US T Bills - an unremittingly fair price

US Tbills are key participants in the manufacture of the risk free yield curve.  That is to say, whatever the yield to maturity of a T-bill priced at $P$ just now, at the tenor $t$, that yield to maturity is also the discount rate to value the future cash flow which constitutes the payment of face value $F$ at time horizon $t$.

So when we come to model the present value of the (singular) cash flow of $F$ at $t$, we realise that the present value of this is simply the current market price of the instrument, $P$.

There may be quibbles.  The yield curve in practice isn't usually a treasury yield curve, but a money markets, futures, swap based yield curve.  Or perhaps one which is blended or fitted.

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.

Anatomy of a convert - Fake Bonds

Why are yield translators relevant to the current thread on understanding convertible bonds?  The reason is that if you want to have a model which gives you an estimate of the price of a convert, you need to have a yield curve in place so that you can find out the value today of a bunch of future payments over the coming years of the life of the convertible you're looking at.  Many of the points of a yield curve are invented or interpolated by a so-called bootstrapping algorithm.  But they're bootstrapped around a few real market facts, real market rates, currently trading that very moment in the market.  From this smattering of real-world points, a whole curve gets magic'ed into existence.  And as I previously mentioned, those real world points, those real world markets - cash markets, government bill and bond markets, swap markets, Eurodollar futures markets (all of which I'll come back to), each has their own history, their own typical loan durations, typical rate quoting conventions.  And where you have a panoply of disparate rate conventions that you'd like to pull together into a single coherent picture of yields, then that's exactly where your yield translators come in.

To flesh this out a bit, I'd like to create a couple of artificial contracts, with many real world details trashed for the purposes of clarity.  Then what I'd like to do is show you how a yield curve works on getting a present value for those made-up contracts of mine.  I'll just initially pluck a yield curve or two out of thin air.  After you see how it is used, then we'll turn our attention to creating a real, honest to goodness, no scrimping yield curve, with a view to having it help us price a convert.  We can use the family of fake bonds to see what a difference the various shapes of yield curve make on valuation, perhaps see when it pays to have accurate yield curves, and when it doesn't really pay to have accuracy.

First up, I'd like in my family of fake bonds a contract which just has a single redemption payment in a year's time.  Then one with a single payment in two, and so on for a ten year horizon.  So we have our first ten family members.  But converts often pay a coupon, so I'd like my eleventh to have twice yearly payments of 4% annualised, running for five years, and ending with a full redemption payment.  Number twelve is the same coupon-bond like payment history, but with 8% annualised.  And finally, I'd like a 6% bi-annual coupon, running for a ten year period, with a redemption at the end.  In all cases, I'd like the face value of these bonds to be £1,000,000.  By the way, this is unrealistic, since usually the face value is £100 or £1,000.  But if we wanted £1,000,000 worth of exposure then we'd just buy 1,000 or them, or 100 of them, respectively.  Why not just make it simple, and let us assume we're buying one of them, and the face value is £1,000,000.

The first ten I'll call fake zero coupon bonds (I'll explain the terminology later, for now it is just a name).  11 I'll call my fully sweetened convert.  12 I'll call my lightly sweetened convert.  13 I'll call my straight bond.

 Now, each of the 13 contracts embody 13 loan you've made (or acquired) to some institution or body who you regard as unimpeachably trust-worthy.  Who do you have in mind?  A family member?  A big bank?  A company with lots of cash?  A company with a long history?  A local state? A government?  A government from a particular time in history?  Perhaps a shell company whose only purpose in life is to fund your coupon payments and your final redemption out of a pot of cash it already have stored safely?  Think about it, and whatever works for you, that's how credit-worthy our fake family of issuers are.  These future cash payments, in other words, are just about as certain is it is possible to be with respect to future cash flows.  This is a pragmatic point I'm making here about certainty.  We're not talking philosophical certainty but a much more contingent and localised certainty.

The final piece of damage I'll inflict upon reality is to assume the world really does operate 24 hours per day, 365 days per year - namely everybody works weekends, and there are no public holidays.

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.