crescet and titubit

The speed with which one's wealth grows, and its absolute level, are tied to one's life style (one's consumption of one's income).  A useful simplification is to assume one's income derives largely from one's wealth.  Economically, this is almost completely unreasonable, since it applies only to a vanishingly small fraction of humanity.  One then needs to spend to live from this wealth.  There are however minimal quality of life spends which may imply several modalities in the relation between the wealth growth process and the spend process.  I assume for simplicity that wealth is sufficiently large that the income spent can be made in a way which still leaves wealth growing.  Put another way, there is an assumption that the wealth process grows faster than both inflation and the daily consumption of your lifestyle.   A second critical threshold is for now also ignored - as with the case where the lifestyle spend significantly impacts the wealth process, transaction costs also can incur a third hurdle to overcome.  These assumptions clear away much of the thrust of the Darst book on asset allocation.

Next, an implicit starting assumption is that wealth at time $t$ may be considered as residing in one or more currency (short term fixed income) buckets.  One then imagines that the mean value theorem can be applied to the act of taking financial risk above this risk free (globalist) position.  That is to say, in equilibrium, the entirety of the job of strategy allocation and capital deployment can be waved away as solved for now, and modelled as a single 'bet' over an appropriate time frame, whose outcome can be a win or a loss.  One then determines the ideal bet size, per unit of time, based on the mathematics of Gambler's Ruin.  That is to say, that the average bet size can be no bigger than some fraction $\delta$ of wealth at point $t$ if volatility (and long term, ruin) is to be avoided.

Of course, in reality, the complete opposite applies with titubit.  Bet sizing is often ignored and instead one's lifestyle generates the major driving constraint to investment returns variance tolerance.  In short, our lack of funds makes us bet too big - this together with transaction costs, destroys our wealth.

What just happened?

To price a convertible, the queen of the capital structure from a pricing point of view, you need to know about volatility, credit, rates.  Indeed, to price volatility you need rates.  And likewise with credit.  So it all points back to rates.  Which is why I decided to start with looking at simple rates products - certificates of deposit, then on to treasuries.  But while looking at CDs, I decided to strip it down to the basic question 'what is a rate?'

Given that the maths associated with rates is uncontroversial, and has been around for centuries, I guess most people don't spend a long time here.  But I want to.  I want to feel what a rate means.  So I invented the 'a man walks into a bar' story to give me space to spell out the elements.  I also have in my mind a slow-camera zoom of rates contexts, starting with macro-economic, then markets, then contractual, then participant based.

The biggest macro-economic variable which provides a context to rates is the likely inflation during the life of the loan.  I'd now like to summarise this in a simplistic mathematical way.

That unpredictable stick of dynamite under every loan

In a previous post I mentioned that knowing the rate offered on a loan by a lender to a borrower isn't enough to know what the lender's fee is.  This is, of course, usually the main element you need to know to decide whether the lend is an attractive proposition for you as a potential lender (and borrower too).  But that's only because inflation is usually well understood.  It has happened often during the last several hundred years  to be well understood and also small in magnitude.  Strictly speaking, it doesn't need to be small, just well understood.  That is to say, predictable.  If inflation was at a rock solid 10% per annum, pretty soon peoples' uncertainty around how to deal with it would be reduced.  The public would factor in this certain knowledge into their future plans.  If inflation was totally unpredictable, but only within a narrow range, say 9.8% to 10.2%, then this pretty much is the same thing.  But if the unpredictability is accompanied by a wide range of values, say from -20% per annum to over 40 quintillion % per month (as it did in Hungary after the second world war), then that's clearly a different matter.  If this is a genuine possibility then it can render trivial all outstanding debt in the hyper-inflated currency.  The above link shows that even major world economies can fall into monthly inflation of over 20,000% per month, as did Germany after Lloyd George and the French lumbered Germany with unpayable levels of war reparation debt after the first world war.  In the Hungarian case, people weren't paying for bread in wheelbarrows full of marks, as in Germany, rather road sweepers would sweep smaller denominations up from the ground where they were discarded.

Inflations of all kinds, from uncontrolled hyper-inflations to strategic and temporary ones, in effect diminish the real value of a borrower's loan.  From the point of view of the lender, they effectively lose their capital.  It  has this effect across the board, since virtually all debt is still contracted in nominal terms.  Another way to view it is that inflations re-distribute real value from lenders to borrowers.  Under those circumstances, the fact that the original contract agreed to pay the lender some rate $r$ pales into insignificance.  So if someone in Hungary after the war agrees to lend 100 pengo to a borrower for a rate $r$ for a year, this was free money for the borrower, since the present value of $100(1+r)$ in a year's time is a very small number of pengos (assuming you had a decent idea what the real discount rate was).  So the lender handed over 100 now for a contractual promise which was worth a tiny fraction of that value now.  The lender would not even bother collecting the payments since the process of trying to collect it would cost thousands of times more than the value.

What's in a rate?

Have you ever looked inside a rate?  In this post I'll try to do so, giving some ideas about how you might slice up a rate in terms of economic forces or financial forces in markets.    Remember, first and foremost a rate is just a fraction of some reference amount.  In other words it describes a unit or quantity in terms of its size relative to some reference quantity.

Let's talk money.  A money rate describes some amount of money with respect to some other reference amount of money.  In the vast majority of cases in fixed income finance, the reference amount of money represents either a starting amount or a closing amount.  Usually the rate summarises some kind of financial promise you're involved in or it represents a post hoc analysis of some investment you made in a security or portfolio of securities.

These rates are also known as returns, yields, and interest.  Return is a nice expression, conjuring up an image of the return of invested capital, with some extra capital too.  Yield is quite an agricultural sounding variant - think of it as expressing the size of a crop with respect to the size of the field.  Interest (interesse) was originally a late payment penalty built into the contracts for loans, which then morphed into contract structures where failure was built in.  This allowed the contracts to side-step Christian usury laws.  Muslims perform a similar piece of arithmetical/contractual engineering in their dealings with returns.

The clearest security with a return is probably a loan by a lender to a borrower for a fixed term, with no intervening days of reckoning, where accumulated interest is rolled into the current capital embedded in the security (compounding).  That is, in cases with just two days of reckoning - at the start day and on the last day - the day of termination of the loan.  The slightly more general case is where the rate as expressed fits into a compound growth formula, as described in another posting.  This implies all such rates must have associated with them implicitly recipes for how they are used.  These recipes are called the rate's time basis, compounding frequency, day count convention.  They flesh out how to operate with the rate.  As noted, again in a previous post, they're all fully interchangeable, so we shouldn't look here to find out what's inside a rate.

 In the next posting I'll explain why I think it is good to categorise of the constituents of a rate as follows: inflation, the market, participant profiles, this deal.  I think of this as a slow zoom camera, first of all, picking up macroeconomic effects, then market (and close-market) effects, then evaluating the states and preferences of the contract participants, before finally looking at the terms of the contract.