## Wednesday, 8 February 2012

### The archaic discount yield and the truer bond equivalent yield

In the world of fixed income there are lots of different kinds of yield.  It can, and does, get confusing.  I'm going to start simply and take it slowly.  First, imagine a simple interest calculation based on two cash values at two different points in time, $C_{t_1}$ and $C_{t_2}$.  This represents a return of $\frac{C_{t_2}-C_{t_1}}{C_{t_1}}$ for that time period $t_2-t_1$.

Now instead of considering these two points as equally important, lets emphasise one, then the other.  In other words, let us leave the realm of mathematics, which doesn't care too much about what these two points in time actually mean to us. First, imagine $C_{t_1}$ is a sum to be invested, your starting capital, as it were. Then we're likely to say we gain $C_{t_2}-C_{t_1}$ (or lose it if negative).  This is what you might call the common-or-garden interpretation of a return: simple interest (no intervening compounding dates internal to the end point dates $t_1$ and $t_2$, an asset starts with value $C_{t_1}$, and over period $t_2-t_1$ it grows to be worth $C_{t_2}$.  This is what the person on the street usually means by a return.
Next switch focus to $C_{t_2}$.  From the perspective of this point, then we might call  $C_{t_2}-C_{t_1}$    the discount we would get off the full price $C_{t_2}$ if we were to have owned it at $t_1$.  The rate $\frac{C_{t_2}-C_{t_1}}{C_{t_1}}$  is called the bond equivalent yield.

Bond equivalent yields are often annualised - to facilitate comparisons.  You can do that in one of many ways, but in all cases, what you're trying to do is find out what the $t_2-t_1$ time period rate would mean if you could continue it for a whole year.  These different ways of scaling rates along the time axis are called day count conventions.

The history of lending naturally segments time into day sized chunks.  In the olden days, there would have been little practical point in getting any more fine grained than a day (except perhaps for periods of hyper-inflation).  And of course, human culture is permeated by the seasonality of the whole year.  So often you're flitting between a days-level view and a year-level view.  By many human calendars, there are also months as in-between  time periods.  So you'll find some day count conventions taking the month time period into consideration too.  For quite some centuries, loans have been made on a multi-year basis, on a month-by-month basis and even, for large amounts to large borrowers, perhaps even on a day by day basis (today's day by day lending capital markets have many overnight lending activities).  It almost always comes down to counting days - either directly, or in assumed 30 day or 360 day or 365 day chunks.

The time unit we typically all settle on to facilitate comparison is the good old fashioned calendar year.  And the most natural of all the ways of scale a $t_2-t_1$ time period rate into a corresponding annualised rate is to multiply by $\frac{365}{t_2-t_1}$.  This is natural in the sense that it quite closely corresponds to reality, since we usually have about 365 days in a year.  Even if you're comparing two loans which only last a few weeks apiece (say, a 2 week and a 3 week), you'd still look to annualise them, based on an agreed or common day count convention.

A second major alternative is to multiply the  $t_2-t_1$ time period rate by $\frac{360}{t_2-t_1}$.  Any why 360?  Well, this is an example of the influence of the month on interest calculations.  Even though there are 12 months in our calendar year, they're quite different in length, ranging at worst from 28 to 31 days.  That's a 10% difference right there.  Lenders don't like this complicated variability so they invented the concept of the 30 day month.  A notional period of time which has the desired advantage of all 12 of these 30 day months being equally 30 days long.  So formulae could be developed which treated any month as the same as any other.  But $12 \times 30 = 360$.  Hence the need, in some markets, for the day count convention which scales the $t_2-t_1$ time period rate to 360 days.  What you're doing is seeing what the corresponding rate would look like for 12 equi-length pseudo-months.

So the two major forms of annualised bond equivalent yield are  $\frac{C_{t_2}-C_{t_1}}{C_{t_1}} \times \frac{365}{t_2-t_1}$ and $\frac{C_{t_2}-C_{t_1}}{C_{t_1}} \times \frac{360}{t_2-t_1}$.  Remember, the fixed income market has been around for a long time, and the shortest time period is the day.  Continuous compounding came much later.  So why don't I just replace time period $t_2-t_1$ with $d$ days.  And, to make the terminology slightly more bond-familiar, lets call $C_{t_2}$ the final or face value $F$ and $C_{t_1}$ the initial price paid, $P$, $F-P$ being my discount.

In other words,  two important varieties of annualised bond equivalent yield are $\frac{F-P}{P} \times \frac{365}{d}$ and $\frac{F-P}{P} \times \frac{360}{d}$.  The measure closest to the true economic return is clearly the 365 day based one.

So what about the discount yield?  Well, it is an inferior measure, dating back to a time when people were doing a lot of these calculations per day, by hand.  Replace the economic return $\frac{F-P}{P}$ at the heart of the bond equivalent yield with a more convenient denominator, but one which makes less economic sense: $\frac{F-P}{F}$.  Usually $F$ is a face value, for example 100 USD.  Clearly dividing by a face value is so much more convenient than dividing by, say 98.34, a current price.  The discount yield, too, could be annualised, and again you could annualise to a real year or to a fake, homogeneous $12 \times 30 = 360$ year.  In the world of short term fixed income, with simple interest, it turns out that the 360 day based annualisation was preferred, certainly in the US market.  This means that the annualised discount yield is often expressed as the formula  $\frac{F-P}{F} \times \frac{360}{d}$.

Once you know the price, the face value and the term, you can calculate either yield directly.  Notice that the discount yield will always result in a smaller yield than the economic return of the bond, since the price of these zero coupon or discount bonds is (almost) always less than the face value.

But why bother with the clearly inferior annualised discount yield when you can have a 365 day based economic return calculation, the annualised bond equivalent yield?  Well, because markets have a history which can't easily be eradicated.  Once certain markets started producing quotes in annualised (360) discount yields, there was no going back.  The convention of quoting a rate in a 360 day annualised discount yield basis pops up in many markets, but none so large and so important as the US Treasury bill market.  But before going on to look at the T bill market, there's one more wrinkle in the bond equivalent yield which needs to be laid out.  That'll be the subject of my next blog post.