The Cost Formula
The cost of a CS is a weighted average of the cost of its debt element and the cost of a call option on the issuer’s shares, since the investor in a CS is a lender and the holder of a call option on the value of the firm. The difference between a conversion right and a regular call option is that a CS holder gets new shares upon exercise. It follows that if the price at which the CS holder is entitled to shares is below market price, then the value of all corporate equity, including the convertor’s, is diluted. This explains why a convertible warrant is worth less than a straight call option on the company’s shares whose exercise leaves existing equity intact.
The market will discount each element to the present using appropriate required rates of return. The cost of CS is an average of the rates weighted by each element’s share of total market value.
The starting point is the textbook formula (see, for example, Copeland, Weston and Shastri, 2004, Chapter 15) which can be summarized as follows.
These are the essential terms: kcv is the cost of convertible debt; B is the value of debt element; W is the value of equity element, being the value of a call option on the company’s shares; B + W is the value of the convertible security; kb is the required rate of return on debt; and kc is the required rate of return on a call option on the company’s shares. See below.
Using the capital asset pricing model,
kc = Rf + [E(Rm) − Rf] βc
where kc is the required rate of return on a call option on the company’s shares with the same maturity as the CS; Rf is the risk-free rate of return for a bond with the same maturity as the CS; E(Rm) is the expected rate of return on a portfolio comprising all the shares in the market; βc is the systematic risk of the call option expressing its correlation with the market. βc is computed by reference to the β of an underlying share of the company adjusted to option using the Black–Scholes option pricing programme:
kcv = kb B ÷ (B + W) + kc W ÷ (B + W)
The basic Black–Scholes option pricing scheme assumes that the issuer pays neither dividends nor tax (see, for example, Berk and DeMarzo, 2007, Chapters 21, 22, 23; Brealey and Myers, 2007, Part 6; and packages like the London Business School’s).
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