# Investment And The Value Of Capital Accounting Essay

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It is a well known proposition that in a world with no taxes, perfect capital markets, unrestricted borrowing and learning and lending at the same riskless rate, and no bankruptcy or agency costs, the firm is indifferent between financing its investments with debt or equity. This result, first proposed by Modigliani and Miller (1958), is an extremely powerful and quite general result which essentially allows the firm's investment and financing policies to be considered separately. Capital structure is said to be irrelevant or indeterminate. The criterion for optimality in investment is the simple net present value rule which yields the "correct" answer independent of financing arrangements. If corporate taxes are then introduced, along with the tax-deductibility of interest payments, the firm should borrow as much as possible to take advantages of the tax shields. Although this implies an optimal capital structure of almost all debt, a separation between investment and financing decisions still obtains.

However, both the preference for debt and the investment/financing separation do not seem to be borne out by the empirical findings. The implications of violating any one of M&M's seven assumptions have been considered by a number of recent papers. The natural question to consider then is: how are investment and financing decisions interrelated? Myres (1997) points out that, in an uncertain environment, there may be perverse investment incentives which arise from the firm's particular financing arrangements. The magnitude of these distortions depends critically on the contractual obligations of the different claims issued and also on the characteristics of the investment project undertaken.

Until recently, however, the investment literature has ignored these financing issues, concentrating only on the optimal investment problem. It should be noted that the term "investment" here does not describe the undertaking of a discrete "project" but rather the continuous accumulation of homogenous depreciating capital. The perfect-foresight models of investment behavior by Eisner and Strotz (1963), Lucas (1967), Gould (1968), and Treadway (1969) solved the firm's optimal capital accumulation problem without any financial considerations since the M&M proposition clearly applied. In a different but related context, Tobin (1969) argued that investment is an increasing function of the ratio of the firm's market-value and replacement cost, known also as q. It has since been shown by Abel (1978) that Tobin's q is in fact an average valuation of capital whereas the relevant variable for investment is the marginal valuation or shadow price of capital. Nevertheless in the context of these q-models of investment, financing decisions are ignored as well.

In this thesis, a q-model of optimal investment is being presented which includes financing considerations under perfect-foresight and the case of uncertainty. Without uncertainty, it is difficult to imagine a situation in which financing considerations are nontrivial. However, before examining the much more complicated stochastic case, at first a formal framework is constructed in which the particular question of interest may be isolated and discard the peripheral issues.

To obtain a non-degenerate financing decision, the theory of M&M and assumptions of no taxes and no leverage costs are relaxed. Specifically, corporate taxation with tax-deductibility of interest payments and individual income and capital gains taxation is introduced. Costs of leverage are then introduced in an ad hoc manner since it clearly cannot be derived rigorously in a deterministic model. It is simply asserted that the level of debt outstanding reduces the firm's real output via an implicit cost function in the production technology, paralleling Lucas' (1967) formulation of implicit costs of adjustment. The motivation behind this formulation is the agency and monitoring costs argument of Jensen and Meckling, as well as the perverse incentives problem of Myres discussed above. It is assumed that, in order for the firm's debt to be held, the stockholders must bear additional costs of monitoring and signaling which take the convenient form of reduced output. No attempt is made to show just how such a representation might capture the gaming which may occur between bondholders and stakeholders; we leave this the uncertainty case.

In this framework then, the optional physical investment and financial policies of the firm are derived. In determining the appropriate objective function for the firm's mergers, we follow Auerbach'(1979) approach closely, extending it to the continuous-time case. It can be shown that most of his results obtain in continuous-time and we include several of the more relevant results in this chapter such as the derivation of the weighted-average cost of capital.

Turning to the question of how the real and financial aspects of the firm's decisions interact, we focus on the specific question of whether or not changes in the shadow value of capital q affect the firm's capital structure. As might be expected, this depends on the properties of the implicit leverage cost function. It will be seen that how it depends on leverage costs suggests a possible line of research in the stochastic case.

**The Model**

Consider a perfectly competitive infinity-lived firm in continuous time which produces a single homogeneous output using capital and labor as inputs. Labor supply is exogenous upward-sloping and physical capital accumulation is the only form of investment the firm may undertake. The firm finances its investment through retained earnings, new issues of common stock, and the sale of short-term riskless debt. It is assumed that the firm seeks to maximize the wealth of existing shareholders and for clarity of exposition we may partition its decisions into two classes; real or production policies and financial policies.

In a world of perfect certainty, fictionless capital markets, and no bankruptcy or agency costs, the Modigliani-Miller theorem clearly applies and an actual separation between real and financial decisions obtains. However, since one of the goals of this chapter is to demonstrate the mechanism through which real shocks affect financial variables, we relax the assumption of no agency or monitoring costs by introducing ad hoc costs of leverage. In particular, it is assumed that output is produced according to the linearly homogeneous production F with both investment I and debt B as arguments indicating implicit costs of adjustment and leverage respectively. Denoting the capital stock by K and labor input by L, we have F=F(K,L, I, B), F is assumed to satisfy the following conditions

Although the costs of investment and leverage are expressed within F for notational convenience, it should be noted that condition (1c) states that these costs are in fact separable, i.e. F (K,L,I,B) =f(K,L)+C(K,I,B). Note also that the adjustment and leverage costs are covex, i.e., the marginal costs of adjustment and leverage are increasing functions of investment and debt respectively. In this chapter we do not attempt to show just how the costs of leverage arise but leave its micro-foundations to the uncertainty case instead and simply assert here that agency and monitoring costs associated with leverage exist and increase with the level of debt outstanding. For convenience, these costs are assumed to be embodied in the function C. The production decisions then involve choosing the time path of investment, labor, and debt issues which maximizes the current wealth of existing shareholders. Note that it is the inclusion of B in the production function which links financial considerations with real variables. It is in fact the only link in this model.

The firm's financial decisions are simply to choose at initial time t the time path of debt and equity B and S which, together with optimal production policies, maximize existing shareholder wealth. It is clear that, with positive costs of leverage and no benefits, it is optimal to hold no debt. The introduction of a corporate income tax T, however, and the tax-deductibility of interest payments provides the firm with positive returns to financing through debt. The tax advantage of debt is assumed to be large enough at the margin to induce a strictly positive debt-equity ratio. Note that B and S are not constrained to be nonnegative, i.e., share repurchased and debt retirement are feasible options. Clearly both B and S must be constrained to be nonnegative.

Given the optimal time paths of I, L, B, and S, the firm's dividend stream D is uniquely determined through the cash-flow accounting identity. It is assumed that the dividend streams are taxed at the personal income tax rate n and capital gains are taxed upon accrual at the rate c, with n > c. Now the following variables are being defined:

E(t) = Ps(t)S(t) = market value of equity

B(t) = market value of debt

X(t) = (1-Tc)[pF(K,L,I,B) -wL - PkI] = after-tax cash flow

D(t) = dividend payment

**The Objective Functional**

It has already been asserted that the objective of the firm is to maximize the current wealth of existing shareholders and, as a partial-equilibrium model of firm behavior, this criterion is not unreasonable. However, it is well known that even in a general equilibrium model with no uncertainty, this criterion is consistent with utility-maximization for any set of arbitrary well-behaved utility functions hence the fact that our analysis is partial-equilibrium in nature does not affect the choice of the firm's objective function.

Since the current wealth of existing shareholders is simply the market value of the equity they own, the firm's objective function is the current share price. In order to make this operational, some method of pricing the firm's equity is required. We use the standard asset-pricing assumption for deterministic models: the share price is, in equilibrium, equal to the present discounted stream of net distributions to which the shareholder is entitled. It will be shown later that this is equivalent to the discounted dividend approach.

**Optimal Investment and Financial Policies**

Given the objective functional (9a), the firm's optimization problem is now well-posed and may be solved subject to the capital accumulation and cash-flow constraints:

K = I-oK

B+PsS = D - (1-Tc)Tb - X

Upon closer inspection, however, it becomes clear that the objective functional is in implicit form. Holding shares S(t) constant for all T > t, it is observed that maximizing the current share price Ps (t) requires knowledge of the entire time path of future share prices Ps(T) which is yet to be determined. A time-varying S(T) may further complicate the problem. With sufficient regularity conditions imposed on the appropriate functions, this simultaneity problem may be resolved (Derzko and Sethi (1982), Lo (1983) but renders the optimization problem intractable. In order to circumvent this technical difficulty, it is maintained that the firm issues no new equity, i.e., S(T) = o for all T > t. it is in fact possible to derive such a result from optimizing behavior without any additional simplifying assumptions.

**The Equivalence of Marginal and Average q.**

Recent paper in the investment and public finance literature have point out the important distinction between marginal and average valuations of incremental additions to the capital stock, concluding that the relevant variable for investment demand should be marginal q. Unfortunately, this variable is unobservable. However, Hayashi (1982) has demonstrated that under certain linear-homogeneity restrictions, marginal and average q coincide. It is no surprise then that, given our homogeneity assumptions, a formal equivalence obtains in our model. Denote average and marginal q by qa and qb respectively. Then by evaluating the share price ps (t) for the optimal (I,B,S,L) policies it is shown in the appendix that:

Note that the above definition of qa differs from the original ''market-value over replacement cost'' definition proposed by Tobin (1965) only through the tax parameter furthermore q will be less then unity in equilibrium due to the presence of tax effects. This is the usual "q-less-then-one" phenomenon.

**Investment and Costs Leverage**

Having solved the firm's optimization problem, we may now consider the effect of changes in real variables on financial variables. In particular, attention will be focused on how changes in the shadow value q affect investment and dept policies. It will become clear that the critical and indeed only link between q and debt policy is FBI. Recall that the first-order necessary conditions of the firm's problem are:

The three cases examined above clearly demonstrate that FB1 is in fact the channel through which changes in q affect the firm's capital structure. For example, if the price of capital pk capital, if the price of capital pk declines exogenously then investment demand rises, but now such investment is financed depends on the sign of FB1. If additional investment leaves marginal leverage costs unchanged then the current capital structure is maintained. If, however, marginal investments change the marginal cost of leverage, capital structure also varies until marginal costs are equated with marginal benefits.

One interpretation of FB1 is that it is measure of the effect of investment risk on agency and monitoring costs. Myers (1977) has demonstrated that if shareholders issue risky debt with maturity exceeding the life of the investment opportunity, a perverse bias against taking good investment projects is created.

Myers points out htat "Honesty is the best policy." However, honesty is not costless. Convincing creditors of one's integrity, etc. is usually best done through protective covenants and other contractual arrangements which legally limit the shareholders' actions. In our simple model, the uncomfortable assumption of riskless debt is slightly more palatable with implicit costs of leverage. It is an attempt to summarize the presence of those monitoring costs mentioned above.

This interpretation must, of course, be qualified in several respects. The fact that our model is deterministic is obviously one of its major drawbacks. Actual costs of leverage depend on the availability of alternative financial instruments, probability of default, the tupbe and magnitude of investment risks, and other stochastic factors. The implicit cost of debt in the production function is at best an ad hoc proxy for all these considerations. It remains to be shown that these costs are in fact related in an interesting way to investment.

**Conclusion.**

In this study, we have presented a framework in which the interaction between real and financial policies may be studies. Taking the costs of leverage as exogenous, it has been show the effect of changes in q on the firm's capital depends essentially on how leverage costs change with investment. This model is clearly not rich enough to show exactly which characteristic model is useful in placing this particular issue in a formal framework. It is on this aspect then of the optimal investment problem that we focus our attention in the stochastic version.

### Chapter II

**Sub-Optimal Investment Policies Under Bankruptcy Risk and the Use of Convertible Debt**

### Introduction

One of the most celebrated results in modern finance theory is Modigiani and Miller's (1958) Proposition which states the market value of the firm is independent of its capital structure. Although many studies since then have focused on specifying conditions which yield an optimal capital structure for the firm[1] little attention has been paid to the Modigliani and Miller (M&M Theory) III which shows that the firm's investment decisions may be separated from its financing decisions.

Several recent studies, however, do consider possible interaction between real and financial policies [2]. In Jensen and Meckling's (1976) seminal paper it is demonstrated that , due to the principal-agent relationship between the firm's owner-manager and new shareholders, there are certain agency costs associated with equity-financing. These costs are in the forth of investment biases which yield a sub-optimal value of the firm. In addition, Jensen and Meckling argue that debt-financing also induces costly investment biases. Although they doe not derive this result formally, Jensen and Meckling appeal to the Black-Scholes (1973) and Merton (1973) option-pricing model and its application to investment decisions by Galai and Masulis (1978). However, they point out that the options-pricing model and its implications for investment incentives do not necessarily obtain when agency costs are present.[3]

In this chapter we propose a formal model of the firm's investment decisions in which the perverse incentives effect of debt-financing may be derived explicitly. In the context of a three-period partial-equilibrium model of the firm it is shown that, due to the asynchronous timing of investment and financing decisions, the firm is biased toward risky and sometimes less profitable investment project. Since in our framework such investment biases do not depend on the special assumptions of the options-pricing model, such as lognormally distributed asset prices or continuous trading , this suggests that Jensen and Meckling's result is a quite general problem with debt-financing. Our results are all the more striking sin e it is assumed throughout this study that agents have rational expectations and share the same information set. In addition, these biases can only be compounded if expectations are not rational or if there are informational asymmetries.

Although Jensen and Meckling also provide an optimal capital structure theory base on balancing agency costs of debt and equity at the margin, we put aside the optimal capital structure problem in this chapter. In order to focus purely on the incentives effect of debt, the firm is assumed to partially finance investment through the issue of a single bond with an arbitrary and fixed face value. The introduction of corporate taxes and the tax-deductibilty of interest payments may induce a determinate debt-equity ratio and will be considered in a future study. Throughout this chapter then, M& M Proposition obtains. However, in our framework M&M's Propositon III does not obtain and the firm no longer bases its investment decisions on the expected net present value criterion. Furthermore, because the firm cannot pre-commit itself to invest in the more profitable investment project, the sub-optimal investment choice is in fact a rational expectations equilibrium.

In this study we also prove Jensen and Meckliong's conjecture that convertible bond may offset some of the incentives effects of debt. In particular it is shown that: financing by convertible debt instead of straight debt allows the firm to pre-commit to the more profitable project, the investment bias is completely eliminated, and the attained rational expectations equilibrium dominates the equilibrium attainable by straight debt-financing. This then clarifies the way in which a convertibility option reduces the agency costs of debt. However, M&M's Proposition I still obtains in this case, so that the firm is indifferent between financing by equity and convertible debt.

In section II, we present the formal model of the firm and demonstrate how asynchronous timing of investment and financing decisions leads to investment biases. The notion of time-consistency will prove useful in analyzing this issue. As in the case in other dynamic decision-making framework, the time-inconsistent solution yields a higher value of the objective function but is rendered infeasible by rational expectations. In Section III we show the time consistent investment choice may be sub-optimal. Section IV demonstrates that convertible debt may eliminate investment biases and allow the firm to precommit to the more valuable project. We conclude in Section V.

The following assumptions are maintained throughout this chapter:

- There are no taxes or transactions costs
- There are no "frictional" costs of bankruptcy, i.e no litigation fees, reorganizational costs, etc
- Agents are risk-neutral, or equivalently,
- Project risk is completely diversifiable.
- There is riskless borrowing and lending at the rate 1+r.
- The firm's management (hereafter called "the firm") seeks to maximize the wealth of current shareholders.
- Equity of the limited-liability type and pays no dividends until the firm is liquidated. In the event of bankruptcy (which we define in Section II), equity holders receive nothing and all proceeds from liquidating the firm go to bondholders.

**Model**

We begin by considering how debt-financing and the possibility of bank-ruptcy affect the objective function of the firm and, consequently, its choice of investment policies.

**Pre-Commitment Through Convertible Debt.**

It has been shown in previous sections that the asynchronization of investment and financing decisions may lead to perverse investment incentives even under the extreme assumptions of rational expectations and no asymmetric information. As we mentioned in the introduction. these biases are worsened by informational asymmetries and expectations which are "less than" rational.

These considerations may be relevant even in relatively efficient financial markets if, as is often the case, over time new investment opportunities arise which were unforseen by both bond holders and equity holders at the time debt was issued. In such circumstances, bond covenants may be of little use in protecting lenders from time-inconsistent behavior by the firm. The most common resolution of this dilemma offered by corporate financial theory is the use of convertible debt. X and Z. the firm may achieve a higher level of shareholder wealth by financing with convertible debt in place of straight debt. The essence of this result is that convertible debt enables the firm to pre-commit itself to the more profitable investment project.

For expositional clarity we first discuss the terms of convertible debt in which only project X is available. It is then shown that when the choice between X and Z is available. there exists an equilibrium convertible debt contract which allows the firm to pre-commit to the more profitable project, i.e., the firm chooses project X and this is in fact a rational expectations equilibrium.

**Conclusion.**

The analysis presented in this study demonstrates the perverse effects corporate debt may have upon the firm's investment decisions, in some cases reducing the present market value of the firm. Although previous studies have shown that similar biases occur when there is asymmetric information or myopic bondholders, we have shown that such biases obtain even with no informational asymmetries and for bondholders with rational expectations.

A conclusion which is often drawn from studying the conflict between bond-holders and equity holders is that "honesty is the best policy." However, as we have shown. even honesty does not eliminate the perverse incentives effect of debt financing. We have established that convertible debt is one way in which the firm may circumvent wealth-decreasing investment biases. Since our results depend essentially on the asynchronization of investment choice and debt issue, this suggests that an alternate method of eliminating undesirable incentives is to synchronize its investment and financing decisions. However, even this may not eliminate the problem entirely since the firm may have previously issued debt outstanding at the time it invests.

Although this chapter does not address the question of how much the firm should borrow. our results may have implications for optimal capital structure theories based on bankruptcy costs. Warner's studies (l977a,1977b) suggest that costs of bankruptcy are only a small fraction of the firm's value, too small to induce a non degenerate capital structure. However, significant bankruptcy costs may not be in the form of third-party fees but may instead be related to losses in intangible assets due to biased investment incentives. These biases may be in the form of unexercised growth options as in Myers' (1977) analysis. or in the form of overly-risky over-priced investments as suggested in this study.

Because the market value of a firm may consist largely of intangible assets such as brand-name or market share. sizable costs may be incurred from such investment biases. Our "non-dynamic" three-period model is unfortunately not suited to explore this possibility. However, the development of a truly dynamic model of optimal investment and financial policies under uncertainty may resolve this issue and will be pursued in future research.

### References

- Abel, A.B., "Investment and the Value of Capital," Report 64, Federal Reserve Bank of Boston, 1978
- Black, F. and M. Scholes, "The Pricing of Options and Corporate Liabilities." Journal of Political Economy, 81(1973), pp. 637-654
- Derzko, N.A. and S.P. Sethi, "General Solution of the Price-Dividend Integral Equation." S.I.A.M. Journal of Mathemtical Analysis, 13(1982), pp. 106-111
- Eisner, R. and R. H. Strotz, "Determinants of Business Investment," in Commission on Money and Credit: Impact of Monetary Policy, Englewood Cliffs, New Jersey: Prentice-Hall, 1963, pp. 59-337
- Galai, D. and R.W. Masulis, "The Option Pricing Model and the Risk Factor of Stock," Journal of Financial Economics, 3(1978), pp. 53-81
- Hayashi, F., "Tobin's Average and marginal q: A Neoclassical Interpretation," Econometrica, 50(1982), pp. 213-224
- Jensen, M. and W. Mechkling, "Theory of the Firm: Managerial Behavior, Agnecy Costs, and Ownership Structure," Journal of Financial Economics, 3(1978). Pp. 305-380
- Lo, A., "A Particular Solution to the Implicit Price-Dividend Integral Equation," mimeo, 1983
- Lucas, R. E. Jr., "Adjustment Costs and the Theory of Supply," Journal of Political Economy, 75(1967), pp. 321-334
- Modigliani, F and Miller, "The Cost of Capital, Corporation Finance and The Theory of Investment," American Economic Review, 48(1958), pp. 281-297
- Myers, S., "Determinants of Corporate Borrowing," Journal of Financial Economics, 5(1977), pp. 147-175
- Tobin, J., "The Theory of Portfolio Selection," In The Theory of Interest Rates, ed. By F.H. Hahn and F.P.R. Brechling, London: Macmillan, 1965, pp. 3051
- Treadway, A.B., "On Rational Entrepreneurial Behaviour and the Demand for Investment," Review of Economic Studies, 38(1969), pp. 227-239