[R-sig-finance] swap to forward rates
Krishna Kumar
kriskumar at earthlink.net
Thu Feb 23 02:29:09 CET 2006
To re-emphasize what David has mentioned and what was in the earlier
West-Hagan paper, depending on the
interpolation technique used one can get completely different set of
rates. And people use smoothness and other criteria
to pick the method. If you play around with different interpolation like
cubic spline in akima and other interpolation methods
that are in CRAN you will notice that the forward is smooth for some and
jagged for others.
What I sent is a very rudimentary - curve-101, perhaps you should also
look at these additional documents as well ,
http://www.fincad.com/newsletter.asp?i=1180&a=2180
http://www.bankofcanada.ca/en/res/wp/2000/wp00-17.pdf
http://www.hfb.de/Dateien/Arbeitsbericht2.pdf
{Most banks have people who are specialized in just building curves..}
The reason why it is good to go via discount factors is that some of the
curves are blended meaning they have a mix of
instrument like repos or muni instruments. In other cases people use E$
futures or other market instruments. The criteria on what
instruments to use in your curve, is typically a) liquidity and b) your
positions in the markets. Because you want to compute the correct
funding for your positions.
Finally you need to get the day counts right that is important and the
recent email on seq.date might help you on this.
Hope this helps,
Kris
Thomas Steiner wrote:
>Again, I want to calculate forward rates from swap rates.
>Krishna Kumar was already very kind to help me here (24.1.06) and he
>provided an algorithm, I did not understand. So I now figured out my
>own one and unfortunately the two results do not coincide but both
>appear very plausible.
>
>How can you help me?
>If you have a lot of time, check my algorithm or my formula.
>If you have already implemented this yourself somewhere, please
>compare my results with yours and with those of Kris. And let me know
>about your results.
>
>Perhaps you can have a quick look at this and verify the results.
>
>I have EUR-swap rates for 2Y, 3Y, 5Y, 7Y, 10Y, 20Y and 30Y, eg for 2005-12-13:
>
>sc=list(
> maturity=c(2,3,5,7,10,20),
> tenors=c(2,1,2,2,3,10), #the difference between two dates (=\tau_i)
> rate=c(0.02982,0.03074,0.03230,0.03354,0.03533,0.03861),
> date="2005-12-13"
> );
>
>#Kris' way from swap via discount to forward rates:
>swap2discount<-function(swapcrv) {
> n<-length(swapcrv$rate)
> dfcrv<-list()
> dfcrv$maturity <- swapcrv$maturity
> dfcrv$discount <- 1/(1+swapcrv$tenors*swapcrv$rate)
> dfcrv$discount[-1]<-
>dfcrv$discount[-1]*(1-(cumsum(swapcrv$tenors[-n]*dfcrv$discount[-n]))*swapcrv$rate[-1])
> dfcrv$date=swapcrv$date
> return(dfcrv)
>}
>discount2fwd<-function(dfcurve,times=dfcurve$maturity) {
> times<-c(0,times)
> n<-length(times)
> fwd<-array(0,n)
> df.tenor<-array(0,n)
> df.tenor<-approx(x=dfcurve$maturity,y=dfcurve$discount,xout=times,rule=2)$y
> df.tenor[1]<-1
> fwd<-(df.tenor[-n]/df.tenor[-1] -1)/diff(times)
> fwd<-c(fwd[1],fwd)
> return(list(maturity=times,rate=fwd,date=dfcurve$date))
>}
>dc=swap2discount(sc)
>discount2fwd(dc)
>#gives rates: 0.02982000 0.03264006 0.03592894 0.04081783 0.04582491 0.05007956
>
>#my function:
># from Brigo/Mercurio, p.15; solve the last equation on this page for F_\beta:
># $$\1+\tau_\beta F_\beta = \frac{1+S_{\alpha,\beta}(t)
>\tau_\beta}{\Pi_{j=\alpha+1}^{\beta-1}(1+\tau_jF_j) -
>S_{\alpha,\beta}(t)\Sum_{i=\alpha+1}^{\beta-1}\tau_i\Pi_{j=i+1}^{\beta-1}(1+\tau_jF_j)
>} $$
>swap2fwd<-function(sc) {
> fwd=array(dim=length(sc$rate))
> fwd[1]=sc$rate[1]
> fwd[2]=( (1+sc$rate[2]*sc$tenors[2])/(1+sc$tenors[1]*fwd[1]-sc$rate[2]*sc$tenors[1])-1
>) / sc$tenors[2]
> n<-length(sc$maturity)
> for (t in 3:n) {
> cumpr=rev(c(1, cumprod(1+sc$tenors[(t-1):2]*fwd[(t-1):2]) ))
> su=sum( sc$tenors[1:(t-1)] * cumpr )
> fwd[t]=(1+sc$rate[t]*sc$tenors[t]) / (
>prod(1+sc$tenors[1:(t-1)]*fwd[1:(t-1)])-sc$rate[t]*su )
> fwd[t]=(fwd[t]-1)/sc$tenors[t]
> }
> return(list(
> maturity=sc$maturity,rate=fwd,date=sc$date
> ))
>}
>swap2fwd(sc)
>#gives 0.02982000 0.03264006 0.03485760 0.03705553 0.04046209 0.04409305
>
>My results are a bit lower then Kris'. What do you suggest me to use
>now? A pro's solution I don't understand or my own answer?
>If I add the 30Y swap rate (0.03922), results are strage again in both
>cases (too high and to low respectively).
>
>What I did not take into account are daycount conventions and the fact
>the the very first point in the swap curve should be a cash rate. This
>will come later...
>
>Any help, hint and comparison is very welcome.
>Thomas
>
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>
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