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There were two sources of uncertainty. First is the price of natural gas, which is known today, but which becomes more uncertain. Second is the uncertainty about the quantity of natural gas in the ground. It has a wide range of uncertainty today, but the range narrows if the company decides to explore the field.
When the analysis was completed the highest value was obtained by making the decision to explore now, by deferring the development decision until year 3, and making an expansion decision in year 11. The value of this set of decisions was 125 percent higher than the value of the base case, which was to develop the field now and wait for three years before deciding to explore.
It is easy to see from these examples that asset options can substantially alter the value of a business. The fact that the options exist does not mean that they are optimally managed. There are two problems. First, managers are not trained to recognize real options. Second, they are usually not familiar with the methodological advances that have made real options easier to apply and to understand. Understanding asset options can provide insight into managing flexibility as a new approach for dealing with uncertainty.
We turn to options implicit in various sources of funding. These liability options are important because they affect the company's weighted average cost of capital.
Plain vanilla approaches to valuation describe the weighted average cost of capital as the simple weighted average of the after-tax opportunity costs of debt and equity. But hybrid securities that have option features are often used as sources of capital. We looked at a random sample of 100 companies listed on the New York Stock Exchange and found that 43 of them had convertible securities outstanding. The yield to maturity on convertible securities is usually much lower than on straight debt with the same maturity and quality. But the yield on convertible securities is a particularly bad estimate of their actual cost of capital.
The first part of our analysis will show how to value callable, convertible debt. Then we will discuss how its cost of capital is estimated.
Valuing Callable, Convertible Securities
Convertible bonds allow their owners to convert them into another security at a predetermined exchange ratio for a fixed interval of time. The ABB 2.75s described in Exhibit 20.15 could be converted into common stock at a price of $112.41 per share anytime during their life. The actual common stock price at
Exhibit 20.15 Terms for ABB 2.75s due 2004
Amount Authorized Amount Outstanding Issued Due
$150.0 million $150.0 million 7/10/97 7/10/2004 7/10
Source: Remco Bos, Fortis Investment Bank.
the time of data collection was $112.43, almost exactly equal to the conversion price. When exercising conversion rights, the bondholder gives up, as the exercise price, the present value of the expected bond payments. For ABB, the bondholders would give up the bond payments in return for 0.8896 shares (per $100 of face value on the bond). Thus, convertible bonds have a changing exercise price.
To show how to value a callable, convertible bond, let's look at a numerical example. The following set of assumptions details the interest rate environment, the way the value of the company varies, and the provisions of the callable, convertible bond.
• The constant risk-free rate is 8 percent per year.
• The company is worth $400,000 right now (no senior debt).
• There is a 62 percent probability that company value will increase by 35 percent and a 38 percent probability that it will decrease by 26 percent as shown in Exhibit 20.16.
• Two securities are outstanding: 150 shares of stock, and 100 callable, convertible bonds that can be converted at a ratio of one-half share per bond.
• If bonds are converted, bondholders will own 50/(150 + 50) = 25 percent of the company. If the bondholders decide to convert, they receive the coupon for that period.
• Each $1,000-face-value bond pays $100 per period coupon.
• Anytime before maturity, stockholders can call the bonds for $1,400. (For simplicity, however, we assume the call decision is made only at the end of the first year.)
• The company pays no dividends.
• The first bond coupon has just been paid.
Exhibit 20.16 Value of a Hypothetical Company That Pays a Constant Coupon
To value the callable, convertible bonds, we start with their final payouts, determine the optimal action, and compute their value at the end of year 1 conditional on the value of the company (illustrated in Exhibit 20.16). Given that the value of the company has gone up the first year, the final value of the company at the end of the second year can be $705,349 or $382,592, ex coupon. If it is $705,349 the bondholders receive $186,337 if they convert and $110,000 otherwise. Obviously, they will convert. If the company value is $382,592, they will not convert, preferring to receive the $1,000 face value per bond plus the last coupon for a total of $110,000 rather than the conversion value of $105,684 (25% of $382,592 plus $10,000). With these facts, we can determine the market value of the bond at the end of year 1. Results of our calculations are given in Exhibit 20.17.