## Saturday, 4 July 2015

### Savings accounts are soil

The term yield has its origin in gieldan, the Germanic word meaning to repay.  It captures not the percentage element - namely that this measure is described relative to some other reference quantity - but to the fact that a repayment is happening.  So immediately chronology is of relevance.  The repayment amount is expressed as a fraction of some reference entity.  Hence the yield being the additional fraction of the base quantity that constitutes the repayment or reward.  It has clear agricultural meaning too - the yield being a measure of the produce which some quantity of land produces in a period of time.  Notice here that the agricultural yield doesn't naturally get expressed as a percentage.  Whilst it is clear that if you own 5 fields you may expect a correspondingly higher yield of crop, you don't really relate the number of carrots to some reference number of carrots.  Rather, you express your yield as a number of carrots per year, perhaps per field too.  If you imagine that there is a cost to having purchased and farmed that field, and that the crop of carrots likewise have a monetary value, then you can express the yield as a cash amount over a cash amount.  Here the yield is a percentage which captures the move from the beginning of the reference period, with your initial investment in the field to the end of the period, e.g. one year later (agriculture is is clearly a cyclical business, so years are one of the commonly used units of repeating time).  A field costs $P$ and then, a year later, you own some more carrots which are worth $y \times P$.  So in total you are now worth $P + y \times P$.  Assumed, of course, that the field's value has neither gone up nor gone down.  That is, $P(1 + y)$ .  Money itself is like a field.  The essence of the theory of the value of money at different points in the near future is that it can have a yield.  Money naturally makes money.  Money's crop is money.  The assumed easy way to do this is to put it in a savings account.  A savings account is soil.  Implicit soil.  The value of one unit of currency in one year's time is $1+y$, where $y$ in this case is the return you'd expect to get on a savings account in that country, for that sum, in that year.

The simplest financial yield to calculate is one where there are two logical observation points - namely the begin date and the end date.  No other financial action happens other than an implicit purchase at open and a reckoning or settling up at the end of the observation period.  With on intervening cash flows, there is no possibility of compounding.  So the simple interest formula can be used. Whatever happens inside those dates is assumed by hypothesis to happen uniformly or homogeneously.  Check out an earlier posting on the inter-relatedness of the discrete compounding formula and continuous and simple interest.  The difference between the simplest $P(1+y)$ above and the simple interest formula $(1+rt)$ can be explained by the fact that in many circumstances the rate or yield $r$ is expressed on a one year basis.  If you're happy to express the rate as a holding period rate, then $y$ is the same thing as $rt$.

Don't forget, in all discussion of the terms of a loan or description of a return, there are four elements which get applied to the principal or starting value, $P$, and they are {$r_{b_{{u}_1}}$,$t_{u_{2}}$,$n$} where r separately has an implicit time basis (usually a rate described in year units, and often expressing a nominal period of 1 year also, but also, especially when calculating return on an investment $b$ is what's known as a holding period yield); $u_2$ being the units in which your time (of loan or of investment) is measured - this is usually in days or years (or months), and $t_{u_{2}}$ then being the holding period or loan period itself.  Finally, $n$ is a unit-less counter which simply adds regular internal structure to the period under investigation, by adding moments  where an earning is recognised, meaning it officially counts as part of your capital from that moment on.  $n$ makes sense in the holding period window and is assumed to run from as small a value as $\frac{1}{t}$ all the way through to $\infty$.

So, when $u_1 = u_2$ namely that the rate basis and the holding period basis is "1.0 means one calendar year" and when $b=t$, namely that the expressed rate the scope of which is identical to the holding period itself (expressed as a number of years) and finally when $n=\frac{1}{t}$ then you have the simplest kind of yield, namely the simple interest one period yield.

If we call the final value $F$ then the equation $F=P(1+y)$ allows you to see clearly , after some re-arrangement, that $F-P=Py$, so $y=\frac{F-P}{P}$.  That is to say, you could express a holding period yield on a holding period ($b=t$) basis and simple interest basis ($n=\frac{1}{t}$) as the money made ($F-P$) expressed as a fraction of the money invested, $P$.

To spell this out a bit.  All you need is a starting value and an end value after a known period of time, and you can begin to enter the world of yields.  Once you have a holding period simple interest yield, you can set about translating this yield into the many other kinds of yield there are.

### Why are bond yields usually less than 10% in the developed capital markets?

What is a yield?  It is a percentage.  It is a relative measure.  What do financiers see or think when they see bond yields?  They compare those with others.  What's the meaning of a 5% coupon?  A 5% return on investment?  And why don't bonds pay, say 100% during normal times?  What kind of world would it have to be for that to be a regular occurrence?  The borrower when a business is more often than not borrowing to finance a line of business (or re-financing a previously financed line of business).  If they decided to borrow on those terms, then for the line of business to make a profit, then the expected annual profitability needs to be somewhere north of 100%.  Businesses simply don't mature that rapidly, nor do they remain so profitably too long before competitors arrive in to bring down average profitability for the sector participants.  So capitalism itself, and the natural difficulty business managers have in finding investible lines of business mean that seeking funding in the first place would not be worth it if the lender expects 100% return or anywhere near that.

Yields on bonds are nominal, in the sense of being unadjusted for inflation.  Loans promise to pay back a future fixed cash sum.  An intervening period of high inflation, for example, will erode the real value of the cash flows, including the final payment.

So I ask again, what is a bond yield?  It is made up of a risk free part, a part shared by the population of peer bonds, and a more firm specific part, which measures the market's judgement on how risky this firm (or sector) is.  A macro-economic context and a firm-specific context.  You get, say, 7% for this bond because the prevailing economic climate dictates that even the least risky lender gets, say, 4% and you receive an additional 3% because of the market's judgement on how confident it feels that the firm will honour its debt obligations.  In the jargon of finance, the yield has a risk free component and a credit component.