Anatomy of a convert - dirty translators

Sometimes you need yield translators for inter-converting meaningful, mathematically valid yields.  But then again, sometimes you need dirty translators,  which switch away from the realm of valid yields and into the realm of historical convenience.  In other words, sometimes the market convention with some yields is simply mathematically not justified, and perhaps that market had its origin in a time before the real mathematics was widely understood.  Once a convention has a real human practice and history associated with it, it is difficult to make a big-bang switch away from the inferior formalism.  Best just to think of these dirty expressions as market quotes.  And the process of packing/unpacking the market quote as a bit like taking your shoes off or putting them on  when entering a house.  Once cleaned, then the rules on yield translation can be applied for your specific purpose.

When we come to looking at specific markets we'll see some of this going on.  But a second point I'd like to make just now about yield conversions is the possibility that the nominal rate time basis - the time period within which you  understand your $n$ and $r$ - isn't yearly.  It can be any period.  It mostly is nominally expressed as a return for a 1 year period, but it could be a rate expressed for a multiple year period, or a half year period, or quarterly.   Day count conventions, which I'll be getting to soon, can be thought of as a digital-to-analogue signal converter for time, embedded at the level of the market quote (taking your shoes off at the door).

Archaeo-Biography

Dateline 2400.  Peta Cube Inc., this year's most successful IPO, has just announced that it has acquired the entire email and calendar records of struggling Data Font Partners, a private equity firm which has in recent years accumulated the full archive of a popular early twenty first century company Google Inc. said to contain personal email records and calendar entries for over three billion human beings who lived around that time.  Data Font also contains the rival  email archive for that period, hotmail, hosted by a maker of contemporaneous operating systems.  In a earlier deal, Peta Cube had previously acquired and structured the mobile telephone geo-location information of a vast number of global mobile phone companies.

A spokesperson for Peta Cube said work was progressing well on a net space project which interfaces with a number of supra-national government public records offices to allow individuals to surf through their ancestry to identify distant relatives of interest.  A notable feature of the project is the use of proprietary network relevance algorithms and story-driven computational language processing modules to construct an automatically written biography for each person, tapping into the public domain Facebook archives.  Experts believe this could revolutionise the genealogy industry.  In addition, it is expected to prove a valuable resource to more traditional biographers and historians of ideas.

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 - 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 - prehistoric rational expectations

I'd love to know the history of the kinds of clause which are typically found in modern day convertibles.  I'm sure each clause would have a fascinating history.  But in the meantime I'd just like to point out first of all that these clauses individually do have a cultural history - somebody invented each and every one, at a specific time, and for a specific purpose.  Either they allowed a potentially failing new issue to go through, or they smuggled in a 'screw you' clause which wasn't well understood by either the issuer themselves, or by the  marketplace.

Second, I'd like to point out just how many of them can be cashed out in terms of algorithms.  Modern convertible pricing systems can turn pretty much every significant clause into a cash-now contribution towards the overall fair value of any convert.  That is quite amazing to me.  

And most interesting of all is that convertibles have existed for quite some time.  Whereas the modern pricing of options traces its origins to the late 1960s only.  How on earth did market participants manage to run that market in the absence of a decent convertible model?  Well, probably profitably.

One way of looking at these 'prehistoric' times for convertibles is to imagine how the disciples of rational expectations would explain what would have been going on back then.  How does a rational agent (or the average rational agent) manage to come up with a market price for a convertible in the absence of a coherent modern convert model.  I guess that the best model available would be the next best target for a prehistoric rational agent?  Certainly it could be the case that the average opinion of prehistoric market participants would be close to the average opinion of current market participants, with perhaps more variance.  But my gut feeling is that this would have been unlikely.  In other words, there was probably some kind of unrecognised persistent bias in the prices of certain prehistoric convert issues.  Really?  Could this be?

And what about the instant that the first step change in convertible pricing occurred?  I guess around the time of E.O. Thorp.  What if he decided not to publish his book on convertibles, but had kept it to himself.   Wouldn't he have an edge?  Wouldn't the market price be inefficient insofar as E.O. Thorp decided to leave money on the table back in the late '60s?  Rational expectations can never be about market efficiency in any absolute sense, but there must surely be an evolution of expectations.  Which means there ought to be a whole series of incrementally more efficient insights and practices when it comes to judging the market's efficiency at any one time, even now.

Robert Lucas attacked by chaos machine

Robert Lucas, a godfather of the rational expectations movement has come in for a lot of criticism recently.  Here's my own brief attempt.

Imagine an artificial economy with just two actors, who each have to guess game theory style at the likely behaviours, economically, of the other.  They both are fully cognisant of each other's economic models.  All they need to do is apply those rules to apply a decent best guess of model parameters - the legendary sloppy assumptions - and we'll sit back and watch well known macro-economic phenomena emerge from their identically specified micro-level models of themselves as economic actors.

Now lets imagine those models shared a similar property (as many many models do) with the logistic function, $X(n+1)=rX(n)(1-X(n))$, namely they are riddled with chaos.  Our two agents might agree perfectly on each other's model and what's more be correct but when it comes to apprximating the model parameters, needless to say, they cannot guess the other's starting value with infinite precision.  The result, over certain wide ranges of the parameter phase space - is utter chaos.  All it takes for rational expectations to be shown to be inadequate is some likelihood of such radical non-linearity in real (no pun intended) sets of micro-founded models of agent interactions within the wider rational expectations movement.

How would a rational exceptions robot respond to the possibility, or even more strongly, the knowledge that their models had 'dark areas'.  I guess the sensible thing to do is to apply probabilistic approximations around those regimes.  And those heuristics too would probably be amenable to the rational expectations approach.  I guess the rational expectations agent can operate under radical uncertainty.  But what if there were clear patterns of information which cry out for some kind of rational expectations model to develop, while an entirely different initial parameter set results in a different rational-seeming system to lure the unsuspecting rational economic agent.  And what's worse, where do you draw the line between uncontroversially certain parts of your model, and the stable-seeming boundaries at the edge of chaos?

Indices of the world, unite!

I'd like to suggest an idea.  Within a short number of years, thanks to Amazon and Google Books, we'll soon be in a position to have available to us in textual format virtually all of the indices of all of the books which have ever been written.  Clearly this is somewhat of an exaggeration, but not too much, especially in the realm of non-fiction.  Imagine a research project which applied the techniques of computational linguistics to automatically link all these indices together, to provide a search and browsing resource between parts of books.  Something like this is already just beginning to happen with citations - but in a sense these are external (but still related) to the content.  Indices get you right in to a particular page.  Once in place, the Big Index, as I'd like to name it, then becomes an intellectual super-highway, a novel way to read (parts of) books, of seeing connections, of hopping, of surfing, of delving freely into the history of a concept, of tracing influence.  On top of this super-highway, us travellers can then leave notes, attach personal commentary (rather like the Amazon kindle shared notes).  In time, we'll be able to follow the paths of great thinkers themselves, what they read and thought about, which connections inspired them.  How they felt about it at the time.  Pieces of text will become cultural monuments, with many transit routes in and out, and with a wealth of personal commentary.  Indices, after all, capture a lot of intelligence and work in abstracting a book - why not put it to use.  Also, it side-steps the famous Borges taxonomy probem, and always remains fully open.   The internet in general, and sites like Wikipedia do some of this for you, but it isn't the same idea.  With my idea we're exploiting a probably well-crafted index designed around the time the book was written, probably with the involvement and approval of the author, which captures some of the structure and flow of the book.  

Soon we'll have an additional option to read through a book, moving at speed.  The idea remains indifferent to one's position on copyright, and of course there'll be nay-sayers who'll decry the already too-disposable approach to reading we have moved to.    I disagree and see it as an enhanced mode of reading, which treat concepts not individual books as the core nuggets you're seeking.  Indices of the world, unite, you have nothing to lose except your identity.

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 - Pioneers

J.J. Hill.  His company issued the first convertible.










Meyer Weinstein.   He was one of the first to hedge convertibles with other securities.  Using heuristic techniques.

E.O. Thorp.  He showed a mathematical approach to valuing a convertible as a bond converting into a warrant.

Fischer Black.  He got a CAPM-friendly solution to the pricing of a call warrant and in return got a call on the Nobel prize for economic science, but it expired just out of the money.






Since then, absolutely nothing of real importance except for a melding of credit and volatility factors into pricing models.

A lot of basics of fixed income modelling already was in place as early as 1913.  See the below book.

Anatomy of a convert - crude terms

Just as a little break form the maths I'd like to consider the major clauses in a convertible and try to come up with memorable catchphrases which describe their effect or purpose.  Some clauses in a convertible prospectus are more straight bond like (i.e. they're often seen in ordinary bonds), whereas others are more specific to the warrant/convertibility side.  Also, since traders are generally a foul-mouthed lot, I'm trying to make them as textually authentic as I can.

First up, the bond like clauses.

The holder's put clause.  "Take your shit and give me my fuckin' money".  The bond holder, on certain named dates, can decide he's had enough and would like his money back.

The  issuer's call clause. "Party's over, get out of my fuckin' house".  The issuer, if its company stock has done particularly well, can decide in short notice to force holders to decide whether or not to convert or else get their money back.

And now a couple of convertibility clauses.

The conversion right.  "First we loan you, now we own you".  The initial investment constituted an income bearing loan to the company.  The bond holder now wants to convert and own a piece of the company, since the share price now makes it desirable to own.

The reset clause.  "We sin, you win".  If the company stock price fails to perform, the bond holder is entitled to get a fatter slice of the company.

The exchangeable. "Pimpin' your daughter".  The exchange property which the issuer refers to isn't the issuer itself, but a related entity - often a majority shareholding in an asset which they now deem non-strategic.

The contingent conversion clause.  "We win, you're in".  The holder's right to convert is itself contingent on the company stock having cleared a number of performance hurdles.

The mandatory structure.  "A chattel forward".  The holder starts off lending the company capital, gets a regular income from it, then ends up getting certain ownership of the entity.

Anatomy of a convert - yield translators

In my last post, I mentioned that I'd examine the ways in which the various levels of rate compounding could be inter-translated.  But first, why is there a need to do this at all?  The answer is to facilitate comparison between various bonds (and swaps, for that matter).

Usually, if two financial instruments can be valued to a present value cash amount, then this explicitly facilitates the comparison.  Good enough for many instruments.  With fixed income instruments like convertibles (actually, any bonds), there's a second dimension people like to compare across disparate instrument types - the yield.  At its most general, you'd like to know the internal rate of return so you can more directly compare one bond to another - including also to a so-called risk free government security.  If you had the internal rate of return of a convertible, then you can directly read off how much greater this bond's internal rate of return is compared with, for example, British gilts by simply subtracting the gilt rate from the bond internal rate of return.  This then gives you an idea of how much additional return you can expect from that bond.

Always remember in what follows, these are just in essence quoting conventions you'll be translating.  And you will need to do this because there are many bond markets out there each with its own historically determined quoting convention which traders in that market obey when quoting rates.  So inter-market comparisons require yield translation.  For any rate quoted with an expectation of the simple interest formula being applied to the principal, there will always be a corresponding other rate which will give you exactly the same final amount, but with $n$ discrete compoundings per year, or with continuous compounding.  Think about this a moment.  I've heard people say that continuous compounding is somehow an approximation when used in academic finance, since real instruments are simple or discretely compounded.  You should now realise, if you didn't already, that for each and every discrete fixed interest instrument on the planet, there's a continuously compounded rate which will present value the cash flows to precisely the same value as the discrete formula.

Remember too, your basic atomic operation on a single bond payment period is that of simple interest.  You're likely to see simple interest in instruments which last a short enough time to have only a single payment period.  This time round, I'll introduce the subscript $s$ to indicate all the variables of this formula relate to simple interest but I'll still assume the principal is £1 and that the rate is annualised over a term running from now and lasting for $t_s$ years.  You will get back $(1+r_st_s)$.  Continuous compounding returns you $e^{r_ct_c}$.  Finally, I'll give two versions of discrete compounding, with different compounding frequencies $n_1$ and $n_2$ and times $t_1$ and $t_2$.  This will allow me to show you how to translate from one discrete convention into a separate one.  Since I'm using 1 and 2 as the subscript, I'll continue for discrete compounding, using $r_1$ and $r_2$ for the discrete-to-discrete case.  If I'm just referring to a single discrete compounding, I'll drop the subscript.  In which case £1 discretely compounfed $n$ times per year for $t$ years will return you $(1 + \frac{r}{n})^{nt}$ in the end.

1. Simple to continuous.   $(1+r_st_s) = e^{r_ct_c}$ which means $ r_ct_c = \ln( 1+r_st_s)$ and as a result  $ r_c = \frac{\ln( 1+r_st_s)}{t_c}$
2. Continuous to simple.  Here you start from the same equation as 1 above but quickly move to $r_s =  \frac{e^{r_ct_c}-1}{t_s}$
3. Simple to simple.  Why not, eh, for completeness?  This is easy,   $(1+r_1t_1) = (1+r_2t_2)$ and the 1s drop off leading you to notice that you're just time scaling one rate into another $r_1=r_2 \frac{t_2}{t_1}$
4. Continuous to continuous.  Start with $  e^{r_1t_1} =   e^{r_2t_2}$, take logs and you're back to the situation of simple to simple, where you're just scaling one rate into another in proportion to the time ratio  $r_1=r_2 \frac{t_2}{t_1}$
5. Simple to discrete.   First as usual the equation, $(1+r_st_s)  = (1+\frac{r}{n})^{nt}$  Then take logs, $\ln(1+r_st_s) = nt \ln (1+\frac{r}{n})$ and divide through by $nt$ so that you get $\ln (1+\frac{r}{n}) = \frac{ \ln(1+r_st_s) }{nt}$.  Raise to the power of $e$ to get $ (1+\frac{r}{n}) = e^{\frac{ \ln(1+r_st_s) }{nt}}$.  After that, take the 1 across, then multiply by $n$ so that $r= n(e^{\frac{ \ln(1+r_st_s) }{nt}}-1)$
6. Discrete to simple, starts at the same place as 5, but is much easier since $ r_s= \frac{(1+\frac{r}{n})^{nt}-1}{t_s}$ right away.  
7. Continuous to discrete.  Equate   $   e^{r_ct_c}  = (1+\frac{r}{n})^{nt}$.  Taking logs you see that $r_ct_c =  nt \ln (1+\frac{r}{n})$  and so $ \ln (1+\frac{r}{n}) = \frac{r_ct_c}{nt}$.  Finally $r=n(e^{ \frac{r_ct_c}{nt}}-1)$
8. Discrete to continuous.  This time the equation in 7 becomes $r_ct_c = \ln{ (1+\frac{r}{n})^{nt} }$ and so straight away $r_c = \frac{ \ln{ (1+\frac{r}{n})^{nt}} }{t_c}$ and if you want to do the conversion as quickly as possible you'll eliminate the power so that  $r_c = \frac{ nt \ln{ (1+\frac{r}{n})} }{t_c}$
9. Discrete to discrete. Start with  $(1+\frac{r_1}{n_1})^{n_1t_1} =  (1+\frac{r_2}{n_2})^{n_2t_2}$.  Take logs.  $  n_1t_1 \ln(1+\frac{r_1}{n_1}) =   n_2t_2 \ln(1+\frac{r_2}{n_2})$.  So  $   \ln(1+\frac{r_1}{n_1}) =  \frac{ n_2t_2}{ n_1t_1 } \ln(1+\frac{r_2}{n_2})$ and when you raise to $e$ again you get  $(1+\frac{r_1}{n_1}) =  e^{\frac{ n_2t_2}{ n_1t_1 } \ln(1+\frac{r_2}{n_2})}$.  In that case   $r_1 =  n_1(e^{\frac{ n_2t_2}{ n_1t_1 } \ln(1+\frac{r_2}{n_2})}-1)$.  As you can see, this is bound to be the most computationally demanding converter.

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.

Anatomy of a convert - in the interest of fairness

You just got a £1,000 bonus and would like to invest it somewhere to help towards your kids' education in the future.  You're naive so don't know about risk adjusted return yet.  You pick up a copy of the Financial Times and read about investment opportunities.  One story tells of a man who bought drilling rights for £10,000,000 and has struck oil.  Those rights are now worth £40,000,000.  Another story tells of last week's lottery winner who, with £10 managed to get a jackpot of £18,000,000.  A chart shows you that the S&P index was at 1,270 at the start of the year and now languishes at 1,155.  Your bank manager tells you if you give her the £1,000 get £5 credited to your account each month.  

If we pretend for a moment that your £1,000 is enough to allow you to participate in any of the above ventures, then you might ask yourself: which one would give me the most money for the investment.  (As noted above, this isn't actually the question you should be asking yourself - you ought to be asking yourself which investment gives me the most new cash given my risk appetite).

Interest rates are just ratios.  Ways of comparing how much additional cash you'd get for a nominal amount of, say £1 invested in a venture.  With the oil rights, you get 400% return, with the lottery 180,000,000% return, with the equity index investment, you get -9% return (you lose that much of your capital, in other words) and with the bank account each month you'd get 0.5%, making approximately 6% in a year.

To really be fair, you'd like to know how long it takes for those investments to pay off as they did.  For the bank account, you get 12 monthly payments.  For the market, it took 10 months to get that far, and for the oil drilling company, ten years.  For the lottery ticket, it took only 1 day.  To re-capitulate,  you started by being fair to the investment by asking how much a nominal amount (£1) invested would return to you, then you try to be fairer still by adjusting for the period of time you need to risk your capital.  This second step is akin to viewing the investments across the same nominal time period (let's say one year).  All you're doing is scaling the final cash returns by assuming a nominal investment for a nominal period.  The annualised return for the oil investment is therefore 40%, for the lottery investment 65,700,000,000%, for the stock market -10.9% and for your bank, 6%.

These simple multiplicative steps in the name of fairness allow you to reduce the noise, to get closer to making an investment decision.

Anatomy of a convert - interested?

Interest has its own interest - cultural, historical, mathematical.  In this post I'd like to point out the vague irrelevancy we like to attribute to it in our everyday lives. Interest rates for loans to the safest bet have most often been somewhere around the 5% level, give or take. I think for many people this is psychologically around the 'fee' scale - we're all used to banks and other money institutions (pension providers, insurance companies, etc.) charging fees with are in the same ball park.  We are also mostly dimly aware of the role of inflation on money - we all dimly know that the meaning of 5% is itself in some general way clouded or constrained by the inflation level, so that gives us an even further excuse to consider the difference between, say 5% and 5.5% as not significant.    

For banks and those dealing with fixed income products, like bond traders, their discrimination levels need to be a lot finer.  The reason is because they're usually applying it to much larger sums.  The only time in most of our lives where we get to play with large sums is when buying a house.  Here, we often come to appreciate the difference in meaning from a 5% payback rate and a 5.5% payback rate.  What we're doing, in our heads, is spelling out the meaning of that 0.5% with respect to a large sum.  Say our house costs £100,000.  Then that 0.5% amounts to £500.  If that was an annual additional payment, then the corresponding monthly payment would be about £42, or a meal for two in a restaurant.  Whereas if you're buying £100,000,000 worth of a convertible issue, then that 0.5% amounts to half a million pounds.  And if you earn less than a pound per day, that 0.5% is the least of your worries.

Anatomy of a convert - the zen homunculus perspective

Before we can understand a convertible bond, we need to understand straight bonds, and for that we need to be confident about fairly comparing cash flows which we make or receive at different points in time.  Many contracts contain such payments, as I alluded to in an earlier posting. Quite often what you do is to find the value now of the receipt or payment at a later date of any number of cash flows.  This process is often referred to as finding the present value of a future payment.  But it doesn't have to be right now - you could just as well decide to use any arbitrary point in time and fine the historical value, or even the future value of a cash flow.  Another term used to describe this process is as discounting the future cash flow.  This is usually because rates used are usually positive, hence the present value is less than the cash flow value at the future date.  But again, this terminology isn't strictly universally correct.  If rates are negative, then the now value will be more than the then value, so describing it as discounting would be a confusing usage.

What we're really doing when we line up cash flows at different times is taking account of the opportunity cost of cash over that time window, from the now to the then.  The referenced Wikipedia article suggests (at least, it did while I was writing this post) that the concept was invented in 1914, which seems unlikely and no double has its origins in the Western world at least as far back as Aquinas but my guess is the idea originated in India or some early Islamic centre of learning.  Anyway, the opportunity cost of a good factors in what you could have done with the money if you hadn't purchased that good.  It is a cost which is often overlooked in everyday thinking, but is essential to economic and financial thinking.

Of course, what I could do with the money is not going to be the same as what you could do with the money. Come to think of it, what I could do with the money now is probably not even the same as what I could do with the money 5 years ago, or perhaps in 5 years' time.

This isn't a point that's often mentioned in discussions on the opportunity cost, so I'll elaborate.  Imagine three fantasy lenders: the Midas lender, one who has a very high opportunity cost, since all his previous projects result in extraordinarily high profits;  the loser lender, all of whose previous projects lose everything; and  the  homunculus with zero appetite for risk.who in their prior utilisation of capital looks for the safest place to put that capital.

In finance, we almost everywhere like to pretend we're the homunculus, even if we think we're the Midas and even if we're actually the loser.  Usually for any given monetary region, the government bonds of that region are considered safest.  (As I write this post, Greece's sovereign debt is close to default, and it must always be borne in mind that the risk free rate may itself contain risks).  A better term for the so-called risk free rate of return might be the least risky investment, and it is usually the respective government bonds for each of the major currencies.  Even this is questionable given recent economic history - I guess your first thought for the least risky investment might be - money in a bank?  cash?  The bank is considered safe only insofar as the government backed deposit insurance partially protects you from the risk of bankruptcy.  Likewise cash is backed by the full faith and credit of the government which controls that currency jurisdiction - since 1971 anyway.  Still, let's pretend Greece wasn't in the Euro to begin with and if faced a sovereign debt crisis.  I can well imagine the credit spread widening so much that  the capital loss on the Greek government's debt that it performed worse than any drachma weakness.  If that is even a logical possibility, then doesn't it become a contingent question which is the true 'risk free rate'? 

In theory we ought to use the most appropriate opportunity cost factor for us personally, when estimating the worth to us of some cash flow.  But we don't tend to do that - we tend to share inter subjectively the zero risked homunculus- we ask ourselves 'what would he do with the money instead'.  Perhaps the only reason is the difficulty in coming to a decent judgement about our own true opportunity cost.   And we always decide that sovereign debt is the risk-avoiding homunculus's investment of choice.

Interestingly there is one place in finance I know of where this subjectivity wins out over the homunculus with  his head in the sand: inside companies.

Certain corporate projects will get green lighted if they are perceived by the management as likely to exceed the hurdle of the opportunity cost (to the company) of having the money placed in some other projects. (or investments)  You're in effect comparing your current project to a weighted average of all the other projects you have started up.  This is usually referred to as the weighted average cost of capital.  Here the weighted average cost of capital is risky and hence higher than the risk-free rate.  But the various sources of funding have heterogeneous expected returns, which is the main point I'm drawing your attention to right now.

There's another sense in which the so-called risk free rate is not risk free - the government debt in question is always of the non-inflation-adjusted variety.  I.e. the debt which feeds into each and every yield curve has inflation risk baked right in there.  In recent times, governments have developed inflation-protected versions of debt instruments (partly as a way of discovering the market's view on inflation), so why not use these securities?

Given we don't, then this risk free rate can be negative in real terms.  It can also be, believe it or not, negative in nominal terms too.  It happened with U.S. dollars for the first time in October 2010.  So if the risk free rate doesn't even have to be positive, why not go the whole hog and claim that cash under a mattress is the most risk free (providing you have a safe house in a nice neighbourhood).

Anyway, practical lesson number 1: if you ever want to build a yield curve engine, make sure negative rates aren't going to throw it a curve-ball.

Finally, the scope of 'risk feee' seems to be related to a homunculus who isn't properly globalised.  It displays a localist bias.  Why only focus on the set of USD denominated, or GBP denominated, or JPY denominated assets when thinking of risk free.  Shouldn't you be thinking of an FX-adjusted GDP-weighted basket of currencies?  Shouldn't the homunculus be a Euro-homunculus?

To really get to the botton of the yield curve, we must start at the beginning, and that means learning to deal with interest rates.  That'll be the subject of my next post.

Anatomy of a convert - I gave the company all my cash and all I got was this lousy security

I'd like to step back a bit.  See the commonality in what is already the beginnings of a complex jungle of security types.  Don't forget, all that's happening here is a company needs money and gets it from people or institutions which have it.  There's a distribution of cash from those who have it now to those who could use it now.  Everything else is secondary detail.  Think of the corporate loan, the straight bond, the convertible, as just exemplars of a grander swap agreement to which all specific security instantiations conform.

In its simplest, most theoretical form, the swap in question is a swap of money.  Imagine a contract in which, for some reason, I agreed to give you right now £1,000 and simultaneously you agreed to give me £1,000.  That's one of the most basic swaps.  Such a like-for-like swap contract doesn't exist, since there's no economic reason for doing it.  But, say I am exchanging £1,000 of coins into £1,000 of notes, then it might make sense.  Or perhaps I like the serial numbers on your cash (or the year of issue, or the portraits or aesthetics of the notes and coins).  I might even be prepared to pay a fee for the 1,000-for-1,000 swap.

Now, imagine it is a swap now of £1,000 for $1,530.  Assuming an FX rate of 1.53 then this has achieved some clearer economic value - I now own dollars and can make a dollar purchase.  Likewise you can quite easily imagine a whole series of increasingly complex, and contingent, swap agreements.

For example, imagine a swap as follows.  Party A gives party B £1,000 now and party B gives party A £1,100 in a year's time.  I could interpret that as a loan, couldn't I?  I could say that this is a loan for a year, with a repayment plus a single annualised interest component of £100.  A loan of £1,000 for a 1 year term with an annualised interest payment of 10%.  Another interpretation could be: this is a purchase of money-in-the-future, the cost of which is £1,000 now.

Cash flow swaps can be unconditional, or contingent on something else in the world happening.  For example,  the swap I've been describing is an unconditional one.  Whereas you could say you're prepared to hand over £1,000 now for the privilege of receiving £1,100 in a year's time, provided interest rates don't exceed 5%.  That proviso means that in some future states of the world, you get £1,000 and in others, you don't.  The claim you have for that £1,100 in a year's time is now called a contingent claim.

Now, specifically with respect to loans and bonds.  While there a lot of historical differences between them, what are the fundamental differences between a loan and a bond.  Well, one difference is that bonds are sliced up into separate chunks - let's say in £100-sized chunks.  So in our example the £1,000 could be created as 10 separate bonds, each of face value £100.  Loans tend to be owed by a much smaller number of lenders - typically one - the bank.   That bank can surely sell its loan on to another bank, so the fact that there's a market in loans or bonds isn't a fundamental distinction between them.  However, bonds are designed to be more liquid.  People have the flexibility to operate in these convenient chunks.  This allows the possibility for a more liquid market - one that might even have its own exchange.  In some general sense, the bond's terms favour or are designed to expect that it might be owned in turn by quite a few different people/institutions.  Loans traditionally haven't been so designed.  The loans market, therefore, tends to be a lot more specialised in terms of the participants than the bond market.

If you were on the executive board of the company in question which needs that £1,000 - which would you prefer - to be beholden to a single bank (or a small number of them), versus to be beholden to a broad set of owners, not one of which feels sufficiently 'in the driving seat' to dictate additional terms to the company?  I would say that, insofar as the company is mature enough to issue bonds, and assuming the fee structures could be made fairly similar, there's still be a reason why corporate managers would prefer bonds.

So liquidity of market and diversity of the debt holder base is one clear fundamental difference between a loan for £1,000 and a corresponding bond issue worth the same.

The second major difference is where you get placed in the queue for pay-off in case the company enters bankruptcy.  Loans appear earlier in the queue than bond holdings.  In other words, in the case that the company gets into trouble and enters bankruptcy, then the loan owners get paid first when liquidating the company's assets, compared to the bond holders.  Equity holders come last in that list.  This pecking order is part of what is defined as the company's capital structure.  This is actually a bunch of laws common to all companies domiciled for legal purposes in that legal jurisdiction.  This legal framework goes a long way to determining the structural detail of the securities industries of that region, and it is arrived at through many decades - centuries in many cases - of legal precedent and case law on contested contracts.

So, whilst there are many reasons why a company's capital structure will look the way it does at any one particular moment in its life, it'll in the end reflect the company's history of decisions about how many terms and conditions it is bound to put up with around borrowing from, say, a small number of lending institutions, versus lending via a liquid bond market with a diverse range of bond-holders.   While in theory there can be international dimensions for both loans and bonds (your 'local' bank for loans can in some cases be an international organisation), it is clear to see how bonds are potentially more globalised - which in turn means a  potentially wider set of possible lenders.  This wider selection should lead to, all other things being equal, a better set of terms and conditions for the loan of the money.

Lending banks get their funds from a number of sources, but the characteristic source of funding is by current accounts and savings accounts.  Savers get a low rate of return when they put their money in a bank, and that bank aggregates all those accounts up, and lends out on a longer term basis, charging a higher interest rate then that which is paid to the current and savings account holders.  The bank makes on the spread.  In practice, banks also go to the capital markets for their short term funding needs.  Doing this too aggressively is what singled out the failing banks on the 2008 crisis.  So here the savers don't make the lending decision directly, in a sense they our-source that credit-allocation decision to the banks.  Whereas with bonds, the investor base gets to make that investment decision themselves.  That's another major fundamental difference - that the original providers of the capital have transferred the credit allocation decision to the lending institution, whereas with bonds, that stays closer to the bond owner - the capital provider.

To summarise, while in the end this is all a case of capital providers striking contracts of varying complexity to lend money to a company, specific differences are along the lines of

1. the diversity of the capital provider base (a small number of concentrated owners versus a wide base)
2. who decides on credit allocation (does the capital provider out-source that decision to a bank, or does he perform the operation himself)
3. the liquidity of the security (local and illiquid versus global, standardised and liquid)
4. the legal pecking order of repayment in the event of corporate default (the capital structure)