Copping a feel of bond floors

It is nice to have some present value experience, or rules of thumb.  Given that one of the long term goals of this blog, and of me personally, is to become very experienced in pricing convertible bonds, then it is  useful to know what the bounds on a bond's present value tend to be.

Here's a rough and ready view.  Convertibles last 5 years.  Interest rates are usually in the 4%  ($r=0.04$) ballpark.   Like bonds, they are usually quoted with their price (and some other analytics) in par format, which for the present purposes is like pretending that they always have a face value $F$ of 100 units of the convertible's currency.

If you pretended that a security was nothing other than a single cash flow at the end of the five years ($t=5$), then the present value would be $100 e^{rt}$ or 82, approximately.  This number is usually referred to as the investment value or the bond floor of the convertible, since it is, in general, the present value of all the fixed income side of the instrument, that is to say, ignoring volatility and optionality components.

Many convertibles have get out clauses, both for the investor and for the issuer.  The net result is that, under favourable market conditions, the convertible might only last 3 years.  Again at 4%, this would give a bond floor of about 89.

When you've only a year left then the bond floor drifts up to 96, on its way to par, which is in this  theoretical and overly simple case, the final repayment price.  The closer in time you get to expiry, the closer the simple bond floor goes toward 100.

If the prevailing discount rate is much lower, say at 1%, then you'd get 5 year present values of 95.  In summary, the five year bond present value for interest rates 1,2,3,4,5,6 and 7 percent are, respectively, 95,90,86,82,78,74,70.  That same range of rates applied to a single cash flow only three years hence, where there isn't so much of a compounding effect, produces these bond floors: 97,94,91,89,86,84,81, which all deviate less from par than the five year instrument.

Below is a somewhat prettier table showing this.  Discount rate on left and years running along the top. There's synmmetry in here since really all iso-values of $-rt$ give the same discount factor.  This is, of course, just a visualisation of how the natural exponential $e^x$ plays out when $x$ is made up of two factors, $-r$ and $t$.

While I'm at it, taking representative discount rates of 1% 4% and 7%, how many years before the present value of 100 drops by half? 70 years, 18 years and 10 years respectively.  Likewise for a drop to one hundredth of its value (that is before the present value of 100 becomes 1)? 420, 105 and 60.

Adding intervening cash-flows by way of interest payments is more of the same, just an extra wrinkle.

If a nation state wanted to get rid of half the public debt in a decade, then one way to achieve it is to have a nominal discount rate of 7%.  If we call that 7% 2% real and 5% inflation, then a central bank just needs 5% inflation for a decade to wipe out half of the debt holders.  If a nation's debt holders are all domestic debt holders, then you've effected a transfer tax from the average lender to the average debtor, a kind of Jubilee.  A foreigner looking at your country might demand more of his currency now for your currency as a result of this worry.  If the debt holders are largely foreign, you are imposing the cost on them, which will have its own macro-economic consequences.

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.

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 - 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)

Anatomy of a convert - why not cut out the bank and go straight to investors?

As I mentioned in a previous post, there are a number of inefficiencies in a company receiving their money via loans exclusively, and if they are sufficiently large, they'll issue longer dated corporate bonds.  Well, convertible bonds are just another step in the evolution of plain old, or straight bonds.  Understanding the straight bond is about 60% of the job of understanding the convertible, so I'll be spending quite some time on them in the coming posts.

In this posting, I'll just give a brief overview of where I'll be going with the subject.  With straight bonds, the bond buyer hands over cash now to the company, and the company then enters into an agreement with the holder to give him regular payments for usually a fixed period of time.  Often the final payment looks like a 'return of capital' payment.  So what we'll need to do is to understand how to compare cash amounts at different times in the future.  We'll also need to get to grips with the maths of a return or a yield, which is a way of comparing one cash amount with another.  This seems almost too easy to bother with, but there a number of wrinkles it is important to understand.  Once done, we follow with a posting on the so-called yield curve.  A secure understanding of the yield curve represents the core of knowledge you need in the world of fixed income investments.

Anatomy of a convert - why not get a loan from the bank?

If a company needs money, it can always go to the bank and ask for a loan.  This is certainly something most companies do, so it is interesting to ask why companies don't just exclusively take out bank loans.  The answer is because the familiar terms and conditions of company loans are not ideal for companies.

Loans can often be variable rate - this allows the bank to cheaply remain neutral to moves in interest rates - they're passing that interest rate or reinvestment risk on to your company, which will need to manage that risk.  At certain times in the business cycle, that risk can be quite substantial.  Regular interest payments could increase by 10%, 25%, 50%, 100% even, on a dramatic upswing from a very low interest rate.  Of course, that could work in your favour if rates are already high and you have a project whose projected return on capital is higher than the high interest rate, and you expect rates to decrease over the following period.

Also, from a cash flow and accounting point of view, having this variable outgoing on your balance sheet can make it hard to maintain that signal of financial stability your chief financial officer would like to present to the world of potential investors in your company.

One talks of the capital markets as the place where companies try to get equity financing, but it is important to realise that your company needs to be of a sufficiently large size to play in the capital markets.  So smaller companies simply may not have this option open to them in a way and on a scale which makes economic sense - investment banking fees have never at any time been called cheap.  They're left to negotiate local deals with local banks.  Those banks operate in a less rarefied environment, perhaps have less of a capacity to make forecasts of your company's true growth prospects over the medium-to-long term (by which I mean over a 5 year repayment horizon).  In other words, your local bank in all likelihood can't afford to make corporate credit assessments beyond this five year horizon.  Result?  The loans they make tend to fall into this maturity horizon.  Bond maturities can be on the twelve year, fifty year, hundred year horizon.  This kind of stability allows corporate finance people to plan with more stable cash-flows and fewer trips to the bank manage to arrange shorter-term loans.

Local banks tend to be in more of a position of power with respect to a local medium sized company and their loan terms surely reflect this.  Often standard business loan contracts have what is known as an early payment premium, which just means that it'll cost you if you are thinking of over-paying on your loan - i.e. giving your bank more money back than was agreed on the primary repayment schedule of the original loan.

Also, a company often has each loan with a single lending institution - its local bank.  Raising debt on the debt markets means in effect there are as many different lenders as there distinct holders of bond or convertible bond certificates out there in the world.  This diversification of the investor base is sometimes considered desirable for the management team of the company, on the assumption that it is less likely that a tiny number of hugely important stakeholders will attempt to bully the management team into modifying their decisions to be more in line with what the debt owners want to see happen.  Still this does happen, especially when the company comes close to bankruptcy.

Banks often will want senior management in the company to offer some kind of personal guarantee on the loan.  With a bond, all the senior managers are in effect personally indemnified in the case of the company subsequently failing to meet its payments.  Nice for them.

It has to be said that loans provide a large proportion of how companies get their hands on capital for projects.  The debt markets provide the next largest fraction, and finally the equity markets provide the third fraction, so I've kind of introduced them in reverse order of importance.

For all of the above reasons, sometimes companies which are large enough will prefer sometimes to get  their capital from the bond markets instead of asking their bank for yet another loan.  And so-called 'straight' corporate bonds are the subject of the next posting.

Anatomy of a convert - why not sell your company in the future for cash now?

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Perhaps you'd only like to sell part of your company conditional on it having made a success some years in advance.  In which case you'd sell warrants to raise your cash.  A warrant is a call option on corporate shares which you haven't created yet.  If the call is on shares which are already outstanding, then no dilution is happening, and instead the call is just a kind of side bet on your company's stock price between two parties, the winner of which gets some pre-existing shares in your company.  Perhaps the loser already owned those shares anyway before he made his side of the bet?  Perhaps not, and he'll have to purchase those shares in the open market to satisfy his bet.  Derivatives exchanges can create markets in call options.

But the company itself can issue these warrants, on a promise to convert the warrant into new shares in the company.  That way, they're potentially diluting the share base by the size of the warrant issue divided by the number of shares outstanding.  Why's this interesting in the context of corporate finance choices for raising capital?  Well, if you get the cash through issuing a warrant issue, there's a chance (for example if the company's share price closes out the option's expiry period under water - i.e. below the strike price of the warrant ) that nobody will want to execute their right to trade in their warrants for shares in your company.  In other words, you get the cash now for a potential sell of a part of your company which only kicks in if your company share price has risen above the strike price.  This has obvious advantages - you only need to sell that fraction of your company in the happy circumstance that the market rates your company's stock price.  Unfortunately, you get less cash for the warrant since you're only selling a possibility - a right to swap for shares.  That warrant is always going to be worth less than the actual share itself.

Interestingly, this has a knock on effect for analysts, who track, among other things, how much money your company makes in each year, divided by the number of shares outstanding.  This measure, called earnings per share (EPS) explains how much earnings a potential purchase of one of your shares would entitle them to, assuming earnings repeated last year's result (which they rarely do).  If you sold new shares and diluted, then your EPS would drop, making you appear less attractive to potential investors who track this kind of thing.  If you issued warrants, then for the life of the warrants, you still have the original number of shares outstanding so don't take this up front hit every time you need capital.  

Analysts and accounting oversight bodies long since have been on to the potential for abuse in this trick, so you often see 'diluted EPS' reported.  This divides your earnings by the sum of all the outstanding shares, plus any shares which could possibly come into existence through the presence of warrant issues.  Well, not just warrant issues, but any other asset class which has the possibility of creating new shares.  So it is a kind of 'worst case' earnings ratio - the denominator being the largest it could possibly be given outstanding derivative issues.

All the same downsides to dilution we came across here also apply when you chose to dilute your company through derivatives.  One upside is that it gives companies whose stock price is somewhat volatile (for example fairly new companies, perhaps in sectors which exploit a new set of technologies) a comparative advantage, since the more volatile the stock, the higher the fair value of a warrant, all else being equal.

Anyway, what's wrong with just going to your local bank manager and asking him for a loan?  The subject of the next posting.

Anatomy of a convert - why not sell your company now for the cash now?

How does a company raise cash for its own corporate needs?  In general, there are two external categories - equity financing and debt financing.  What follows  is all quite a lot easier to understand if you assume firstly it is a small company faced with the funding choice; perhaps it owned entirely by one person.  Then imagine a second company which is larger (or perhaps the same company at a later stage of its development).

I mention two external categories but there's always the possibility of funding new project internally, through the profits that the company itself makes.  Not a lot of companies are in this fortunate position.  Certainly some mature companies get to end up like this.  But it is in the nature of many corporate projects that first you have to invest to build a potentially profitable line of business, and then slowly pay down the initial investment over the months and years that the line of business is operational (maybe over a longer or shorter time horizon too).  Rarely do you find lines of business which have low up front costs and immediate positive profit.  You can also sell off various assets your company owns in order to raise capital, with the approval, explicit or implicit, depending on local conditions, of the owners of the company.  For example you might sell off assets from pre-existing lines of business whose profits don't meet a certain threshold, within the context of a re-structuring of your business.  Kind of like selling  a kidney to buy some weights to build up your muscles.  You might think of these options as organic and cannibalistic internal funding.

Now, let's look at equity financing.  Well, the new money could be self-funded - the owners could just inject capital directly into the company.  Imagine a small one man operation, perhaps the owners of a crepes cart which serves up tasty French style crepes at local places where people gather.  The owners may need a power generator, to power their oven and fridge on the go.  The owners may just pay for it themselves and consider it an asset of the company.  By virtue of them being the only owner of the company and all its assets, then still own 100% of the power generator.  Of course the generator might depreciate in value, but that's another story.  The cash they might otherwise have had in their pocket might be depreciating too through inflationary effects.  The large-company equivalent of this is to go the share holders of that company and initiate a rights issue.  This is where they go to the owners - i.e. the set of all holders of their shares - and ask them to contribute more cash to the company.  This is disguised as the purchase of new shares, but assuming all the current owners take up 100% of their allocated rights, all that's happened is they seem nominally to own more numbers of shares, but just exactly the same percentage stake in the company.  Rights issues in reality can get complicated, when there isn't 100% take-up, but that too is another story.  If 100% is taken up, the net result is the set of original owners of the company still own the same fractions of the company they once did, but they've transferred additional cash into the company.

After you've tapped your owner (or owners) multiple times this way, eventually they become reluctant to hand over the required cash to the company for its ongoing new projects; then perhaps you can initiate that other kind of equity financing - the issue of new shares to the wider investor community.  Here you really are diluting the original owner's fraction.  Again, start with a one man band company.  The crepes cart.  He's the sole owner.  But a friend from the local village offers to pay for the power generator, if he can become a 20% owner of the company.  So with that cash, he in effect re-designs the ownership structure, so that there are now 5 shares.  Whereas there was 1 share, 100% owned by the owner, there are now 4 shares owned by the original owner and 1 share owned by the new investor.  So the original investor base gets diluted by the issuance of these new shares.  The new owner has bought himself a fraction of that company by paying cash.  

That's equity financing.  And it happens a lot with growing companies.  Private equity funds are specialist lenders in these circumstances.  Angel investors another form.  That's what happens with IPOs, when you market new shares to the wider investing public.  But there must surely come a point where the owner becomes reluctant to dilute the ownership fraction of the original investor base (and in particular, his own ownership fraction) any more.  

To see this dilution in action, let's see how many times he can dilute his original 100%, assuming there's a 20% dilution happening every time  (you keep 4 for every 1 given away).  You can decide yourself at what point you think selling fractions of your company is the best way to get a hold of some money for your company's projects.

1. 100% owned at start
2. 80% owned
3. 64% owned
4. 51% owned
5. 41% owned
6. 33% owned
7. 26% owned
8. 21% owned 
9. 16% owned
10. 13% owned

Companies are assumed always have an ongoing  need for cash to fund their various projects.  Whilst this is not always true it is certainly true, almost by definition, with growing companies.  It looks like you'd only need to make requests for corporate funding a measly ten times using this method before you'd given away a large fraction of the company, even assuming you find ten generations of new investors willing to invest in your company.  

Anatomy of a convert - why?

Why is there a convertible asset class in the first place?  Don't we have enough variety already?  First, a very high level explanation of what a convertible bond is.  An issuing institution (a company) creates a new convertible bond  issue because first and foremost it needs money.  So in that sense a convert is like a standard corporate bond, or a bank loan.  But if you decided to loan the money to the institution, then you get an additional right, namely the right to hand in your fraction of the convertible issue in exchange for a contractually agreed number of shares - usually shares in the issuing company itself.  Notice that this is usually a 'right but not an obligation', which means that it is a kind of call option on the company's shares.  So the lender transfers a cash amount up front to the issuing institution in return for a convertible security which entitles it to many things, usually including a regular interest payment, just like a bond or a loan, the option to convert into shares, under conditions agreed up front, and the right to ask for your money back, again under certain circumstances; there are also many entitlements accruing to the issuer wrapped up in the legal package too - the right under certain circumstances to hand the money back to the lender earlier.  In short, a convertible is a bond with a bunch of additional derivatives buried in the package.

So why not just get a loan?  Why go to so much fuss, pay a bunch of investment bankers and lawyers hefty arrangement fees, potentially give part of your beloved company away in form of share options?  It all seems ridiculously, fiendishly complex.  The answer lies in the domain of corporate finance, which will be the subject of the next posting.

Anatomy of a convert - introduction

I thought I'd try to understand how securities get priced by taking one of the most tricky of the asset classes, the convertible bond, and breaking it down into its constituent valuation parts, slowly building up my understanding to a stage where it is possible to price a real convertible bond.

This will take time, and along the way, I'll post on rates, yield curves, corporate bonds, equities, options, warrants, exotic contractual clauses, all of which will need to come together to get the fair price of a convertible bond.  This will be a long term project but is a well-targeted way of exploring some of the main  points of quantitative finance without it becoming a general introduction on quantitative finance.  Also textbooks, virtually without exception, are written in too dry a manner.  I'd like this exploration of the elements which go into the pricing of a convertible bond to be fun, slowly paced, and capable of taking the time to flesh out any side-track subjects which happen to take my fancy along the way.  If I can't write clearly about it then that's a clear sign I don't understand it enough myself yet.

I'll include Anatomy of a convert somewhere in the post title each time I make a contribution to this thread, so it should be, in time, easy to pull all these postings together, though if the posting seems like it could be useful as a stand-alone article, then I'll give it its own title, then link to it from some other Anatomy of a convert posting, so that they could still all be pulled together.

I shall be pitching it at a mathematically literate reader, though not one who knows a lot up front about finance.  I don't want each posting to be too onerous so I'll try to make them short.