## Sunday, 9 October 2011

### Anatomy of a convert - on the interest of interest

In my last post, I glossed over one extra possibility - that in your sequence of cash payments strung out over a number of back to back time periods (for example your £5 per month over 12 consecutive months), after having received the first payment by the end of the first month, then during the second and subsequent months not only do you earn a return on the £1,000 initially invested, but you also earn interest on the £5 which by rights is now yours.  The presence of this additional method of accruing returns is called compound interest.  The compound moment is the moment when your interest payment comes due and is immediately available to earn interest for you.  The more frequently that compounding occurs, the more valuable its effect.  It can happen not at all (referred to as simple interest), with a certain finite frequency, or in the limit, with infinite frequency.

In all cases I've come across, when you drill down to the most atomic interest payment period, then that interest calculation period is always simple, never compound.  Only when you have a string of two or more interest periods is the possibility of compounding even possible.  So think of all kinds of compounding as the application of simple interest, but with a changed amount of principal at  the start of the later simple interest period.  You can see this clearly from the maths.

Simple interest expressed as an annualised $r$ applied for $t$ years on a nominal £1 amount results in $(1+rt)$ at the end of the period.  So if a bank gives you a promise to return 6% to you for a month, if you give them your £1,000 then you should expect $1000 \times (1+0.06 \times \frac{1}{12})$ back, which is £5.  Compound interest is just this simple interest repeated with a new principal of £1,005.

If you compound $n$ times per year over the period $t$ then your return on £1 for an annualised $r$ will be $(1+\frac{r}{n})^{nt}$  Why not just consider $n$ to be the number of compounding periods, and drop the $t$ - you could, but don't forget the $r$ is usually expressed on an annualised basis, and if you made $n$ be the entire number of compounding periods and $r$ be the full term rate, not an annualised rate, then you'd get the easier to understand equation $(1+\frac{r}{n})^{n}$ and this is clearer because you see it is just the product of $n$ separate applications of a simple interest formula, where the time period simple interest is just $\frac{r}{n}$.   Imagine a juicy deal where you get 100% return annualised, and compounded for $n$ time perdiods as before.  The cash back on £1 would then be $(1+\frac{1}{n})^n$.  Now $\lim_{n\rightarrow \infty}(1+\frac{1}{n})^n = e$, the Euler constant (approximately 2.718).  In other words, something very useful occurs - you get to use $e$ instead of discrete compounding.  Why is this useful - well, as you'll see later, this kind of compounding is often assumed in the academic literature since the operations of integration and differentiation are well understood on $e$ and are noticibly easier to work with than integrations of awkward polynomials.  If you compound more and more frequently you eventually reach a limit.  Compounding in the limit is called continuous compounding.  So if someone gave you that juicy deal of 100% annualised but didn't tell you how often the compounding was, then he's underspecified the contract - since you could be getting anything from £2 to £2.71 back at the end of the year.  Quite a difference.  The moral is, unless you know the degree of compounding on a multi-period interest payment, then the contract is underspecified.

Continuous compounding will appear again when we talk about yield curves.  Simple interest is more likely to be seen with very short duration kinds of bond - mostly short term government bonds and so-called cash instruments.  Finally, corporate bonds - convertible and otherwise - are often paid out twice a year.  But they go as cash to the bond holder, who's free to do anything they want with the cash - for example re-invest it in this bond, invest it in a so-called risk free government bond, put it in a savings account or stuff it under the mattress, to name but a few.  So in valuing a bond of any kind, this needs to be taken into consideration.

Next up, I'll show different kinds of direct translation from one rate regime to another, all of which will be practically useful when it comes finally to valuing a convertible.