I have seen this approach in my work and I would like to understand the theoretical justification behind this approach.
To calculate the interest rate PV01 of a floating rate note. A synthetic bond is created that pays the next coupon (which was fixed already during the previous coupon payment date) and the face value (say 100) at the next coupon payment date. The price of this bond is equated to the price of a hypothetical treasury bond that pays 100 at the next coupon payment date. The yield of this synthetic bond is calculated and the yield based sensitivity (pv01) is termed as the pv01 of the original floater.
I haven’t been able to understand how the two prices are equal. Can anybody help me out here?
It's because the FRN is equal to a portfolio of (a) the shorter dated bond you described and (b) a forward bond, purchased on the next coupon payment date, featuring a floating rate coupon and a maturity date the same as the FRN. If the bond in (b) has zero dv01, the result follows. We argue that indeed it has zero dv01, because when interest rates move, both the return on the bond and the discount rate move identically. Like any rule of thumb, it's an approximation. If there is a spread on the FRN over the floating interest rate, this will create a small dv01.