## I. The Valuation of Corporate Securities

In a recent paper Black and Scholes [3] presented an explicit equilibrium model for valuing options. In this paper they indicated that a similar analysis could potentially be applied to all corporate securities. In other papers, both Merton [8] and Ross [11] noted the broad applicability of option pricing arguments. At the same time Black and Scholes also pointed out that actual security indentures have a variety of conditions that would bring new features and complications into the valuation process.

Our objective in this paper is to make some general statements on this valuation process and then turn to an analysis of certain types of bond indenture provisions which are often found in practice. Specifically, we will look at the effects of safety covenants, subordination arrangements, and restrictions on the financing of interest and dividend payments.

Throughout the paper we will make the following assumptions:

- a1) Every individual acts as if he can buy or sell as much of any security as he wishes without affecting the market price.
- a2) There exists a riskless asset paying a known constant interest rate
*r*. - a3) Individuals may take short positions in any security, including the riskless asset, and receive the proceeds of the sale. Restitution is required for payouts made to securities held short.
- a4)Trading takes place continuously.
- a5)There are no taxes, indivisibilities, bankruptcy costs, transaction costs, or agency costs.
- a6)The value of the firm follows a diffusion process with instantaneous variance proportional to the square of the value.

This last assumption is quite important and needs some amplification. Until very recently this was the standard framework for discussions of contingent claim pricing. Increasing evidence, however, indicates that it may not be completely. Appropriate.^{1} The instantaneous variance may be some other function of the firm value, and possibly dependent on time as well.^{2} It may also depend on other random variables. Furthermore, discontinuities associated with jump processes may be important.^{3} Nevertheless, this assumption provides a useful setting for the points we want to make and facilitates comparison with earlier results.

With these assumptions, the standard hedging or capital asset pricing arguments lead to a valuation equation. For the process we are considering here, it is derived in its most general form in Merton [9] as

where *f* is a generic label for any of the firm's securities, *V* is the value of the firm, *t* denotes time, *f*.

Suppose the firm has outstanding only equity and a single bond issue with a promised final payment of *P*. At the maturity date of the bonds, *T*, the stockholders will pay off the bondholders if they can. If they cannot, the ownership of the firm passes to the bondholders. So at time *T*, the bonds will have the value min(*V, P)* and the stock will have the value max(*V* – *P*, 0).

Now this formulation already implicity contains several assumptions about the bond indenture. The fact that *P* were assumed known (and finite) implies that the bond contract renders them determinate by placing limiting restrictions on, respectively, the firm's investment, payout, and further financing policies.

Furthermore, it assumes that the fortunes of the firm may cause its value to rise to an arbitrarily high level or dwindle to nearly nothing without any sort of reorganization occurring in the firm's financial arrangements. More generally, there may be both lower and upper boundaries at which the firm's securities must take on specific values. The boundaries may be given exogenously by the contract specifications or determined endogenously as part of an optimal decision problem.

The indenture agreements which we will consider serve as examples of a specified or induced lower boundary at which the firm will be reorganized. An example of an upper boundary is a call provision on a bond.^{4} Also, the final payment at the maturity date may be a quite arbitrary function of the value of the firm at that time” ξ(*V*(*T*)).

It will be helpful to look at this problem in a way discussed in Cox and Ross [5, 6].^{5} The valuation equation (1) does not involve preferences, so a solution derived for any specific set of preferences must hold in general. In particular, the relative value of contingent claims in terms of the value of underlying assets must be consistent with risk neutrality.^{6}

If we know the distribution of the underlying assets in a risk-neutral world, then we can readily solve a number of valuation problems.^{7} We can in our problem think of each security as having four sources of value: its value at the maturity date if the firm is not reorganized before then, its value if the firm is reorganized at the lower boundary, its value if the firm is reorganized at the upper boundary, and the value of the payouts it will potentially receive. Although the first three sources are mutually exclusive, they are all possible outcomes given our current position, so they each contribute to current value. The contribution to the total value of a claim of any of its component sources will in a risk neutral world simply be the discounted expected value of that component.

For any claim *f* let *f*, as given by the contract, if the firm is reorganized at the lower (upper) boundary *τ*. Denote the distribution in a risk neutral world of the value of the firm at time

and

where *κ*(*·*) denotes the interval (

The contribution of the potential value at the reorganization boundaries is somewhat different. Formerly we knew the time of receipt of each potential payment but not the amount which would actually be received. Here the amount to be received at each boundary is a known function specified by the contract, but the time of receipt is a random variable. However, its distribution is just that of the first passage time to the boundary, and the approach taken by Cox and Ross can still be applied.

Let

This development also disposes of uniqueness problems, since economically inadmissible solutions to the valuation equation are automatically avoided by the probabilistic approach. However, it cannot be applied directly to situations where the boundaries must be determined endogenously as part of an optimal stopping problem.

Actual payouts by firms, of course, occur in lumps at discrete intervals. In many situations it is more convenient and perfectly acceptable to represent these payouts as a continual flow. Many other times, however, it is preferable to explicitly recognize the discrete nature of things. This is particularly true in optimal stopping problems when the structure of the problem dictates that decisions will be made only at these discrete points. An example in terms of options would be an American call on a stock paying discrete dividends. Restrictions on the financing of coupon payments to debt, which we will discuss later, provides an example in terms of corporate liabilities. To solve these problems we could work recursively, with the terminal condition at each stage determined by the solution to the previous stage. Start at the last payment date. If a decision is made to stop at this point, the claimholder receives a payoff given by the terms of the contract. If he does not stop, his payoff is the value of a claim with one more period to go, given that the value of the firm is its current value minus the payment. This value is determined by the payment to be received at the maturity date. The claimholder can then determine his optimal decision rule. With the optimal decision rule specified, we can find the value of the claim as a function of firm value at the last decision point. At the next-to-last decision point we would face an identical problem except that the value function we just found would take the place of the function giving the payment to be received at the maturity date, By working backward we can find the value of the claim at any time. Note that this gives only an approximate solution when the optimal decision points are actually continuous in time. However, we could always get a better approximation by adding more discrete decision points, even though no payouts are being made at these additional points.

Throughout the paper we will make use of the relationship between the equilibrium expected return on any of the individual securities of the firm, *v*, and the (exogenously determined) equilibrium expected return on the total firm, *μ*. As given in Black and Scholes [3] and Merton [9], this is