# A contingent claims analysis of trade credit.

Paul D. Adams is an Assistant Professor of Finance, Steve B. Wyatt is
an Associate Professor of Finance, and Yong H. Kim is a Professor of
Finance, all at the University of Cincinnati, Cincinnati, Ohio.

Trade credit involves the exchange of goods for a promised payment in some future period and accounts receivable represent a major proportion of corporate assets. However, little attention has been paid by the academic profession, as compared with other financial decisions, to the valuation of trade credit, i.e., the decision to grant credit and set credit limits. Furthermore, much of the limited literature has examined the credit decision from the perspective of the selling firm, without regard to financial market equilibrium or product market equilibrium.

Since trade credit involves a simultaneous transaction in both financial and product markets, it is important to make clear the assumption we are making about these markets when evaluating trade credit. If we assume that financial markets are competitive and complete, then all financial transactions are zero NPV in equilibrium, including the decision to grant trade credit. Firms cannot earn economic rents from simply granting trade credit.

With competitive and complete capital markets, the problem facing the firm that decides to offer trade credit is to grant credit to all firms whose discounted risky promise (DRP) is equal to (or greater than) the marginal cost (C) of goods sold on credit. If we further assume that the industry of the credit-granting firms is also perfectly competitive, then in equilibrium, the cash price for each firm's product equals the marginal cost of production. Since the cash price (|P.sup.*~) of goods is also equal to C, as in any competitive market, DRP = C = |P.sup.*~ in equilibrium. With competitive capital markets and competitive product markets, firm's product sales and trade credit extensions are zero NPV transactions.

If the product market is not perfectly competitive, then the product's cash price will be above marginal cost. Hence, firms can earn economic rents in the product market. For example, in the monopoly case, the optimal price |P.sup.*~ is greater than the marginal cost from a sale, hence |P.sup.*~ |is greater than~ C. Therefore, it is possible for the NPV of the credit sale to be positive when compared to C, just as the NPV of a cash sale would be positive as compared to C. This, however, does not imply that the NPV of the credit decision (the granting of credit) is a positive NPV decision because the marginal cost of a sale is not the correct comparison variable for determining the NPV of a credit decision. For the NPV of a credit decision to be positive, the seller must be in a position to earn economic rents from credit extension. The firm offering trade credit should not accept a credit sale over a cash sale unless the present value of the promised payment is at least as great as the cash sale |P.sup.*~ (i.e., DRP |is greater than or equal to~ |P.sup.*~). In this paper, we assume that the firm has some market power in the product market, but no market power in financial markets. This market power may be exploited in setting cash sale terms and credit sale terms that result in price discrimination. If this is the case, buyers may not be indifferent to cash versus credit purchases. If a credit policy can be used as such a tool, then granting credit may appear to be a positive NPV decision as compared to the cash price. Because antitrust considerations preclude outright price discrimination, the credit price performs this function. The correct comparison price for determining the NPV of the credit sale is not the current cash price, but the cash price that would be charged if price discrimination were allowed. Relative to this price, the NPV of the credit-granting decision must be zero. In this paper, we will take the optimal pricing decision as given.

The paper is organized as follows. Section I provides a brief review of the trade credit limit literature and motivates the current study. Section II develops a contingent claims model of trade credit valuation which is consistent with capital market equilibrium. Section III discusses detailed simulations of trade credit values and buyer equity values based on the model. Finally, Section IV addresses the issue of establishing trade credit limits and Section V concludes the paper.

I. A Review of the Literature

Several early publications in credit management(1) (notably Bierman-Hausman |8~) claim that the probability of default of the buyer firm is positively related to the amount of credit granted. The logic is as follows: as the buyer purchases more from the seller, its debt service capacity declines, raising the probability of default. The tradeoff between increased profits from greater sales volume and the increase in default probability leads to an optimum in the amount of credit granted (a credit limit). The process by which the probability of default rises is not specified by Bierman-Hausman. The model developed in this paper incorporates the buyer's probability of default as an endogenously determined variable. This paper therefore represents an improvement over previous papers that treat this probability as exogenous.

Kim and Atkins |3~ and |17~ view trade credit policy as an investment decision, and thus, suggest a net present value (NPV) framework to determine changes in the credit policy. They argue that the basic problem faced by the credit grantor is to change credit policy in terms of timing (credit payment date) and amount (credit limit) of cash flows whenever such a change increases the value of the firm via a positive NPV. Sartoris and Hill |26~ elaborate on the timing dimension of such credit policy changes, while the credit limit aspect has been explored by Schwartz and Whitcomb |28~ and Copeland and Khoury |11~. The focus of credit limit analysis has been on valuing the buyer's nominal promised payment using the appropriate risk-adjusted discount rate.

However, Miles and Varma |23~ note practical difficulties with the discount rate approach, and have developed an option pricing solution to the valuation of the buyer's promise. Their work is an important departure from previous studies. Their approach follows the assumption set of the original Black and Scholes |9~ model, where no cash disbursements are assumed on the underlying asset prior to expiration.

Our study generalizes the Miles-Varma |23~ framework in two important ways. First, although the basic mathematical form of the two models is identical, the interpretation of variables differs considerably. Miles-Varma assume a Modigliani and Miller |24~ and |25~ world where there is only a capital structure change, while we allow for a capital structure change and a change in asset structure of the buyer firm. Thus, our model allows for the effects of changes in the size and asset composition of the buyer as a result of a credit purchase. Second, Miles-Varma assume that the addition of trade credit in the buyer firm's capital structure has no effect on the standard deviation of equity. In contrast, we develop a framework which allows us to distinguish between the pre- and post-standard deviation of equity, and show that, in addition to leverage and scale effects, the change in the standard deviation of equity is, in part, dependent on the magnitude of the NPV of the project which is financed by trade credit.(2) Theoretically speaking, if the NPV of the project is sufficiently positive, the risk of the buyer could decline with the purchase. If this is true, no finite wealth-maximizing credit limit would result. This is also an improvement on Bierman-Hausman's |8~ logic, because they assume no benefits from the credit purchase with regard to default risk. Finally, we develop an illustrative trade credit limit example, incorporate dividend payouts on the part of the buyer firm, and provide detailed simulations through which the practical usefulness of the approach is illustrated.

II. The Basic Model

Trade credit involves the exchange of goods with a present value of C, for a promised payment of M dollars in T periods. The market value (that is, present value) of this promise will be denoted by: P(V, T, M, ||sigma~.sup.2~), where V is the market value of the firm, T is the maturity of the promised payment, M is the level of the promised payment, and ||sigma~.sup.2~ is the variance of the debtor (or buyer) firm's assets. At maturity, (T = 0) there are two possible outcomes. First, if the value of the debtor firm's assets is greater than M, the debtholder will receive the full promised payment of M. If, however, the value of the firm's assets is less than M, then the debtholder will receive V, the value of the debtor firm's assets. If other debt is present, the holder of trade credit will receive some |alpha~V, where |alpha~ is a fraction between zero and one, depending on the level of preexisting debt, and priorities in bankruptcy.

Valuation of the promised payment P(V, T, M, ||sigma~.sup.2~) is a fairly straightforward application of option pricing theory if certain standard assumptions are maintained.

A. Case I -- Buyer Has No Preexisting Debt in Its Capital Structure

The valuation of the promised payment in the case of no debt in the capital structure is a straightforward application of theorems of Black and Scholes |9~, Merton |21~, and Cox and Ross |12~. Casting the problem in a Cox-Ross framework, the value of equity, E, for a firm with debt is simply

|Mathematical Expression Omitted~

where PDF(V) = 1/|(2|pi~||sigma~.sup.2~).sup.1/2~ exp|-1/2|{|ln(V) - |mu~~/|sigma~}.sup.2~ and |mu~ = ln(V) + rT - (1/2)||sigma~.sup.2~. Notice that PDF(V) is the risk-neutral transformation of the log normal distribution. Solving Equation (1), by a theorem provided by Smith |29~, the value of equity can be expressed as

|Mathematical Expression Omitted~

where N() is the cumulative normal distribution,

|Mathematical Expression Omitted~,

|Mathematical Expression Omitted~.

Here, equity represents a call option on the assets of the firm in the absence of any cash dividend payments. If the firm pays cash dividends, then, an adjustment to these equations can be inferred from Merton's |21~ dividend model. Let |delta~ be the continuous dividend yield as in Merton |21~. The adjustment to the model then becomes

|mu~ = ln(V) + (r-|delta~)T -(1/2)||sigma~.sup.2~. (3)

Equation (4) is the discrete analogue to Equation (3). That is, Equation (4) adjusts for any pattern of known discrete cash dividends:(3)

|Mathematical Expression Omitted~

where |D.sub.i~ is the cash dividend payment at date |t.sub.i~, |t.sub.i~ |is less than~ |t.sub.i+1~ |is less than~ |t.sub.n~ |is less than~ T, and n is the total number of ex-dividend dates. Using Equation (4), the equivalent continuous yield can be computed as

|Mathematical Expression Omitted~.

Since we've assumed that |t.sub.n~ (the last dividend date) is less than T, the expression for |delta~ reaches zero before T reaches zero if |D.sub.i~ is zero for all i. Hence, the limit of |delta~ equals zero as T becomes infinitesimally small. This limit rules out the possibility of |delta~ tending to infinity as T becomes arbitrarily small. The value of equity, Equation (2), generalizes to Equation (6) with dividends:

E(V, T, M, ||sigma~.sup.2~, |delta~) = V|e.sup.|delta~T~ N(|d.sub.1~) - |Me.sup.-rT~N(|d.sub.2~), (6)

where N() is the cumulative normal distribution,

|Mathematical Expression Omitted~,

|Mathematical Expression Omitted~,

||sigma~.sup.2~ = Variance of the firm value,

|delta~ is given by Equation (5).

Having solved for the value of equity given the payment of dividends, the value of the promised payment M at maturity (T = 0) is simply the minimum of M or V. This leads to the value of the promised payment as

P(V, T, M, ||sigma~.sup.2~, |delta~) = |Ve.sup.-|delta~T~N(-|d.sub.1~) + |Me.sup.-rT~N(|d.sub.2~). (7)

To be able to use this equation to evaluate the trade credit decision, several assumptions must be made to determine V and ||sigma~.sup.2~. First, an expression which relates observable variables must be derived. One way to do this is to relate the observed value of equity and the standard deviation of equity to derive an expression for V, and ||sigma~.sup.2~. If a firm has already taken on some promised payment (debt) M, the relation between the standard deviation of equity, ||sigma~.sub.e~, and the standard deviation of firm value is given by

||sigma~.sub.e~ = |N(|d.sub.1~)|Ve.sup.-|delta~T~/E~|sigma~. (8)

To use this expression, we must note that the observed standard deviation of equity is before the debt of trade credit is added. Therefore, to relate the observed standard deviation to the expression in Equation (8), we must make some assumptions about the effect of the asset acquired by the buyer firm relative to the extension of trade credit. In effect, the question becomes, what is the effect of adding M dollars worth of assets on the standard deviation of equity. If we assume for a moment that the riskiness of the buyer firm is unaffected by the asset acquired and the dividend payout is unaffected, the total derivative of ||sigma~.sub.e~, with respect to the righthand side of Equation (8), must be found. In its full form, Equation (8) is a complex equation involving M. Substituting for |d.sub.1~ and |d.sub.2~, and restating the expressions for E(V, M, T, ||sigma~.sup.2~, |delta~), and ||sigma~.sub.e~, we obtain

|Mathematical Expression Omitted~,

|Mathematical Expression Omitted~

Rewriting ||sigma~.sub.e~ as

||sigma~.sub.e~ = f{N(M), E(M), V(M)}, (11)

and taking the derivative of ||sigma~.sub.e~ with respect to M, the level of the promised payment yields

|Mathematical Expression Omitted~

where |f.sub.x~ is the partial derivative of f with respect to the argument x.

As can be seen from Equation (12), the change in the risk level of the equity is a function of three conditions: a pure leverage (equity risk) effect represented by dN(M), an asset scale effect of dV(M), and an equity effect of dE(M). The term dN(M) is the pure equity risk effect of adding a marginal dollar of leverage in the capital structure. The term dV(M) is the asset scale effect of adding more assets to the firm without changing the level of equity. There is a possible mitigating factor dE, or the NPV of the asset added to the firm.(4) If the effect of dE is sufficiently large, it is possible that the risk of equity could decline rather than increase as we would normally expect to happen (|f.sub.E~ |is less than~ 0 by inspection). Intuitively, the NPV of the asset added by the buyer in the credit sale could offset the risk effect of adding more debt to the firm's capital structure. To be able to bridge between the observed standard deviation prior to extending credit (pre ||sigma~.sub.e~) and the standard deviation given in Equation (8), Equation (12) must be used. To use Equation (12), however, some assumptions need to be made about dE. In general, it is highly unlikely that the standard deviation of equity observed previous to the transaction will remain the same after adding both an asset and liability to the buyer firm, unless the scale of the transaction is very small relative to the buyer firm.

Equation (12) highlights the dual impact that trade credit may have on the buyer firm. First, the extension of trade credit may increase the amount of debt the buyer firm has and second, the level of the buyer's assets increase as well. In effect, trade credit simultaneously alters the capital and asset structure of the buyer firm. By contrast, most analysis of other types of debt contracts considers shifts in the capital structure alone without any shifts in asset structure. With trade credit, both the asset and capital structure change.(5)

If it is assumed that the NPV of the asset added by the buyer is zero, then the expression simplifies to

d||sigma~.sub.e~/dM = |f.sub.N~dN(M)/dM + |f.sub.V~dV(M)/dM. (13)

Throughout the rest of this paper, it will be assumed that the NPV of the asset added by the buyer firm is zero,(6) though any alternative assumption presents little problem for the model if the correct NPV is known in advance of valuing a trade credit promise. Without making an assumption about this NPV, it is impossible to link the pre-sale standard deviation of equity to the correct post-sale standard deviation of equity, and hence, use Equation (7) in practice.

B. Valuation of Trade Credit With Preexisting Debt: Three Variations on a Basic Theme

Equations (8) and (12) can be easily extended to cover the cases when the buyer firm has preexisting debt which may have lower (Case II),(7) higher (Case III), or equal priority (Case IV) in bankruptcy. (Case I being the situation where there is no previous debt in the capital structure.) To analyze these additional scenarios requires derivations of equations similar to Equations (12) and (13). This will enable us to link any observed standard deviation of equity to the new standard deviation of equity resulting from additional debt and additional assets. When a firm has preexisting debt, the determination of the post new debt standard deviation of equity changes slightly from Equation (8). The relation becomes

||sigma~.sub.e~= |N(|d.sub.3~)|Ve.sup.-|delta~T~/E~|sigma~. (14)

Notice that N(|d.sub.3~) differs from N(|d.sub.1~) by the new debt M which is added to the existing level of debt B. The problem of inferring the change in the standard deviation of equity may be mitigated somewhat because the level of existing debt may be fairly large as compared to new debt. The valuation equation for equity when a firm has preexisting debt becomes

E(V, T, M, B, ||sigma~.sup.2~, |delta~) = |Ve.sup.-|delta~T~N(|d.sub.3~) - (M + B)|e.sup.-rT~N(|d.sub.4~), (15)

where:

|Mathematical Expression Omitted~,

|Mathematical Expression Omitted~.

C. Case III -- Existing Debt Has Priority Over Seller's Trade Credit

This is the most likely scenario faced by sellers extending trade credit since it is usually the most junior form of debt of a buyer. Most bonds are not secured by specific assets (that is, most bonds are not mortgage bonds). Instead, most bonds that are not debentures are secured by seniority provisions which give them first claim on all assets, including those acquired in the future. In this scenario, the value of the promised payment is given as

|Mathematical Expression Omitted~

where

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~.

In this situation, the value of the trade credit is seriously affected by the presence of existing debt in two ways: the first concerns the priority effect -- the fact that trade credit is junior to existing debt, and the second is the leverage effect of adding more debt to the capital structure.

D. Case IV -- Existing Debt and Seller's Trade Credit Have Equal Priority

Finally, consider a scenario where the firm sells to buyers whose debt consists of trade credit or other debt with the same priority in bankruptcy. Let |alpha~ be the ratio of the seller's credit to the total debt of the firm, |alpha~ = M/(B + M), then the value of the promised payment is given as

P3(|alpha~V, T, M, B, ||sigma~.sup.2~, |delta~) = |alpha~|Ve.sup.|delta~T~N(-|d.sub.3~) + |Me.sup.-rT~N(|d.sub.4~). (17)

Here, the total class of creditors divide the remaining assets of the firm, in the event of bankruptcy, with the seller receiving |alpha~ share of the proceeds. Because Cases II and III are extensions of Case IV, the latter can be viewed as our base case.

III. Simulation Analysis

Exhibits 1 through 3 plot the general tradeoffs between buyer firm value, the present value of a future promised payment (trade credit), the level of preexisting debt and the value of buyer firm equity. Each exhibit varies parameters across a wide range of values. To standardize the results, the present value of the future promised payment (computed from Equations (7), (16) and (17), respectively) is divided by the corresponding risk-free value and the value of buyer firm equity (computed from Equations (9) and (15), respectively) is divided by the corresponding total buyer firm value. Thus, the present value of the promised payment in Exhibits 1 and 2 is measured as a percentage of the corresponding risk-free value, and the value of buyer firm equity in Exhibit 3 is measured as a percentage of total buyer firm value.(8) Exhibits 1 and 2 show very interesting results as the level of prior debt is varied. As the graphs indicate, the percentage value of the promised payment is very sensitive to the level of prior debt. Rather suddenly, there is a significant drop-off in the percentage value of the promised payment as the level of prior debt is increased. Thus, at some critical point, a small change in the prior debt to value ratio can have a large influence on the percentage value of the promised payment. Finally, Exhibit 3 shows that the value of buyer firm equity as a percentage of total buyer firm value falls (rises) as the level of preexisting debt rises (falls) and rises (falls) as total buyer firm value rises (falls).

Although not reported in this paper, we also developed comparable exhibits without dividends (|D.sub.i~ = 0 for all i). The greatest contrast was Case III, where preexisting debt has priority over trade credit. Exhibit 1 shows that the value of trade credit as a fraction of the corresponding risk-free value ranges from 80% to 100% for the assumed parameters. However, in the absence of cash dividends, the value of trade credit as a fraction of the corresponding risk-free value ranges from 94% to 100% for the same set of parameters. Hence, in the absence of cash dividends, trade credit is fairly risk-free and in the presence of cash dividends, trade credit becomes much more risky. This is particularly interesting since Case III is the most realistic scenario. Thus, the explicit incorporation of cash dividends in the trade credit decision can lead to results that differ dramatically from the case with no dividends.(9)

IV. Determining Trade Credit Limits -- An Illustration

Earlier we discussed how to use our model to determine the present value of a future promised payment of the buyer firm with and without preexisting debt. In the former case, we analyzed the following priority arrangements: (i) trade credit that is senior to preexisting debt, (ii) trade credit that is junior to preexisting debt, and (iii) trade credit that has equal priority with preexisting debt. A numerical search procedure was used to simultaneously find the pair of buyer firm value (V) and buyer firm standard deviation (|sigma~) that satisfies the appropriate valuation equation for buyer firm equity (E) and corresponding standard deviation equation for equity (||sigma~.sub.e~). These numerically solved values of V and |sigma~ were then used to determine the present value of the promised payment. Of course, the NPV is the residual, that is, the present value of the future promised payment minus the marginal cost of the goods sold on credit for a given firm. The observables are the risk-free rate (r), schedule of dividend payments (|D.sub.1~, |D.sub.2~,..., |D.sub.n~), future promised payment (M), preexisting debt (B), maturity of the promised payment (T), marginal cost of goods sold on credit (C), and the buyer firm's value of equity while the unobservable variables are V, |sigma~ (which is computed from a well-known relationship with ||sigma~.sub.e~), and ||sigma~.sub.e~ (which can be estimated from historical common stock data). The seller firm will extend trade credit to a given buyer firm up to the point where the NPV is zero. The seller firm will not extend trade credit beyond this point since to do so would be a negative NPV decision. Thus, the trade credit limit for a given firm size is established at the point where the marginal cost of goods sold equals the promised payment (i.e., NPV is zero). What follows is an example for illustrative purposes.

Let's assume the following: r = 9%; E = $1; ||sigma~.sub.e~ = 1.0; T = 1 year; B = $0.20; C (the marginal cost of the trade credit sale) = $0.87; a $0.125 cash dividend one month prior to the contract date; and trade credit is junior to preexisting debt (the most realistic scenario). Applying the model (as described above) results in the following solution to the trade credit limit problem: V = $2.169; |sigma~ = 0.525; M = $1. If the future promised payment is $1, then, the present value of the payment is $0.87 and the NPV is zero. Thus, the trade credit limit is established at $1, that is, the seller would not extend trade credit beyond $1 since to do so would be a negative NPV decision. Theoretically speaking, if the NPV of the project is sufficiently positive, the risk of equity of the buyer firm could decline with the purchase. As noted earlier, if this is true, no finite wealth-maximizing credit limit would result.

V. Conclusion

The framework developed in our study is substantially more general than that proposed by Miles and Varma |23~. First, although the basic mathematical form of the two models is identical, the interpretation of variables differs considerably. They assume a Modigliani and Miller |24~ and |25~ world where there is only a capital structure change, while we allow for the effects of changes in the size and asset composition of the buyer as a result of a credit purchase. Second, Miles and Varma |23~ assume that the addition of trade credit in the firm's capital structure has no effect on the standard deviation of equity. In contrast, we develop a framework which allows us to distinguish between the pre- and post-standard deviation of equity and show that, in addition to leverage and scale effects, the change in the standard deviation of equity is, in part, dependent on the NPV of the project which is financed by trade credit. Thus, if the NPV of the project is sufficiently positive, the risk of the buyer firm could decline with the purchase. In addition to being an improvement over previous papers that treat the buyer's probability of default as exogenous, our study also improves on the Bierman-Hausman |8~ type logic which assumes no possible benefits from the purchase. Finally, we develop an illustrative trade credit limit example which incorporates dividend payouts on the part of the buyer firm and provide detailed simulations through which the practical usefulness of the approach is illustrated.

1Theoretical expositions that either explore the economic rationale for the existence of trade credit (e.g., Lewellen, McConnell and Scott |19~, Ferris |16~, Emery |13~, |14~, |15~, and Smith |30~) or integrate the product-pricing and credit-policy decisions (e.g., Kim and Atkins |17~, and Lam and Chen |18~) are beyond the scope of this study. From a descriptive perspective, however, recent attempts by Srinivasan and Kim |31~ and |32~ to replicate the credit analyst's judgement in setting credit limits may prove to be useful exposition since, by far, the most popular method of setting credit limits appears to be the analyst's judgement (Besley and Osteryoung |7~, and Beranek and Scherr |5~). Also, for a recent discussion on managerial policy alternatives to trade credit, see Mian and Smith |22~.

2Another way of thinking of this effect is as the profitability of the additional business that the purchase will bring to the buyer.

3We've assumed that dividends are known for purposes of simplicity. The methodology employed in this paper is a variation of the Adams and Wyatt |1~ and |2~ model used to price foreign currency options.

4Notice that these partial derivatives involve the cumulative normal distribution which does not have a closed form solution. Because of this and the relatively complex interaction of M with other variables, this derivative must be solved by numerical methods.

5Miles and Varma |23~ do not make this distinction in their paper.

6An interesting research question remains, although it is beyond the scope of this paper, i.e., empirically determining the point at which the equity effect begins to mitigate the combination of pure leverage and asset scale effects. This can best be achieved via detailed simulations assuming various NPVs. This point was suggested by an anonymous referee of this journal.

7According to Case II, the extension of trade credit has priority over the buyer firm's existing debt. Such a case could arise when the buyer firm has preferred stock as the only debt-like instrument in its capital structure. Besides preferred stock, some forms of convertible bonds are junior to trade credit. When such a case arises, the value of the promised payment to the trade creditor is

P1(V, T, M, ||sigma~.sup.2~, |delta~) = |Ve.sup.|delta~T~N(-|d.sub.1~) + |Me.sup.-rT~N(|d.sub.2~).

Notice that the value of the promised payment is the same as when the firm has no preexisting debt. Intuitively, this arises because the selling firm has first claim on assets over the other debtholders in the event of bankruptcy. Hence, unless some other variable (such as firm value) is affected by the extension of trade credit, the fact that the selling firm has existing debt is irrelevant to the valuation of the promise to repay the trade credit.

8Note that E, |sigma~, T, B, r, schedule of dividend payments (|D.sub.1~, |D.sub.2~, |D.sub.n~), C (the marginal cost of the goods sold) and M are directly observable or easily obtained. This information is used by the seller to infer the value of the buyer's assets (V) and the return variance on these assets (||sigma~.sup.2~). Since V and ||sigma~.sup.2~ cannot be solved for explicitly, a single computer numeric search algorithm is used to obtain V and ||sigma~.sup.2~. These values are then used to solve for the value of the promised payment (P, P1, P2, and P3). The NPV of the trade credit sale is merely the difference between the value of promised payment and C.

9Moreover, we developed simulated graphs of Cases I through IV across present values of nominal promised payments as a percentage of corresponding risk-free values, firm standard deviations and firm values. These plots were identical for Cases I and II. The reason for this result is because preexisting debt is equivalent to equity as long as this debt is junior to trade credit. In practice, the only debt and debt-like instruments that are junior to trade credit are income bonds and preferred stock. In each of the four cases, the present value of the future promised payment as a percentage of the risk-free value falls (rises) rapidly as the standard deviation of buyer firm value rises (falls) and rises (falls) slowly as the value of the buyer firm rises (falls). There is a clear and smooth rise (fall) in the percentage value of the promised payment as the value of the buyer firm rises (falls). Finally, we analyzed simulated graphs of Cases I through IV across values of buyer firm equity as a percentage of corresponding firm values, firm standard deviations and firm values. The results show that the value of buyer firm equity as a percentage of total buyer firm value is positively related to both buyer firm standard deviation and total buyer firm value.

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19. W. Lewellen, J. McConnell, and J. Scott, "Capital Market Influences on Trade Credit Policies," Journal of Financial Research (Summer 1980), pp. 105-113.

20. D. Mehta, "The Formulation of Credit Policy Models," Management Science (October 1968), pp. B30-B50.

21. R.C. Merton, "Theory of Rational Option Pricing," Bell Journal of Economics and Management Science (Spring 1973), pp. 141-183.

22. S.L. Mian and C.W. Smith, Jr., "Accounts Receivable Management Policy: Theory and Evidence," Journal of Finance (March 1992), pp. 169-200.

23. J.A. Miles and R. Varma, "Using Financial Market Data to Make Trade Credit Decisions," Journal of Business Finance and Accounting (Winter 1986), pp. 505-518.

24. F. Modigliani and M.H. Miller, "The Cost of Capital, Corporation Finance, and the Theory of Investment," American Economic Review (June 1958), pp. 261-297.

25. F. Modigliani and M.H. Miller, "The Cost of Capital, Corporation Finance, and the Theory of Investment: Reply," American Economic Review (September 1959), pp. 655-669.

26. W. Sartoris and N.C. Hill, "Evaluating Credit Policy Alternatives: A Present Value Framework," Journal of Financial Research (Spring 1981), pp. 81-89.

27. F.C. Scherr, "Optimal Trade Credit Limits," presented at the Fifth Symposium on Cash, Treasury and Working Capital Management, 1989.

28. R. Schwartz and T. Whitcomb, "The Trade Credit Decisions," in Handbook of Financial Economics, J. Bicksler (ed.), Amsterdam, North Holland, 1980, pp. 257-273.

29. C.W. Smith, Jr., "Option Pricing: A Review," Journal of Financial Economics (January/March 1976), pp. 3-51.

30. J.K. Smith, "Trade Credit and Informational Asymmetry," Journal of Finance (September 1987), pp. 863-872.

31. V. Srinivasan and Y.H. Kim, "Credit Granting: A Comparative Analysis of Classification Procedures," Journal of Finance (July 1987), pp. 665-681.

32. V. Srinivasan and Y.H. Kim, "Designing Expert Financial Systems: A Case Study of Corporate Credit Management," Financial Management (Autumn 1988), pp. 33-43.

33. J.D. Stowe, "An Integer Programming Solution for the Optimal Credit Investigation/Credit Granting Sequence," Financial Management (Summer 1985), pp. 56-76.

Trade credit involves the exchange of goods for a promised payment in some future period and accounts receivable represent a major proportion of corporate assets. However, little attention has been paid by the academic profession, as compared with other financial decisions, to the valuation of trade credit, i.e., the decision to grant credit and set credit limits. Furthermore, much of the limited literature has examined the credit decision from the perspective of the selling firm, without regard to financial market equilibrium or product market equilibrium.

Since trade credit involves a simultaneous transaction in both financial and product markets, it is important to make clear the assumption we are making about these markets when evaluating trade credit. If we assume that financial markets are competitive and complete, then all financial transactions are zero NPV in equilibrium, including the decision to grant trade credit. Firms cannot earn economic rents from simply granting trade credit.

With competitive and complete capital markets, the problem facing the firm that decides to offer trade credit is to grant credit to all firms whose discounted risky promise (DRP) is equal to (or greater than) the marginal cost (C) of goods sold on credit. If we further assume that the industry of the credit-granting firms is also perfectly competitive, then in equilibrium, the cash price for each firm's product equals the marginal cost of production. Since the cash price (|P.sup.*~) of goods is also equal to C, as in any competitive market, DRP = C = |P.sup.*~ in equilibrium. With competitive capital markets and competitive product markets, firm's product sales and trade credit extensions are zero NPV transactions.

If the product market is not perfectly competitive, then the product's cash price will be above marginal cost. Hence, firms can earn economic rents in the product market. For example, in the monopoly case, the optimal price |P.sup.*~ is greater than the marginal cost from a sale, hence |P.sup.*~ |is greater than~ C. Therefore, it is possible for the NPV of the credit sale to be positive when compared to C, just as the NPV of a cash sale would be positive as compared to C. This, however, does not imply that the NPV of the credit decision (the granting of credit) is a positive NPV decision because the marginal cost of a sale is not the correct comparison variable for determining the NPV of a credit decision. For the NPV of a credit decision to be positive, the seller must be in a position to earn economic rents from credit extension. The firm offering trade credit should not accept a credit sale over a cash sale unless the present value of the promised payment is at least as great as the cash sale |P.sup.*~ (i.e., DRP |is greater than or equal to~ |P.sup.*~). In this paper, we assume that the firm has some market power in the product market, but no market power in financial markets. This market power may be exploited in setting cash sale terms and credit sale terms that result in price discrimination. If this is the case, buyers may not be indifferent to cash versus credit purchases. If a credit policy can be used as such a tool, then granting credit may appear to be a positive NPV decision as compared to the cash price. Because antitrust considerations preclude outright price discrimination, the credit price performs this function. The correct comparison price for determining the NPV of the credit sale is not the current cash price, but the cash price that would be charged if price discrimination were allowed. Relative to this price, the NPV of the credit-granting decision must be zero. In this paper, we will take the optimal pricing decision as given.

The paper is organized as follows. Section I provides a brief review of the trade credit limit literature and motivates the current study. Section II develops a contingent claims model of trade credit valuation which is consistent with capital market equilibrium. Section III discusses detailed simulations of trade credit values and buyer equity values based on the model. Finally, Section IV addresses the issue of establishing trade credit limits and Section V concludes the paper.

I. A Review of the Literature

Several early publications in credit management(1) (notably Bierman-Hausman |8~) claim that the probability of default of the buyer firm is positively related to the amount of credit granted. The logic is as follows: as the buyer purchases more from the seller, its debt service capacity declines, raising the probability of default. The tradeoff between increased profits from greater sales volume and the increase in default probability leads to an optimum in the amount of credit granted (a credit limit). The process by which the probability of default rises is not specified by Bierman-Hausman. The model developed in this paper incorporates the buyer's probability of default as an endogenously determined variable. This paper therefore represents an improvement over previous papers that treat this probability as exogenous.

Kim and Atkins |3~ and |17~ view trade credit policy as an investment decision, and thus, suggest a net present value (NPV) framework to determine changes in the credit policy. They argue that the basic problem faced by the credit grantor is to change credit policy in terms of timing (credit payment date) and amount (credit limit) of cash flows whenever such a change increases the value of the firm via a positive NPV. Sartoris and Hill |26~ elaborate on the timing dimension of such credit policy changes, while the credit limit aspect has been explored by Schwartz and Whitcomb |28~ and Copeland and Khoury |11~. The focus of credit limit analysis has been on valuing the buyer's nominal promised payment using the appropriate risk-adjusted discount rate.

However, Miles and Varma |23~ note practical difficulties with the discount rate approach, and have developed an option pricing solution to the valuation of the buyer's promise. Their work is an important departure from previous studies. Their approach follows the assumption set of the original Black and Scholes |9~ model, where no cash disbursements are assumed on the underlying asset prior to expiration.

Our study generalizes the Miles-Varma |23~ framework in two important ways. First, although the basic mathematical form of the two models is identical, the interpretation of variables differs considerably. Miles-Varma assume a Modigliani and Miller |24~ and |25~ world where there is only a capital structure change, while we allow for a capital structure change and a change in asset structure of the buyer firm. Thus, our model allows for the effects of changes in the size and asset composition of the buyer as a result of a credit purchase. Second, Miles-Varma assume that the addition of trade credit in the buyer firm's capital structure has no effect on the standard deviation of equity. In contrast, we develop a framework which allows us to distinguish between the pre- and post-standard deviation of equity, and show that, in addition to leverage and scale effects, the change in the standard deviation of equity is, in part, dependent on the magnitude of the NPV of the project which is financed by trade credit.(2) Theoretically speaking, if the NPV of the project is sufficiently positive, the risk of the buyer could decline with the purchase. If this is true, no finite wealth-maximizing credit limit would result. This is also an improvement on Bierman-Hausman's |8~ logic, because they assume no benefits from the credit purchase with regard to default risk. Finally, we develop an illustrative trade credit limit example, incorporate dividend payouts on the part of the buyer firm, and provide detailed simulations through which the practical usefulness of the approach is illustrated.

II. The Basic Model

Trade credit involves the exchange of goods with a present value of C, for a promised payment of M dollars in T periods. The market value (that is, present value) of this promise will be denoted by: P(V, T, M, ||sigma~.sup.2~), where V is the market value of the firm, T is the maturity of the promised payment, M is the level of the promised payment, and ||sigma~.sup.2~ is the variance of the debtor (or buyer) firm's assets. At maturity, (T = 0) there are two possible outcomes. First, if the value of the debtor firm's assets is greater than M, the debtholder will receive the full promised payment of M. If, however, the value of the firm's assets is less than M, then the debtholder will receive V, the value of the debtor firm's assets. If other debt is present, the holder of trade credit will receive some |alpha~V, where |alpha~ is a fraction between zero and one, depending on the level of preexisting debt, and priorities in bankruptcy.

Valuation of the promised payment P(V, T, M, ||sigma~.sup.2~) is a fairly straightforward application of option pricing theory if certain standard assumptions are maintained.

A. Case I -- Buyer Has No Preexisting Debt in Its Capital Structure

The valuation of the promised payment in the case of no debt in the capital structure is a straightforward application of theorems of Black and Scholes |9~, Merton |21~, and Cox and Ross |12~. Casting the problem in a Cox-Ross framework, the value of equity, E, for a firm with debt is simply

|Mathematical Expression Omitted~

where PDF(V) = 1/|(2|pi~||sigma~.sup.2~).sup.1/2~ exp|-1/2|{|ln(V) - |mu~~/|sigma~}.sup.2~ and |mu~ = ln(V) + rT - (1/2)||sigma~.sup.2~. Notice that PDF(V) is the risk-neutral transformation of the log normal distribution. Solving Equation (1), by a theorem provided by Smith |29~, the value of equity can be expressed as

|Mathematical Expression Omitted~

where N() is the cumulative normal distribution,

|Mathematical Expression Omitted~,

|Mathematical Expression Omitted~.

Here, equity represents a call option on the assets of the firm in the absence of any cash dividend payments. If the firm pays cash dividends, then, an adjustment to these equations can be inferred from Merton's |21~ dividend model. Let |delta~ be the continuous dividend yield as in Merton |21~. The adjustment to the model then becomes

|mu~ = ln(V) + (r-|delta~)T -(1/2)||sigma~.sup.2~. (3)

Equation (4) is the discrete analogue to Equation (3). That is, Equation (4) adjusts for any pattern of known discrete cash dividends:(3)

|Mathematical Expression Omitted~

where |D.sub.i~ is the cash dividend payment at date |t.sub.i~, |t.sub.i~ |is less than~ |t.sub.i+1~ |is less than~ |t.sub.n~ |is less than~ T, and n is the total number of ex-dividend dates. Using Equation (4), the equivalent continuous yield can be computed as

|Mathematical Expression Omitted~.

Since we've assumed that |t.sub.n~ (the last dividend date) is less than T, the expression for |delta~ reaches zero before T reaches zero if |D.sub.i~ is zero for all i. Hence, the limit of |delta~ equals zero as T becomes infinitesimally small. This limit rules out the possibility of |delta~ tending to infinity as T becomes arbitrarily small. The value of equity, Equation (2), generalizes to Equation (6) with dividends:

E(V, T, M, ||sigma~.sup.2~, |delta~) = V|e.sup.|delta~T~ N(|d.sub.1~) - |Me.sup.-rT~N(|d.sub.2~), (6)

where N() is the cumulative normal distribution,

|Mathematical Expression Omitted~,

|Mathematical Expression Omitted~,

||sigma~.sup.2~ = Variance of the firm value,

|delta~ is given by Equation (5).

Having solved for the value of equity given the payment of dividends, the value of the promised payment M at maturity (T = 0) is simply the minimum of M or V. This leads to the value of the promised payment as

P(V, T, M, ||sigma~.sup.2~, |delta~) = |Ve.sup.-|delta~T~N(-|d.sub.1~) + |Me.sup.-rT~N(|d.sub.2~). (7)

To be able to use this equation to evaluate the trade credit decision, several assumptions must be made to determine V and ||sigma~.sup.2~. First, an expression which relates observable variables must be derived. One way to do this is to relate the observed value of equity and the standard deviation of equity to derive an expression for V, and ||sigma~.sup.2~. If a firm has already taken on some promised payment (debt) M, the relation between the standard deviation of equity, ||sigma~.sub.e~, and the standard deviation of firm value is given by

||sigma~.sub.e~ = |N(|d.sub.1~)|Ve.sup.-|delta~T~/E~|sigma~. (8)

To use this expression, we must note that the observed standard deviation of equity is before the debt of trade credit is added. Therefore, to relate the observed standard deviation to the expression in Equation (8), we must make some assumptions about the effect of the asset acquired by the buyer firm relative to the extension of trade credit. In effect, the question becomes, what is the effect of adding M dollars worth of assets on the standard deviation of equity. If we assume for a moment that the riskiness of the buyer firm is unaffected by the asset acquired and the dividend payout is unaffected, the total derivative of ||sigma~.sub.e~, with respect to the righthand side of Equation (8), must be found. In its full form, Equation (8) is a complex equation involving M. Substituting for |d.sub.1~ and |d.sub.2~, and restating the expressions for E(V, M, T, ||sigma~.sup.2~, |delta~), and ||sigma~.sub.e~, we obtain

|Mathematical Expression Omitted~,

|Mathematical Expression Omitted~

Rewriting ||sigma~.sub.e~ as

||sigma~.sub.e~ = f{N(M), E(M), V(M)}, (11)

and taking the derivative of ||sigma~.sub.e~ with respect to M, the level of the promised payment yields

|Mathematical Expression Omitted~

where |f.sub.x~ is the partial derivative of f with respect to the argument x.

As can be seen from Equation (12), the change in the risk level of the equity is a function of three conditions: a pure leverage (equity risk) effect represented by dN(M), an asset scale effect of dV(M), and an equity effect of dE(M). The term dN(M) is the pure equity risk effect of adding a marginal dollar of leverage in the capital structure. The term dV(M) is the asset scale effect of adding more assets to the firm without changing the level of equity. There is a possible mitigating factor dE, or the NPV of the asset added to the firm.(4) If the effect of dE is sufficiently large, it is possible that the risk of equity could decline rather than increase as we would normally expect to happen (|f.sub.E~ |is less than~ 0 by inspection). Intuitively, the NPV of the asset added by the buyer in the credit sale could offset the risk effect of adding more debt to the firm's capital structure. To be able to bridge between the observed standard deviation prior to extending credit (pre ||sigma~.sub.e~) and the standard deviation given in Equation (8), Equation (12) must be used. To use Equation (12), however, some assumptions need to be made about dE. In general, it is highly unlikely that the standard deviation of equity observed previous to the transaction will remain the same after adding both an asset and liability to the buyer firm, unless the scale of the transaction is very small relative to the buyer firm.

Equation (12) highlights the dual impact that trade credit may have on the buyer firm. First, the extension of trade credit may increase the amount of debt the buyer firm has and second, the level of the buyer's assets increase as well. In effect, trade credit simultaneously alters the capital and asset structure of the buyer firm. By contrast, most analysis of other types of debt contracts considers shifts in the capital structure alone without any shifts in asset structure. With trade credit, both the asset and capital structure change.(5)

If it is assumed that the NPV of the asset added by the buyer is zero, then the expression simplifies to

d||sigma~.sub.e~/dM = |f.sub.N~dN(M)/dM + |f.sub.V~dV(M)/dM. (13)

Throughout the rest of this paper, it will be assumed that the NPV of the asset added by the buyer firm is zero,(6) though any alternative assumption presents little problem for the model if the correct NPV is known in advance of valuing a trade credit promise. Without making an assumption about this NPV, it is impossible to link the pre-sale standard deviation of equity to the correct post-sale standard deviation of equity, and hence, use Equation (7) in practice.

B. Valuation of Trade Credit With Preexisting Debt: Three Variations on a Basic Theme

Equations (8) and (12) can be easily extended to cover the cases when the buyer firm has preexisting debt which may have lower (Case II),(7) higher (Case III), or equal priority (Case IV) in bankruptcy. (Case I being the situation where there is no previous debt in the capital structure.) To analyze these additional scenarios requires derivations of equations similar to Equations (12) and (13). This will enable us to link any observed standard deviation of equity to the new standard deviation of equity resulting from additional debt and additional assets. When a firm has preexisting debt, the determination of the post new debt standard deviation of equity changes slightly from Equation (8). The relation becomes

||sigma~.sub.e~= |N(|d.sub.3~)|Ve.sup.-|delta~T~/E~|sigma~. (14)

Notice that N(|d.sub.3~) differs from N(|d.sub.1~) by the new debt M which is added to the existing level of debt B. The problem of inferring the change in the standard deviation of equity may be mitigated somewhat because the level of existing debt may be fairly large as compared to new debt. The valuation equation for equity when a firm has preexisting debt becomes

E(V, T, M, B, ||sigma~.sup.2~, |delta~) = |Ve.sup.-|delta~T~N(|d.sub.3~) - (M + B)|e.sup.-rT~N(|d.sub.4~), (15)

where:

|Mathematical Expression Omitted~,

|Mathematical Expression Omitted~.

C. Case III -- Existing Debt Has Priority Over Seller's Trade Credit

This is the most likely scenario faced by sellers extending trade credit since it is usually the most junior form of debt of a buyer. Most bonds are not secured by specific assets (that is, most bonds are not mortgage bonds). Instead, most bonds that are not debentures are secured by seniority provisions which give them first claim on all assets, including those acquired in the future. In this scenario, the value of the promised payment is given as

|Mathematical Expression Omitted~

where

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~.

In this situation, the value of the trade credit is seriously affected by the presence of existing debt in two ways: the first concerns the priority effect -- the fact that trade credit is junior to existing debt, and the second is the leverage effect of adding more debt to the capital structure.

D. Case IV -- Existing Debt and Seller's Trade Credit Have Equal Priority

Finally, consider a scenario where the firm sells to buyers whose debt consists of trade credit or other debt with the same priority in bankruptcy. Let |alpha~ be the ratio of the seller's credit to the total debt of the firm, |alpha~ = M/(B + M), then the value of the promised payment is given as

P3(|alpha~V, T, M, B, ||sigma~.sup.2~, |delta~) = |alpha~|Ve.sup.|delta~T~N(-|d.sub.3~) + |Me.sup.-rT~N(|d.sub.4~). (17)

Here, the total class of creditors divide the remaining assets of the firm, in the event of bankruptcy, with the seller receiving |alpha~ share of the proceeds. Because Cases II and III are extensions of Case IV, the latter can be viewed as our base case.

III. Simulation Analysis

Exhibits 1 through 3 plot the general tradeoffs between buyer firm value, the present value of a future promised payment (trade credit), the level of preexisting debt and the value of buyer firm equity. Each exhibit varies parameters across a wide range of values. To standardize the results, the present value of the future promised payment (computed from Equations (7), (16) and (17), respectively) is divided by the corresponding risk-free value and the value of buyer firm equity (computed from Equations (9) and (15), respectively) is divided by the corresponding total buyer firm value. Thus, the present value of the promised payment in Exhibits 1 and 2 is measured as a percentage of the corresponding risk-free value, and the value of buyer firm equity in Exhibit 3 is measured as a percentage of total buyer firm value.(8) Exhibits 1 and 2 show very interesting results as the level of prior debt is varied. As the graphs indicate, the percentage value of the promised payment is very sensitive to the level of prior debt. Rather suddenly, there is a significant drop-off in the percentage value of the promised payment as the level of prior debt is increased. Thus, at some critical point, a small change in the prior debt to value ratio can have a large influence on the percentage value of the promised payment. Finally, Exhibit 3 shows that the value of buyer firm equity as a percentage of total buyer firm value falls (rises) as the level of preexisting debt rises (falls) and rises (falls) as total buyer firm value rises (falls).

Although not reported in this paper, we also developed comparable exhibits without dividends (|D.sub.i~ = 0 for all i). The greatest contrast was Case III, where preexisting debt has priority over trade credit. Exhibit 1 shows that the value of trade credit as a fraction of the corresponding risk-free value ranges from 80% to 100% for the assumed parameters. However, in the absence of cash dividends, the value of trade credit as a fraction of the corresponding risk-free value ranges from 94% to 100% for the same set of parameters. Hence, in the absence of cash dividends, trade credit is fairly risk-free and in the presence of cash dividends, trade credit becomes much more risky. This is particularly interesting since Case III is the most realistic scenario. Thus, the explicit incorporation of cash dividends in the trade credit decision can lead to results that differ dramatically from the case with no dividends.(9)

IV. Determining Trade Credit Limits -- An Illustration

Earlier we discussed how to use our model to determine the present value of a future promised payment of the buyer firm with and without preexisting debt. In the former case, we analyzed the following priority arrangements: (i) trade credit that is senior to preexisting debt, (ii) trade credit that is junior to preexisting debt, and (iii) trade credit that has equal priority with preexisting debt. A numerical search procedure was used to simultaneously find the pair of buyer firm value (V) and buyer firm standard deviation (|sigma~) that satisfies the appropriate valuation equation for buyer firm equity (E) and corresponding standard deviation equation for equity (||sigma~.sub.e~). These numerically solved values of V and |sigma~ were then used to determine the present value of the promised payment. Of course, the NPV is the residual, that is, the present value of the future promised payment minus the marginal cost of the goods sold on credit for a given firm. The observables are the risk-free rate (r), schedule of dividend payments (|D.sub.1~, |D.sub.2~,..., |D.sub.n~), future promised payment (M), preexisting debt (B), maturity of the promised payment (T), marginal cost of goods sold on credit (C), and the buyer firm's value of equity while the unobservable variables are V, |sigma~ (which is computed from a well-known relationship with ||sigma~.sub.e~), and ||sigma~.sub.e~ (which can be estimated from historical common stock data). The seller firm will extend trade credit to a given buyer firm up to the point where the NPV is zero. The seller firm will not extend trade credit beyond this point since to do so would be a negative NPV decision. Thus, the trade credit limit for a given firm size is established at the point where the marginal cost of goods sold equals the promised payment (i.e., NPV is zero). What follows is an example for illustrative purposes.

Let's assume the following: r = 9%; E = $1; ||sigma~.sub.e~ = 1.0; T = 1 year; B = $0.20; C (the marginal cost of the trade credit sale) = $0.87; a $0.125 cash dividend one month prior to the contract date; and trade credit is junior to preexisting debt (the most realistic scenario). Applying the model (as described above) results in the following solution to the trade credit limit problem: V = $2.169; |sigma~ = 0.525; M = $1. If the future promised payment is $1, then, the present value of the payment is $0.87 and the NPV is zero. Thus, the trade credit limit is established at $1, that is, the seller would not extend trade credit beyond $1 since to do so would be a negative NPV decision. Theoretically speaking, if the NPV of the project is sufficiently positive, the risk of equity of the buyer firm could decline with the purchase. As noted earlier, if this is true, no finite wealth-maximizing credit limit would result.

V. Conclusion

The framework developed in our study is substantially more general than that proposed by Miles and Varma |23~. First, although the basic mathematical form of the two models is identical, the interpretation of variables differs considerably. They assume a Modigliani and Miller |24~ and |25~ world where there is only a capital structure change, while we allow for the effects of changes in the size and asset composition of the buyer as a result of a credit purchase. Second, Miles and Varma |23~ assume that the addition of trade credit in the firm's capital structure has no effect on the standard deviation of equity. In contrast, we develop a framework which allows us to distinguish between the pre- and post-standard deviation of equity and show that, in addition to leverage and scale effects, the change in the standard deviation of equity is, in part, dependent on the NPV of the project which is financed by trade credit. Thus, if the NPV of the project is sufficiently positive, the risk of the buyer firm could decline with the purchase. In addition to being an improvement over previous papers that treat the buyer's probability of default as exogenous, our study also improves on the Bierman-Hausman |8~ type logic which assumes no possible benefits from the purchase. Finally, we develop an illustrative trade credit limit example which incorporates dividend payouts on the part of the buyer firm and provide detailed simulations through which the practical usefulness of the approach is illustrated.

1Theoretical expositions that either explore the economic rationale for the existence of trade credit (e.g., Lewellen, McConnell and Scott |19~, Ferris |16~, Emery |13~, |14~, |15~, and Smith |30~) or integrate the product-pricing and credit-policy decisions (e.g., Kim and Atkins |17~, and Lam and Chen |18~) are beyond the scope of this study. From a descriptive perspective, however, recent attempts by Srinivasan and Kim |31~ and |32~ to replicate the credit analyst's judgement in setting credit limits may prove to be useful exposition since, by far, the most popular method of setting credit limits appears to be the analyst's judgement (Besley and Osteryoung |7~, and Beranek and Scherr |5~). Also, for a recent discussion on managerial policy alternatives to trade credit, see Mian and Smith |22~.

2Another way of thinking of this effect is as the profitability of the additional business that the purchase will bring to the buyer.

3We've assumed that dividends are known for purposes of simplicity. The methodology employed in this paper is a variation of the Adams and Wyatt |1~ and |2~ model used to price foreign currency options.

4Notice that these partial derivatives involve the cumulative normal distribution which does not have a closed form solution. Because of this and the relatively complex interaction of M with other variables, this derivative must be solved by numerical methods.

5Miles and Varma |23~ do not make this distinction in their paper.

6An interesting research question remains, although it is beyond the scope of this paper, i.e., empirically determining the point at which the equity effect begins to mitigate the combination of pure leverage and asset scale effects. This can best be achieved via detailed simulations assuming various NPVs. This point was suggested by an anonymous referee of this journal.

7According to Case II, the extension of trade credit has priority over the buyer firm's existing debt. Such a case could arise when the buyer firm has preferred stock as the only debt-like instrument in its capital structure. Besides preferred stock, some forms of convertible bonds are junior to trade credit. When such a case arises, the value of the promised payment to the trade creditor is

P1(V, T, M, ||sigma~.sup.2~, |delta~) = |Ve.sup.|delta~T~N(-|d.sub.1~) + |Me.sup.-rT~N(|d.sub.2~).

Notice that the value of the promised payment is the same as when the firm has no preexisting debt. Intuitively, this arises because the selling firm has first claim on assets over the other debtholders in the event of bankruptcy. Hence, unless some other variable (such as firm value) is affected by the extension of trade credit, the fact that the selling firm has existing debt is irrelevant to the valuation of the promise to repay the trade credit.

8Note that E, |sigma~, T, B, r, schedule of dividend payments (|D.sub.1~, |D.sub.2~, |D.sub.n~), C (the marginal cost of the goods sold) and M are directly observable or easily obtained. This information is used by the seller to infer the value of the buyer's assets (V) and the return variance on these assets (||sigma~.sup.2~). Since V and ||sigma~.sup.2~ cannot be solved for explicitly, a single computer numeric search algorithm is used to obtain V and ||sigma~.sup.2~. These values are then used to solve for the value of the promised payment (P, P1, P2, and P3). The NPV of the trade credit sale is merely the difference between the value of promised payment and C.

9Moreover, we developed simulated graphs of Cases I through IV across present values of nominal promised payments as a percentage of corresponding risk-free values, firm standard deviations and firm values. These plots were identical for Cases I and II. The reason for this result is because preexisting debt is equivalent to equity as long as this debt is junior to trade credit. In practice, the only debt and debt-like instruments that are junior to trade credit are income bonds and preferred stock. In each of the four cases, the present value of the future promised payment as a percentage of the risk-free value falls (rises) rapidly as the standard deviation of buyer firm value rises (falls) and rises (falls) slowly as the value of the buyer firm rises (falls). There is a clear and smooth rise (fall) in the percentage value of the promised payment as the value of the buyer firm rises (falls). Finally, we analyzed simulated graphs of Cases I through IV across values of buyer firm equity as a percentage of corresponding firm values, firm standard deviations and firm values. The results show that the value of buyer firm equity as a percentage of total buyer firm value is positively related to both buyer firm standard deviation and total buyer firm value.

References

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Title Annotation: | Special Issue: Corporate Control |
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Author: | Adams, Paul D.; Wyatt, Steve B.; Kim, Yong H. |

Publication: | Financial Management |

Date: | Sep 22, 1992 |

Words: | 6104 |

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