# What is the yield when a floating-rate note is issued above/below par?

I am new in this area so all help is much appreciated!

Let's say a 3-year floating rate note pays a coupon of LIBOR+100 bps, and is issued at a premium with price = 100.5.

I understand that this must mean that the total yield for the investor is lower than LIBOR+100 bps. But how would you calculate what the yield is? I assume you must make guesses about the future LIBOR fixings?

Yes, you would make "guesses", but fortunately these guesses are derived from market-observed rates.

Assuming a semi-annual coupon rate and discrete compounding, the price of a bond ($P$) is given by:

$$P=\sum_{i=1}^{2T} \frac{CF_i}{(1+\frac{Y}{2})^i}$$

where $CF_i$ is the cashflow at time $i$, $Y$ is the annual yield, and $T$ is the number of years. The cashflows are linked to LIBOR such that for all cashflows (except maturity):

$$CF_i = N(LIBOR_i +0.01)$$

where $N$ is the notional of the bond, and $LIBOR_i$ is the zero-coupon LIBOR rate at tenor $i$. Recall that LIBOR is a combination of rates that generate a curve at a variety of tenors. In order to determine the yield ($Y$), we must first determine the coupon payments that are based off of this curve. For example, a cashflow payment 1.5 years from now will be determined by the LIBOR zero-curve for the 1.5 year tenor. Once we solve for all of the cashflows ($CF_i$), the only remaining variable is $Y$.

• Thanks! That explains it. I'll add a follow-up question here if it's OK: I keep on reading that on the reset date, the value of the FRN should return to par. But isn't that only true if the FRN was issued at par and the perceived credit quality of the issuer has not changed since the issue? – user2078515 Dec 3 '15 at 18:27
• A semi-annual FRN can be replicated with rolling 6-month bonds selling at par. So in theory, yes, the FRN should return to par. However, I see your concern with the credit quality of the issuer and their ability to repay the notional and coupons. Perhaps someone who actively trades FRNs can chime in for this. – dmanuge Dec 3 '15 at 20:14