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CFA_2012, you are exactly right that if interest falls below the floor then the floating rate borrower is actually worse off because he/she needs to make up for the additional payments.

The sole advantage of selling a floor in this case is to use the proceeds to offset the cost of buying a call option even though selling a floor is disadvantageous for the borrower if interest rate falls.

In a nutshell, you capped the interest rate you have to pay at the upper end (via the call option/ceiling) and at the same time capped the interest rate you have to pay at the lower end (through the put option/floor), hence the interest rate "collar".

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