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I have 2 questions that probably are related. Suppose there is an IRS that pays a 2% fixed rate every 6 months and receives the Libor 3 months (but paid every 6 months). The swap starts today (March 5th) so the first payment is on Sept 5th (in 6 months). But the Libor is a 3 months rate in this case so the fixing should be every 3 months, I guess. If that's the case, there are 2 important dates from now and the first payment: the first fixing date (in 3 months time... i.e. June 5th) and the second fixing date which is also payment date on Sept 5th. But I have read that it is convention to have the float rate to be fixed in advance (2 days before the period) and then paid in arrears at the end of the period. But how does it work in my example? The swap starts on March 5th and pays the Libor 3 months in 6 months time. So in 6 months I should have 2 different fixings for the Libor 3 months. But when? One is happening in 3 months time on June 5th but what about the other? Is it done today?

Is there a simple explanation how the stub rate works? How is ot calculated? Is it related to my first doubt about the very first fixing for the Libor 3 months?

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Libor is indeed usually fixed in advance (and paid in arrears). Thus, in your example the first fixing date will be 2 business days before March 5th, and the second fixing date will be 2 business days before June 5th. Usually, therefore, the first fixing is already known when the swap is traded.

You say that the Libor leg is paid semi-annually - that's not the most common convention, but certainly exists. Usually, even if the fixed leg pays semi-annually, the Libor leg will be paid at the natural frequency of the Libor index (i.e. quarterly in your example). In your semi-annual example I would expect the two Libor fixings would be compounded to form the final payment amount made on September 5th.

By the way, market practitioners would not describe this as a "stub" period. That term is reserved for the case when the first period is not the usual length. For example if the swap generally has semi-annual payments, but the first period is only four months long, that would be described as a front stub period.

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