Jovan Medford Mathematical Finance graduate who is passionate about using math and data to solve problems.

Convertible Arbitrage – What it is and does it Still Work?

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Performance index of convertible arbitrage hedge funds by Barclays. Source

What is Convertible Arbitrage? It is a trading strategy that consists of shorting overpriced warrants while buying the underlying stock.

Does it still work?

It is very difficult to be profitable with it. Convertible arbitrage started in the 1960s. Many hedge funds have been deploying this strategy and have innovated on top of it.

That said, learning this strategy is useful as it hones your foundation for spotting opportunities between different financial products in the market.

Table of Contents

General idea of Convertible Arbitrage

The Basic System consisted of shorting warrants while buying the underlying stock.

Warrants are a derivative that gives the buyer the right to buy a stock at a certain price. It is similar to an option. More info here and here.

It turned out that warrants were often overpriced, so if you shorted the warrant and bought the underlying stock, then we would have a trade that minimizes losses while still providing the opportunity to make money regardless of whether stock prices went up or down. 

This is called a warrant hedge is a very good example of many other simple hedging strategies, so it would be useful to see it in action with a simple example.

Example:

For now, we will ignore transaction and conversion costs. We’ll also assume that one warrant can be converted to one share. Consider a company SMP under the following situation:

Common Share Price: $6

Exercise Price: $10

Warrant Price: $3

We are going to buy 100 SMP common shares and short 100 SMP warrants with hopes of converting both positions to cash before the expiry date. 

If the common share price is at or below the exercise price then the warrant will only be worth a few cents. Otherwise, the warrant will sell for about 10$ less than the stock price.

Profit and Loss:

If the share price rises to $20 then the warrant would be worth about 10$. Therefore, on one hand we should make 20–6 = 14 from the rise in the stock price and lose 10–3 = 7, for the rise in value of the warrant. This gives us a profit of $700. 

If the share price stays between 10–6, then the warrant will sell for a few cents. Therefore, the buyer of the warrant will not execute and we would still make $3 per share on the short sale. 

If the common falls below 6, we will lose the amount below 6. Unless the common falls below 3, these losses will be more than offset our short-sale profits and we will still have net profit. 

In the very worst case if the stock price goes to 0, we would still only have lost $300. This is much less than the possible $700 we could have gained, so the probability of success is higher.

Extending to Convertibles

We can extend the basic system to a convertible with the key thought that any convertible security contains a warrant in disguise. For the purposes of this article, we will focus on convertible bonds since these will be most important in the follow up discussions surrounding the performance of convertible arbitrage.

What is the Convertible Arbitrage?

What is a convertible bond?

A bond is a debt instrument, i.e. when a company sells a bond it is actually borrowing money. When an investor buys a bond, they receive a contract that has an agreed upon set of interest payments (called coupons) which would be received upon a schedule such as annually, quarterly and so on. At the maturity date stipulated in the contract, the investor will also receive a lumpsum called the redemption value which is normally $1000. More info on bonds here.

A convertible bond is a special contract that gives the owner the right to exchange the bond for a fixed number of shares. Based on this right, the price of a convertible bond is usually higher than that of a regular bond. In this way we can think of these types of bonds as regular bonds that come with implicit call options. Therefore, in simple terms, we can think of the value of the convertible bond as the sum of the value of the bond and the value of the options. [Note here that we are switching warrants with call options based on the similarities of their functions]. More info on convertible bonds here.

Convertible bond price, parity and bond floor — Source: Credit Suisse (2014)

The diagonal dotted line is showing parity, which is just the value of the shares of stock which the bond could be converted into (i.e. the conversion value). The graph is showing that the convertible bond price (red-line) is made up of the parity plus the conversion premium.

Functionally speaking, issuing convertible bonds allow corporations to access funds quickly relative to traditional equity and debt offerings. On the other hand, investors can use convertible bonds to hedge the underlying equity, credit, and interest rate risk.

Another way of thinking about it, is that corporations that are high risk are able to monetize their volatility by selling convertible bonds. This is possible since long-term call options on volatile stock can be valuable. So, the latent call option in a convertible bond still makes the bond valuable to investors even at lower coupon rates. Even so, to attract buyers and raise capital quickly for the firm, convertible bonds are often issued at prices below the fundamental value of the debt and call option.

After they are issued, the lack of liquidity of the convertible bond encourages further discount from the fundamental price on trades occurring before the maturity date. Once the bond matures however, an investor will receive the current fundamental value of the bond.

It is this difference between the fundamental price and the discounted prices before maturity that allows for the possibility of arbitrage. 

Pricing Models

There are many different models used to price convertible bonds including both closed-form and numerical solutions. As always, closed-form solutions carry more assumptions and can be restrictive in practice. On the other hand, even simpler numerical solutions such as lattice methods, or Monte Carlo simulations, include path-dependent payoff structures allowing more flexibility and generalizations with convertible bond features. Furthermore, Monte Carlo and lattice methods are relatively easy to implement.

The lattice method we’ll briefly mention here is the CRR binomial pricing model, which is always good for picking up intuition, especially for visual learners. It is a discrete form of the Geometric Brownian Motion for modelling stock prices. 

Below we have:

S = initial stock price

u = up factor; d = down factor

q = probability stock goes up (risk neutral probability); 1-q = probability stock goes down

Where u, d and q are all calculated according to the conditions of a “risk neutral world.”

2-period Binomial Tree

2-period Binomial Tree

*CRR is short for Cox-Ross-Rubinstein

In quick terms, the goal is to develop a lattice of possible stock prices and split the bond into its debt and equity components. For the equity component, calculate the value of a parent node by taking the present value of the expectation over the values of the children nodes. In this manner, backtrack all the way through the tree and then repeat the same for the debt portion. Add the two final results for the price. This is a very rough explanation of the Tsiveriotis and Fernandes approach. (1998)

There is several different software that you can use to calculate the price of a convertible bond using this method. 

For instance, MATLAB has a built-in function cbondbycrr that allows you to calculate convertible bond prices using a CRRTree, you can follow the example in the link below, or go to the actual MathWorks website if you prefer.

Price = cbondbycrr(CRRTree,CouponRate,Settle,Maturity,ConvRatio)

Additionally, both QuantLib and RQuantLib have built in functions for valuing convertible bonds, you can follow these to the documentation:

Rquantlib: https://cran.r-project.org/web/packages/RQuantLib/RQuantLib.pdf

Python quantlib example: http://gouthamanbalaraman.com/blog/value-convertible-bond-quantlib-python.html

Convertible Bond Arbitrage

Arbitrageurs can exploit the discount on convertible bonds while limiting exposure to unwanted risks, through the strategy of longing the convertible bond and shorting the underlying assets

To hedge the equity portion, we can short stock as the price rises, or cover additional stock if the price falls. In doing this, we are readjusting the hedge dynamically as the price changes. This is called delta hedging. The straight debt is more difficult to hedge since there would be less prices available on the market. However, you can still hedge credit risk by shorting additional equity or by using credit swaps.

Convertible arbitrage is popular because of the relatively predictable hedge that can be established as we described above. Furthermore, the profit potential of the strategy is largely made up of the inefficiency in price between the convertible and stock, in addition to the cash-flows that can be derived from the hedge. This is instead of relying on the market moving in a particular direction. However, there are many additional techniques that can build from the convexity of the bond and any other expertise that the arbitrageur may have.

Arbitrageurs looking to profit from the discounted price of bonds, prefer the situation where most of the value of the convertible bond is within the equity option. 

What Are We looking for?

To list some of the characteristics in a convertible that make this type of arbitrage worthwhile we can consider the following:

  • The Convertible Should Be Underpriced/Overpriced

The main point of convertible arbitrage is to take advantage of convertible bonds that have prices different from their fundamental value. Therefore, establishing that this difference exists is the most important thing to look for in a convertible bond.

  • High Volatility

When we combine the volatility of the stock with the length of time it may take to execute a convertible arbitrage strategy, it makes sense that at high volatility, the probability increases that the convertible bond will eventually be in the money. 

  • Low Dividend 

Since we are shorting the underlying stock, we are responsible for paying any dividends to the lender of the stock.

  • Low Conversion Premium

The conversion premium is the difference between the current price of the bond and the value of the common stock, if the bond were to be converted. For example, if we have a bond valued at $1500, which can be converted to 50 common stock shares of $25, then we would have

conversion premium = 1500 — (50*25) = 250

This amount is usually expressed as a percentage of the bond value. Generally speaking, it is preferable to have a convertible with a conversion premium of 25% or lower. A lower conversion premium indicates lower interest rate risk and credit sensitivity.

Additional Strategies 

Reverse Hedging

Also referenced as convertible arbitrage, this variation in strategy involves the selling short of the convertible bond and buying the underlying security. If the implicit warrant or option becomes very under-priced, then the expected return from short selling might significantly decrease. In this case reverse hedging may provide consistently better returns.

Gamma Trades

As the stock price rises our delta increases, so we would need to short additional shares to restore the delta neutral hedge. The same is true if the price falls, then we would have to buy more stock. If there is higher volatility in the market then stocks will rise and fall more often. So we will have opportunity for profit as we continually sell as the stock price rises — sell high — and buy as a stock price falls — buy low. 

Vega Trades

Convertibles with higher vega are more sensitive to changes to volatility, since this strategy takes advantage of this, they are sometimes called volatility trades. For vega trading we long the convertible bond and short appropriate options on the underlying stock that are trading at high levels of volatility. 

What are the risks of convertible arbitrage?

It is terribly important to understand that while arbitrage strategies are designed to mitigate volatility (changes in asset prices), there are still many other types of risk that we must consider. This is especially true in the case of convertible arbitrage, because there may be stipulations in the contract that you will have to hold the convertible bond for a length of time before converting to common stock. Therefore, within that time, the arbitrageur must be able to assess and plan for any significant market and economic changes that may occur within the time conversion is allowed.

  • Credit Risk

Since most convertible bonds may be ungraded or below investment grade, there is the very real threat that the issuers of the bond may default. 

  • Foreign Exchange Risk

Convertible Arbitrage portfolios may include positions across multiple territories and would therefore involve different currencies. This exposes some of the positions to the risk of a change in currency value. One can buy forward contracts on currency to mitigate this risk.

  • Management Risks

Since the strategy is rooted in the difference between the fundamental value of the convertible bond, if the manager calculates this value incorrectly, then the strategy may be unprofitable from the start. This situation is easily compounded by the other risks mentioned and can quickly become a disaster.

  • Interest Rate Risk

Generally speaking, bonds with longer maturity dates are sensitive to potential changes in interest rates. While shorting more stock is a good way to hedge these risks, lower hedge ratios may require additional protection.

  • Short Selling Ban

The banning of short sales is rare, but it can occur in times of crisis. If such is the case then it would be impossible to get into or re-balance convertible arbitrage portfolios.

Is convertible arbitrage profitable?

In his book “Beat the Market”, Dr. Thorp says that using convert arb techniques he was able to make annualized returns of 25% a year with virtually no risk between the years of 1961–1965. In fact, he went on to suggest that if $10,00 had been invested in the basic system alone for any 15 years between 1945–1965, then it would have grown to over $500,000.

Even in the years between 1990–2007, the Hedge Fund Research (HFR) convert arb index was able to garner annualized returns of 10 percent, with annualized volatility (based on quarterly returns) of 5 %. However, this was no longer quite “Beating the market” since the S&P 500 had annualized returns of 11% but at a much higher annualized volatility of 15%. 

The drop from astonishing profitability of the strategy can be attributed to a few things. For one investing has become something of a worldwide sport in recent times. Take the CFA exams for instance. Back in 1963 when the test started, there were only 183 candidates, we can compare this number with 256,000 candidates in 2018. With that being said, it is easy to see that it is significantly harder to get an edge of that size. 

In fact, Dr. Thorp himself decided to throw in the towel on his arbitrage fund in 2002. In doing so he remarked that the growth in hedge fund assets and competition had undercut many of the profit opportunities.

2008 

The market crash of 2008 was perhaps the most significant event in recent financial history — up until this Covid-19 situation. In 2007, over 81% of fund managers had been able to achieve positive returns, and many funds had been able to remain positive through the beginning of 2008. However, with the collapse of the 150+ year old Wall Street Firm — Lehman Brothers — the indirect losses stemming from Lehman issued shares, resulted in devaluations of convertible bonds amongst many other things. Furthermore, the ban of short sales globally prevented fund managers from being able to properly rebalance their positions. To add even more insult to injury, managers were unable to sell the underlying securities since the potential buyers were not able to hedge their positions. In the end fund managers experienced a near 40% drawdown as a result of the crisis.

Post 2008

The graphs in this section were taken from the Man Institute, with the source data coming from the Barclays Hedge Fund Research company.

AuM Decline in Convertible Arbitrage Funds

AuM Decline in Convertible Arbitrage Funds

Here we can see that there was a sharp decrease in the Assets under Management (AuM) of convertible arbitrage funds despite making a comeback in the years immediately following the 2008 crash.

Decline in Long-Side Leverage 

The above graph shows there was a steady descent in the amount of leverage employed in order to buy convertible bonds. In 2008 the leverage stood at 8 times the underlying asset but by 2018 it was down to less than 2. Normally, the amount of leverage used in a strategy is relative to the amount of liquidity of the assets underlying the strategy. However, the events of 2008 showed convert arb investors, that overly high leverage can make them far too vulnerable to the effects of forced liquidation in a depressed market. Indeed, the size of respective short selling positions have also shrunk.

Another change since the crash, is that convertible arbitrage funds are no longer consuming as big of a lion share with respect to convertible bonds. As of 2019, convert arb funds only held 45% of all convertible funds as opposed to 75% back in 2008. The last point is that there has been a slowdown in the issuance of convertible bonds themselves post the crash. It was the combination of these things that led to the sharp shrinkage in convert arb AuM.

Who created convertible arbitrage?

Dr. Edward Thorp.

This man is considered to be the father of quantitative trading.

He developed the system of counting cards by which he proved that a player could beat a casino in a game of Blackjack. He literally wrote the book on “How to Beat Your Dealer” — it went on to sell millions of copies and there was even a movie made about the strategy. 

After conquering the gambling world, he turned his attention to investing. Equipped with the experience and knowledge of gambling success, he was able to establish serious parallels between the two worlds. Thorp understood that to succeed in both worlds, you need to be able to measure probabilities and place the magnitude of bets according to the likelihood of success. If you risk too much when the probabilities aren’t in your favour, you will lose all of your money. If you bet too little when you are more likely to win, then you also end up leaving money on the table. This was how he was able to beat Blackjack.

After reading everything he could find, Thorp developed a mathematics-based approach for hedging warrants after he realized that they were often mispriced. In his book “Beat the Market”, Dr. Thorp outlines the forethought, development and execution of these strategies which we refer to as convertible arbitrage.

On a final note regarding Ed Thorp, his hedge fund Princeton Newport Partners, went on to earn a 19.1% annualized rate of return for more than two decades. Based on the success of his strategies, and him being the first person to apply this level of mathematics to trading, he is widely considered to be the father of Quantitative investing.

The Growth of The Hedge Fund Industry

Perhaps we can make the case that the entire reason there is a hedge fund industry today could be attributed in part to convertible arbitrage. Starting with the first hedge fund in 1949, the first wave of hedge funds started to pop up subsequent to the publication of “Beat the Market”. After this came the macro investment funds made popular by George Soros and the bull markets of the 1980’s and 1990’s. The 1990’s saw the biggest growth as the hedge fund industry moved into mainstream. In fact, according to Hedge Fund Research Inc, in 1990 there were less than 200 funds possessing around $20B in assets but by the year 2000, this had grown to over 4500 funds with almost $500B in assets. 

From this history, we can see that the forethought behind convertible arbitrage techniques carry a lot of weight in the development of investing as we know it. So, it would make sense for us to go through the very basic system and get a good foundation.

Works Influenced

The techniques outlined in “Beat the Market” way back in 1967 were light years ahead of their time. Perhaps we might even say that the early and continued success of these convertible arbitrage strategies, are much of the cause for the respect given to the application of mathematics to finance. Keeping this in mind, the number of significant persons in finance that have since been influenced by the techniques outlined in “Beat the Market”, is astounding in its own right. Coming from Ed Thorp’s website here are some of the people that the book inspired:

  • Fischer Black, co-creator of the Nobel Prize winning Black-Scholes Formula. This is one of the most important discoveries in financial history and is still used in option pricing today.
  • Ken Griffin, founder of Citadel Investments
  • Bill Gross, who revolutionized bond trading and founded PIMCO
  • Blair Hull, blackjack player and founder of Hull Trading and Hull Tactical Asset Allocation
  • Frank Meyer, founder of Glenwood Investments
  • Paul Singer, founder of Elliot Management Corporation

Appendix

Fundamentals

Arbitrage: This is the situation that occurs due to market inefficiencies where you can make a profit without risk. It can be described as:

The simultaneous purchase and sale of the same or equivalent securities, commodities, or foreign exchange in different markets to profit from unequal prices.

Convertibles: Any securities that can be changed into other securities; the addition of cash may be required. These include: warrants, convertible bonds, puts and calls to name a few.

Warrant: A warrant entitles the holder to purchase common shares at the exercise price until the expiration date of the warrant. This right is worthless if everyone “knows” the common will trade at or below exercise price until after expiration. But if there is some chance the common will rise above the exercise price before expiration, the warrant offers a possible profit.

Short Selling: The process of selling a security before you own it and buying it in the future. The process is as follows:

  • Sell at the current price a security not owned.
  • Borrow the security, leaving the proceeds of the sale with the lender as collateral.
  • Buy the security in the market at a later time.
  • Return the newly purchased certificate to the lender, who returns the original

Greeks

Delta: the rate of change of price in the underlying asset vs the change of price of the derivative

Gamma: The rate of change of the delta 

Vega: rate of change in the value of the bond per unit change in volatility

Jovan Medford
Jovan Medford Mathematical Finance graduate who is passionate about using math and data to solve problems.