Mini Chapter Four
Duration and PVBP for a
Floating Rate Bond
Duration
Where,
PV: Present value of floating rate bond
P: Principal Amount or Face Value
Rates in the notation are what’s currently ‘implied’ or ‘projected’ by market:
R01: current projected zero rate for 1y, R11 : current projected zero rate for 1y forward 1y, R21: current projected zero rate for 2y forward 1y
Algebraic expansion of the right-hand expression reduces it to principal value P i.e. the present value of a floating rate bond equals its initial principal value or par. The notation above can be generalized for any tenor floating rate bond and the result would be the same.
The result PV = P proves that a floating rate bond would always have a value of Par at the start with no resets having occurred. As soon as the first reset occurs we can see that the numerator of the first term gets fixed but the denominator can continue to change as the underlying interest rate (discount rate) changes with passage of time and the bond starts behaving like a fixed rate bond out to the first reset tenor. By the same corollary, we can also say that a floating rate bond will pull to par in the run-up to the next reset period.
At any point in time therefore a floating rate bond is a package of a short dated fixed rate bonds – starting with the residual tenor of the current reset period in case the underlying index has fixed – with coupons based on the index at the start of the current period and/or the implied forward starting floating rates.
At the start of every reset period then, our example of the 10y floating rate bond would behave like a 6m fixed rate bond exhibiting the same duration and convexity properties. PVBP of this 10-year floating rate bond will be equal to the PVBP of the 6m fixed rate bond with a coupon equal to the current 6m floating rate fix.
