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If I equalize the discount factor at the same point in time between a fixed-rate payer and a variable-rate payer, I will have the following problem when referencing the data above the swap curve. Let's say you have a swap rate curve of 3M and 6M, with a fixed rate payout semi-annually and a floating rate payout quarterly.

$((1+\frac{r_6}{2})DF_6=(\frac{r_3}{4})DF_3+(1+\frac{r_6}{4})DF_6)$

Here, for DF3, there is a contradiction.

$\frac{1}{1+r_3}$

In this case, there are two ways : either by using different discount factors DF6 at the same point in time, or by using different floating rates at 3M on the interest rate curve instead of adopting the rate at 3M. Which method should I use?

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