# Eonia swap calculation of floating rate

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.

• I have updated my answer. Commented Jan 23, 2020 at 14:28

## 1 Answer

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.

• Thanks for your answer, so let’s assume today is 22/01/2020 we can use the eonia of the 20/01/2020 and the eonia of 21/01/2020 to find the rate of today ? Or we need all rates of between today and next payment in one year ? Especially those future eonia rates are unknown in the market ... Commented Jan 22, 2020 at 7:45
• Umm not really. The overnight rate from day $d_i$ to the "following" day $d_{i+1}$ (where following could be 2-3 days apart due to weekends or holidays) is normally published on the same day $d_i$ before 7pm (see Wikipedia). You cannot use past rates to deduce the rate from today to tomorrow. You need all the forward rates from today till next payment. These are observable in the market, probably through Bloomberg, though I do not use it so I am not sure how to extract the information. I have updated my answer. Commented Jan 23, 2020 at 14:21
• Thank you, this very helpful. Commented Jan 24, 2020 at 8:10