I don't understand why the duration of a floating rate note equal to the time to the next coupon payment? Please, look at my calculations.
Here: P - is price at moment 0.
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Let the first coupon be fixed at c, and consider the duration of the bond immediately thereafter. At this point $L_(0,6)$ can move. Now in your notation you should find that $$P=N(1+c/2)/(1+L_(0,6)/2)$$. Now if you calculate $(1/P)dP/dL$ you get $1/2* (1/(1+L/2))$ which is 1/2, discounted for 6 months, where $L=L_(0,6)$.