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I am trying to learn how to value interest rate swap through portfolio of FRA's(forward rate agreement).But I have got stuck in calculation of floating leg.

Here is the scenario as given below for which I need help.

  • The swap starts at 05-Jan-19 for which the zero coupon discount factor is 1.The 1st cashflow period is from 05-Jan-19 to 05-Jul-19.

  • The start date and end date of cashflow for 2nd period is from 05-Jul-19 to 05-Jan-20.

  • By linear interpolation (zero coupon discount factors are given);I got zero coupon discount factors at 05-Jul-19 and 05-Jan-20 (2nd period). Assume these zero coupon discount factors to be df1 and df2 respectively.

Questions - How can I find forward discount factor for this 2nd period(05-Jul-19 to 05-Jan-20).Also how can I find forward reference rate for this 2nd period.

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Let $df\left(t_1, t_2\right)$ represent the discount factor between the two periods. You then have:

$df\left(t_0, t_2\right) =df\left(t_0, t_1\right) \,df\left(t_1, t_2\right) $

So

$df\left(t_1,t_2\right) =\frac{df\left(t_0, t_2\right)}{df\left(t_0, t_1\right) }$

The forward rate between the two periods as at time 0 is as follows:

$F\left(0, t_1,t_2\right)=\frac{1}{t_2-t_1} \left(\frac{df\left(t_0, t_1\right)}{df\left(t_0, t_2\right) }-1\right)$

Which you can easily verify by noting that:

$df \left(t_0,t_2\right)=\frac{df \left(t_0, t_1\right)}{1+\left(t_2-t_1\right)\, F\left(0, t_1,t_2\right)}$

Re-zero rate comment below, if you assume annual compounding then the discount factor for t years is:

$df(t)=\frac{1}{\left(1+r\right)^t}$

Which means

$r=\left(df(t)\right)^{\frac{1}{t}}-1$

And if you assume continuous compounding then

$df(t)=e^{-r\,t} \Rightarrow r=-\frac{1}{t} \ln df(t)$

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  • $\begingroup$ Thanks.Really helped.One more question is how can I find zero rates from given zero coupon discount factors? $\endgroup$ – user2967440 Apr 29 at 3:15
  • $\begingroup$ If you assume continuous compounding, then minus log of discount factor divided by maturity in years (6 months =0.5, 1 year=1) $\endgroup$ – Magic is in the chain Apr 29 at 7:42
  • $\begingroup$ I have read the formula of zero rate to be (1/df-1)/t or (1/df)^(1/t)-1.Please suggest when to apply which formula.Also for example if discount factor for 05-Jan-19 is 1 and 05-Jul-19 is 0.988.How can we calculate zero rate for these 2 dates? $\endgroup$ – user2967440 Apr 29 at 11:49
  • $\begingroup$ Added text in the answer above $\endgroup$ – Magic is in the chain Apr 29 at 17:21

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