How OIS Floating Leg Compounding Becomes a Money Market Cash Flow
Summary
The discussion clarifies how to calculate the floating and fixed cash flows on an overnight indexed swap. The fixed leg is generally a simple interest amount under the applicable day-count convention, rather than a compounded notional repayment. For the floating leg, daily overnight rates are compounded across their accrual intervals; the accumulated return is then converted into an annualized money market rate and applied to the accrual fraction and notional.
An example works through a short USD OIS, accounting for daily accruals and a multi-day final interval. It illustrates how the compounded floating rate can differ slightly from the fixed rate and how that difference determines the net payment. The explanation notes that the quoted rate may be rounded. Conventions and accrual schedules matter, so the example’s ACT/360 assumptions should not be generalized to every currency or contract.
Key ideas
- OIS swaps normally exchange interest cash flows rather than principal.
- The fixed leg is usually calculated as a simple rate times the accrual fraction and notional.
- The floating leg compounds overnight fixings using their individual accrual periods.
- The compounded return is annualized as a money market rate before calculating the floating cash flow.
- Day-count conventions, accrual schedules, and rounding affect the final payment.
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Full text
# OIS example in Hull's book
# OIS example in Hull's book
In Hull's book (9th edition), on pages 202-203, there is an example for computing the payoff of an OIS that I am confused about. It says suppose in a US 3-month OIS the notional principal is \$100 million and the fixed rate (i.e. the OIS rate) is 3% per annum. If the geometric average of overnight effective federal funds rates during the 3 months proves to be 2.8 per annum, the fixed-rate payer has to pay 0.25*(0.030-0.028)*\$100 million. In my understanding, if the overnight interest rates over the period are $r_1,...,r_n$, and the swap rate is $q$, then isn't the fixed-rate payer paying $$(1+\frac{q}{360})^n$$ and receiving $$(1+\frac{r_1}{360})(1+\frac{r_2}{360})...(1+\frac{r_n}{360})?$$ In which case, if the geometric average of the overnight interest rate is $$\frac{r^*}{360}=[(1+\frac{r_1}{360})(1+\frac{r_2}{360})...(1+\frac{r_n}{360})]^{\frac{1}{n}}-1$$ and the cash flow of the fixed-rate payer is actually $$(1+\frac{r^*}{360})^n-(1+\frac{q}{360})^n$$ Is my understanding correct? I can't make sense of the example in the book.
## Answer by user35980 (score 1)
https://quant.stackexchange.com/a/60918
I'm not clear on the algebra you've given, but I think all Hull is doing is saying the annualised fixed rate is 3% and the 3m OIS float rate fixing is 2.8% - thus the cashflow is the difference of this adjusted for the accrual period (0.25 in this case), times the notional. He doesn't say anything about how the 2.8% comes about. Using your notation: if the daily fixings in the 3 month period (which, for the sake of argument, say has 66 good business days) are $r_i$ and the (business daily) accrual periods are $\delta_i$, for $i=1,...,66$, then that 2.8% comes from something like: $$ ((\prod_{i=1}^{66} (1+\delta_i r_i))-1)/0.25.$$
## Answer by Adam N. (score 0)
https://quant.stackexchange.com/a/74770
> isn't the fixed-rate payer paying $(1+q/360)^n$
Interest rate swaps typically don't involve exchange of notional. Also, fixed leg of a swap is usually quoted in money market convention, not as a compounded rate. So the fixed cash flow would be $q\cdot n/360$, assuming ACT/360.
> and receiving $(1+r_1/360)(1+r_2/360)...(1+r_n/360)$
Floating (compounded) leg of an OIS is usually converted to a money market rate convention as well, so one would calculate $r=((1+r_1/360)(1+r_2/360)...(1+r_n/360)-1)\cdot 360/n$. The result is usually rounded, and floating cash flow amounts to $r\cdot n/360$.
As a concrete example, consider a 100M USD OIS, traded on 09/02/2023 for 1 week (VD 13/02/2023, MD 21/02/2023), with a fixed rate of 4,5%. Fixed payment would be 100M*4,5%*8/360=100 000,00 USD. Floating rate would be ((1+4,55%/360)*(1+4,55%/360)*(1+4,55%/360)*(1+4,55%/360)*(1+4,55%*4/360)-1)*360/8=4,55158% (rounded to 0,00001%). Floating payment would be 100M*4,55158%*8/360=101 146,22 USD. Net, the swap buyer would receive 1 146,22 USD on 23/02/2023.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.