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How the Value of United Warrants Is Affected by Dilution

Unfortunately, our calculations for United warrants do not tell the whole story. Remember that when investors exercise a traded call or put option, there is no change in either the company’s assets or the number of shares outstanding. But, if United’s warrants are exercised, the number of shares outstanding will increase by Nq 100,000. Also the assets will increase by the amount of the exercise money (Nq EX 100,000 $10 $1 million). In other words, there will be dilution. We need to allow for this dilution when we value the warrants.

Let us call the value of United’s equity V:

Value of equity V value of United’s total assets value of debt

If the warrants are exercised, equity value will increase by the amount of the exercise money to V NqEX. At the same time the number of shares will increase to N Nq. So the share price after the warrants are exercised will be

V NqEX

Share price after exercise N Nq

At maturity the warrant holder can choose to let the warrants lapse or to exercise them and receive the share price less the exercise price. Thus the value of the warrants will be the share price minus the exercise price or zero, whichever is the higher. Another way to write this is

Warrant value at maturity maximum 1share price exercise price, zero2

This tells us the effect of dilution on the value of United’s warrants. The warrant value is the value of 1/(1 q) call options written on the stock of an alternative firm with the same total equity value V, but with no outstanding warrants. The alternative firm’s stock price would be equal to V/N—that is, the total value of United’s

4In Chapter 21 we saw that the Black–Scholes formula for the value of a call is

3N1d1 2 P 4 3N1d2 2 PV1EX24

where d1 log 3P/PV1EX24/ 2t 2t/2 d2 d1 2t

N1d1 2 cumulative normal probability function Plugging the data for United into this formula gives

d1 log 312/110/1.14 24/1.40 242 .40 24/2 1.104 and d2 1.104 .40 24 .304 Appendix Table 6 shows that N1d1 2 .865, and N1d2 2 .620. Therefore, estimated warrant value

.865 12 .620 110/1.14 2 $6.15.

equity (V) divided by the number of shares outstanding (N).5 The stock price of this alternative firm is more variable than United’s stock price. So when we value the call option on the alternative firm, we must remember to use the standard deviation of the changes in V/N.

Now we can recalculate the value of United’s warrants allowing for dilution. First we find the stock price of the alternative firm:

Current equity value of alternative firm V value of United’s total assets

value of loans

18 5.5 $12.5 million

Current share price of alternative firm V 12.5 million $12.50

N 1 million

Also, suppose the standard deviation of the share price changes of this alternative firm is * .41.6

The Black–Scholes formula gives a value of $6.64 for a call option on a stock with a price of $12.50 and a standard deviation of .41. The value of United warrants is equal to the value of 1/(1 q) call options on the stock of this alternative firm. Thus warrant value is

This is a somewhat lower value than the one we computed when we ignored dilution but still a bad deal for United.

5The modifications to allow for dilution when valuing warrants were originally proposed in F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy 81 (May–June 1973), pp. 637–654. Our exposition follows a discussion in D. Galai and M. I. Schneller, “Pricing of Warrants and the Valuation of the Firm,” Journal of Finance 33 (December 1978), pp. 1333–1342.

6How in practice could we compute *? It would be easy if we could wait until the warrants had been trading for some time. In that case * could be computed from the returns on a package of all the company’s shares and warrants. In the present case we need to value the warrants before they start trading. We argue as follows: The standard deviation of the assets before the issue is equal to the standard deviation of a package of the common stock and the existing loans. For example, suppose that the company’s debt is risk-free and that the standard deviation of stock returns before the bond–warrant issue is 38 percent. Then we calculate the standard deviation of the initial assets as follows:

Notice that in our example the standard deviation of the stock returns before the warrant issue was slightly lower than the standard deviation of the package of stock and warrants. However, the warrant holders bear proportionately more of this risk than do the stockholders; so the bond–warrant package could either increase or reduce the risk of the stock.

It may sound from all this as if you need to know the value of United warrants to compute their value. This is not so. The formula does not call for warrant value but for V, the value of United’s equity (that is, the shares plus warrants). Given equity value, the formula calculates how the overall value of equity should be split up between stock and warrants. Thus, suppose that United’s underwriter advises that $500,000 extra can be raised by issuing a package of bonds and warrants rather than bonds alone. Is this a fair price? You can check using the Black–Scholes formula with the adjustment for dilution.

Finally, notice that these modifications are necessary to apply the Black–Scholes formula to value a warrant. They are not needed by the warrant holder, who must decide whether to exercise at maturity. If at maturity the price of the stock exceeds the exercise price of the warrant, the warrant holder will of course exercise.

23 . 2 WHAT IS A CONVERTIBLE BOND?

The convertible bond is a close relative of the bond–warrant package. Many companies choose to issue convertible preferred as an alternative to issuing packages of preferred stock and warrants. We will concentrate on convertible bonds, but almost all our comments apply to convertible preferred issues.

In 1999 Amazon.com issued $1.25 billion of 4 3/4 percent convertible bonds due in 2009.7 These could be converted at any time to 6.41 shares of common stock. In other words, the owner had a 10-year option to return the bond to the company and receive 6.41 shares of stock in exchange. The number of shares into which each bond can be converted is called the bond’s conversion ratio. The conversion ratio of the Amazon bond was 6.41.

To receive 6.41 shares of Amazon stock, the owner had to surrender bonds with a face value of $1,000. This means that to receive one share, the owner had to surrender a face amount of $1,000/6.41 $156.01. This figure is called the conversion price. Anybody who bought the bond at $1,000 to convert it into 6.41 shares paid the equivalent of $156.01 per share.

At the time of issue the price of Amazon stock was about $120 so the conversion price was 30 percent higher than the stock price.

Convertibles are usually protected against stock splits or stock dividends. When Amazon subsequently split its stock 2 for 1, the conversion ratio was increased to 12.82 and the conversion price dropped to $1,000/12.82 $78.00.

The Convertible Menagerie

Amazon’s convertible issue is typical, but you may come across more complicated cases. For example, in November 2000 Tyco raised the record sum of $3.5 billion from a convertible issue. The Tyco issue was an example of a LYON (liquid yield option note). A LYON is a callable and puttable, zero-coupon convertible bond (and you can’t get much more complicated than that).

7The Amazon issue consisted of convertible subordinated notes. The term subordinated indicates that the issue is a junior debt; its holders will be at the bottom of the heap of creditors in the event of default. Notes are simply unsecured bonds. Therefore there are no specific assets that have been reserved to pay off the holders in the event of default. There is more about these terms in Section 25.3.

The Tyco issue was a 20-year zero-coupon bond that was convertible at any time into 10.3 shares. The bonds were issued at a price of $741.65, which gave bondholders a yield to maturity of 1.5 percent. When Tyco issued the convertible, corporate bonds were yielding roughly 8 percent. So an investor who converted immediately would be giving up a bond worth $1,000/1.0820 $215. Investors who waited 20 years to convert would be relinquishing a bond worth $1,000 (as long as the firm is solvent). So the value of the bond that they give up increases each year.

The Tyco LYON contains two other options. Starting in 2007 the company has the right to call the bond for cash. The exercise price of this call option starts at 82.34 percent and increases by 1.5 percent each year until it reaches 100 percent in 2014. The bondholders also have an option, for there are five days between 2001 and 2014 on which they can demand repayment of their bonds. The repayment price starts at 75.28 percent and then is increased by 1.5 percent a year. These put options help to provide a more solid floor to the issue. Even if interest rates rise and prices of other bonds fall, LYON holders have a guaranteed price on these five days at which they can sell their bonds.8 Obviously, investors who exercise the put give up the opportunity to convert their bonds into stock; it would be worth taking advantage of the guarantee only if the conversion price of the bonds was well below the exercise price of the put.9

Mandatory Convertibles

In recent years a number of companies have issued preferred stock or debt that is automatically converted into equity after several years. Investors in mandatory convertibles receive the benefit of a higher current income than common stockholders, but there is a limit on the value of the common stock that they ultimately receive. Thus they share in the appreciation of the common stock only up to this limit. As the stock price rises above this limit, the number of shares that the convertible holder receives is reduced proportionately.

Valuing Convertible Bonds

The owner of a convertible owns a bond and a call option on the firm’s stock. So does the owner of a bond–warrant package. There are differences, of course, the most important being the requirement that a convertible owner give up the bond in order to exercise the call option. The owner of a bond–warrant package can (generally) exercise the warrant for cash and keep the bond. Nevertheless, understanding convertibles is easier if you analyze them first as bonds and then as call options.

Imagine that Eastman Kojak has issued convertible bonds with a total face value of $1 million and that these can be converted at any stage to one million shares of common stock. The price of Kojak’s convertible bond depends on its bond value and its conversion value. The bond value is what each bond would sell for if it could not be converted. The conversion value is what the bond would sell for if it had to be converted immediately.

8Of course, this guarantee would not be worth much if the company was in financial distress and couldn’t buy the bonds back.

9The reasons for issuing LYONs are discussed in J. J. McConnell and E. S. Schwartz, “The Origin of LYONs: A Case Study in Financial Innovation,” Journal of Applied Corporate Finance 4 (Winter 1992), pp. 40–47. For a discussion of how to value an earlier LYON issue by Waste Management see J. McConnell and E. S. Schwartz, “Taming LYONs,” Journal of Finance 41 (July 1986), pp. 561–576.

investors became disenchanted with dot.com companies and Amazon’s stock price fell by 75 percent to $15. This was well below the conversion price of $78.03. Convertible bondholders might have hoped that the bond value would provide a secure floor to the value of their investment. Unfortunately, by early 2001 Amazon’s bonds no longer looked as safe as they once had, and Moody’s placed its convertible in the junk bond category Caa. By the spring of that year the price of the convertible had fallen to about $400 and offered a promised yield to maturity of 20 percent.

Figure 23.2(b) shows the possible conversion values at maturity of Kojak’s convertible. We assume that Kojak already has one million shares of common stock outstanding, so the convertible holders will be entitled to half the value of the firm. For example, if the firm is worth $2 million,11 the one million shares obtained by conversion would be worth $1 each. Each convertible bond can be exchanged for 1,000 shares of stock and therefore would have a conversion value of 1,000 1 $1,000.

Kojak’s convertible also cannot sell for less than its conversion value. If it did, smart investors would buy the convertible, exchange it rapidly for stock, and sell the stock. Their profit would be equal to the difference between the conversion value and the price of the convertible.

Therefore, there are two lower bounds to the price of the convertible: its bond value and its conversion value. Investors will not convert if bond value exceeds conversion value; they will do so if conversion value exceeds bond value. In other words, the price of the convertible at maturity is represented by the higher of the two lines in Figure 23.2(a) and (b). This is shown in Figure 23.2(c).

Value before Maturity We can also draw a picture similar to Figure 23.2 when the convertible is not about to mature. Because even healthy companies may subsequently fall sick and default on their bonds, other things equal, the bond value will be lower when the bond has some time to run. Thus bond value before maturity is represented by the curved line in Figure 23.3(a).12

Figure 23.3(b) shows that the lower bound to the price of a convertible before maturity is again the higher of the bond value and conversion value. However, before maturity the convertible bondholders do not have to make a now-or-never choice for or against conversion. They can wait and then, with the benefit of hindsight, take whatever course turns out to give them the highest payoff. Thus before maturity a convertible is always worth more than its lower-bound value. Its actual selling price will behave as shown by the top line in Figure 23.3(c). The difference between the top line and the lower bound is the value of a call option on the firm. Remember, however, that this option can be exercised only by giving up the bond. In other words, the option to convert is a call option with an exercise price equal to the bond value.

Dilution and Dividends Revisited

If you want to value a convertible, it is easiest to break the problem down into two parts. First estimate bond value; then add the value of the conversion option.

When you value the conversion option, you need to look out for the same things that make warrants more tricky to value than traded options. For example,

11Firm value is equal to the value of Kojak’s common stock plus the value of its convertible bonds.

12Remember, the value of a risky bond is the value of a safe bond less the value of a put option on the firm’s assets. The value of this option increases with maturity.

(a) Before maturity the bond value of Eastman Kojak’s convertible bond is close to that of a similar default-free bond when firm value is high, but it falls sharply if firm value falls to a very low level. (b) If investors were obliged to make an immediate decision for or against conversion, the value of the convertible would be equal to the higher of bond value or conversion value. (c) Since convertible bondholders do not have to make a decision until maturity, (b) represents a lower limit. The value of the convertible bond is worth more than either bond value or conversion value.

dilution may be important. If the bonds are converted, the company saves on its interest payments and is relieved of having to eventually repay the loan; on the other hand, net profits have to be divided among a larger number of shares.13 Companies are obliged to show in their financial statements how earnings would be affected by conversion.14

Also, you must remember that the convertible owner is missing out on the dividends on the common stock. If these dividends are higher than the interest on the

13In practice investors often ignore dilution and calculate conversion value as the share price times the number of shares into which the bonds can be converted. A convertible bond actually gives an option to acquire a fraction of the “new equity”—the equity after conversion. When we calculated the conversion value of Kojak’s convertible, we recognized this by multiplying the proportion of common stock that the convertible bondholders would receive by the total value of the firm’s assets (i.e., the value of the common stock plus the value of the convertible).

14These “diluted” earnings take into account the extra shares but not the savings in interest payments.