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.
2 Answers
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: $$ ((\prod_{i=1}^{66} (1+\delta_i r_i))-1)/0.25.$$
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.
