Note on Drafting Mathematical Formulas
Since business is usually about numbers, most business documents contain mathematical formulas of one kind or another. Not surprisingly, formulas are often the most critical part of the document and carry the gravest consequences if misinterpreted or incorrectly drafted.
How should mathematical formulas be drafted to avoid misinterpretation and error? For simple formulas, writing the formula out in words within the sentence suffices:
Examples:
- “The Temporary Ratio is equal to three-fourths the Fixed Ratio.”
- “The Disbursement Amount will be multiplied by the sum of the Base Percentage and the Supplemental Percentage.”
More complicated formulas quickly become very unwieldy. The example below is from a real stock warrant and is a common formulation of a “net exercise” provision. (Net exercise is the term given to a provision that allows one to receive fewer shares upon exercise of a warrant rather than paying the exercise price in cash.)
Upon exercise of the Net Exercise Right, the Company shall deliver to the holder of this Warrant, that number of shares of Common Stock equal to the quotient obtained by dividing: (x) the difference between (i) the product of (A) the Current Market Price of a share of Common Stock multiplied by (B) the number of Converted Warrant Shares and (ii) the product of (A) the Exercise Price multiplied by (B) the number of the Converted Warrant Shares by (y) the Current Market Price of a share of Common Stock on the Conversion Date.
The relative simplicity of the above formula is completely obscured in the language. Most people would not be able to calculate the formula without writing it out first in typical mathematical notation. So why not do it for them in the first place? Most people would find the formula quicker to read, not to mention easier to understand. Mistakes in transcription are avoided, and time is saved. So let’s rewrite it in the standard style used in academic writing:
Upon exercise of the Net Exercise Right, the number of common shares to be received, CS, is determined by the following formula:
Where:
W = the number of Warrant Shares converted;
MP = the Market Price of the Common Stock at the time of conversion; and
EP = the Exercise Price.
When drafting mathematical formulas, don’t be afraid to do a little rearranging of terms. Often a different arrangement will make it easier for readers to intuit the parts, thereby making it easier to understand the whole.
The net exercise equation can be rewritten as:
CS = W
The term,, is simply a term that reflects a percentage change (expressed in decimals) in two prices -- the Market Price and the Exercise Price.
The above equation simply says: to determine the shares you will receive upon net exercise, multiply the number of shares that you would have otherwise received by the percentage difference in the Market Price and the Exercise Price. When written this way, the net exercise formula can be rewritten in a sentence that may be followed without too much difficulty. Consider:
If all or a portion of the Warrant is exercised on a cashless basis pursuant to this subsection, the number of Common Shares received at conversion will be determined by multiplying the number of shares otherwise issuable on conversion of the Warrant by the percentage difference between the Market Price and the Exercise Price. The percentage difference between the Market Price and the Exercise Price is calculated by subtracting the Exercise Price from the Current Market Price and dividing the result by the Current Market Price.
I think this language is easier to follow than the original provision but I would probably use the mathematical formula if I were drafting the warrant. I find formulas quicker to grasp and less subject to misinterpretation.