Compounding fictions

Behind the mathematics of compound interest are a series of layers of thought experiment, fiction upon fiction, compounded.  Interest itself as an idea has its modern origin in the act of lending or borrowing money.  In general, it measures the growth (or diminution) of a quantity across an observation period.  

The starting point for the fiction is that you can predict the future.  You can't of course.  But laying out your expectations about the future is a kind of fictional account of what you hope will happen.

Remember also that all interest is compounded, with per year compounding periods of $n=\infty, \frac{1}{365}, \frac{1}{12}, \frac{1}{4}, \frac{1}{2}, 1,\frac{1}{t}$.  So with compounding, you identify a number (1 or more) of regularly occurring, evenly spaced way points along the journey from observation start point (investment initiation, loan beginning) to observation end point (maturity, investment exit).  No cash flow needs to accompany these marker points.  Though cash flows often do.  They can be merely notional.  But at these way points, ownership is transferred from one party to another of some fractional value - cash or something else valuable.

Then, within the boundary of these fictional way points, come even more fine grained way points, where some linear fraction of the sum to be earned in the current way point period is measured out.  This fine grained linear fraction often measures so-called accrued interest.  Again it is often expressed as simple interest on the coupon you're in the process of compounding, but in reality it is often $n=\frac{1}{t}$ compounding within a semi-annual bond coupon compounding $n=2$.  It is done so that when a bond is traded on any day which doesn't fall on a compounding period end date (i.e. on most days), the buyer and seller break up the intermediate value of the current coupon betwen buyer and seller - a bit like Pascal's problem of points, where a game is ended at a point where the rules haven't fleshed out the value.  The presumption of fairness drives a buyer or seller of the bond to agree to apply time-weighted $\frac{1}{t}$ compounding to get a micro read on how most fairly to split up the current coupon.  

When you express $\frac{1}{t}$ compounding for a period, what you're doing is implicitly saying that no further micro structure is worth considering between the two dates - in this case the dates which bound the current coupon.  That being so, then you can use a linear fraction of days (or years, seconds, whatever, as it is a ratio after all) to work out the fair division of the currently in play coupon.  That is, compounding only at the end allows you to treat the rate as a linearly scalable ratio.

The human scale on interest payment doesn't go further than these two levels, namely coupon compounding and accrued interest.  The sums involved in any one deal are usually of a scale that further kinds of compounding don't seem worth it in analysis.

But the mathematics doesn't care about that mere human constraint.  You could keep on carrying on this micro analysis to the sub-day level, in theory to the continuously compounded level.  This might become more relevant in centuries to come when the world of high frequency trading comes to the world of sovereign bond relative value trading.

These layers of fiction exist in different places.  For example, the bi-annual compounding frequency of many bonds exists as a fact of the bond prospectus.  But the accrued interest fiction exists between buyers and sellers of that asset.  It is a form of market practice, that is to say, a sub-prospectus descriptor.  Said differently, the writer of a bond prospectus could imagine a world where a different daily accrual convention could apply.  

Every use of the compounding formula so far in these recent postings concerns discovering something about a trans-temporal analysis of a series of cash flows wrapped up in the conventional legalistic dressing of a bond.   It is a single algorithm which explains how the quantity changes at moments within the observation range (internal period ends and internal period middles and starts).

Formulae aesthetics: compound interest with a holding period nominal rate or with an annualised rate

$(1+r)^m$ versus $(1+\frac{r}{n})^{nt}$

I've often in books found two formulations for the compound interest rate formula, which, remember, I see as inclusive of the simple and compounded formulae too.  One expression of the compound interest rate - the more messy one - implicitly assumes that $r$ is given to you as an annualised nominal rate, and the other assumes it is simply the nominal rate which applies to the compounding period $m$.  The first has more parts to it, due to the initial step of transforming the annualised rate to the compounding period rate.  The second doesn't have this step.  I prefer this second formula.  

It is more basic.  We annualise for human purposes.  To compare rates across a standardised time horizon.  There's nothing inherently mathematically important about the human centric time horizon of a year.  For the maths, all that matters is the implicit compounding period, and the number of periods you roll the compounding operation.

So in the formulae at the top of this posting read the first formula as saying the following:  Take a dollar and a rate $r$ which has, as all rates do, an implicit compounding basis of some time period.  Lay out $m$ of these time measures end to end.  Walk that dollar forward and at the end of each of those $m$ time periods, grow the value of your dollar pot by $1+r$.    It is a series of $m$ salaries which are paid at the end of some implicit time period, implied in the compounding basis of the nominal rate $r$.  The rate doesn't give you a full picture of itself unless accompanied by the compounding basis, a time measure.   It is also nominal in the sense that it isn't quite the return you will see on your investment.  For example, with a nominal rate of 10% on an annualised compounding basis, run over two years, a dollar becomes 1.21, meaning your holding period return is 21%, not 20% (2 times 10%).  Likewise 10% over four years gets you a holding period return of 46.41%, not 40%.  Think of 'nominal' as meaning a rate which is used internally in the production of your final holding rate return.  It is referenced at $m$ points on this journey.  But when someone asks you how much your journey profited you, the answer only emerges after having run the algorithm, made the journey, evaluated the compounding formula.

Simply, if we prefer to see $r$ quoted on an annualised basis, an initial translation needs to happen to make annualised $r$ into the equivalent compounding period rate.  Compounding periods are almost always less than a year, so you see $m$ expressed as $m=t \times n$, meaning that the $m$ compounding periods in question run $n$ times a year for $t$ years.  So you get to an equivalent compounding period rate by chopping down your annualised rate $r$ to $\frac{r}{n}$

To put this all another way, virtually every time you come to use the compounding formula you'll probably use $(1+\frac{r}{n})^{nt}$ but the mathematics doesn't care about years, and $(1+r)^m$ is the cleaner, clearer formulation.  They both do exactly the same job.

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.

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.