Duration. Floating rate note

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. • The definition of duration for these purposes is the Price sensitivity to a bump in interest rates. Try to calculate that , assuming that the first coupon has been fixed. – dm63 Jul 13 at 10:27
• In my calculations first coupon is fixed and equals L_0,6. Where L_0,6 is half-year LIBOR rate, which is fixed in the moment 0. – wormeer Jul 13 at 10:50
• So, in my calculations the duration equals 1 (not 1/2 - time to the fitst coupon payment).. Why is so? – wormeer Jul 13 at 10:56
• Shouldn’t be. Let the first coupon be c, and let all other rates move, including L(0,6) – dm63 Jul 13 at 11:27
• Ok. What wrong in the formula? I used the simple definition of a duration, which you can find in Wikipedia.. – wormeer Jul 13 at 11:47

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 $$L=L_(0,6)$$.