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OK, re-reading that, it strikes me that I should probably break down the jargon a bit...

Basically, you can offer people a loan from today to tomorrow (where "tomorrow" means "a month to a quarter from now") at today's rate and from tomorrow to the day after tomorrow at the rate you think will obtain tomorrow, plus a payment for the risk that the rate you think will obtain tomorrow is not, in fact, the risk that turns out to actually obtain tomorrow.

If you know what the central bank is doing today (which you do, because the CB tells you) and you can predict the upper limit on how much it is going to change what it is doing tomorrow (which you can for small enough values of "tomorrow," because the CB makes changes in a gradual manner) then there exists an equation which will tell you what the arbitrage-free price of that risk you took on is.

But a loan from today to tomorrow and another loan from tomorrow to today (at a rate agreed upon today) is just a loan from today to the day after tomorrow. So they should have the same total interest cost.

Now repeat this entire setup to add a loan from the day after tomorrow to the day after the day after tomorrow.

Now, in principle you can do this for the whole yield curve, and it does hold very well for the short maturity end. But there are three places it can break down as you go further out the yield curve:

First, the equation that tells you what the fair markup of a forward rate is depends on knowing the boundaries of the variation in what the CB is going to do. You probably know that a month into the future. You might know it a year into the future. Five years? Not so much. Ten years? Fuggetaboutit.

Second, the equation that tells you the fair markup only holds if people are actually able to borrow and sell forward contracts that are overpriced and borrow cash to buy forward contracts that are underpriced. Now, if you were a loan officer at a major bank and some City puke came along and told you that he had spotted an opportunity for making free money, but it required that you let him borrow money for eight years, and that he was right about how much the central bank would change the interest rate during those eight years...

... then you'd tell him to go play on the highway.

So for the long maturity end of the yield curve the arbitrage-free price may not actually obtain, even if there is a consensus on what the volatility would likely be. Because people's credit lines are limited.

And third, even if the first arbitrage relation were in force (if nothing else then because every bond trader has read the same sort of books I have, and the only unpardonable sin for a bond trader is to fuck up while everyone else does not), the arbitrage relation between forwards and annuities may not hold. Because if you tell a banker that you've found an arbitrage opportunity between annuities and forwards, then he'll go "yeah, that's what Long Term Capital Management thought too," and tell you to go play on the highway.

(Incidentally, this is why I do not usually unpack the jargon until someone asks me to. I didn't even manage to get rid of all the financespeak in this one.)

- Jake

Friends come and go. Enemies accumulate.

by JakeS (JangoSierra 'at' gmail 'dot' com) on Fri Sep 7th, 2012 at 11:58:00 AM EST
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