Svensson.rate[5,],main="Fitting Svensson yield curve",
 xlab=c("Pillars in years"), type="l", col=3)
lines( maturity.ECB, ECBYieldCurve[5,],col=2)
legend("topleft",legend=c("fitted yield curve","observed yield curve"),
col=c(3,2),lty=1)
grid()