Valuing Corporate Securities: Some Effects of Bond Indenture Provisions

In a recent paper 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 8 and 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. 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. 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 σ 2 , p ( V , t ) and p ′ ( V , t ) , and 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 5.5 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. 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. 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 3 and 9, this is ν − r = ( V f υ / f ) ( μ − r ) . Furthermore, since the process followed by any individual security is a transformation of that governing the total value of the firm, its instantaneous variance will be σ 2 V 2 f v 2 . Thus we can write the ratio of the instantaneous standard deviation of the rate of return on any individual security to that of the firm as V f v / f . Another way to say this is that in equilibrium the excess expected return per unit of risk must be the same for all of the firm's securities. The elasticity V f υ / f thus conveys the essential information about relative risk and expected return. In subsequent use of the term elasticity, we will always be referring to this function. In this section we will consider the effects of safety covenants on the value and behavior of the firm's securities. Safety covenants are contractual provisions which give the bondholders the right to bankrupt or force a reorganization of the firm if it is doing poorly according to some standard. One standard for this may be the omission of interest payments on the debt. However, if the stockholders are allowed to sell the assets of the firm to meet the interest payments, then this restriction is not very effective. In this situation a natural form for a safety covenant is the following: if the value of the firm falls to a specified level, which may change over time, then the bondholders are entitled to force the firm into bankruptcy and obtain the ownership of the assets. In this form of agreement, interest payments to the debt do not play a critical role, so we will assume that the firm has outstanding only a single issue of discount bonds. We will, however, assume that the contractual provisions allow the stockholders to receive a continuous dividend payment, aV, proportional to the value of the firm. With a continuous time analysis, it is quite reasonable for the time dependence of the safety covenant to take an exponential form, so we will let the specified bankruptcy level, C 1 ( t ) , be C e − γ ( T − t ) . This formula holds for all C e − γ ( T − t ) ⩽ P e r ( T − t ) . An interesting choice is C e − γ ( T − t ) = ρ P e − r ( T − t ) , with 0 ⩽ ρ ⩽ 1 , so that the reorganization value specified in the safety covenant is a constant fraction of the present value of the promised final payment. For clarity in making comparisons, we will use only this form below. 9 has extensively studied in this setting the properties of discount bonds when there are no safety covenants and no dividends. Rather than repeat parts of his analysis, we will focus on properties which are particular to the existence of safety covenants. The most basic properties, such as the fact that B is an increasing function of V and t and a decreasing function of σ 2 ,r, and a remain the same. It is easy to verify that B is an increasing function of ρ. Contrary to what is sometimes claimed, premature bankruptcy is not in itself detrimental for the bondholders. It is in their interests to have a contract which will force bankruptcy as quickly as possible. If bankruptcy occurs, the total ownership of the firm will pass to the bondholders, and this is the best they can achieve in any circumstances. A second look shows that B is a convex function of ρ, going to P e − r ( T − t ) , the riskless value, as ρ goes to one. The elasticity of B is a decreasing concave function of ρ, going to zero as ρ goes to one, so a higher bankruptcy level always makes the debt safer. The elasticity of the stock is an increasing convex function of ρ. Safety covenants provide a floor value for the bond which limits the gains to stockholders from somehow circumventing the other indenture restrictions. For example, as either σ 2 or a goes to infinity, the value of the bonds goes to ρ P e − r ( T − t ) rather than zero. Similarly, if we compare the riskiness of bonds of firms differing only in investment policy or dividend policy, we find important differences for large values of a and σ 2 . If ρ = 0 , the elasticity is an increasing concave function of a, going to one as a goes to infinity. If ρ > 0 , the elasticity has an initial increasing concave segment, but then reaches a maximum, followed successively by decreasing concave and convex segments going to zero as a goes to infinity. The behavior of the elasticity with respect to the variance is for small values of σ 2 qualitatively the same as the case with no safety covenant, but as σ 2 it zero rather than The behavior of the with respect to the value of the firm is also interesting and is in the stock is entitled to receive as the value of the firm we find that the riskiness and expected return of the stock first then and as the value the bankruptcy we could think of this in the For values of V the boundary it is quite that the stockholders will and their claim is quite As V we a stage where bankruptcy is no but it is most that will be for the stockholders at the maturity date. The value of the stock from the value of the it is entitled to and these are proportional to the value of the firm and have As V the part of the value to the amount it may receive at the maturity date, and the riskiness as V reaches a very high level, it certain that the bonds will be in and the stock to a in the firm as a with of V / ( V − P e − r ( T − t ) ) . of the Another form of indenture the subordination of the claims of one of debt the bonds, to of a second the bonds. At the maturity date of the bonds, payments can be made to the debt only if the promised payment to the debt has Suppose that both of bonds are discount bonds, and let the promised payments to and debt respectively, P and at the maturity date the value of each of the firm's securities will be as in The in the first section that the values of and discount bonds, and of options with could be given a the case with no payouts and no safety covenants. the distribution function ( V ( T ) , T V ( t ) , t ) . as in the values of the firm's securities can be as the distribution when these are by the discount e − r ( T − t ) . as the which is by the indicated does achieve its of giving the bonds a value than they would have if they were the fraction of an bond the value of the bonds will be than P / ( P ) the value of a single issue with promised payment P . This directly from the of discount bonds in the final payment. The effects of a safety covenant on the debt are just as we would is a decreasing convex function of ρ, a when ρ = 1 . For ρ > 1 , it is an increasing convex a when ρ = P . For values of ρ 1 , the of the safety covenant to the bondholders and are at the of the bondholders as as the As ρ the bondholders to receive as and the falls the In the of this section we will let ρ = the debt, the value of the debt can be an increasing function of σ 2 . of the function shows is an increasing function of σ 2 for V than V . This that the bondholders as a may some have interests with respect to in the total riskiness of the firm's investment To the value of their the bondholders must on the right to investment policy which will the risk of the firm. As we can be an increasing function of time to the debt, it is possible for the debt to be at and if such a development is the bondholders would find it in their interests to to the maturity date of the bond Although it is possible for the value of the bonds to be either a decreasing or increasing function of the interest it is always a decreasing function of the dividend The behavior of the elasticity with respect to time maturity for the firm and values is in Until 1 = V P , 2 = P V P , 3 = P V = ( r − a ) 1 2 σ 2 , . = ( r − a ) 1 2 σ 2 Suppose that the firm has interest paying bonds In this section we will that it is quite important the stockholders are allowed to the to make the payments to the bondholders. of interest paying bonds have assumed that the stockholders are allowed to sell the assets of the firm to make these payments. Many bonds have contractual provisions which the to which this can be To focus on the effects of these that the of assets for this is in fact payments, and any dividend payments, must be by new securities. To the value of their claim the bondholders must also that the new securities be equity or bonds. For the bonds have a promised final payment of P and make interest payments of = P e r t ′ , where t ′ is the between payments. If an interest payment is not the firm is in and the promised payment P The bonds would then be ( V , P ) . this is the value the bonds can possibly the bondholders would always be to a payment and the stockholders would always want to make the payment if there is any way they possibly can. However, they may not be This would the value of the equity the payment is if it is would be than the value of the payment. if the present stockholders an equity issue which would their interest to they would still find no for of this can occur when the assets of the firm still have value. It provides one with the safety covenants discussed of the fact that many firms in bankruptcy and reorganization even though their total value may be quite these conditions the use of debt, and the terms of the debt, have important Suppose that of restrictions or of ownership the bondholders are to play a cannot at some date to a change in their contract or take an part in the firm. To in these the bondholders must that any subsequent debt be to their However, any debt at all in this situation would actually the bondholders and the This is it would then be more that a payment will be and the bondholders will take over the firm. To consider the value of the claims a payment has In an to the to in fact make that payment the stockholders were to for the total value of the firm the value of the bonds, they can only the total value the value of both the and bonds. The bondholders would be better and that the debt was at a the would have to out of the of the If it is possible for the to change their will be different. may find it in their interests to the issue of additional debt rather than allow a payment to be In the of debt could be by a contract of the Suppose that in the debt indenture it is specified that if the stockholders find that they cannot make a payment by new they will their equity interest over to the bondholders. The bondholders could then the firm as one having only equity and bonds. If such an is there would then be no to debt, since the firm would in to equity at the the debt would have a We have the in terms of in to the there may be only restrictions on the of such as the of assets by current or the bondholders may be to change their these would have a the would not be The solution can be by the discussed in the first For example, consider the situation before the last payment is ( V , t ) be the value of the firm's stock if the payment is This is the solution to the standard problem with terminal condition ( V − P , 0 ) . the value of the firm at which the payment can be V , is the of ( V , t ) = . The value of the stock just before the payments is ( V , t ) , will be ( V , t ) − if V V and zero if V V . The value of the bonds will be V − ( V , t ) . For the situation just before the payment is we the same analysis with ( V , t ) ( V − P , 0 ) . By working in this we can obtain a solution to the but in general no form will be our earlier we know that there will be some point at which no more equity can be and the bondholders will take over the firm. To find this point, think of in the In equilibrium new equity financing must sell at a so it makes no we think of it as being by new or by the So we can think of this as a situation where the stockholders will make payments into the firm to the interest payments to the bondholders, but at any time they have the right to stop making payments and either turn the firm over to the bondholders or pay them It is that the critical value of the firm at which they will do V , is of the current value of the firm and will be by the stockholders to the value of the bonds and the value of their of the solutions shows that is always than so the financing restrictions do the value of the bonds. V is is to in V than is and it is in the of having a lower elasticity, but when V is small the are The to the restrictions its at V and is a decreasing convex function of For the case with financing we find that the value at which the stockholders would the firm is a increasing function of and a decreasing convex function of σ 2 and shows that > . At V , and have the same value by As V the between them at first and then to zero as the value of each claim that of riskless debt, The and riskiness of to is qualitatively the the same as its comparison to of the shows that both and are increasing concave of V and are both decreasing of σ 2 , having an initial concave followed by a convex Similarly, both are increasing of and σ 2 . In this paper we first discussed some general in the valuation of contingent We some solution which could be applied even when the problem and discussed an way of the We then the effects of three specific provisions often found in bond were safety covenants, subordination arrangements, and restrictions on the financing of interest and dividend payments. We found that these provisions do the value of bonds, and that they may have a quite on the behavior of the firm's securities. The most important to our involve the assumptions about the of bankruptcy and about the probabilistic process governing the value of the firm. of our general hold for other but of the specific and would be different. It be noted that if the value of the firm a jump the value of a safety covenant may be since the value of the firm could then points the bankruptcy level without first The of bankruptcy have a more important This would depend on the specific form of the bankruptcy and also on the of other such as which would have to be into the analysis to the existence of debt in a world with bankruptcy However, their on our analysis not be We are bankruptcy as simply the of the ownership of the firm to the bondholders. The of the firm not be The bondholders may not want to the but the stockholders not The bondholders could the or new or they could the firm and sell all or part of their may be in the of but if are specified in the first place with an these then their may be