In the text below no credit risk is taken into account
Surprisingly there’s very little to read on plain vanilla FRNs. Maybe the reason is that they are generally considered to be covered in swaps section. Indeed, IRS is effictively straight bond + FRN, hence where there are some words on how to handle a floating leg.
Yet, most interesting thing they do is they show that FRN must price at par at issue date, and believe that is is perfectly enough. And no words on FRN yield or even duration.
Hence I did much effort to find comrehensive FRN description. I failed, but yet I have now much data and I will try to aggregate it as follows.
So, what about price
Now, every good boy and girl knows that FRNs must price at par. Why? I have several arguments all leading to same result.
Here’s a picture.
is a one-period interest rate effective at time .
- Now, at time we buy one-period FRN. It’s payments are already known: it is . Hence, its price is
- At time we buy a two period FRN. Here’s a picture:
Proceeds from two-period FRN can be split into proceeds fron two one-period FRN, first is one-period FRN starting immediately, second is one-period FRN starting one period from now. Look at this:
This means, that price of our two-period FRN must also be . Let us see: we invest in first FRN, and in the end of the first period we obtain . From this amount we invest into second FRN, and in the end of the second period we obtain . This replicates cash flows from two-period FRN. Voila.
Just some calculations
Frankly I hate the way of the proof above. It’s all about the words, you know. I love old plain formulas. Today, after having broken two pencils up and tearing twenty sheets of paper apart I have finally obtained my long-awaited Nobel.
Let me arrange them so that to have a common divisor. The GCD is essentially .
Hence, the numerator now is a sum of the following terms:
- First term becomes
- Second term becomes
- Following terms become
- -th term now is
- The last but one term is
- And the last term doesn’t change and stays
Now we start adding terms together from last to first. Summing two last terms we get .
Next, we add -th term. The sum now is . Can you see it? Adding all the other terms up we have in numerator, and it is exactly our denominator. All terms cancel out, and we have perfect without any excessive words.
That’s all for today. Next sessions will be devoted to spreads, durations and yields.