# How Convertible Bond Arbitrage Works II

Yesterday, Preet Banerjee from WhereDoesAllMyMoneyGo.com explained the basics behind convertible bonds.  Today he will continue to explain how convertible bond arbitrage works along with some case studies.

Okay, so yesterday we explained the CONCEPT of convertible bonds. Let’s quickly define Arbitrage:

Arbitrage

Arbitrage is when you are long and short two related securities (or the same security in some cases) in order to make a gain when the price changes.  I realize that it sounds counterintuitive – the best way to explain is to show an example.  Hmmm… I wish I had a security that lends (no pun intended!) itself well to arbitrage… Hold the phone! I do: The Convertible Bond!!!!  Let’s take a look:

Convertible Bond Arbitrage

Like I said the best way for me to explain is to dive right in with an example – and then you can work backwards to make sure it makes sense.

Let’s start out with our convertible bond at issue time.  It is being issued at \$1000, with a coupon payment of 5% and let’s say that it is a 2 year bond (not common, but for the sake of the example).  It has a conversion rate of 50:1 – so you can exercise your conversion privilege anytime and receive 50 shares of the underlying common stock. Right now the share price is \$20.

To engage in Convertible Bond Arbitrage you have to buy and sell (at the same time) two related securities – and in this case you are buying the convertible bond for \$1000 and at the same time you are shorting 25 shares of the common stock – you’ll note that you are shorting HALF the amount of shares that the bond could be converted to, which is 50. (If you are not familiar with shorting – stop now and look it up – then come right back here.)  Now we will look at three cases: 1) No change in the share price for 1 year. 2) Share price goes up after 1 year. And 3) Share price goes down after 1 year.  In all three cases you MAKE a gain (all else being equal).

Case 1: Share price does not change for the year

You have received \$50 in interest for the year (coupon payments on the bond).  You have also received interest on the proceeds of the short sale for one year (when you short a stock you are selling it which means you have cash on hand that you can gain interest on – we’ll say your brokerage account pays 2% on cash balances).  Since you “sold” 25 shares at \$20 you have \$500 earning the 2% interest – which equals \$10.  There is also the interest you have to pay for borrowing the shares from your broker to sell – let’s say this is 1% – so you have a COST of \$5 to borrow the shares you shorted. Your running total so far is \$55 in your pocket.

Now, remember nothing has happened to the share price, so your convertible bond is still worth \$1000.  At day 366, you cover your short and you sell your convertible bond. In other words you have completely unwound the strategy.

So at the end of the day you earned 5.5% for the year (Interest on the bond plus the interest on the short sale proceeds less the cost of borrowing the shorted stock). Not bad for 5% bond, right?

Case 2: Share price rises to \$25 after 1 year.

50 Shares at \$25 is worth \$1250 – so your convertible bond has made you money (\$250 gain).  You have to subtract your loss on your short position (you lose money if a shorted stock goes up in value). In this case, when you unwind your short you have essentially sold it for \$20 at the beginning and bought it back for \$25 when you cover your position at the end of the year.  Therefore you have a loss of \$5 per share (\$20 Shorting price less \$25 Covering price) which for 25 Shares equals a \$125 loss on your short position.  We still have the interest on the proceeds of the short position (\$10) and the interest charged to borrow the stock from your broker to establish a short position (-\$5).  This leaves you with \$250 (Capital Gain on the convertible bond) + \$50 (Interest earned on the bond) – \$125 (Loss on the short) + \$10 (Interest earned on the proceeds of the short sale) – \$5 (Interest charged on the borrowed securities for the short) = \$180 at the end of the year.

So in this case you have earned 18.0% for the year because the share price went up.

Case 3: Share price decreases to \$15 after 1 year

Okay, so your convertible bond is worth \$750 since 50 shares x \$15 = \$750? WRONG – Remember – the value of the convertible bond won’t fall below it’s intrinsic value – or what it would trade like if were just a bond.  So in this case it is still worth \$1000. You are break even on the bond.  BUT you have made money on your short position.  Since the stock has gone down in value you have effectively sold it at \$20 and bought it for \$15 when you cover your short position for a gain of \$5 per share.  Multiply that by 25 shares are you have a gain of \$125 on your short sale.  We still have the same interest on the short sale proceeds for the year of \$10 and a cost of interest on the borrowed securities from your broker of \$5.  Add all that up and we have \$0 + \$125 + \$10 – \$5 = \$130.

So in this case you have earned 13% for the year because the share price went down.

The End – Now don’t go out there and do it

This concludes my guest article on Convertible Bond Arbitrage – I would suggest not rushing out and immediately implementing this strategy just yet – do a LOT more research – because the conversion rates can be complex formulas for convertible bonds and you need to examine the effects of each of the three scenarios I just went through with the conversion math of the convertible bond issue you are looking at.  I made up a fictitious convertible bond example with an accompanying fictitious stock example.

If you found this article of interest – please take a look at my own blog and consider signing up for my RSS feed there: WhereDoesAllMyMoneyGo.com

I’d like to thank to FrugalTrader for asking me to write this article – it is an honour for a new blogger like myself to receive such a request!

### I've Completed My Million Dollar Journey. Let Me Guide You Through Yours!

Sign up below to get a copy of our free eBook: Can I Retire Yet?

Posted in

### FT

FT is the founder and editor of Million Dollar Journey (est. 2006). Through various financial strategies outlined on this site, he grew his net worth from \$200,000 in 2006 to \$1,000,000 by 2014. You can read more about him here.
Subscribe
Notify of

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Inline Feedbacks
ct
8 years ago

Hey the 50 dollar coupon payment in case 3 is missing, but Great article though! I find it very informational!

Nobu
10 years ago

What happened to the 5% coupon payment in Case #3? You would still get the \$50 regardless of the share price of the stock, so your theoretical gain would be \$180 or 18%.

J. Stanford
11 years ago

Meridien – There are unfortunately barriers of entry for individual investors. Generally commission paid on the bond is high, and its difficult for individual investros to borrow stock at cheap rates. You can hedge the bonds with CFDs (contracts for differece) but you will not be able to leverage much. You can get pricing models that will turn out all the greeks you need, but you will have to pay for them… If you can do the math yourself (optionality, bi-nomial or PDEs) you could save a lot of money! You would also need access to credit market makers if you want to hedge the bonds via CDS or asset swaps… not likely that a private investor would have a prime brokerage setup.

Meridien
11 years ago

I will leave here a question I have posted on a forum:

I am wondering if an individual investor can engage in convertible bonds arbitrage or if there are barriers of entry impeaching one to do so (such access to greeks data, material needed, software, database, or else).

I will greatly appreciate your insights.

Meridien
11 years ago

Prior,

you will have to get familiar with the greek term “rho”. It tells you the price sensitivity to changes in interest rates.

Piotr
12 years ago

How would changing interest rates play into this strategy?

Jonathan Stanford
13 years ago

Joe – Actually there are opportunities, although they are rare. Usually CBs have rather long maturities and often the longest quoted option market you will find is out 2 or so years. So its difficult to put a “market” price the that kind of volatility. Also credit spread assumptions may differ among market makers, so the CB market isn’t as efficient as you may think. Sometimes you can buy “cheap” implied volatility, and then hope for increased realised volatility in the future.

Another important use of CB arbitrage are carry trades. If you setup the trade as discussed above, and strip the credit (by means of asset swaps) you actually use very little of your original cash. Therefore you can put on a large degree of leverage, and your original coupon becomes massive if compared to your original cash investment. Obviously, the borrow costs, and dividend costs will also be leveraged.

Its tricky. As you said above, there if no free lunch, but if you are willing to put on the risk, then the rewards could follow.

Joe Lavely
13 years ago

J Stanford –

Thanks.

Yes, your comments help. If I understand you correctly, the hedge [shorting the stock] is inordinately profitable only if the conversion privilege [Call option] is mispriced – in your explanation, actual volatility exceeds the level that is priced into the CB.

Anytime any security is mispriced, there is an opportunity for inordinate returns. But, I presume that the Call options on almost all CBs are not mispriced; I presume that they are properly priced.

Accordingly, I still have the question as to why most CB investors do, in fact, short the stock.

Jonathan Stanford
13 years ago

Joe L. you are absolutely correct. Many oversimplifications have been made in these examples, and make it sound a lot easyer that it actually is. First of all they are assuming the CBs trade on parity (same price as the value of the underlying stocks) which is not correct, as they trade on a premium to parity (conversion premium). They also assume zero delta on the down and 100 delta on the up. CBs actually start off at issue at-the-money (close to 50% delta). As you mention the bond floor is below the market price, so if the stock falls, your CB will fall to its bond floor with a delta shift decreasing as you go down. Another important omission in their examples is the cost of dividends of the short stock. If you have a short stock position, you have to pay up the dividend that the stock has payed. If there is no dividend protection in the prospectus, this can be expensive…

I will try to give you the explanation you are looking for as to why they do it:

When you buy a CB you pay an implied level of future volatility in the stock. You hedge the delta (say 50% to start off with). At this point you are market neutral as far as the stock movement is concerned. If the stock falls the delta sensitivity of the CB will be lower, but you still have your original 50% hedge on, so you will be making more money on the hedge than the money you are loosing on the CB. At this point the delta may be 45% so in order to rebalance your positon you buy back 5% hedge. If the stock goes back up, you are short 45% hedge but the CB will be back to a 50% delta, so you will make more money on the CB than you lose on the hedge.

In essence you are long volatility, because you have to continue to do this buy low sell high trades (delta hedging) in order to make the same money you will lose on time value (conversion premium). So in order to have many of these stock movements to rebalance, and therefore to make profits, your stock has to trade at a realised volatility higher than the implied volatility you payed on day one. If future realised volatility is lower, you WILL lose money.

Not to mention interest rate exposure, and above all, credit risk exposure…

Does that help?

Joe Lavely
13 years ago

The cases contain several erroneous assumptions. I will focus on only one.

In Case 3, there is a statement that, if the stock price falls, the price of the convertible bond [CB] remains constant at \$1,000 because that remains the intrinsic value of the bond portion of the CB. But, the price of the CB WOULD FALL because 5% is NOT the rate that would exist on a straight [non-convertible] bond; \$1,000 is NOT the instrinsic value of the bond. The rate on the CB is 5% only because it carries the conversion privilege. The rate on a straight bond would be higher than 5%.

Indeed Comment 11 acknowledges that the rate on a CB is lower than the rate on a straight bond.

The price of a CB consists of the price of a straight bond plus the price of the conversion privilege [similar to a Call Option]. As the price of the stock falls from \$20 to \$15, the value of the conversion privilege [Call Option] falls and so does the value of the CB. In Case 3, the decrease in the value of the conversion privilege would exactly offset the other gains so that the net inordinate gain (above the 5% interest) would equal zero. [The math here is pretty complex and there is an assumption that the stock hedge was appropriate (dependent on the “delta” of the stock) and that everything occurred instantaneously.]

It is true that CB investors usually hedge by shorting the stock. However, I have never seen an adequate explanation of why they do it – unless, of course, they believe that the conversion privilege is mispriced.

Remember Friedman’s dictum – There is no such thing as a free lunch.

If the proposed scheme worked, we would all be rich.