Hi everyone!

I have encountered a problem related to valuation of a floored leg paying quarterly compounded interest rates. The floor strike is zero and the rates are negative throughout the cashflow period of interest. The software used for valuation models the floor using a floating rate and a floorlet for each compound reset, with the strike applied to each fixing.

In my opinion, it would be the most intuitive to model a floor as a European option on the componded interest rate with strike 0 stretching over the full cashflow period, rather than several compounded floorlets. This given that the leg pay-off becomes max(floating rate, 0), i.e. the leg becomes perfectly floored, which is what I would expect a dealer to want to accomplish when flooring the leg.  

Our software provider suggests a solution as follows in this example:

Nominal
1,000,000

Cashflow period, rate:
2016-06-30 - 2016-09-30, -0.5%
2016-09-30 - 2016-12-30, -0.5%
2016-12-30 - 2017-03-30, -0.5%
2017-03-30 - 2017-06-30, -0.5%

Day Count Convention: Act / 360

The interest becomes compounded as:
Nominal x ( (1 + rate x day count fraction) x ... x (1 + rate x day count fraction) -1 )

The software provider reason that the rate is -0,5 for the floating rate and (0 - (-0,5)) = 0,5 for the floorlets. Therefore, the floating rate component is worth:
1,000,000 x ( (1 - 0.005 x 92 / 360)(1 - 0.005 x 91 / 360)(1 - 0.005 x 90 / 360)(1 - 0.005 x 92 / 360) - 1) = 5059,82

The floor value becomes the following according to this logic:
1,000,000 x ( (1 + 0.005 x 92 / 360)(1 + 0.005 x 91 / 360)(1 + 0.005 x 90 / 360)(1 + 0.005 x 92 / 360) - 1) = 5079,09

The leg is therefore worth 5059,82 - 5079,09 = -19,27, i.e. the floor is worth more than the rate it is supposed to be off-setting. I find this very strange as I regard the floorlets to be a modeling tool rather than an actual cashflow. Thereby, I would expect the result to be 0 no matter how the strike is applied (compounded rate vs. each fixing), as the leg always pays max(interest rate, 0), which is always zero in this example.

I would like to ask you what market praxis is for floored legs with compounded resets, to apply the strike to the compounded rate or to each fixing, and if the reasoning above implying a negative leg pay-off is really correct.

Any input on this topic is highly appreciated.

Hi MAARE,

I once wrote a paper about FIReD (Floored Instrument on Rolled DEposit), which is basically the instrument you describe with the floor on the composition. This was written before the multi-curve became the standard, and thus in a single Libor curve framework. The paper is available on IDEAS at https://ideas.repec.org/p/pra/mprapa/1534.htmlMaybe you will find some useful information in it.

Marc