Ah, convertible bonds. On the trading floor, a converts-trading colleague had a Post-It note on his monitor:

"Interest rates up: converts down

Interest rates down: converts down

Volatility up: converts down

Volatility down: converts down

Apple strudels up: converts down

et cetera"

Converts are complicated. For one, nobody is fighting for you post-issuance. You don't (yet) hold stock, so the Board doesn't think you're pretty. Yet your bonds will be subordinated, making them swim like equity. This leads to all kinds of fun [2] when markets fail to maintain monotonicity.

Monotonicity means if a line is going up it keeps going up; never down (and vice versa) [3]. A graph for the Empire State Building with the floor number on the X axis and the height of said floor on the Y axis would be monotonically increasing. If I bent the building into a U shape, that graph would not be.

If you graph pay-offs for people in your company as a function of the stock price, you want it to be monotonic. That means interest are aligned. If your CFO makes a million dollars when the stock tanks, he's going to want the stock to tank.

Let's consider HappyCo. HappyCo has issued 10,000 shares of Common Stock. They trade at $100 per share (a $1MM market capitalization [4]). HappyCo issues 1,000 convertible bonds that can be turned into one share of Common Stock in exchange for $120 per share. To keep this easy for now, let's say HappyCo issue these converts secretly, i.e. the market can't price them in before exercise.

If the stock price is way above 130 everyone wins. If the stock price is below 130, converts lose. (Stockholders also lose.)

But what if the price is exactly 130? The market, thinking there are 10,000 shares outstanding, weighs the company in at a $1.31MM market cap. But then--dun dun dun--everyone converts. Holy shit, 1,000 new shares! At what price does this new share count (11,000) yield a $1.31MM market cap? 119.09 per share. At 129 pre-conversion, the $1.29MM company trades at 129 per share after accounting for the converts (nobody converts). At 130 pre-conversion it trades at 118.18 post-conversion. It takes us until 141.90 pre-conversion to get back to 129 post-conversion. The price goes monotonically up, but the stockholders do okay, then worse, then better again. Monotonicity was broken. An evil shareholder learning of the converts might prefer a pre-conversion price of 129 over 140.

This may seem silly. Nobody secretly issues stock [5]. The market would start pricing the converts in as the stock price approached the conversion price. How does it do that?

Options! A convert is a bond married to an option. (People once made a lot of money arbitraging converts against the issuer's stock, bond and options markets [6]). You get all the legal complexity of bonds [7] entangled with the mathematical complexity of options [8]. As I said, fun [2].

But bankers would just price things properly at the outset to ensure they maintain monotonicity, right? Well, they try to. But look at the variables in a common option valuation equation [9]. Rates, volatility, dividends, et cetera. Each of these changes the balance for converts holders. (For example, if dividends go up one might want to convert sooner, i.e. at a lower price. What if taxes go up at the same time? Who knows! Fun [2]!)

In a perfect world, each variable would iterate, one by one, like an Excel spreadsheet that isn't complaining about circularity. Investors would, one by one, plug new numbers into their models to get a clean answer. Unfortunately, we inhabit a reality where more than two aspects of it can change simultaneously.

Upshot: converts commonly ram through their boundaries all the time. When this happens, everyone converts. Actually, not everyone since each investor has a different break-even conversion price. An insurance company, with lower borrowing costs, will convert differently than an individual investor on margin. Similarly, an invest trading out of a tax-deferred IRA will convert differently than one trading straight. Fun [2]!

TL; DR Converts a lots of fun [2] for financial theoreticians, fun for market makers and hedge funds, a little less fun for bankers and an affordable source of entertainment for all.

[2] http://dwarffortresswiki.org/index.php/DF2012:Losing

[3] https://en.wikipedia.org/wiki/Monotonic_function

[4] http://www.investopedia.com/terms/m/marketcapitalization.asp

[5] http://sharesleuth.com/investigations/2012/12/small-companie...

[6] http://www.institutionalinvestor.com/article/1027772/boy-won...

[7] http://www.treasurer.ca.gov/cdiac/debtpubs/handbook.pdf

[8] https://en.wikipedia.org/wiki/Black–Scholes_model

[9] https://en.wikipedia.org/wiki/Black–Scholes_model#Black.E2.8...