# LMM & multiple curves

I was reading through a paper that attempted to present a theoretical explanation for the divergence in value of different LIBOR tenors (and thus for the use of different curves for different tenors). The author's framework explained differences in terms of FRA and basis swap spreads (in Conjecture 6, p.19), and I was wondering:

1. have any comprehensive theoretical frameworks been developed that are used in practice?
2. are there any other reasons (beyond empirical fitting) for using multiple curves (inspired by the answers to this question)?

Finally, in the framework of multiple curves, is it then appropriate to say that 1Y LIBOR is above 1Y swaps because the swaps use OIS discounting (and are collateralized)? If so, does this extend to explaining any of the difference in long term rates (10Y swaps below 10Y treasuries), or are these dynamics better explained by regulatory environments and repo markets?

Thanks

As an example: 6 month libor is typically higher than 3 month libor because of the extra credit risk in lending to banks for an additional 3 months. Derivatives on 6 month libor have to take this into account. For example, a 5yr basis swap exists where 6 month libor can be swapped for 3 month libor plus a spread. Credit modeling techniques would need to be applied to model the term structure of this spread, and other tenor spreads.

In your example, 1 year libor is higher than the 1 year swap rate because the latter is equivalent to a chain of four 3 month Libors, so has less credit risk. Note that we are talking about the credit risk of the underlying rate index, not of the swap itself which is negligible.

10 year swaps being below 10 year Treasuries is mostly explained by the current regulatory environment, as you suggest. Specifically, the cost of holding assets such as Treasuries on balance sheet.

• Thanks -- to make sure I follow your first paragraph: would it be appropriate to describe the pre-crisis LIBOR yield curve (singular, right?) as something like "LIBOR was LIBOR was LIBOR" in the sense that the only difference between 3m and 6m LIBOR was time-value and NOT default probability (credit risk)? And now, the reason we model separate curves is to capture default probability coming from the additional risk of receiving payments less frequently? Jul 28, 2016 at 18:30
• Also, I think I see how in the spot case of lending for 3m vs 6m, the 6m loan is inherently riskier. However, I’m hung up on what I think is “cumulative” credit risk. What am I missing in thinking that a chain of four 3m LIBORs should have similar credit risk to a single 12m LIBOR? Or that a chain of two 3m LIBORs should give us 6m LIBOR? Could you explain this sentence: "1 year libor is higher than the 1 year swap rate because the latter is equivalent to a chain of four 3 month Libors, so has less credit risk"? Jul 28, 2016 at 18:31
• Your first comment I agree with. Pre crisis bank credit spreads were tight and basis swap levels were close to zero.
– dm63
Jul 29, 2016 at 8:31
• The second comment is a good question. Basically you are asking why credit spread curves are usually upward sloping. I guess it's because defaults happen gradually, so that if a bank is currently healthy, it's chances of defaulting next week are less than its chances of defaulting during a weekly period a year from now.
– dm63
Jul 29, 2016 at 9:13