# Valuation Measuring and managing the value ofpanies - Koller T.

ISBN 0-471-36190-9

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Exhibit 20.17 Valuation of a Callable, Convertible Bond

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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 payoff iiim(1/4)(529.94) +1.08B = 186.34 Down state payoff -M(V4)(529.94) + 1.08B-110.00]

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(l/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

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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:

H0 = SI, 152,61 =$100/{1 +}/) + SI, 100/(1 +yf

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.

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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.

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