# Convertible Arbitrage. Thought of the Day 108.0

A convertible bond can be thought of as a fixed income security that has an embedded equity call option. The convertible investor has the right, but not the obligation, to convert (exchange) the bond into a predetermined number of common shares. The investor will presumably convert sometime at or before the maturity of the bond if the value of the common shares exceeds the cash redemption value of the bond. The convertible therefore has both debt and equity characteristics and, as a result, provides an asymmetrical risk and return profile. Until the investor converts the bond into common shares of the issuer, the issuer is obligated to pay a fixed coupon to the investor and repay the bond at maturity if conversion never occurs. A convertible’s price is sensitive to, among other things, changes in market interest rates, credit risk of the issuer, and the issuer’s common share price and share price volatility.

Analysis of convertible bond prices factors in three different sources of value: investment value, conversion value, and option value. The investment value is the theoretical value at which the bond would trade if it were not convertible. This represents the security’s floor value, or minimum price at which it should trade as a nonconvertible bond. The conversion value represents the value of the common stock into which the bond can be converted. If, for example, these shares are trading at \$30 and the bond can convert into 100 shares, the conversion value is \$3,000. The investment value and conversion value can be considered, at maturity, the low and high price boundaries for the convertible bond. The option value represents the theoretical value of having the right, but not the obligation, to convert the bond into common shares. Until maturity, a convertible trades at a price between the investment value and the option value.

A Black-Scholes option pricing model, in combination with a bond valuation model, can be used to price a convertible security. However, a binomial option model, with some adjustments, is the best method for determining the value of a convertible security. Convertible arbitrage is a market-neutral investment strategy that involves the simultaneous purchase of convertible securities and the short sale of common shares (selling borrowed stock) that underlie the convertible. An investor attempts to exploit inefficiencies in the pricing of the convertible in relation to the security’s embedded call option on the convertible issuer’s common stock. In addition, there are cash flows associated with the arbitrage position that combine with the security’s inefficient pricing to create favorable returns to an investor who is able to properly manage a hedge position through a dynamic hedging process. The hedge involves selling short a percentage of the shares that the convertible can convert into based on the change in the convertible’s price with respect to the change in the underlying common stock price (delta) and the change in delta with respect to the change in the underlying common stock (gamma). The short position must be adjusted frequently in an attempt to neutralize the impact of changing common share prices during the life of the convertible security. This process of managing the short position in the issuer’s stock is called “delta hedging.”

If hedging is done properly, whenever the convertible issuer’s common share price decreases, the gain from the short stock position should exceed the loss from the convertible holding. Equally, whenever the issuer’s common share price increases, the gain from the convertible holding should exceed the loss from the short stock position. In addition to the returns produced by delta hedging, the investor will receive returns from the convertible’s coupon payment and interest income associated with the short stock sale. However, this cash flow is reduced by paying a cash amount to stock lenders equal to the dividend the lenders would have received if the stock were not loaned to the convertible investor, and further reduced by stock borrow costs paid to a prime broker. In addition, if the investor leverages the investment by borrowing cash from a prime broker, there will be interest expense on the loan. Finally, if an investor chooses to hedge credit risk of the issuer, or interest rate risk, there will be additional costs associated with credit default swaps and a short Treasury position. This strategy attempts to create returns that exceed the returns that would be available from purchasing a nonconverting bond with the same maturity issued by the same issuer, without being exposed to common share price risk. Most convertible arbitrageurs attempt to achieve double-digit annual returns from convertible arbitrage.

# Delta Hedging.

The principal investors in most convertible securities are hedge funds that engage in convertible arbitrage strategies. These investors typically purchase the convertible and simultaneously sell short a certain number of the issuer’s common shares that underlie the convertible. The number of shares they sell short as a percent of the shares underlying the convertible is approximately equal to the risk-neutral probability at that point in time (as determined by a convertible pricing model that uses binomial option pricing as its foundation) that the investor will eventually convert the security into common shares. This probability is then applied to the number of common shares the convertible security could convert into to determine the number of shares the hedge fund investor should sell short (the “hedge ratio”).

As an example, assume a company’s share price is \$10 at the time of its convertible issuance. A hedge fund purchases a portion of the convertible, which gives the right to convert into 100 common shares of the issuer. If the hedge ratio is 65%, the hedge fund may sell short 65 shares of the issuer’s stock on the same date as the convertible purchase. During the life span of the convertible, the hedge fund investor may sell more shares short or buy shares, based on the changing hedge ratio. To illustrate, if one month after purchasing the convertible (and establishing a 65-share short position) the issuer’s share price decreases to \$9, the hedge ratio may drop from 65 to 60%. To align the hedge ratio with the shares sold short as a percent of shares the investor has the right to convert the security into, the hedge fund investor will need to buy five shares in the open market from other shareholders and deliver those shares to the parties who had lent the shares originally. “Covering” five shares of their short position leaves the hedge fund with a new short position of 60 shares. If the issuer’s share price two months after issuance increases to \$11, the hedge ratio may increase to 70%. In this case, the hedge fund investor may want to be short 70 shares. The investor achieves this position by borrowing 10 more shares and selling them short, which increases the short position from 60 to 70 shares. This process of buying low and selling high continues until the convertible either converts or matures.

The end result is that the hedge fund investor is generating trading profits throughout the life of the convertible by buying stock to reduce the short position when the issuer’s share price drops, and borrowing and selling shares short when the issuer’s share price increases. This dynamic trading process is called “delta hedging,” which is a well-known and consistently practiced strategy by hedge funds. Since hedge funds typically purchase between 60% and 80% of most convertible securities in the public markets, a significant amount of trading in the issuer’s stock takes place throughout the life of a convertible security. The purpose of all this trading in the convertible issuer’s common stock is to hedge share price risk embedded in the convertible and create trading profits that offset the opportunity cost of purchasing a convertible that has a coupon that is substantially lower than a straight bond from the same issuer with the same maturity.

In order for hedge funds to invest in convertible securities, there needs to be a substantial amount of the issuer’s common shares available for hedge funds to borrow, and adequate liquidity in the issuer’s stock for hedge funds to buy and sell shares in relation to their delta hedging activity. If there are insufficient shares available to be borrowed or inadequate trading volume in the issuer’s stock, a prospective issuer is generally discouraged from issuing a convertible security in the public markets, or is required to issue a smaller convertible, because hedge funds may not be able to participate. Alternatively, an issuer could attempt to privately place a convertible with a single non-hedge fund investor. However, it may be impossible to find such an investor, and even if found, the required pricing for the convertible is likely to be disadvantageous for the issuer.

When a new convertible security is priced in the public capital markets, it is generally the case that the terms of the security imply a theoretical value of between 102% and 105% of face value, based on a convertible pricing model. The convertible is usually sold at a price of 100% to investors, and is therefore underpriced compared to its theoretical value. This practice provides an incentive for hedge funds to purchase the security, knowing that, by delta hedging their investment, they should be able to extract trading profits at least equal to the difference between the theoretical value and “par” (100%). For a public market convertible with atypical characteristics (e.g., an oversized issuance relative to market capitalization, an issuer with limited stock trading volume, or an issuer with limited stock borrow availability), hedge fund investors normally require an even higher theoretical value (relative to par) as an inducement to invest.

Convertible pricing models incorporate binomial trees to determine the theoretical value of convertible securities. These models consider the following factors that influence the theoretical value: current common stock price; anticipated volatility of the common stock return during the life of the convertible security; risk-free interest rate; the company’s stock borrow cost and common stock dividend yield; the company’s credit risk; maturity of the convertible security; and the convertible security’s coupon or dividend rate and payment frequency, conversion premium, and length of call protection.