Tuesday, 25 October 2011

Anatomy of a convert - time and time again

In a previous posting I showed all the ways simple discrete compounded and continuously compounded rates could be inter-converted.  However, now I'd like to state that all these types of rate can really be seen as variants of the discrete compounding $(1+\frac{r}{n})^{nt}$.  It is good that this can be seen as the most fundamental representation, in my mind, since it works so well as a contributory definition of capital as a property which produces other properties.

You already know, thanks to Jacob Bernoulli, that the limit of discrete compounding is $e^{rt}$.  All you need to see is that, when you let $n=\frac{1}{t}$ then the discrete compounding formula $(1+\frac{r}{n})^{nt}$ becomes $(1+rt)^{\frac{t}{t}}$ which is identical to the simple interest formula $(1+rt)$.  Great; so we have a continuum of compounding frequencies, running from 1 to $\infty$.

What are you doing when you set $n=\frac{1}{t}$?  Well, remember $t$ is the term of the loan or bond and $n$ is usually the number of times per year (i.e. per unit t=1) you compound.  So in the general case you will have discretely compounded $nt$ times.  But if you only want to compound once at the end, then setting $n=\frac{1}{t}$ is the way to do it.  Also notice the similarity with geometric and arithmetic means here.  With arithmetic means, between the start observation $t_0$ and the end observation $t_N$ you have a series of in between observations $t_1, t_2,...$ and you can work out a series of returns $t_{i+1}/t_i-1$ and then calculate the arithmetic average of these returns.  Likewise you can calculate the single value $t_N/t_0-1$.

Anyway, back to the fixed income analysis of converts.  As you'd imagine with any loan, both parties probably have in mind some notional loan term.  You can imagine a retail client approaching a bank manager.  One of the first questions is bound to be for how long do you need this loan?  After that some haggling will result in a rate.  Understand that the rate in question is actually the remaining pair $r$,$n$.  You always need to know the $n$ to understand the value of $r$.  If you really wanted to lay this out sequentially, then you could say, in any negotiation about a loan you first set the term, $t$, then set the compounding framework $n$ for understanding finally the rate $r$.  A more prosaic interpretation is to say that $n$ selects the formula you use to plug your $r$ and $t$ into.

Notice the choice of $n$ here doesn't have any implication for actual cash flow transactions - the borrower could actually keep a hold of the interest until a final payment.  Or on the same analysis, he could pay out on the compounding dates into your account and you'd be free to do with the interest anything you wanted, including spending it unwisely.  It doesn't alter the theoretical analysis.   Payment dates are just a best considered on a different schedule to the compounding schedule implied by $n$.  

The world of fixed income is overwhelmingly interested in $n={\frac{1}{t},1,2,4,12,365,\infty}$, namely simple interest, annualised compounding, bi-annual compounding, quarterly, monthly, daily and continuous compounding.  At a pinch you could reduce it further to $n={\frac{1}{t},1,2,\infty}$

But this is not all.  It would be if all fixed income markets quoted securities in one of the rate formalisms covered by $n={\frac{1}{t},1,2,4,12,365,\infty}$.  They don't.  Often they quote some other market variable, and you need to do some unpacking of that market quote.  That unpacking is, naturally, a function of the various fixed income markets themselves - and there is additionally some regional variation in conventions/usage patterns.

In the next posting, I'll talk about the day count conventions, the digital-to-analogue converters of time and the final wrinkle we need to iron out before we can move on to look at real market quotes and begin to get into the details of building a yield curve.