YieldCurve (version 4.1)

Svensson: Estimation of the Svensson parameters

Description

Returns the estimated coefficients of the Svensson's model.

Usage

Svensson(rate, maturity )

Arguments

rate

vector or matrix which contains the interest rates.

maturity

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

Value

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

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()