## Monday, 5 December 2011

### Anatomy of a convert - analytics should facilitate comparison

Bonds, equity derivatives, and therefore convertibles have a number of measures of value and risk which can be applied to them at any one moment in time.  Luckily, for bonds and convertibles, there's a handy monetising fixed point called the face value.  Many of the contract terms of bonds are described in terms of this monetiser (i.e. are expressed as a ratio to the face value or denomination).   The coupon is the most obvious contract feature which is expressed as a ratio to the face value.  Originally, the face value of the bond represented the final 'redemption' payment - where the borrower finally pays back the loan (assumed to be the denomination) made to the company.  These days, this is still often the case, though the final redemption payback amount can accrete to some value above or below the face value.  Even here, the final redemption value is often expressed in terms of a ratio to the face value.

The great thing about expressing all these terms with respect to the face value is that the bond analyst is already beginning the task of facilitating comparisons of one bond with another.  This helps the investor to compare many bonds with each other.

From a cash flow point of view, as a bond holder you need to know which absolute cash flows you'll be getting, and when.  From that point of view, you'll receive, to make up an example, £100 once a year, for 19 years, followed by a payment of £1,100 (made up of the final £100 coupon, plus the redemption amount, equal to the face value, of £1,000).

But from the perspective of an investor who would like to compare these amounts with other bonds, it is great to transform them into a series of 19 annual payments of $0.1 \times D$, where $D$ is the denomination, followed by a final payment of $1.1 \times D$.  More convenient still for us to drop the reference to the denomination, and move to percentages rather than ratios, stating that the bond pays a coupon of 10% annualised for 20 years (and, redeems at face value).  Likewise the traded price of this bond, on the open market can be quotes in currency - for example £960 - or it can be quoted as a ratio of the denomination - $0.96 \times D$.  Again, dropping reference to the denomination and moving to percentage terminology, the bond price is quoted as 96.  This quoting style is referred to as percentage of par quoting or simply the par convention.  Certain bond markets prefer the par convention, others prefer to see a real cash amount quoted for the price - this convention is called the unit trading convention; since it is popular in French markets, you'll often hear of it as French style quoting.  Clearly, this is just a quoting style - for every unit trader price, there's a par quoted number which is its equivalent, and vice versa.

In practice, rarely is a bond actually trading precisely at par, even when it is first issued.  They usually start off pricing not too far away from par.  In the end, though, if the bond is still around by the time of redemption, it'll eventually trader closer and closer to par - to the redemption amount.  An instant before it redeems, when all the coupon payments during its entire life have already been made, then you can see that the fair value ought to converge to par, since that's what it represents at that point - a promise to pay the face value to the bearer in the next instant.

The original terminology for face value comes from coinage, where this number would be printed or embossed on one or two of the faces of the coin.  Denomination captures the sense, again probably originally from the world of coinage, that there is a graded set of related values within a singular system.  This makes more sense with money than it does with bonds, since for any given issue, there usually is only one face value.  But from its original latin, you get the sense that it is an amount which is fully named - and perhaps in a sense fully defined by the arbitrary act of naming its value.  Maybe this term caught on in a world where fiat currencies were being born.  The Chinese in the tenth century first issued fiat paper currency, but it was first attempted in the eighteenth century in the western world, and this lines up quite nicely with the word's entomology, where it took on the monetary sense around the 1650s.  Finally, lurking behind the technical definition of face value is the implication that the value expressed on the face is not the real value.  For coins, movement in inflation, or the price of precious metals, or the degree to which the coin's edge has been clipped, all explain why the face value is not the same as the real price.  Likewise this carries over well into bond terminology where, as I noted earlier, during most of the life of a bond, it won't be worth precisely the face value.