libor - Curve building dates overlapping impact on discount factor

I'm building a short end of the libor curve using deposit & fra due to overlapping in dates I get wrong values of Discount factor, here's the data i'm working with:

• My today date is : 23/10/2019


• Start of my deposit 6m contract is 25/10/2019 end date is 27/04/2020,day count is act/360 with rate 5%


• Start of my fra 6x12m contract is 27/04/2020 end date is 27/10/2020,day count conv is act/360 with rate 5.2%

Can someone please explain how to manage that overlapping between deposit and fra? and how to get the right discount factor ?

thanks

Answer

There is no overlapping, the first instrument is tied to the LIBOR rate starting at $25/10/2019$, the second one is tied to the LIBOR Rate at $27/04/2020$.

For the sake of clarity, let assume that the spot date and today's date are the same, that there is only one curve (LIBOR Curve).

WE use the definition of the forward rate starting at $T$ and ending at $U$ as $$F(0,T,U)=\frac{1}{U-T}\left(\frac{P(0,T)}{P(0,U)}-1\right)$$

where $P(0,T)$ is the zero-coupon bond paying one unit at time $T$

$T_0= 25/10/2019$,$T_1= 27/04/2020$ , $T_2= 27/10/2020$

We have that $$0.05=\frac{1}{0.5}\left(\frac{1}{P(0,T_1)}-1\right)$$, therefore

$$P(0,T_1)=\frac{1}{1+0.5\times0.05}$$

As for the FRA : $$0.052=\frac{1}{0.5}\left(\frac{P(0,T_1)}{P(0,T_2)}-1\right)$$

Thus, $$P(0,T_2)=P(0,T_1)\frac{1}{1+0.5\times0.052}$$