Revisiting Optimal Call Policy for Convertibles

Abstract

When a company calls its convertible bonds, it typically must give the bondholders a notice period of approximately 30 days to decide whether to convert the bonds. This important institutional detail substantially affects the optimal call policy for convertible bonds. When the company calls the bonds, it fixes the price at which bondholders can redeem them, effectively giving bondholders a 30-day put option. The optimal time to call the convertibles minimizes the value of the conversion option net of the put option. This optimization problem is solved here, and a simple decision rule for the company results. This solution contains those of previous researchers as a special case.

When a company calls its convertible bonds, it typically must give the bondholders a notice period of approximately 30 days to decide whether to convert the bonds. This important institutional detail substantially affects the optimal call policy for convertible bonds.

By incorporating this detail, this article provides a new benchmark for the appropriate point at which to call a convertible bond issue. Academics and practitioners have long advocated a “safety cushion” for calling convertible bonds that is above the point at which the stock price equals the conversion price. I rationalize and quantify this safety cushion.

Specifically, the safety cushion is rational because when a company calls a convertible bond issue, it is effectively giving a put option on the underlying stock to the bondholders. That is, by calling the convertibles, the company fixes the price at which bondholders can redeem the bonds, which gives bondholders a put option for the notice period (in practice, usually 30 days). The optimal time to call the convertibles minimizes the value of the conversion option net of the put option. I analytically solve the problem of the optimal call policy: Companies call so as to maximize the value of their existing equity; to do so, they minimize the value of the bond and warrant net of the stock and put options that must be given up by calling the convertible bonds. I derive a closed-form solution to the optimal call policy for the special case in which exercise of the warrants has no dilutive effect on the existing equity. The more general case of arbitrary dilution can be readily solved numerically.

I find that companies will delay calling their convertible bonds when there is a notice period, and this delay increases monotonically as the length of the notice period increases. The instance in which the notice period is of zero length (subject of previous research) is a special case of the optimal call policy found in my approach.

I document the sensitivity of the optimal call policy to changes in key parameters. The results indicate that, all else being equal, optimal call policy is most sensitive to changes in the volatility of the underlying stock. It is less sensitive to the length of the call period, the dilutive effect the convertibles will have on the existing equity, and the maturity of the bonds. Furthermore, optimal call policy is quite insensitive to interest rate environments.

When several variables are allowed to change simultaneously, optimal call policy can be much higher than for the base case described in the article. For instance, with moderate potential dilution but high volatility, a long maturity, and a long call period, the optimal call policy can easily be above a 50 percent premium (that is, the company in this case should call when the stock price is more than 50 percent higher than the conversion price). Similarly, with a moderate call period but high volatility, a long maturity, and low potential dilution, the optimal call policy can easily be above a 70 percent premium.

These results are important for both financial managers who wish to correctly minimize the value of their companies' liabilities and bond traders who wish to correctly anticipate strategic call behavior by those financial managers.

I am grateful for the helpful comments of Tom Arnold, Craig Holden, Shane Johnson, Ted Moore, and seminar participants at the 1999 Frank Batten Young Scholars Conference at the College of William and Mary and the 1999 Financial Management Association meeting. Any errors are my own.