Forever

A very simple model of the firm's value is the dividend growth model, perhaps better called the dividends-with-growth model.  In it, a firm's value, market capitalisation, P, is given as next year's cash dividend divided by an appropriate discount rate minus a growth rate.  P=D/(r-g).

Separately, no longer in the world of equity but in the world of fixed income, if I chose to lend somebody money with a view never to get the principal back, only annual coupons, then the value of that infinite stream of coupons is a perpetuity, whose value is given by C/r, the coupon divided by an appropriate discount rate.

I just want to find a set of simplifying assumptions about the firm which unite these two valuations.  Imagine there are N companies in the world, and the average company produces a dividend stream D with an average corporate cost of capital (weighted average of all sources of funding for the average company) was r.  Let's further pretend that all N companies actually have this D,r set-up - i.e. you momentarily pretend there's no variation in the N companies.  In those N companies there'd be financial institutions too.  So they too have a weighted average cost of capital of r and a dividend yield of D.

Let's further pretend that the only line of business these financial institutions were in was making a market for perpetuities.  Namely that they only made loans or took money on the basis that they'd never get/give the capital back, merely that they participated in a perpetuities market.

You have some money and you want a return.  You look at the bank's perpetuity market making.This bank would not be wise, given the modern theory of value creation and destruction, to engage in any lending or borrowing activity which somehow didn't return at the very least the average corporate WACC, r.  So, ignoring bid ask spread, and the imposition of healthy profit margins, you'd expect your principal investment to be worth the equivalent in perpetuity coupon terms  of C/r, namely that you'd receive a regular annual payment, in today nominal terms, of C each year.

If companies in general were perfectly efficient then the growth component, g, of the dividend discount model might reduce in the limit to the expected long term inflation rate.  That is to say, in a fictitious super-competitive corporate environment, the only dividend growth would be inflation based.  So let's live in an inflation free world, where g=0 and no innovation occurs (at least with respect to these perpetuity banks).

The dividend yield is a fraction of capital made, on an annualised basis, by the firm.  So the only way in which D can be made is through the one line of business, namely perpetuity market making.  So in general it must make D on the bid ask spread, that is, it must transact a sufficient number of deals.  Assuming If no defaults ever happen and it lends out all of its capital it must charge in aggregate D as the perpetuity coupon. The tighter the actual bid ask spread, the wider must be its balance sheet.

Coase defenestrates Knight on firms

Coase dismisses Knight's view that firms exists to serve the purpose of concentrating income uncertainty to the owners of the firm, not the workers.  He does this by hypothesising precisely the opposite - a firm which presents a fixed income service interface whilst internally paying workers on a profits basis.  While I agree that this isn't an essence of the firm, I do disagree that Coase's argument challenges that - just because it is possible to create a firm which pays its workers on a profits basis assuming fixed income service costs, this isn't a convincing argument against the idea.

After all, in a sense, all companies create a fixed income cost base with a view to making variable (and hopefully excess) profits.  Just like marketers, in fact.  Though with marketers, the uncertainty is even lower due to the reduced complexity in not having a firm and employees.

I think Coase is right that firms aren't the sine qua non of providing fixed income for workers - financial contracts can do this, in theory, which you can pick up in a market, independent of firms per se.  Insurance policies do this kind of thing.  These could, in theory, exist in a pre-firm world.

Everyday lower prices!

One advantage which Coase explicitly mentions in his classic paper on why a firm might provide a lower cost solution to direct bilateral markets contracts has to do with long term contracts.  It may be cheaper to replace a series of spot or futures contracts with an in-house longer term one which fixes in the costs of the factor of production.

Is this a real advantage?  Assume for the moment that the futures price of the factor of production in the market place is the fair value of the factor.  

The argument goes like this.  We need some factor F to make one of the corporate products.  There is a spot price $S_t$ for this product right now.  We need it for a series of t periods in the future.  We might look to buy it on the futures market for expiry t.  There we assume (naively) a fair value of $S_te^{rt}$, using the risk free rate.  But, the futures market isn't infinitely deep - it peters out after a number of months or quarters or years.  Wouldn't it be better, more possible, cheaper, for a firm to internally charge for this?

My first thought is if those longer term contracts are needed by companies, then one of two things would happen. First, there might be a deeper (temporally) futures market.  Second, there is an OTC market which can (and does) service those needs which might exist, as it does currently for financial derivatives.  

My second thought: even ignoring that point, how does the company achieve a better price, all costs considered, than the market?  What is the company doing, other than taking a chance.  The hurdle it has to overcome is the cost of that futures contract, assuming it exists, for the corresponding time period, including all costs.  Companies in general can't take a chance and persistently win, on average.  The efficient markets hypothesis would see to that.  So on average, the average company will be no better and no worse off than the futures market would imply.  Otherwise the company is merely subsidising that particular factor of production within the the corporate balance sheet.  If this is persistent, how can that company survive long term? Luck is not on the side of the average company, so if it is internally charging $S_te^{rt}$ minus some benefit based on reduced contract costs, isn't this just putting the extra cost on the firm generally, socialising the cost within the firm?  Doesn't this just put the firm at a disadvantage, at the margin, to the degree that this is an economically significant benefit?  All this assumes that the average company can't produce the factor any cheaper than the spot price $S_t$ implies.

It leaves untouched the general argument for the deal costs - contract creation costs, the costs in general of getting the deal to 'the market'. 

Also with large corporates there's a real internal communication problem.  It is a version of the search cost all over again.  Large companies are a bit like 'the world out there' insofar as finding the right internal market has its own costs. And the larger the firm, the larger these costs.  This would be a drag in general on the size of the firm.

Corporate crack filling

In my recent post, I described a simple model of firms.  I'm now reflecting on what I left out and how to situate the main areas of study of the company.  First of all, there's no concept of spatial location between companies.  Second, there's no concept of what the companies do.  Third, their value is a single number which compresses down to a single point the whole balance sheet and income statement, and, more generally still, the set of internal actions which constitute the life of a company, including representation of its debt and outstanding securities (the behaviour of the company's finance department).  Fourth, there's only one kind of ownership style, whereas in reality there are many.

In general the system doesn't move.  Modelling 120,000,000 companies, I'm going to have to make some simplifications. But that's OK and is done often in analysing the company from an economic point of view.  I briefly explained how it could be initially calibrated so that value is distributed and ownership is distributed according to the Zipf/Pareto power laws we see in our own economies.

Also, just as it can be said that all humans are the final owners of all company value, so too in general with economic value. It would be good to know how much extra-corporate wealth there is and to similarly arrange it.  This at the very least would put a constraint on the scope of purchases.

There also needs to be a random element (a stochastic process) to the growth and contraction of wealth within any one particular company.  Clearly, the singular 'value' of a company is not entirely randomly driven, but is expressed as an algorithm, whose parameters determine the current 'health' of that company.

Adding a spatial constraint will guide which companies to 'interact' with which others.

Everything is expressed in one currency.  Having a spatial element will allow there to be a number of regions with their own currency domains, and with the possibility of calculating macro-economic measures, like balance of trade, foreign investment, currency controls.

The shape and behaviour of a modern company is also hugely a function of the policy and legal framework within which it works, and that itself is enormous and complex.  The availability of credit also drives some of the key parameters around risk taking within companies.

Insofar as the market capitalisation (value) of a company is a function of the equity, then all those options models (Heston etc) might be useful for building an equity-as-call Merton model of the company.

I also have a long-standing idea to model news arrival as an inhomogeneous Poisson process, but that's  all going to have to wait until much later.

The simplest model of the firm

Coase highlights, as did Adam Smith, that the firm makes sense in only in a wider economic environment.  Let me immediately jump to the situation of firms in a wider economic context.  There's a fairly easy, though not initially very useful way to model them.

How many companies are there in the world?  That's clearly a temporally bounded question.  Right now (April 2014) a casual search on the web reveals that there are 120 million companies in the world, of which 45,000 are listed on various exchanges, and, separately, about 65,000 are classed as multi-national.  (I got this from Quora).  Forbes tells us that there are 147 companies which 'control everything'.  While I appreciate the journalistic hyperbole, it isn't going to be too much of a surprise to me that there's a power law distribution in ownership of the companies in the world, and in their size.  This is based on research at the Swiss Federal Institute of Technology, which performed an ownership analysis of 37 million of the hypothesised 120 million and discovered that 147 companies owned 40% of the net value represented in the 37 million.  Apparently 737 companies in their research owned 80% of the value.  

There are N humans and M companies.  Companies can be owned by humans or other companies.  The final owners of all value in all companies are always humans, though in specific cases, the mth company is owned by a number of humans and a number of other companies.  The only constraint at the individual company level is that a company cannot own itself.

A single 32 bit word in a computer can represent over 4 billion distinct things.  This would be more than enough to represent distinct companies.  But let us be generous.  Let us say we are looking to capture the ownership relationship.  With 7 billion people on the planet (a large fraction of whom own nothing), together with our 120 million companies, we'd like more than 4 billion distinct things.  We could just represent each human or company by a 64 bit word, and we'd be good to go for millennia.  If we plan to model it on computers, we might want to economise.  We could do this by assuming that no more than 4 billion people own shares.  This is eminently reasonable for this year.  That's 4 bytes per human/company.  Pew research recently (2013) claimed that 53% of Americans have no stocks, even including retirement account holdings.  This from the world's leading stock owning nation.

Let's embed 120 million companies as a series of 120 million 32 bit words, and worry about the ownership identities later, knowing that our 32 bit word is probably more than capable of capturing this.  That's 457 megabytes.  This can model all the companies in the world.  With 27 Gigabytes of memory I can start to model all the people and all the companies in the world, just in the memory of a computer costing less than £1,500.

What, in addition to its identity, would we like to minimally model with a company?  For me, two things which jump out above all others are who owns it, and how valuable it is.  Or put another way, what the company does isn't being modelled at this stage.  Let me get a handle on this.  Upper limits first.  The current high water mark was Apple, a while back, which topped $463,000,000.  Companies are said to be bust if their net equity is negative, so I could imagine again a 32 bit word could capture this number, just about.  Precise values are not needed.  Nor negative numbers.  So make it integral, giving values from 0 to 4,294,967,295.  If we say the number represents the value in thousands of dollars, this is a fair compromise to getting small values but capturing larger ones.  The maximum value in any one company would top out at 4.2 trillion dollars, which is plenty of headroom above 0.4 trillion.  That's another 457 Mb for the value of each company.  

Finally, the ownership relation.  This has the possibility to be a big one. For example, Apple has by implication of its market cap divided by its share price about 890 million shares of a free float.  Clearly, at worst, this could be distributed to 890 million separate individuals.  Times 120 million companies, worst case, this is quite intractable if your goal is to have it all in RAM.  In a couple of years it ought to require no cleverness and we will be able to model each share as a separate object.  I think it might be useful to own fractions of a share, a share of course already being a fraction of ownership of the value of the company.

In the real world, a company has a register of holders of shares, and from the individual perspective, an owner of a collection of shares has their own list of the things owned and their amounts.  Having both is redundant information, though it might allow faster two way lookup (what companies does this person own and who owns this company).  Also a company can be modelled as a unitary whole, with fractional ownership.  That is, the number of shares per se is not essential.  All companies are fully owned by someone.  Transformations of the ownership unit will be applied if a dilution event occurs.

The company will have a list of N owners, together with their fraction.  Ownership can change.

Later, I think it is possible using techniques for automatic classification to encode the similarity between companies as patterns in the 32 bit word.  This will allow industry modelling.  Finally, it is possible to imply a power law distribution of company value and distribution of ownership using real data.  

The billionaires' t-shirt

A story.  Off the coast of some fantasy country are a thousand tiny islands, owned by a thousand owners.  On each island was a single shop selling t-shirts, each one with a different message: "I came to Tikka Moa and all I got was this lousy t-shirt" or "I came to Tima Takkoa and all I got was this lousy t-shirt".  Each t-shirt costs $1.  The shop owner on each island owns the single ship which takes you from the mainland to the island.  Return tickets vary from $1,000,000 to $10,000,000.  When you arrive, along the path to the shop you are offered a series of increasingly costly admittance tickets and legal proofs each of which entitles you to move on up the path.  You then hand over your pound for the t-shirt and make your way back, again enduring a costly series of checks and duties.

You get to hear about the islands only by virtue of being a member of one of a small number of exclusive members clubs which have high entry fees.  These island owners employ a vast number of quasi-policemen on the mainland who vet and confiscate and pursue anyone who tries to wear one of their t-shirts without the appropriate documentation.

The t-shirt operates as some kind of ridiculous Veblen good.  Well, assuming you take the total cost of the the transaction, and not the sticker price.  The sticker itself is also sold with the t-shirt, at a cost of $100.

At a rough approximation, the cost of the t-shirt is neither here nor there, yet still you need to be a millionaire to pay for all of the ostensible transaction costs.  In short the full price is nothing but the transaction costs.  How would the story change economically if the seller had rolled up everything into the t-shirt cost, making all the so-called transaction costs seemingly 'free'.  Well, not much.  The members' clubs would probably not be so exclusive.  There would be a lot more phony trips.  But a simple 'show me the money' rule could cut most of that out.