# 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.

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