Valuing Floating-Rate Bonds Between Coupon Dates
Summary
The document explains why a floating-rate bond is commonly valued near par just after a coupon reset or payment. It starts with a loan whose interest rate sets both its coupon and its discount rate: the discounted repayment equals principal, leaving the loan with zero value at inception. Chaining successive loans across coupon periods gives the same par result when borrowing and lending rates match the rate used for discounting.
Between payment dates, accrued interest and the difference between linear accrual and exponential discounting can create a small deviation from par. The answer cautions that the textbook result relies on treating Libor as a rate at which arbitrary amounts can be borrowed and lent, an assumption that does not hold in practice. It therefore distinguishes the idealized floating-rate bond from a real floating-rate note, whose value can depart from par. It does not work through the question’s specific dates and rates or develop a multi-curve valuation method, so the explanation is conceptual rather than a full pricing calculation.
Key ideas
- A floating-rate loan has zero initial value when its coupon rate also determines its discount rate.
- Rolling such loans across coupon periods gives a floating-rate bond a par value at reset under the same-rate assumption.
- Between coupon dates, accrued interest and the accrual convention can cause a small difference from par.
- The textbook argument assumes borrowing and lending are available at the reference rate without constraint.
- Real floating-rate notes can depart from par because market funding and discounting conditions differ from that idealization.
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Full text
# What is the value/price of a bond paying floating rate
# What is the value/price of a bond paying floating rate
I am going through J.C.Hull for swaps. Where he says we can value a swap using bonds. Let $B_{fl}$: value of floating rate bond, $L$ notional principal. Why is $B_{fl} = L$ just after a payment ? What about between payments ? In nutshell how can I calculate the value of such a bond ?
eg. ( notional principal $L=100$ ) If a floating rate bond has expiry in 15 months. And coupon payments are in next 3,9 and 15 months. The LIBOR at preceeding coupon payment date was $3$% (semi annual compounding rate ). Also say the current 3 month LIBOR is $2$% in continuous compounding. The why is bond value/price = $(100+1.5)*e^{-2*(3/12)}$ ? Why aren't the rest of cash flows taken in for pricing, with coupon rates taken to be forward rates ? Is this an approximation ? Or is it the same as even if I take all cash flows into account ? If so how do I prove it ?
## Answer by Phil H (score 3)
https://quant.stackexchange.com/a/42236
Let's look at a much smaller piece of the puzzle: What is the value of a 3m loan at Libor? Cashflows:
```
t0 -P
t3 +P(1+r.f)
```
Where `f` is the year fraction for `t0`-`t3` and `r` is the rate on the loan. If `r` is the funding rate, then the discount factor (price of a ZCB maturing at t3) is `1/(1+r.f)`:
```
dt cf df pv
t0 -P 1 -P
t3 +P(1+r.f) 1/(1+r.f) +P
```
This makes sense: if I borrow an amount and repay at Libor, then the overall value is zero. If it weren't, we would be seeing an arbitrage.
Now imagine we bolt a second loan onto the end of the first, still with a principal P:
```
dt cf df pv
t0 -P 1 -P
t3 +P(1+r03.f03)-P 1/(1+r03.f03) +P-P/(1+r.f)
t6 +P(1+r36.f36) 1/(1+r06.f06) +P(1+r36.f36)/(1+r06.f06)
```
Where `03` and `36` refer to 0x3 and 3x6 etc.
That looks more complex, but we know it's just a 0x3 loan and a 3x6 loan. And we know that as long as the borrowing and lending rates are the same (i.e. interest matches discounting), then both loans have zero net present value.
So extending this to arbitrarily many rolls always yields the same thing: The fair value of a (Libor) floating rate bond at inception is zero.
Between cashflows, assuming the linear accrued interest calculations usual in bond trading, there will be some small value due to the difference between a straight line and an exponential decay, but that is all.
## But Libor
Well, sort of. In textbooks you can borrow and lend arbitrary amounts at Libor, risk free. In reality, that's not risk free, you can't borrow and lend arbitrary amounts at Libor, and your boss wouldn't let you if you could.
So these days, a floating rate note (flat) does not have zero value, and all the textbooks are wrong.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.