# How to express the price of an FRN using its duration and interest rate difference?

Where:

$$P_F$$: Current value of the floating rate bond

$$D$$ : Duration of the floating rate bond

$$R_F$$: Interest rate of the floating rate bond

$$R_I$$: Interest rate of the fixed-rate bond

$$P_I$$: Value of the fixed-rate bond issued on the same day = PAR

I heard in a lecture about FRN that using interest rate swaps (IRS) in practice, the value of FRN can be estimated approximately, if not accurately, using its duration, interest rate difference, and the value of a fixed-rate bond. However, I couldn’t understand this method on my own, even after considering it from various angles.

My questions regarding this method are:

1. How is this formula derived?
2. Why do we assume a fixed-rate bond issued on the same day?
3. Which interest rates should be used to determine the difference? Both for fixed and floating rates.

I would really appreciate it if you could answer these questions for my understanding.