Friday, 21 October 2011

Anatomy of a convert - dates of interest

There are so many things I'd like to say about the interest calculations whose inter-translation I covered.  First up is the step up in complexity when moving from simple interest to compounded interest (discrete or continuous).  I would imagine the maths for simple interest has probably been understood and practised for at least a couple of millennia.  And while compounded loans are surely not much younger, their mathematics, that is, showing what the fair price ought to be, is quite recent.  The definitive book about discrete compounding came out in 1613, by Richard Witt.  No doubt Indian mathematicians probably cracked it 600 years earlier, but in our Western dominated tradition, we like to 'reset the clock' on important intellectual discoveries like this, unfortunately.  There's a nice temporal recapitulation here - the mathematics for fairly valuing certain future cash flows was first published in a Western book a mere 41 years before Pascal and Fermat opened up the way for estimating the fair value of uncertain future events.  Likewise we're spending time on understanding the fixed income side of convertibles before looking at their optionality, which requires more probability theory to understand.  Also, whilst Napier first talked about the exponential constant in 1618 - a mere 5 years after the Witt book -  Jacob Bernoulli, in working on the compound interest problem, identified that $e^x = \lim_{x}(1+{\frac{1}{x})}^x$, namely that if someone came to you and offered you a 100% annualised, continuously compounded rate of return for a year, if you lend them £1, then you'd get back £2.72 approximately.

I've come to realise how Christian, Islamic and Jewish arguments against money generally and the practice of usury in particular (which to many an ancient mind was strongly associated with compound interest, often regarded as grossly unfair) tainted - and still does taint - the Western world's view, so perhaps it is no wonder that we have to wait until 1613 for a full book on the subject. Compare that with a modern definition of capital as a property which creates other properties.  All you need to do is realise the recursive nature of this definition and you have a compelling need to assume compounding as the basis for understanding how capital works.  We've broken through with the mathematics, but we retain much of the moral disgust which accompanies lending and interest generally.  Even those ancient loan makers who only lend out on a simple interest basis, assuming that when they get their payback, they lend it out again (namely lend out their repaid capital).  This practice of a sequence of simple interest based loans it itself a compounding operation when viewed from the perspective of the loan maker's business over time.  So any attempt to distinguish on moral grounds simple versus compound interest must surely be bogus.  It isn't the compounding frequency that's the problem in usury, it is the rate of return.  Any fair simple interest rate has a corresponding fair compounded rate.

There's too great a temptation to rush forwards in my overview of the anatomy of a convert, but I'll hang around a while on the subject of yields.  Remember where we are right now.  I'm seeing how to model the value of cash accruing to us in a future date so that we get a handle on the value now. In this world of rates, I started with so-called risk-free rates.  This allows me to ignore how to model credit, for now.

I'd like to spend some time on the general concept of a yield curve.  But even when I restrict for now my attention to maximum-creditworthiness borrowers, there can still be a confusing jungle of rate forms  (often called rate bases in the financial jargon).  The reason for this is we get those rates from several disparate actual markets.  And each of those markets has its own culture - its own quoting convention, time horizon.  If we ever want to imply anything from real rate market data, we'll need to understand each of those markets' quoting conventions.When you can do that, you can feed the rates into a homogeneous view, the yield curve.  And just as there are multiple conventions for quoting bonds or money market rates, or swap rates for market data quote interpretation, so too there are multiple ways of expressing the output yield curves.

In the next posting I'd like to develop a general purpose and fairly simple framework for placing rate formalisms in a context which makes the operations seem totally sensible.