The Valuation of Convertibles

By Markus Zeder, Prof. Dr. R. Volkart, 31 March 2000

The thesis of Markus Zeder under Professor R. Volkart at the Institute of Swiss Banking of the University of Zurich explains and analyzes different valuation approaches. It additionally discusses the characteristics of convertibles and, through practical examples, illustrates their sensible use in portfolio management.

Executive Summary

1. Definition of the problem

A convertible consists of a customary bond (without any conversion right) and a conversion right that provides the investor with the right to convert the convertible into a specific number of shares of the issuing company during a specific time period. These two components are thus inseparably linked with each other and must therefore be incorporated into the valuation as a portfolio. This corresponds to the classical interpretation of this instrument. By means of put-call parity, a convertible can also be transformed into and viewed as a portfolio consisting of a put option with the same parameters as the call option plus an equity. This interpretation of convertibles opens up new insights into the characteristics of this rather complex instrument. It thus appears plausible that a convertible corresponds to an investment in the underlying security - the stock - plus a hedge against downward price movements, which is provided by the put option. This hypothesis may hold true without restriction in the theoretical world of modeling, assuming perfect and efficient capital markets. But in the real world, certain assumptions must be relaxed or even discarded completely. These changes can cause the complexity and mathematical effort involved to increase immensely, and they also mean that the parities no longer have unrestricted validity, but only approximative validity.

2. Organization of the paper

The paper is divided into three sections.

The first section looks at the characteristics of convertibles. The value boundaries within which prices must move are documented formally as well as graphically. Observations on the simplest form of a convertible without additional rights are followed by more complex examples in which individual, additional influences are built in step by step. This section also contains thoughts about and observations on the reasons for issuing convertibles, from the perspective of the issuer as well as of the investor. The final part of this section explains the differences between convertibles and a similar instrument - warrant bonds.

The second section forms the bulk of this paper. It is devoted to the question of valuing such a complex hybrid instrument. First of all, the section looks at various classical valuation models that view a convertible as a portfolio consisting of an equity plus a conversion right, and then summarizes this information in an overview. A detailed example with accompanying diagrams is provided to make it easier to follow the mathematically complex process. In a next step in the question of valuing convertibles, a convertible is viewed from a new perspective through put-call parity, i.e. as a portfolio consisting of a put option plus an equity. The characteristics and valuation of this put element are analyzed and illustrated.

The third section aims to document the characteristics of convertibles through practical examples based on an empirical study. It illustrates how a convertible generates a return similar to that of the underlying equity when stock prices are rising, whereas a convertible records a significantly lower loss than that of the underlying equity when stock prices are falling. An investor who fears that a price drop may occur following a sharp rise in the stock price should sell the stock and buy a convertible with a low premium. This switch strategy allows the investor to continue participating in rising stock prices and simultaneously provides a hedge against the feared price drop. Moreover, this strategy is less expensive for the investor than hedging his equity position through a customary put option. Based on these examples, this section also graphically illustrates the different value zones that a convertible can pass through, and their characteristics (see figure 1). The dashed line depicts the value boundaries, while the continuous line represents the convertible.

3. Results

In the second section, it was possible to confirm the hypothesis that a convertible may also be viewed as a put option plus an equity, thus leaving behind an implicit option after deducting the equity position. In the case of perfect and efficient capital markets, this put option corresponds precisely to the conversion premium. In practice, given the imperfect capital markets of the real world, the conversion premium consists of a put element plus a swap that hedges the exchange of dividends for coupon payments.

In the third section, the empirical study confirmed the dual behavior of convertibles and thus their asymmetrical behavior. A convertible participates in rising prices of its underlying stock (equity component) while simultaneously hedging against setbacks in the underlying stock (bond component). It was shown that the return of a convertible is often not far behind that of the underlying stock and that a convertible can provide a certain degree of hedging against a downward trend of the stock price. However, if the price of the underlying stock falls very sharply, thus seriously jeopardizing the future of the company, this hedging effect fails and the convertible suffers heavy losses. Furthermore, the section examined and confirmed the assertions about the value zones that a convertible can pass through, and thus the various characteristics regarding participation in the performance of the stock price. Correlation was applied for this purpose.

4. General assessment

This paper presented valuation models that either maintain a constant interest rate or consider the interest rate structure through forward rates (time-based interest rates). In an additional step, it would be interesting to observe the quantitative change in value when stochastic interest rates are employed. This would be even more consistent with the circumstances that prevail in practice today.

An empirical study on the arbitrage described in this paper with respect to the implicit put option of the convertible versus customary put options in the market would shed more light on the actual efficiency in the market for convertibles.