Day Count Convention & Compounding Frequency Assumption in Interest Rate Swaps and Discount Factors

Question

This question concerns old LIBOR Swaps where their fixed legs are based on 30/360, and floating legs on Act/360.

Q1. Let's assume the simple self-discounting case where spot rates are obtained directly from the LIBOR swap curve itself. Can differences in day count conventions of the two legs lead to different "discount factors" to use for each leg? (I am asking this because I've seen some educational websites that showed slightly different discount factors (for the same time period) used for different legs of the same swap.)

Q2. When bootstrapping the LIBOR swap curve to derive discount factors, what compounding frequency is normally assumed in the industry? continuous? daily? quarterly? semi-annually? Is it correct to assume that the same compounding frequency is assumed for both legs? (or can it be different?)

Q3. If the OIS curve is bootstrapped instead of the LIBOR swap curve to derive spot rates and discount factors for LIBOR swaps, would an answer to the Q1 above change?

Q4. Would an answer to the Q2 above be different if we assumed an OIS discounting?

Answer 1

Q1. Let's assume the simple self-discounting case where spot rates are obtained directly from the LIBOR swap curve itself. Can differences in day count conventions of the two legs lead to different "discount factors" to use for each leg? (I am asking this because I've seen some educational websites that showed slightly different discount factors (for the same time period) used for different legs of the same swap.)

For the purpose of calculating discount factors, the day count convention should ensure one-to-one mapping between a date and the day count fraction (i.e., a unique time fraction for each date). You do not need to use the day count convention of the underlying swap and really shouldn't, because conventions such as 30/360 does not guarantee unique mapping. The most commonly used industry convention for discount factors is simply ACT/365 (e.g., Bloomberg), although some shops use ACT/365.25. You pick one and then stick with it for consistency.

You could also choose not to use a day count convention at all and map each discount factor to a date directly.

Q2. When bootstrapping the LIBOR swap curve to derive discount factors, what compounding frequency is normally assumed in the industry? continuous? daily? quarterly? semi-annually? Is it correct to assume that the same compounding frequency is assumed for both legs? (or can it be different?)

Discount factors are basically the price of a "bond" with a notional value of 1. Price is not affected by compounding frequency.

Q3. If the OIS curve is bootstrapped instead of the LIBOR swap curve to derive spot rates and discount factors for LIBOR swaps, would an answer to the Q1 above change?

No, the answer would be the same.

Q4. Would an answer to the Q2 above be different if we assumed an OIS discounting?

Also no.