# 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 ?

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.