Here's a weird chart:

Herbalife was up more than 5 percent from the previous close in the morning, on decent earnings and news of a big stock buyback financed by a $1 billion convertible bond offering. It then faded and was down as much as 3.5 percent, as the market sold off and as people started feeling a bit more tepid about the buyback. Then it rallied in the last hour of trading, closing up 7.2 percent. It's down about 3 percent so far today. Why?

I will tell you my hypothesis, though it is just that, a hypothesis. It starts with the fact that Herbalife bought back $686 million worth of stock yesterday. Remember, as we discussed yesterday, that it bought that stock in the form of forward transactions: Herbalife enters a forward with its banks, which enter into swaps with the hedge funds buying the convertible. Herbalife gets the shares in five years (though it gets to retire them for earnings-per-share purposes today); the hedge funds get the ability to short Herbalife and hedge their convertibles without having to actually go and borrow Herbalife stock.

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Quick Interlude on Convertible Arbitrage: Convertible hedge funds buy convertible bonds and sell short the underlying stock. The aim is to hedge out stock-price risk and make money on credit and volatility. A convertible is sort of like a bond, in that Herbalife has to pay you back in 5 years and pays you interest until then, but also sort of like stock, in that if Herbalife's stock price goes up you will convert the bond and own Herbalife stock. People use the term "delta" to measure how stock-like a convertible is. A convertible with a 100 percent delta is just like stock; a convertible with zero percent delta is just a straight bond that will never convert into shares. Most newly issued convertibles have a delta between, I don't know, 50 and 90 percent.

If you're looking to hedge out stock price risk, you should sell short an amount of stock equal to the delta: Roughly speaking, if a convertible is 70 percent stock-like, and you sell 70 percent of the underlying stock, you're left with zero stock price risk. Why do you do this? Because it turns out that the math requires you to adjust your hedge over time. If the stock price goes up, the convertible becomes more stock-like -- it gets more likely to convert -- so the delta goes up. If your delta was 70 and now it's 75, you need to sell 5 percent more stock. As the stock goes down, your delta goes down, and you buy back some of the stock you shorted.

Notice that you're buying when the stock goes down and selling when it goes up. Buy low, sell high is a good strategy; I recommend it. The more the stock bounces around, the more you get to buy low and sell high, while always keeping your price risk hedged. What this means is that the convertible is just a bet on volatility: The more often the stock goes up and down, the more money a convertible arbitrage investor makes. If the stock stays flat, it's a bad deal. If it goes up and down and up and down, you make money. You don't care whether it ends up or down at the end of the day, just if it bounces around a lot on the way there.

To do this strategy, you need to sell the stock short. To do that, you need to borrow the stock; short selling without borrowing is illegal. Some stocks are hard to borrow. Herbalife is not, especially,1 but it is risky to borrow, because some people are very noisily shorting Herbalife, and some other people are very noisily talking about engineering a short squeeze. If Carl Icahn were to launch a tender offer, say, it might get a lot more expensive to short Herbalife, and the convertible trade would become considerably less fun. Selling stock to Herbalife -- through its banks -- and not having to deliver the stock for five years -- because you're selling via swaps and forwards -- is a way around that problem. You have, in effect, five years of guaranteed stock borrow.

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OK, back from the interlude! Remember, the point was, Herbalife is buying some $686 million of stock from hedge funds, via these forward/swap transactions. In my post yesterday, I guessed that Herbalife would be buying about $590 million worth of stock, so I was almost $100 million too low. My guess was based on the following assumptions:

  1. I assumed that the convertible would price with a 1.75 percent coupon and 27.5 percent conversion premium, the midpoint of the range at which it was marketed.
  2. I assumed that Herbalife wanted to buy as much stock as it efficiently could.
  3. I assumed that convertible arbitrage hedge funds were going to buy 100 percent of the convertible. (Usually, convertible offerings are about 50/50 hedge funds and outright long-only convertible investors, but it makes some sense to sell mostly to hedge funds here, because they'll sell you stock efficiently.)
  4. I assumed that convertible arbitrage hedge funds would want to sell about 75 percent of the shares underlying the convertible, because their pricing models would tell them that that's the correct theoretical hedge -- the correct delta -- for the convertible. They get long the convertible, short the stock, and have no price risk to the stock and just make money on volatility.

Assumption 1 was wrong; the convertible priced at the cheap end (for buyers -- expensive for Herbalife), with a 2 percent coupon and 25 percent conversion premium, suggesting that the convertible was not especially popular with investors. That leaves 11.6 million shares underlying the convertible, so if my other assumptions were right, Herbalife would be buying 75 percent of 11.6 million, or 8.7 million shares, or $600 million worth.

So my other assumptions were wrong, too. I mean, assumption 2 seems to have been right -- Herbalife bought even more stock than I expected. Assumption 3 was never going to be entirely correct, but it seems to have been close. I've heard that around 85 percent of the bonds went to hedge funds, with the rest to outright investors; I can't confirm that but as you'll see from the below it seems reasonable.

Assumption 4 was wrong. Herbalife bought $686 million worth of stock (along with spending $111 million on capped call transactions). That's about 9.94 million shares, or about 86 percent of the shares underlying the bonds.

If you believe that about 86 percent of the bonds went to convertible arbitrage hedge funds, that means that the hedge funds sold Herbalife 100 percent of their underlying shares. One hundred percent is a nice round number, so it makes a kind of sense, but it's not exactly the "right" number. Because the hedge funds' models were telling them to sell less. My rough guess yesterday was that the delta was around 75 percent; others have told me it might have been more like 65 percent. It certainly wasn't 100 percent.2

Let's say it was 70 percent. So hedge funds were selling Herbalife 9.94 million shares -- synthetically, by entering into swaps with banks who were then entering into a forward with Herbalife. But they were only supposed to sell about 6.96 million shares to hedge their convertibles. So they had to buy back almost 3 million shares.

About 3.7 million Herbalife shares traded in the last hour yesterday, or almost 3 million more than average.3 Mighty convenient!

Here is a diagram-esque thing:

So what happened? Well, best I can tell, convertible arbitrageurs didn't want to sell Herbalife just the 7 million-ish shares that their models told them to. One, because that leaves them open to a short squeeze in the future: If their delta goes up to 9 million shares, then they'll be short 7 million shares through swaps to Herbalife, but will need to go borrow and short the other 2 million shares. In a stock such as Herbalife, there's no guarantee that that will be easy; maybe the stock is up because someone is tendering for it. The only guarantee is to sell Herbalife all of the 10 million shares underlying the convertible, since the most you could ever need to be short is 100 percent of the underlying stock.4

Oh, and two, because this is free money! Look what (seems to have) happened:

  1. Convertible arbitrageurs agree during the day to sell 10 million shares to Herbalife at the closing price.
  2. They know that they and their friends need to buy back 3 million of those shares.
  3. They do that in the last hour of trading, pushing up the closing price.

The result is sort of obvious: Hedge funds pounded the close, because they knew that the higher they could get the closing price the more money they'd make.5 And so they sold Herbalife, in the aggregate, almost 10 million shares, at a price of $69.02 per share. And they bought 3 million of those shares at a price of -- well, less than $69.02. The volume weighted average price in the last hour of trading yesterday (including the close) was $66.73. Hedge funds made about $2.29 a share -- a little under $7 million -- by buying in the last hour, pushing up the price, and selling at the closing price.

Not a huge amount of money, I guess, but not bad for an hour's essentially risk-free work.

Mostly this is just technical fun, and an illustration the way that convertible hedge funds occasionally stumble into free money to reward them for their usually thankless hard work. But I suppose, financial markets being what they are, someone could complain. Who? Herbalife, maybe, though they have mixed motives: They were selling a convertible, so the higher the closing price the higher the conversion price (and the less dilutive the convertible). But they were buying stock, so the higher the closing price the more they were paying for the stock (and the less accretive the buyback). On balance I guess you'd prefer to buy the stock back cheaper, but this doesn't seem like an awful result for them.

Who else? Well, Herbalife shareholders who sold at $63 during the day yesterday, only to see the stock jump to $69 at the close, were presumably blindsided by these goings-on.6 If my theory is right, some people knew that pushing Herbalife up at the close was a way to get free money from Herbalife, and they knew it because Herbalife more or less told them. Regular shareholders, who didn't have access to Herbalife's briefly operational free money machine, might feel a little aggrieved.

1 Borrow costs are not well disclosed but anecdotally it seems like borrowing Herbalife will run you about 1 percent a year more than borrowing the easiest-to-borrow stocks. That's not free, but it's not very expensive either.

2 My guess squares with Bloomberg OVCV's guess, using default assumptions. You can get to a 65 percent delta with like a 425 credit spread and 36 volatility, so, sure. Let's say 65-75 percent.

3 Here's the 3.7 million. The average, I don't know, IVAT seems to tell me that average trading in the last hour is around 750,000 shares. Or last Wednesday, a slightly-above-average but mostly normal day (3.7 million shares traded versus a 3.4 million average) saw 843,969 shares traded in the last hour, including market-on-close.

4 Here is where a real convertible dork would talk about negative gamma and hedging credit with equity, but I will hold off.

5 The convert arbs don't really care what stock price is used to price the convertible, as long as it matches their hedge price. So, if the conversion price is set at a 25 percent premium over a $69.02 close, and they enter their swaps at $69.02, that's as good as if it's set at a 25 percent premium over a $60 close, and they enter their swaps at $60. (Roughly speaking.) The fact that the stock is down today isn't even bad: It's more volatility for them. (Plus, you know -- you could probably have guessed that this would happen, and leaned a little short.)

6 Especially because they are ... not especially well disclosed? The press release announcing the deal says "The Forward Transactions are generally expected to facilitate privately negotiated derivative transactions between the Forward Counterparties and holders of the Convertible Notes, including swaps, relating to the common shares by which holders of the Convertible Notes will establish short positions relating to the common shares and otherwise hedge their investments in the Convertible Notes concurrently with, or shortly after, the pricing of the Convertible Notes."

That doesn't tell you that hedge funds will actually be buying shares at the end of the day, and in droves, as part of their hedge. I guessed yesterday that that might happen, but I used to do this for a living, and anyway my guess was in a footnote because I figured it was the less likely result. I thought it was more likely that the hedge funds would be neither buying nor selling in the market -- just shorting their delta to Herbalife via swaps -- and the press release didn't tell me much one way or the other.

To contact the writer of this article: Matt Levine at mlevine51@bloomberg.net.

To contact the editor responsible for this article: Tobin Harshaw at tharshaw@bloomberg.net.