Negative duration of FRN

Floating Rate Note (FRN) usually has duration very close to zero. An FRN is reset to par on every reset date when the floating coupon is paid out. However it could have negative duration when it is traded at substantial discounts. Since, as we have already mentioned, the coupon of an FRN is floating, the discount is most probably due to credit instead of interest rate. Suppose then that an FRN is traded at a discount because of credit concern. Then we can write

FRN = FRN' + X

where FRN' is a note at a discount, FRN is an otherwise identical par note and X is some instrument that can be constructed so that the above expression holds. The point is that X has positive duration (as can be shown easily if we assume that the cash flow of FRN' is that of FRN with an extra spread S). Since the duration of FRN is close to zero, and the duration of X is positive, we must have that the duration of FRN' is negative.