Valuing a Floating-Coupon Perpetual Bond and Estimating Its Yield
Summary
The document derives the value of a noncallable perpetual bond whose coupon floats with a reference rate plus a fixed spread. In a single-curve framework, it discounts each projected forward-rate coupon and the spread component. The floating-rate portion telescopes to principal less the discount factor at the infinite horizon; when that discount factor vanishes, the value simplifies to principal plus the spread multiplied by the infinite-horizon annuity factor.
With a flat yield curve and simple compounding, the answer further reduces price to a function of the spread and rate, then rearranges that relationship to infer yield from observed market value. These formulas depend on the simplifying curve and compounding assumptions, and the infinite-horizon result requires suitable discounting. A second answer notes a practical limitation: very long-dated floating-rate projections are uncertain and long-end curves may be thinly quoted. It suggests bracketing distant cash flows, while observing that their discounted contribution may be small.
Key ideas
- Value floating coupons by discounting projected forward rates and the fixed spread across payment dates.
- In the single-curve setup, the floating-rate component telescopes to principal less the terminal discount factor.
- The spread contributes an infinite-horizon annuity value when the bond has no maturity.
- A flat curve with simple compounding yields a simplified price-to-yield relationship.
- Long-horizon coupon projections are uncertain, and distant cash flows depend on curve extrapolation.
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Full text
# How to calculate the yield of a perpetual bond that pays a floating coupon payment?
# 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.
## Answer by Kermittfrog (score 1)
https://quant.stackexchange.com/a/70679
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
$$ y = \frac{s}{M-1} $$
## Answer by Dimitri Vulis (score 0)
https://quant.stackexchange.com/a/70672
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.