Eonia swap calculation of floating rate

Question

I'm new to swaps, I've a question about how to calculate the floating rate of an EONIA Swap from market quotation, so that we can keep an eye on the evaluation of our contract Market Value, DV01, etc..

The formula for the EONIA swap floating rate is: $$r=\frac{360}{n}\left(\prod_{i=t_s}^{t_e-1}\left(1+\frac{d_i}{360}r_i\right)-1\right)$$ where:

• $r$ is the variable rate taking compound interest into account;
• $t_s$ the start date of the EONIA swap;
• $t_e$ the end date of the EONIA Swap;
• $r_i$ the EONIA fixing rate on the $i$-th day;
• $d_i$ the number of days that the value $r_i$ is applied (normally one day, three days for weekends)
• $n$ Total number of days.

Let's say we entered a 20y EONIA Swap on the 20/01/2020, 1y fixed payment frequency, 1y float payment frequency. How can we calculate our float rate $r$? Do we do a future projection of the floating rate? Can someone breakdown this on a simple example please?

Thanks.

Answer 1

Let the EONIA schedule be $\{t_i\}_{0\leq i\leq n}$ with $t_0=t_s$ and $t_n=t_s$. The short answer is that the value $V$ of the cash flow: $$r=\frac{360}{n}\left(\prod_{i=0}^{n-1}\left(1+\frac{d_i}{360}r(t_i,t_i,t_{i+1})\right)-1\right)$$ where we have made explicit the fact that the floating EONIA rate $r(t_i,t_i,t_{i+1})$ is observed at a date $t_i$, for the period going from $t_i$ to $t_{i+1}$ (where $t_{i+1}-t_i=d_i$), is given by: $$V(r)=\frac{360}{n}(P(0,t_s)-P(0,t_e))$$ where $P(0,t)$ is the discount factor from $t$ up to the present from the prevailing EONIA rate curve. For more details you can check this answer, in particular the last formula with the forward rates $r(0,t_i,t_{i+1})$, noted $L(\dots)$ there, can be of interest to you.

Edit: as explained in the hyperlinked answer above, the value of the EONIA cash flow can be represented as: $$V(r)=\frac{360}{n}(P(0,t_s)-P(0,t_e))=\frac{360}{n}\left(\prod_{i=0}^{n-1}\left(1+\frac{d_i}{360}r(\color{blue}{0},t_i,t_{i+1})\right)-1\right)$$ where $r(\color{blue}{0},t_i,t_{i+1})$ is the forward EONIA rate today ($t=0$) for the future period $[t_i,t_{i+1}]$. The forward EONIA rate should be observable today through a market data service such as Bloomberg or Reuters.