# How to calculate the yield of a perpetual bond that pays a floating coupon payment?

I know that perpetual bonds are becoming a rare phenomenon and that ones that pay a variable coupon are even rarer. However, I believe that there are such bonds out there, and I'm hoping that someone can explain the mathematics behind calculating the yield of these types of bonds. Assume that the bond is not callable and does not have any other features.

Thank you.

Let's stick with first principles and assume a single-curve world. Assume a discount factor curve $$D_i\equiv D(t_i), t\geq 0, D(0)=1$$. The risk-neutral expected forward rate from $$t_i$$ to $$t_{i+1}=t_i+\Delta$$, i.e. for a tenor $$\Delta$$, is $$F(t_i,t_{i+1}|t)=\frac{1}{\Delta}\left(\frac{D_{t_i}}{D(t_{i+1})}-1\right)$$. Given some fixed spread level $$s$$, the present value of the floating rate bond is then

$$PV=\sum_{i=1}^{\infty}\Delta(F(t_{i-1},t_i|t)+s)D_i=1-D(t_{\infty})+s\Delta\sum_{i=1}^{\infty}D_i=1+s\Delta A_{\infty}$$

where $$A_{\infty}$$ is the annnuity factor. If we simplify further and assume a flat yield curve ($$r_t=r\forall t$$) and simple compounding, we arrive at

$$PV = 1 + \frac{s}{r}=\frac{r+s}{r}$$

Using this formula, you can compute ytm (sic!) $$y$$ given some market value $$M$$ of your floating bond as

$$y = \frac{s}{M-1}$$

• Doesn't this assume that the spread is fixed? My question is, what if the spread is a floating variable factor. Commented Apr 29, 2022 at 13:48
• No, it assumes a variable rate coupon plus a spread. You can set the spread to zero (which would not be of any interest then, no?), or you can add the first fixed cash flow. Hth? Commented Apr 29, 2022 at 15:57

Projecting what the the market thinks the 3Mo LIBOR will be in 50 years is a little iffy. USD and EUR swap curves are liquid to 30 years. People mark swap rates up to 50 years but they don't print that often. Still, assume you can project the index and therefore your coupon for 50 years. You could extrapolate beyond the last 50 year quote assume flat forward, but it won't do much because the present value of the coupons past approximately 50 years is close to 0. To bracket, you can pretend that in 50 years the cash flows just stop; or that you receive a large cash flow, like 2x principal; and solve for yields; and the two won't be materially different.

• I was thinking of that method, but I was wondering if there is any other method that steers away from arbitrary numbers (ex: 30, 50 years etc...) Commented Apr 29, 2022 at 13:51
• 30 is the longest treasury yield is most countries. 50 is the longest more-or-less obserable swap rate in USD and EUR. Extrapolating beyong that, you're guessing your projection and discount curves. But bracketing like I wrote will reassure that you're not adding much noise. Also: a yield of a floater is not a very useful number because it changes whenever the projecton curve changes. but you can use the same approach to get discount margin, or Z-spread, OAS, etc. Commented Apr 29, 2022 at 17:11