## Sunday, 20 January 2013

### What just happened?

To price a convertible, the queen of the capital structure from a pricing point of view, you need to know about volatility, credit, rates.  Indeed, to price volatility you need rates.  And likewise with credit.  So it all points back to rates.  Which is why I decided to start with looking at simple rates products - certificates of deposit, then on to treasuries.  But while looking at CDs, I decided to strip it down to the basic question 'what is a rate?'

Given that the maths associated with rates is uncontroversial, and has been around for centuries, I guess most people don't spend a long time here.  But I want to.  I want to feel what a rate means.  So I invented the 'a man walks into a bar' story to give me space to spell out the elements.  I also have in my mind a slow-camera zoom of rates contexts, starting with macro-economic, then markets, then contractual, then participant based.

The biggest macro-economic variable which provides a context to rates is the likely inflation during the life of the loan.  I'd now like to summarise this in a simplistic mathematical way.

### That unpredictable stick of dynamite under every loan

In a previous post I mentioned that knowing the rate offered on a loan by a lender to a borrower isn't enough to know what the lender's fee is.  This is, of course, usually the main element you need to know to decide whether the lend is an attractive proposition for you as a potential lender (and borrower too).  But that's only because inflation is usually well understood.  It has happened often during the last several hundred years  to be well understood and also small in magnitude.  Strictly speaking, it doesn't need to be small, just well understood.  That is to say, predictable.  If inflation was at a rock solid 10% per annum, pretty soon peoples' uncertainty around how to deal with it would be reduced.  The public would factor in this certain knowledge into their future plans.  If inflation was totally unpredictable, but only within a narrow range, say 9.8% to 10.2%, then this pretty much is the same thing.  But if the unpredictability is accompanied by a wide range of values, say from -20% per annum to over 40 quintillion % per month (as it did in Hungary after the second world war), then that's clearly a different matter.  If this is a genuine possibility then it can render trivial all outstanding debt in the hyper-inflated currency.  The above link shows that even major world economies can fall into monthly inflation of over 20,000% per month, as did Germany after Lloyd George and the French lumbered Germany with unpayable levels of war reparation debt after the first world war.  In the Hungarian case, people weren't paying for bread in wheelbarrows full of marks, as in Germany, rather road sweepers would sweep smaller denominations up from the ground where they were discarded.

Inflations of all kinds, from uncontrolled hyper-inflations to strategic and temporary ones, in effect diminish the real value of a borrower's loan.  From the point of view of the lender, they effectively lose their capital.  It  has this effect across the board, since virtually all debt is still contracted in nominal terms.  Another way to view it is that inflations re-distribute real value from lenders to borrowers.  Under those circumstances, the fact that the original contract agreed to pay the lender some rate $r$ pales into insignificance.  So if someone in Hungary after the war agrees to lend 100 pengo to a borrower for a rate $r$ for a year, this was free money for the borrower, since the present value of $100(1+r)$ in a year's time is a very small number of pengos (assuming you had a decent idea what the real discount rate was).  So the lender handed over 100 now for a contractual promise which was worth a tiny fraction of that value now.  The lender would not even bother collecting the payments since the process of trying to collect it would cost thousands of times more than the value.