# Calculate OIS rate 3 months, 1 year

I don't know if there are any similar posts in this forum but I’m trying to describe below all things that I understand about OIS rates and Libor rates.

As I understand, OIS rate 3 months is calcultated as a fixed rate for a period 3 months which is exchanged for the geometric average (floating rate) of the overnight rates during this period.

1. First, Overnight rates in this case are interest rates where a group of banks are agreed to pay in the interbank market, during 1 day. Regarding those rates, we have EONIA for Euro zones, SONIA for UK or Fed funds rate for US. These rates are fixed day-by-day by Central Bank. I'm not sure how they are calculated but it seems like they are the average of lending rates between a group of selected banks, at a given date. It means that each of a number of selected banks (for example 60 banks in Euro zones) will contribute their own lending rate everyday, and we will take the average of them as the overnight rate at that day. Am i right until now?
2. Based on Overnight rates we have, we are able to calculate the OIS rate 3 months by the following formula:

$$\text{OIS Rate 3 months} = \prod_i \left( 1 + \frac{n_ir_i}{D}\right) -1$$

with:

• $r_i$: is overnight rate at date $i$,
• $D$: number of days during 3 months,
• $n_i$: number of open days between date $i$ and date $i+1$.

Is it right?

1. For OIS Rate more than 3 months (ex 1 year or 5 years), we usually subdivide the period of maturity into every 3 months because of quarterly payment. It means that at the end of sub-period 3 months, we will exchange fixed flux with floating flux.
So how to calculate OIS rate 5 years? we do the same (based on OIS rate 3 months as we do with overnight rates in the formula above?
• i think the D will be number of days in the year and not in 3 months. This is because the rate give in generally annual rate – onkar May 24 '17 at 12:08

It is not fixed by the central bank. The rate is the average of the - literal overnight borrowing and lending (deposits and borrowings) between high credit worthy banks (also known as the Overnight group). This is the proxy for risk free rate used in several Bond Math Calculations and has been adopted by financial institutions.

To put it in simplest terms, take the current effective overnight fed funds rate. Lets Say today its 2.38, and lets say the market is projecting that at next months FOMC meeting in 30 days the fed is going to cut rates .25 bp and then leave rates unchanged there after.

That leaves the projected fed funds rate over the next 90 days to be roughly 30 days at 2.38 percent and 60 days at 2.13 (From 2.38-.25 bp cut).

In the most basic terms and not accounting for compounding, think of it as the fixed rate receiving 2.38 for a 3rd of the lending period and 2.13 for 2/3rds of the period = [(2.38 x .3333) + (2.13 x .6666)]/3 = 2.2132 or [(1.0238)(1.0213)(1.0213)]^(1/3) -1 = 2.2132

so, if 90 day Libor is currently 2.33 and the 90 day OIS rate is 2.2132, the Libor OIS spread is .1168 (2.33-2.2132)

• And how do you interpret the value of the Libor OIS spread? .1168 = (2.33-2.2132) – Max Aug 8 '19 at 14:35
1. i'd say you are about right

2. the formula you gave shows the cashflow you receive, per unit notional, on the ois leg of an ois swap in 3 months time. The ois rate is a rate whose term is just 1 day. The 3m ois rate is actually a Swap rate , where you pay a fixed rate , and receive ois with daily compounding. the value of this 3m ois swap rate can be observed in the OTC market. Similarly for other ois swap rates.

3. i am not sure exactly what you want to calculate?

• So when we talk about OIS rate 3months on 23/01/2017, how can we calculate it precisely pls? the same question for OIS rate 1 year on 23/01/2017? – Lucky_Luf Jan 23 '17 at 13:31
• those 2 swap rates are equilibrium market rates determined by demand and supply and are quoted by banks/brokers – Randor Jan 24 '17 at 16:31
• I think what you mean is Overnight rate (EONIA, SONIA or fed funds rate). they are determined by the contribution of banks member which is a match of demand and supply. But for 3m OIS rate or 1 year OIS rate, they can be valuated by overnight rates. but i don't know exactly how? that's my question – Lucky_Luf Jan 30 '17 at 11:35
• the 3m and the 1y ois rates are equilibrium market rates , just as the underlying ois rate itself is. – Randor Feb 2 '17 at 20:39
• @Randor but if the ois rates are equilibrium market rates, then how could it related to the overnight rates, since OIS rate has already gave a restriction of overnight rates or OIS rate is calculated from forward overnight rate? – user6703592 Oct 14 '18 at 11:13

if you want an equation connecting 3m ois swap rate with the current ois 1d rate and projected future ois 1d rates, then the eqn is just like in ibor swaps, keeping in mind that the float leg of the ois swap is always par because of the daily resets