ABA Section of Business Law
May/June 2001 (Volume 10, Number 5)
Legal-Ease
Fun - and grief - with algebra
By Howard Darmstadter
A couple columns ago, I argued that legal documents would be much
clearer if numerical manipulations were presented in the standard
arithmetical notation we learned in childhood. (Since I wrote that
column, I've discovered that the Revised Model Simplified Indenture,
which appeared in the May 2000 Business Lawyer, also favors algebraic
equations over words. Way to go!)
At the end of my column, however, I warned that arithmetical notation can be drafted as poorly as any other provision, and I gave a little example. For this column, I'd like to expand that thought by presenting an example from a recent prospectus for medium term notes. Some of those notes are issued at a "commercial paper rate," plus or minus a spread. The prospectus defines the commercial paper rate for a note as the money market yield for that note, which it in turn defines as (stand back, folks):
D x 360
360 - (D x M)
where "D" is the per annum rate for commercial paper quoted on a bank discount basis (expressed as a decimal), and "M" is the number of days in the period for which interest is being calculated.
Well, that's unenlightening. The formula isn't very complicated,
and the term "money market yield" and the formula come
straight out of a standard text, but that doesn't make it any easier
to understand. Consider the numerator of the fraction: Even if you
know what the "bank discount rate" is, what do you get
when you multiply it by 360? A number, no doubt, but not a meaningful
one - it's not a price, or a discount, or anything else with a natural
meaning. It's just a number that inexplicably appears in a formula.
Nor is the denominator, 360 - (D x M), any clearer. As we'll see,
360 is the number of days in the short year adopted for lots of
money market instruments. But why are we deducting D x M from that
number? The result is not going to be a number of days, or anything
else that is familiar.
This kind of thing could give algebra a bad name.
Let's try to straighten this out, beginning with a little introduction to money market terminology.
Suppose Irma Investor buys a Prestodigimatics Inc. note for $1,000. A year hence, Presto pays Irma $1,000 along with $60 interest. Then Irma's investment has an annual yield of
$60
$1,000
which is 0.06. To change the decimal yield to a percentage, we multiply by 100:
$60 x 100
$1,000
which gives 6 percent.
The example is simple, not least because I assumed that Irma received her $60 in interest after exactly one year. Suppose, however, that Irma received her $60 in interest after only 300 days. What is the annual yield on her investment?
To calculate Irma's annual yield, we first calculate her daily yield by dividing the 6 percent yield by 300 days - the term of her investment - and then multiplying by 365, the number of days in the year. The formula looks like:
$60 x 100 x 365 days
$1,000 300 days
which gives an annual yield of 7.3 percent.
Generalizing, we can say that annual yield is
$ interest received x days in year x 100
$ investment term of instrument
As is well known, our unruly planet makes its annual circuit of the sun in a bit more than 365 days, with the result that every fourth year has to have 366 days rather than 365. To smooth out this awkwardness, and to simplify some other calculations, the yield on many instruments is calculated on a year of 360 days. (Today, when our children are born with silver HP-12s clutched in their little fists, it may be difficult to appreciate that there was a time when the calculations were done by hand, usually with the aid of elaborate tables. In our computerized age, the 360-day year remains, perhaps because it gives lenders a bit of extra interest - sort of like the quaint British pre-decimal practice of quoting prices in guineas rather than pounds.)
If we calculate yield based on a year of 360 days, our annual yield
formula becomes
$ interest received x 360 x 100
$ investment term of instrument
Annual yield on an investment over a year of 360 days is referred
to in the markets as money market yield.
Commercial paper and certain other short-term instruments - most
notably, Treasury bills - do not pay interest in the conventional
sense. If Irma buys a $1,000 face amount commercial paper note,
she will only receive the face amount - $1,000 - at maturity. No
interest. For Irma to make a profit, she must pay less than $1,000
for the note.
Suppose Irma pays $980 for a $1,000 note that will mature in 90 days. Then the $20 difference between the price Irma pays and the $1,000 face amount of the note that she will receive at maturity is the discount. These types of instruments are usually referred to as discount paper.
Our money market yield formula also works for discount paper. We just have to realize that the discount - in our example, the $20 difference between the price Irma pays and the amount she receives at maturity - is the interest, while the $980 Irma pays is the investment. Plugging the numbers in to our formula gives
$20 x 365 x 100
$980 90
which (trust me) works out to a money market yield on Irma's commercial paper note of about 8.163 percent.
We'd be finished now, except that commercial paper prices are seldom quoted in terms of dollar discounts. Instead, they are quoted in terms of annual discount rates. For example, Irma's $20 discount on a $1,000 face amount commercial paper note is obviously a 2 percent discount rate (nonannualized). To get the annual discount rate, we divide by 90 (to get the daily discount rate), and then multiply by 360. This gives
2% x 360
90
or 8 percent. The annual discount rate for a 360-day year is the bank discount rate. (The bank discount rate is always lower than the money market yield because the former divides the discount by the face amount, while the latter divides by the face amount less the discount.)
Our problem now is how to convert the bank discount rate (in our example, 8 percent) into the money market yield (in our example, 8.163 percent).
No problem. Since we computed the bank discount rate from the dollar discount and the investment, all we have to do is reverse the process. In particular,
$ discount = $ face amount x bank discount rate x term of instrument
360
All that is left now to derive the formula in the prospectus are a few algebraic manipulations. The details aren't too messy, but neither are they interesting; if you must see it all spelled out, the sidebar will let you in on the secret.
My point, however, is that things are clearer if you don't do the manipulations. The only reason for the sidebar is to persuade you (and myself) that our earlier money market yield formula works. For prospectus purposes, we might be better advised to simply say that:
Annual yield (over a year of 360 days) for a commercial paper instrument
("money market yield") may be computed by the formula
$ discount x 360 x 100
$ face amount of instrument - $ discount remaining term of instrument
where the $ discount is computed by the formula
$ face amount of instrument x bank discount rate x remaining term
of instrument
360
and the bank discount rate is the percentage discount on the instrument, annualized over a year of 360 days.
This isn't a formula found in finance texts, and it takes longer to punch into a calculator, but it is easy to understand. For one thing, it explains "bank discount rate." More important, it doesn't group items in ways that are inscrutable; numerators and denominators correspond to well-understood concepts.
The messy details
The money market yield formula is
$ interest received x 360 x 100
$ investment term of instrument
and the dollar discount formula is
$ discount = $ face amount x bank discount rate x term of instrument
360
Since we're interested in the money market yield, which is a rate, not a dollar amount, we can compute the dollar discount on a standard face amount of $1 - there are obvious simplifications in multiplying or dividing by one. Substituting our $ discount formula for "$ interest received" in our market yield formula, with a $1 face amount, and replacing "$ investment" with the face amount ($1) minus the formula for dollar discount, gives
$1 x bank discount rate x term of instrument x 360 x 100
360
$1 - ( $1 x bank discount rate x terms of instrument ) term of instrument
360
Canceling the "$" signs and the extraneous multiplications
by one, and substituting "D" for "bank discount rate"
and "M" (think "Maturity") for the term of the
instrument gives
D x M x 360 x 100
360
$1 - ( D x M ) M
360
Multiplying the two fractions gives
D x M x 360 x 100
360
( $1 - ( D x M ) ) x M
360
Canceling the "360"s and the "M"s gives
D x 100
$1 - ( D x M )
360
Now multiply the numerator and denominator by 360 to get
D x 360 x 100
360 - ( D x M )
which is the textbook and prospectus formula.
