Monday, 15 June 2015

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.

No comments:

Post a Comment