## Liability Options

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

Exhibit 20.15 Terms for ABB 2.75s due 2004

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

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.

Exhibit 20.17 Valuation of a Callable, Convertible Bond

To value the callable, convertible bond at the end of the first year, given that the value of the firm has increased to $539,900 before the coupon and $529,000 ex coupon, we form a replicating portfolio that is composed of "m" units of the company (divided by 4 because the bond holders get 1/4th of the company) plus B risk-free bonds. This portfolio will have exactly the same payoffs as the bond in the second year:

Up State pnyo/f mu( !/4)(529.94) + 1 .t)SB = lHft.34 Down stjitL' payoff -[mii(l/4)(52M4) + 1.08B - ] 10.00)

Solving, we find that m = 0.946, and B = 15.87, therefore the market value of the callable, convertible bond is the same as the market value of the replicating portfolio,

Market value = m (1/4) (529.94) + B- 141.25 Plus the dividend, $10, for a total of $151.25.

Unfortunately for the bond holders, this market value is higher than the value if called, $140, therefore the firm will call the bonds. As a preventative measure, the bond holders will convert before the firm can call, and will receive 25 percent of $529.90 plus a coupon of $10, a total of $142.50. Thus, their expected payout in this state of nature is $142,500.

To value the bond in other states of nature, we repeat the replicating portfolio approach to estimate the market value of the bond and compare it with the value if converted or called. For example, in the down state in the first year the market value, $101.90, is higher than the value if called or converted. Working backward, we find that the market value today is $115,261 for all of the callable, convertible bonds, or $1,152.61 per bond. Exhibit 20.18

Exhibit 20.18 Values of the Callable Convertible Bond and Implied Interest Rates

shows the value of the callable, convertible bond in each state of nature, and the discount rates between them. Note that these rates are all greater than the 8 percent risk-free rate.

Whenever the enterprise approach for valuing a company is used, the market value of equity is estimated by first valuing the whole company, the enterprise, and then subtracting the market value of debt to estimate the value of equity. Having a good estimate of the market value of convertible securities is often crucial. In the example, the value of the company, $400,000, less the market value of callable, convertible debt, $115,261, is equal to the value of equity, namely $284,739. Had we used the face value of the debt, $100,000, we would have overestimated the equity value by 5.4 percent.

The Cost of Capital for Callable, Convertible Securities

Professor Eugene Brigham once surveyed the chief financial officers of 22 companies that had issued convertible debt. Of those surveyed, 68 percent said they had used convertible debt because they believed their stock price would rise and that convertibles would provide a way of selling common stock at a price above the existing market. Another 27 percent said that their company had wanted straight debt but had found conditions to be such that a straight bond issue could not be sold at a reasonable rate of interest.

Neither reason makes sense. Convertible bonds are not cheap debt. Because convertible bonds are riskier, their true cost of capital is greater (on a before-tax basis) than the cost of straight debt. Also, convertible bonds are not equal to deferred sale of common stock at an attractive price. The uncertain sale of shares at $28, each at some unknown future date, can hardly be compared directly with a current share price of $25.

The risk of convertible debt is higher than that of straight debt and lower than that of equity, so its true opportunity cost lies between these limits. The yield to maturity on convertible debt (often lower than on the company's senior debt) has nothing to do with its opportunity cost, because convertible debt has an option embedded in it, and options are much riskier than debt. Going back to our numerical example, if we naively estimate the cost of capital on the callable, convertible bond by using the observed price of the bond, $1,152.61, to calculate a yield to maturity, we come up with an estimate of 2.13 percent:

This is obviously wrong because it is less than the risk-free rate of 8 percent. If we use the true risk-adjusted rates in Exhibit 20.18 the implied geometric average required rate of return on the callable, convertible fond is 14.74 percent pretax.

Three broad categories of information are needed to value a callable, convertible bond and to determine its cost of capital:

1. The interest rate environment. Ideally, we would capture the entire term structure and its expected variability. But our model can handle only one random variable at a time, and the variability of the company's common stock is the most important element. Consequently, the interest rate environment is captured by the yield to maturity on a Treasury bond with the same maturity as the convertible bond.

2. Characteristics of the bond. We need to know the amount outstanding, the face value, the number of months to maturity, the conversion price, the number of months until the first coupon date, the time between coupons, the annual coupon rate, and the call provisions (the call prices and their timing).

3. Characteristics of the common stock. Since the bond is convertible into common stock, we need to know the current stock price, the equity beta, the expected dividend per share, the ex dividend dates, the number of shares outstanding, the equity volatility, and the amount of senior debt outstanding.

Exhibit 20.19 shows our estimate of the value and the before-tax cost of capital for a sample of seven callable, convertible bonds. The results were provided by McKinsey's convertible securities pricing model. In every case, the before-tax cost of capital for the callable, convertible bond is higher than the coupon rate, and in all but one case the difference is substantial.

Exhibit 20.19 Valuation and Cost of Capital for Convertible Bonds

March 2000

Company |
Theoretical Value1 |
Market Price1 |
Difference (percent) |
Coupon Rate (percent) |
Cost of Capital (percent) |

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