US T Bills - an unremittingly fair price

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

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

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

Unbalanced sheet

To understand a subject I need to hang around with it for a long time.  Take balance sheets.  The context: regulatory bodies force companies to expose themselves four times a year, with an annual big exposure.  they impose a partial form and partial meaning on the semantics which go into this exposure.  There are other so-called extraordinary (read: irregular) requirements on a firm's reporting obligation.  But the most detail you will get, and in a time series as regular and as stable as the regulatory framework which underpins this obligation, happens with quarterly reporting.

A company, on 4 equally spaced out dates during the 260 day working year (on about 1.5% of the days) will need to go to the expense of producing the quarterly report.

First up is the balance sheet.  This is an atemporal list of assets and liabilities, with the remainder of assets minus liabilities being called the equity of the company.  The owners of the company (the shareholders, the slave owners) own whatever's left if a sale of all assets happened and all the bills (liabilities) are paid.  Unfortunately, however, the assets are recorded on a balance sheet with value in play at purchase.  That is to say, the assets are marked on a historical cost basis.  Not ideal.  Kind of like reading the ancient price sticker on an item.

 One way to think of accounting per se is a land where sophisticated models are banished.  It is in many ways a discipline which operates on pure reads from documented (and as uncontroversial as possible, provable, unquestioned) numbers.  So in many ways it is a bit like compliance (and indeed it is driven by the same kind of regulatory exposure which goes on in a typical compliance department).  

Take the case of assets.  In general, it is a non-trivial exercise to judge the value of an asset at any moment in time.  Accounting is the discipline which has found a way to side-step this rather difficult question but answers it always with the asset's purchase price.  Likewise the liability is reflected on an at-the-time notional basis.  Remaining interest payments are taken into consideration but on a non discounting basis.  This of how simple a model of the movements of asset prices it is.  The asset has an observed purchase price.  It remains at this value with depreciation applied over the reasonable life of the asset.  The asset itself may or may not generate cash flow too.  Whilst this rather simplistic 'price tag and wither' model is implicitly attached to the asset, real life assets can be driven by a complex market-inspired array of inputs.

Take a company which owns real estate property.  Over the years, the price of a property will rise and fall with market conditions.  But the financial balance sheet statement will only show the purchase price with income and amortisation adjustments.  The income generated by the asset will itself either have gone into funding the asset base or reducing the borrowing of the firm.  Continuing the housing analogy, the market value of the asset is estimated in a role akin to the surveyor.  Accountants by analogy would simply consult the most recent value in recorded in the land registry record and dispense with the need for a surveyor.  Another way to think of the accounting approach to valuing assets is to realise that the approach makes calculating any capital gain or loss easy.

Small business reporting of quarterly statements (or its precursor) can be construed as being all about laying out your taxable obligation.  Indeed, the small business owner gives his accountant the set of transaction dates partly to make sure the firm's tax obligations are,  Historical cost accounting is also as a result in line with the cash flow statements.

Firm emancipation

The firm can be a transient entity.  Rather like a human being in a way.  It can live, it can die, it can thrive and struggle, it can experience ups and downs in cycles.  Novelists tell us that each human is individual, unique, one of a kind.  But sociologists, psychologists and anthropologists see us in our commonalities, examine the motives and cultural pressures which form us.  One approach does not necessarily obviate the other.  So it is with firms.  We all have multiple purposes in our own lives, but with the firm, there is one predominating purpose, which is the creation of value (that is, capital, that is, wealth) for the owners of the firm.  So the firm is in this sense monochrome - the caricature of a slave - a human whose current purpose is in making money for its owner.

How this came to be is itself an interesting part of American corporate history. In short, a rail road company liked the idea that it too, like citizens, could get a tax benefit on the mortgages it had.  To do this, they successfully argued that firms are sufficiently like people that they can avail themselves of individual tax benefits.  It relies on the equal protection clause of the 14th amendment of the US Constitution, which came into law in 1868, a couple of years after Lincoln's anti-slavery victory.  This amendment was designed to solidify the 1866 civil rights act.

The equal protection clause says:

All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside. No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws.

The issue had been that many beaten confederate states, post Lincoln's victory, has enacted state legislation preventing black people from enjoying rights as full as the victors wanted.  The 'equal protection' provision was thus born.  

Roll forward to 1886 and the Southern Pacific Railway Company managed to get companies to be considered under this equal protection clause, hence allowing it to get the tax rebate it wanted.

Think about it.  A company managed to use civil rights legislation to win rights for itself and hence pay fewer taxes.  The 'equal protection' perspective goes like this: the constitution prevents states enacting laws which enable unequal protection of persons.  And persons gets interpreted to mean corporate bodies.  The rail road companies got to keep their taxes.

Was Homo Ecomomicus a former socialist?

Marshall and Mill and the deinition of economic man.  John Neville Keynes' book on "The Scope and Method of Political Economy" - it name checks the economic man, and the model/assumptions debate beautifully clearly.I want to see how far I can make the point that this wealth maximiser had Comtian socialist leanings.

Cournot was also instrumental in defining the mathematical model which underlies economic man

US T-Bills - the bank discount basis quote versus the yield to maturity

US Treasury bills are securities which represent a loan you make to the U.S. government.  There are only two cash flows, the first when you lend the U.S. government the money, and the second, some days, weeks or months later, when they pay you back, hopefully.  They're considered one of the safest assets on the planet, certainly one of the safest assets denominated in the current reserve currency, USD.  They have no internal dates or compounding points within the bounds of the begin and end date. The US Treasury borrows money at all sorts of time maturity.  Bills are loans for a year or less.  Following much older traditions, loans this short have their own way of being quoted - the so called bank discount basis - which bears only a loose relationship to more strict, economically meaningful measures like yield to maturity.

These instruments are created quite regularly.  Different batches can have different face values - 100 USD to 1,000,000 USD.  They are an example of what's known as a discount bond.

Why would you want to lend money to the U.S. government and what do they do with the money?  That's a subject for another posting.  This posting concerns just one thing - the semantics around the quoting of a U.S. Treasury Bill.

You are mostly there in your understanding of the bank discount basis when you realise it was used in the days before computers.  It quotes the simple  (non compounded) interest rate using not the sum invested, but the face value (the value at maturity).  So it isn't an internal rate of return, not a yield to maturity. It has the advantage that face value is usually a nice round number of units of the currency, say 100, 1,000 or 1,000,000.  In the days before computers, these calculations had to be done in peoples' heads, and dividing  12/1,000 (bank discount basis 1.2%)  is so much easier than, say, 12/988 (yield to maturity 1.214574899%).  If face values weren't nice round numbers, then the bank discount basis just wouldn't have been calculated by anyone.  The bank discount basis is the fixed income equivalent of wearing your jumper inside out.  

It isn't just a story of human laziness or a story about our problem with mental arithmetic.  It is a story about finding a way of working which minimises the likelihood of error.  Remember also that back at the beginning of this quoting style (maybe in the money markets of London or Amsterdam) and there  either were no futures and options markets or they were weakly developed, meaning less liquidity, less of a need for granular price moves.  The the coarser the granularity of the ticking market, the less a concern for the difference between 1.2 and 1.21.  

Compounding (anatocism) has a special history in the Western Christian tradition.  If was for long periods considered a form of usury and anti-Christian - due to some poor thinking on money by Aristotle.  That is definitely another posting too, but it reinforces the conviction I have that simple interest and compounded interest have distinct cultural histories.  The full formulation of compounded interest was most clearly spelled out in modern times by Richard Witt in 1613.

I mentioned in an earlier post why when annualising a holding period-yield (as we do with the bank discount basis) it makes sense to use 360 days.  In short, by assuming all 12 months are 30 days long, rather than 364.25/12, with some significant variance, calculations become clearer, simpler to perform.  Mathematical models make simplifying assumptions.  There is probably some variability in the time it takes for the earth to circle the sun, and for the earth to rotate once.  The mathematics of interest makes simplifying assumptions and reflects back to us how differently we chose to morally appraise compounding of interest.