## Thursday, 9 July 2015

### Some yields are meaningful, some aren't, but they can all be used, and serve a purpose

Finally I'm beginning to see a few commonalities in the subject of yields.  First, think of all yields as either economically/financially meaningful, or not quite meaningful.  I suggested historical reasons why these non-financially meaningful yields exist.

Second, think of simple interest as merely compounding with a frequency of once per holding period.  In that sense, all rates have some kind of time basis, and a compounding frequency.  Separately, those rates can be applied to periods of time different again from the time basis.  The simplest analysis is when the compounding frequency is once per holding period, and the holding period is the same length of time as the usual rate time basis, namely a year.  When the holding period is not a year, then there's always a question of how to convert.   At its most stripped down, the time basis is usually year based and/or day based, with an optional additional month counting algorithm.

Third, there are a number of markets out there with their own quoting conventions (some of which are financially meaningful, some not), so there's a need to use yield translators to facilitate easy comparison across disparate market quoting conventions.

Take the case of the humble six month Treasury bill, with price now $P$ and redemption or face value $F$ and six months left to run, $t=0.5$ or $d=180$.  That market likes to repackage the price as the bank discount basis, being $(F-P)/F \times 360/d$.  This is clearly a non meaningful yield. Notice that the second part of the expression is an attempt at annualising the holding period.

It is possible to make it comparable to another, somewhat more meaningful yield, the money market equivalent yield, as might be seen with certificates of deposit.  This yield would be $(F-P)/P \times 360/d$.  Two changes make this much more economically meaningful.  First, the denominator is your initial investment, and second, the annualising factor, makes a more realistic assumption about the regularity of years and (implicitly)  months.  In doing such comparisons, you can compare the return more fairly with money market instruments. Ignoring the annualising factors for a moment, you can see that the money market equivalent yield is basically saying that $r = F/P -P/P$ that is, $r = F/P -1$ so that $F/P = 1+r$.  The rate in question is a holding period rate, and we chose one compounding period.  Ie $(1+r)^1 = F/P$; alternatively $P=F/(1+r)^1$.  So we have taken the heart of the money market yield equation and walked it around to look just like the formula for a zero coupon bond.  If we then go the final step and say that the holding period rate $r$ can be expressed in any number $n$ of notional compounding periods within the holding term, then $P=F/(1+r)^n$, which actually is the formula for pricing a zero coupon bond.  If $r$ is understood to be an annualised basis rate, you get to $P=F/(1+\frac{r}{n})^{nt}$.  In the world of bonds, $n=2$ is a hugely significant compounding frequency - namely twice a year.  If I have a rate (e.g. $r=0.2$ for a holding period, say half a year, to find out the equivalent annualised rate $r_e$ I'd use this formula: $r_e = (1+r)^2 -1$, i.e. 0.44.  But there's another of these non-financially meaningful market conventions which says that the equivalent annualised rate is $2 \times 0.2 = 0.4$.  This approach is called the bond equivalent annualised yield method and its main advantage is ease of calculation - you're pretending that this rate linearly scaled up (or down) to a 1 year basis; that is, you're ignoring compounding.