# Calculating Implied rates from OIS and Futures

I've been trying to figure out how to calculate the implied rate for interest rate decisions by central banks using OIS and came across an explanation that I can't quite wrap my head around:

Apart from futures, you can also look at OIS swaps, which for some countries even have directly quoted central bank meeting date swaps. If not, you can still rely on the following equilibrium:

$$1 + \frac{r*n}{360} = \prod_{i=1}^n \left(1+ \frac{r_i*d_i}{360}\right)$$

where the left hand side is the fixed part (r is the quoted OIS price / fixed rate), and the RHS the floating part, with ri denoting the expected floating rate on the ith day, di the number of days ri applies for (1 for weekdays, 3 for weekends) and n is the total number of days for the swap. Since r, n and di is known, you can solve this.

Let's say I have a OIS contract with a duration of 1 year and the OIS rate for that duration is 3%. The current interest rate is 2.5%. The next central bank meeting is in 30 days and I want to figure out what the market is pricing in for the implied rate of that meeting? According to the explanation I can calculate ri for the ith day but isn't ri changing every night depending on what the overnight market does? What would the calculation look like for deducing what the implied rate (and hence the central bank decision) would be at that date? Please show your working out as that is where my understanding gets me lost.

Moving onto a different example with the 3 month short-term interest rate futures. Current interest rate is 2.5%. The next central bank meeting is in 30 days. Let's say the 3 month STIR future is currently priced at 3%. What would the calculation look like to try and find what the market expects the central bank to do with interest rates in 30 days?

Thank you

• What is the source for the formula and paragraph you have quoted? Commented May 4 at 8:07
• @Alper here is the link quant.stackexchange.com/questions/71778/… Commented May 5 at 3:42

This sounds like something I would write.

Firstly, you can't price central bank meeting expectations from a 1Y OIS rate, it does not contain sufficient information. Its like trying to forecast the weather with a thermometer.

Lets say that you can collect enough information. E.g. OIS rates at monthly intervals (1m, 2m, 3m, etc.) or what is even better, the OIS rates between the central bank meeting dates, which are common prices in major currencies.

Here is an example:

The 1st ECB dates in this case run from 12-Jun-24 to 24-Jul-24.

Using the formula (where I have pre-solved it):

$$1 + 0.03681 \times \frac{42}{360} = (1 + 0.03673 * \frac{1}{360}) ^{42}$$

So the daily ESTR fixing is expected to be 3.673% in this period. If you compare this with the historical ESTR fixing relative to the central bank rate of 4% you can see that this has reliably averaged 9.5bps below the central bank rate:

Thus we conclude that an average ESTR rate of 3.673% over that ECB period, gives rise to an expected ECB central bank rate of 3.768% or in probabilities and 92.8% chance of 25bps cut and a 7.2% chance of no cut.

If you do not have central bank meeting dates available you have to do extra work to model or imply from the data you do have, and with that modelling with come model uncertainty.

• Thank you for taking the time to answer. This was an explanation from AKdemy on another post here on the forums. May I ask how you arrived at the average rate of 3.768? What was your calculation for that? I don't have access to financial services (such as bloomberg) and was looking for OIS rates on free online sources for the major currencies and, for example for the EURO, only found 3 month OIS EURIBOR swaps that have a minimum duration of 1 year and was trying to figure out how to derive the implied rates from there. Commented May 3 at 12:06
• You take the calculated 3.673% ESTR rate and add the historical basis average of 9.5bps = 3.768%. You cannot really do this without some modelling expectations. Sure I got this from Bloomberg but generally SOFR or ESTR rates are available from the appropriate authorities website.
– Attack68
Commented May 3 at 12:25
• Why would I need the current ESTR rate to calculate the implied rate at a future central bank meeting? From what I understood, all I need is the OIS rate(according to the equation above) and the only one available for free are the 3 Month EURIBOR swaps that have a minimum duration of 1 year and with those I am not able to model my implied rates according to you. Correct? Commented May 3 at 13:01
• I dont think you are understanding. The central bank rate is different to the ESTR rate. But there is a very stable basis between them. There is no market or instrument that prices central bank rates directly, therefore in the ideal scenario you use the closest rate which is an ESTR swap but you need to add back the basis to imply the central bank rate.
– Attack68
Commented May 3 at 14:36
• Thank you for clearing that up. How would you suggest one proceed if they don't have all the information necessary to calculate the implied rate? I see currently that 3 month EURIBOR swaps with a 1 year duration have an OIS of about 3.5%. The 1 year duration is the min swap tenor I could find. There were also 2,3,etc year tenor. But nothing sub 1 year.What extra work would I have to do to model the implied rate? Commented May 5 at 3:33