The payoff to the bondholder as a function of the value of the company's assets is shown in Figure 1. The profile is similar to someone holding a share of stock for which they have sold a call at one exercise price and bought a fraction of a call at a higher exercise price.
| . | _____________._______ | / | / | / |/________________________________ X X(1+1/q) Market Value of Assets of Corp. Figure 1
Suppose the number of shares outstanding is n and let V be the value of the compay assets. Let X be the face value of all the m convertible bonds and let c be the conversion ratio. The market price S of the stock on the day of maturity of the bonds would be S = (V-X)/n, if the bonds are not converted. If they are converted then S = V/(n+cm). The bonds will be converted only if
X/m < cS X/m < cV/(n+cm) V > X(n+cm)/cm V > X(1+q)/q,where q = cm/n is the dilution ratio.
The value of the portfolio consisting of all the convertible bonds at the date of maturity is equal to the maximum of:
As an equation this condition is:
Suppose a company's debt consists of $20 million of convertible bonds with a maturity date two years from now. The value of the assets of the company is $30 million. The risk-free interest rate is five percent per year and the volatility of the value of the company's assets is 0.4 per year. There are 200,000 shares of stock outstanding and 20,000 convertible bonds with a conversion ratio of 20. The dilution ratio q is then (20)(20,000)/200,000=2. Assuming that the Black-Scholes formula for call value applies, then the value of all the convertible bonds is:
Value of Value of Value of Value of Company Stockholder Debt Convertible Assets Equity Bonds ($mill) ($mill) ($mill) ($mill) 5.000 0.020 4.980 4.982 10.000 0.559 9.441 9.534 15.000 2.320 12.680 13.239 20.000 5.238 14.762 16.352 25.000 8.958 16.042 21.223 30.000 13.178 16.812 22.060 35.000 17.701 17.299 24.949 40.000 22.405 17.595 27.916
The optimal strategy for the bondholders is to postpone conversion if the current dividend is low in comparison with interest received on the bonds. This would enable the bondholders to obtain the benefits of the capital gains without giving up the interest and relative security of bond ownership. For this reason, convertible bond issuers often include a call provision that can be used to force early conversion. A call provision gives the company the right to buy back the bonds. If convertible bonds are called the holders have a short time to convert to common stock rather than sell their bonds back to the company.