Calculate forward discount factors and forward reference rate when discount factors are known

Question

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.

Answer 1

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

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