Avoiding Distortion in Corporate Valuation Litigation: An Application of Discounted Cash Flow

Abstract

A methodology used widely by economic experts to determine a company's value is discounted cash flow (DCF).1 The most common application of this valuation methodology requires the firm's free cash flow to be discounted at the firm's cost of capital. This article deals with an inherent tautology in the determination of value using the DCF analysis, namely that the firm's value is initially assumed in order to calculate the value of the firm. This “Catch 22” may well result in a firm value that deviates significantly from the firm's true value and in an erroneous valuation presented to the court. The article describes scenarios that result in distorted DCF values. In particular, it identifies those situations where firm value deviates markedly from that predicted by conventional DCF analysis.A variety of techniques are used to value a firm, yet discounted cash flow is one of the more frequently applied methodologies. It is generally considered a particularly reliable indicator of value because it employs specific cash flows of the company in question rather than relying on characteristics of comparable companies or comparable acquisitions.The financial expert valuing a firm using DCF typically projects the firm's debt-free cash flows and discounts these cash flows using the firm's weighted average cost of capital (WACC). The value obtained is generally presented to the court as the firm's Business Enterprise Value (BEV). The firm's debt is then subtracted from this value resulting in a calculated value of the firm's equity. This methodology is used in valuation matters by both plaintiff and defendant experts. For example in the 2007 Highfields Capital, Ltd. v. AXA Financial, Inc. matter,2 the court states that “[t]ypically, Delaware courts tend to favor a DCF model over other available methodologies in an appraisal proceeding.”3 Similarly, in other valuation matters, courts ordinarily rely on the DCF methodology as a standard approach to determine value.4 Yet, while DCF has gained credence in the courtroom, frequently, the calculated equity value using DCF is incorrect.The DCF methodology incorporates a systemic flaw which has the potential to significantly affect the quality of a valuation. The flaw arises because of the innate tautology in calculating the firm's value. Namely, in order to calculate the value of a stream of cash flows, the expert must first assume the firm's value. In other words, in calculating the firm's DCF, one of the key inputs is the firm's weighted average cost of capital. This is comprised of the proportions of debt and equity multiplied by the respective costs of the firm's debt and equity. But to determine the proportions of debt and equity in the firm's capital structure, one must divide the assumed values of debt and equity by the value of the firm. These assumed values or assumed proportions of debt and equity may well differ from the values obtained by performing the standard DCF calculations. This inconsistency may lead to significant under or overestimates of the true value of the firm.The WACC incorporates the assumed proportions of equity and debt in the firm's capital structure. These proportions result from the firm's equity to value (E/V) and debt to value (D/V) ratios. To obtain these ratios, one generally uses either the firm's own market value of equity and debt, target values of equity and debt, or values based on comparable firms in the industry. While any of these starting points of equity and debt may be appropriate, they may vary significantly from the proportions implied by the DCF calculation. Given that an analysis assumes initial proportions of debt and equity as a percentage of firm value, there is no assurance that the use of DCF will result in the identical final proportions of debt and equity and in the correct value of the firm determined by the economic expert.This article discusses a resolution of this circularity problem. The resolution is based on the degree to which the firm rebalances its debt. The traditional view of rebalancing has the firm trading off the costs and benefits of different degrees of leverage to determine an optimal proportion of debt and equity. Whenever the debt and equity shift from this proportion, the firm responds by rebalancing its debt and equity such that the proportions are returned to an optimal level. A firm that rebalances its debt continuously keeps its proportions of debt and equity to value constant. As such, the WACC does not shift and the value calculated using the DCF will reflect the true value of the firm. Thus, to the extent that a firm continuously rebalances its capital structure, the traditional DCF provides the appropriate value of the firm. On the other hand, if the firm rebalances only periodically or has no plans to re-balance its debt, the value of the firm calculated by the application of the DCF results in a shift in the proportion of debt to value and equity to value. This in turn causes the WACC to shift, driving a change in value. Thus, if firms do not continuously rebalance, the calculated value of the firm will be distorted from the firm's true value. This may well result in an inappropriate value of the firm presented to the court and possibly used as a wrong basis for a damage calculation.In the literature there are several earlier investigations of the circularity problem resulting from the common use of DCF. Wood and Leitch (2004) focus on finite projects. They offer a valuation procedure which allows for capital structure to change over time and in which the cost of debt and equity depend on the capital structure. The procedure is recursive and starts from the terminal period. First, the weights of debt and equity for the WACC of the last period are calculated based on the last cash flows to each of these financing sources. Then, working backwards, the second to last period's debt ratio and WACC are calculated based on the second to last period's cash flows and the last period's valuation of debt and equity, etc. Because of the circularity the solution is obtained in an iterative process.5 Valez-Pareja and Tham (2005) show that for infinitely lived firms the iterative solution offered in Wood and Leitch (2004) yields erroneous valuations. They improve and simplify the recursive procedure suggested by Wood and Leitch. In their revised procedure, as they admit, the circularity problem is not completely resolved.Both the iterative solutions offered by Wood and Leitch and by Valez-Pareja and Tham are backward looking. Because the procedures are built recursively based on the last period, they require good forecasting of the long-term cash flows to equity and debt. Our approach does not build on the exact cash flow path to debt and equity. Instead we seek only the resolution of the circularity problem under fixed debt and provide a practical and relatively simpler solution. When accurate information about future cash flow to debt and equity is available, the procedures suggested by Wood and Leitch and refined by Valez-Pareja and Tham will provide good solutions to the circularity problem. Most practical DCF valuations, however, consider long-lived firms or projects where it is impossible to forecast remote cash flows with accuracy, suggesting a better fit with our approach than with a backward-looking approach.Mohatney (2006) also considers the circularity problem and offers an iterative solution. His solution, however, assumes an exogenous growth of debt and equity financing. He does not consider the issue of rebalancing.The difficulty caused by the circularity embedded in a standard DCF calculation was recently highlighted by Delaware Chancery Court Judge Jacobs in his In re Emerging Communications, Inc. Shareholders Litigation opinion. He discusses the “ultimate circularity inherent in the WACC” and mentions, but does not describe, the use of an iterative solution proposed by one of the experts to resolve the problem.6 He goes on to describe an iterative solution proposed in Cornell (1993). Cornell, however, does not address the effect of the firm's capital structure policy on the severity of the circularity problem.Whether firms rebalance their debt or not and the implications for valuations are still open questions. At one extreme firms are assumed to continuously rebalance their debt (e.g., Ruback 1998, 2002). At the other extreme firms do not adjust their debt capacity at all (e.g., Modigliani and Miller, 1963). The truth is probably somewhere in between. Lemmon, Roberts, and Zender (2006) find that firms consistently rebalance over a long period of time. However, many other recent articles argue that firms do not rebalance continuously, but slowly adjust their capital structures, if they do so at all. Fama and French (2002), for example, suggest that firms slowly adjust to target proportions of debt and equity in their capital structures. Baker and Wurgler (2002) argue that market timing has a persistent impact on firms' capital structures. Welch (2004) also concludes that equity price shocks have a long-term impact on capital structures, and that changes in a firm's debt to value ratio originate in fluctuations in equity prices rather than rebalancing. In sum, these studies suggest that firms often do not rebalance and their debt ratios increase and decrease with market cycles. When markets are booming, IPOs and secondary equity offerings frequently result in high market values, whereas during recessions, market valuations are low and firms hardly issue any equity. Even if there is some rebalancing, it happens with a significant delay (e.g., Roberts, 2002). Leary and Roberts (2005) find that adjustment costs drive a “persistent effect on leverage” with these costs determining the speed at which firms respond to leverage shocks. They also conclude that the rebalancing process is “slow” and firms, when they do tend to rebalance, do so to a targeted range rather than a specific debt to equity level. This supports the survey results of Graham and Harvey (2001) who indicate that 71% of the CFOs in their sample had a targeted debt/equity range, while only 10% had a specific targeted debt/equity ratio. There is, unfortunately, no available research that assesses whether a specific firm pursues a rebalancing strategy or not. In order to determine if the traditional DCF methodology is appropriate, one must make a judgmental assessment for each firm being valued as to whether debt is being rebalanced. The closer the firm is to continuously rebalancing its debt, the more appropriate the use of the traditional DCF. This paper will show that the application of traditional DCF may well distort the calculated value. Moreover, after more than two decades of providing expert testimony on corporate valuation we have seen no cases in which experts have incorporated debt rebalancing into the valuation analysis.Let us first consider the traditional DCF valuation using the free cash flow method (FCF). In this method, cash flows over the investment horizon are estimated and discounted at the weighted average cost of capital (WACC). The WACC is represented by Here rD and rE are the firm's cost of debt and cost of equity, respectively, τC is the corporate tax rate, and and are the debt ratio and the equity ratio, respectively. The firm value is calculated as where cfi is the free cash flow in period i. The debt and equity ratio inputs used are generally obtained from market data portraying the firm in question or from comparable firm data. The debt ratio is calculated as where EM and DM are the market values of the equity and debt, respectively. Frequently the debt is not traded in which case the book value of debt is used as an estimate for the market value of debt. The following example demonstrates the standard valuation process.Consider the following data obtained by an economic expert (Expert A) in the process of valuing Globus Industries. Globus stock trades in an efficient market where the firm has a cost of debt and equity of rD = 6% and rE = 14.15%, respectively. The corporate tax rate is τC = 35%. The market value of the firm's equity (observed) is EM = $650 million, and the market value of its debt is DM = $975 million, resulting in a firm value of $1.625 billion. The debt ratio assumed for the valuation is thus and the firm's weighted average cost of capital is thusThe future cash flows of the firm are estimated to be $250 million per year in perpetuity.7 Each year's after tax cash flow is thus $250 * 0.65 = $162.5 million, with the value of the firm, VA, calculated as and the value of the firm's equity, EA, calculated asTypically the analysis would stop where the example above ends, with the results suggesting that the value of the firm is VA = $2.031 billion and that the value of the firm's equity is EA = $1.056 billion. The analysis would then conclude that the equity is traded at a discount, specifically at of its true value. A more careful look at the above example, however, raises the following problem: if the value of the firm is VA = $2.031 billion, the actual debt ratio is only The assumption underlying the WACC valuation method is that debt accounts for 60% of the value, and because the assumption does not hold, this leads to distortion in the expert's calculated value. Namely, given that debt financing is only 48% of the value, the conclusion that VA = $2.031 billion does not follow. The WACC method requires that debt financing would account for 60% of the value, i.e., DA = 60% * $2.031 = $1.219 billion. Since debt is only DM = $975 million, not the imputed value DA = $1.219 billion, a conundrum is posed by a $2.031 billion valuation of the firm.In what follows we demonstrate that the WACC valuation is correct only under a very restrictive assumption, namely, that the firm continuously rebalances its debt. We develop an iterative valuation methodology to correct the standard valuation method for firms that are not expected to adjust their debt. We show that the valuation differs substantially from that of firms assumed to rebalance.A. Valuation: The Policy of Debt RebalancingIf indeed, the true value of the firm is $2.031 billion, then several observations are likely. First, assuming efficient markets, if the expert is indeed correct in his estimate, the market would sooner or later make an upward correction in the firm's equity valuation from $650 billion to $1.056 billion, while increasing the firm value from $1.625 billion to $2.031 billion. This revised equity value increases the value of the firm to $2.031 billion and lowers the ratio of D/V to $975/$2,031 = 48%. Since the nature of the firm's business suggests a debt capacity of 60%, as is also reflected in the WACC, the firm's debt capacity is now revealed to be much higher than the actual amount borrowed of $975 million. Namely the firm's debt capacity is 60% * $2.031 = $1.219 billion. Thus, over time, the firm presumably could increase its debt. To the extent that the firm rebalances its debt, the firm would maintain the same debt/value ratio that it had under the prior valuation since the nature of the business has not been altered. Thus, under rebalancing, the firm would increase its D/V ratio from $975/$2,031 = 48% to $1,219/$2,031 = 60% by issuing $244 million of debt and using these funds to retire $244 million of the equity. Following the rebalancing process, the firm is left with equity having a value $1,056-$244 = $812 million and indeed E/V = $812/$2,031 = 40%, which is consistent with the expert's valuation. In fact, for the valuation to be correct, the firm must rebalance (increase) its level of debt to keep the debt ratio fixed because this is a central assumption underlying the valuation using the WACC.The process by which the firm shifts from the equity value of $650 to an equity value of $812 million can be seen in the following T-accounts (millions of dollars).The initial market value balance sheet of the firm is as follows:Information revealed in the market place causes the stock price to increase eliminating the undervaluation:The firm rebalances its debt ratio by issuing $244 million of debt and retiring $244 million of equity (e.g., through a dividend or a stock buyback):Thus, to the extent that the information revealed to the market is believed to be accurate and to the extent that the firm rebalances its debt, several observations can be made. As the market value of the equity of Globus increases, the firm's debt capacity increases. The firm's actual debt should also increase over time as the firm uses its debt capacity, and the firm's value will thus increase. In other words, given that the proportions of debt and equity to value remain unchanged, the firm's WACC does not change and the value determined by the expert is appropriately determined.B. Valuation: The Policy of Fixed DebtWhile rebalancing appropriately resets the firm's debt/equity proportions following equity price shocks, the evidence, as described earlier, suggests that continuous rebalancing is rare. Moreover, even in those situations where the expert's projections are reasonable, the firm may still not adjust its debt based on the value projected, resulting in lost tax shields which impact the calculated DCF value. Complicating the firm's ability to rebalance its debt, even after a possible correction of the market pricing of the equity, is the fact that DCF valuation is about discounting projected cash flows whereas debt holders frequently require collateral. While some firms have the stability and track record to borrow based on reasonably projected cash flows, many firms turn to asset-based lenders which require collateral. Rebalancing when debt is constrained by collateral requirements may well become impossible. At the heart of the problem is that the WACC that the expert uses to perform the valuation assumes a debt ratio which may differ significantly from the WACC implied by the value that the expert calculates.Thus, if the firm being valued (in this case, Globus) does not rebalance continuously, several alternative observations can be made:Without rebalancing, the firm will not change its debt financing despite any correction in the market valuation of the equity. Thus, the weights of debt and equity would be based on the actual debt of the firm with the debt/value ratio equaling $975/$2,031 = 48%, and the equity/value ratio equaling 52%. Since these ratios differ from those initially assumed (debt/value ratio of 60% and equity/value ratio of 40%) the true WACC must be calculated in an iterative process as follows: The new debt and equity ratios are used to calculate a second WACC:This WACC is significantly different and higher than that predicted when re-balancing occurs (i.e., $8%). Namely, under the expert's estimates of the future cash flows WACC2 implies that the value of the firm isHowever, a valuation of $1.761 billion again implies a third WACC. This is because the new debt ratio is $975/$1,761 = 55.37%. This becomes a process which continues until convergence, Namely, the third WACC is calculatedUnder WACC3 Globus' value is calculated to beThis iteration process must be repeated until convergence. The WACC in this example has converged on the 14th iteration to 8.762% with the value of Globus converging to Indeed the implied debt ratio is $975/$1,854.5 = 52.57% which is consistent with Thus, for the firm that does not rebalance its debt, the value is $1.8545 billion, with the debt ratio calculated at $975/$1,854.5 = 52.57%. Table 1 summarizes the results of the iteration process. While WACC is initially assumed at 8%, the final WACC is increased by 9.5% to 8.762%. Driving this increase in WACC was a reduction in the assumed proportion of debt in the capital structure from 60% to 52.57%, a reduction of 12.4%. Note that in the iteration process, Expert A initially determined that VA, the value of the firm, is equal to $2.031 billion and ended with VA = $1.8545 billion, 91% of the initial estimate, whereas the value of the equity resulted in only 83% = ($1,854.5 — $975)/($2,031 — $975) of the initial estimate.10The tedious iteration process can be avoided using an analytic approach in the case where all the cash flows are identical. The analytic solution is demonstrated in section A of the Appendix. In section B of the Appendix we consider the convergence criterion for the iterative process.The example above demonstrates that under the assumption that firms do not adjust their debt financing, the standard WACC method can significantly distort the valuation and lead to inappropriate results and inaccurate damage estimates. In the example, we considered a firm that was undervalued based on the expert's analysis. Note that the only case in which the standard DCF valuation is correct for the non-rebalancing firm arises when the expert's valuation is identical to the market valuation. When the expert's valuation deviates from the market's valuation and the firm is not expected to rebalance its debt, he can use the iteration process on the firm's WACC to determine the proper value of the firm.The problem will also exist when the expert's valuation results in a value lower than the firm's market value. Consider, for example, a second expert (Expert B) who believes that the estimated cash flows are only $150 million per year in perpetuity. The expert will calculate the value of the firm as For this valuation to be correct, the value of the equity should decrease to $1,219 — 975 = $244 million. Assuming now, instead, that the second expert's estimate of the cash flows (rather then the first expert's estimate) is the correct estimate, and that the financial markets are generally efficient, the market price should fall to reflect the cash flow information. As discussed earlier, based on the firm and industry characteristics, the firm's debt and equity to value ratios remain at 60% and 40% respectively under rebalancing. Since the equity value becomes 40% of the firm value, or 40% * $1,219 = $488 million, the firm should issue equity in the amount of $244 million (to obtain $488 million) and retire debt in the same amount. However, while firms may or may not rebalance their debt following an increase in the market value of their equity, they are less likely to rebalance their debt following a decrease in their market value.11For firms that do not rebalance, using the standard DCF rather than the appropriate iterative process causes the expert to overestimate the true value of an undervalued firm and underestimate the true value of an overvalued firm. For example, in the numerical example considered above, the first expert concluded that the firm value is $2.031 billion, which is higher than the market valuation of $1.625 billion ($975 million debt + $650 million equity), whereas we have shown that if the firm does not rebalance its debt, the firm's value given this expert's forecasted cash flows is only $1.855 billion. In other words, when the expert's estimate is greater than the market valuation (i.e., the firm is undervalued), the standard DCF overestimates the firm's value. Similarly, the second expert we considered had lower estimates of the cash flows than the first expert and concluded that the firm's value is only $1.219 billion, which is lower than the market valuation of $1.625 billion. However, it can be shown (using the iteration process) that if the firm does not rebalance its debt, the firm's true value given the second expert's forecasted cash flows is actually $1.395 billion. In other words, when the expert's estimate is lower than the market valuation (i.e., the firm is overvalued), the standard DCF underestimates the firm's value. In sum, for firms that do not rebalance their debt, the standard DCF valuation rather than the iterative procedure always exaggerates the market mispricing, that is, it always exaggerates the difference between the true value of the firm and the market price.To understand why the standard DCF valuation always exaggerates the market mispricing for non-rebalancing firms, consider its inherent tautology. The standard DCF valuation calculates the discount rate based on the debt ratio implied by the current valuation of the debt and equity in the financial markets. However, for firms that are currently mispriced in the financial markets, the true debt ratio is different. In the example, this is evident by recalling that the firm's value in the financial markets is $1.625 billion, and the value calculated using the standard DCF analysis by the first expert was higher than the true value given his cash flow forecasts ($2.031 billion vs. $1.855 billion, respectively), whereas the standard DCF value calculated by the second expert was lower than the true value given his cash flow forecasts ($1.219 billion vs. $1.395 billion, respectively).If the first expert's forecasted cash flows are correct and the firm is undervalued according to the standard DCF valuation ($1.625 billion vs. $2.031 billion), its true debt ratio is lower than the debt ratio in the financial markets. Indeed for a non-rebalancing firm the first expert's valuation implies a debt ratio of $975/$2,031 = 48% rather than the debt ratio of $975/$1,625 = 60% implied by the market valuation. Thus, the weight of the debt used by the first expert in his valuation (60%) is too high, resulting in a cost of capital which is too low and consequently a valuation which is too high. Following the iteration process, the true value of the non-rebalancing firm based on the first expert's forecasted cash flows is found to be a lower $1.855 billion. Conversely, if the second expert's forecasted cash flows are correct and the firm is overvalued according to the standard DCF valuation ($1.625 billion vs. $1.219 billion), its true debt ratio is higher than the debt ratio in the financial markets. Indeed, for a non-rebalancing firm, the debt ratio implied by the second expert's valuation is $975/$1,219 = 80% rather than the debt ratio of $975/$1,625 = 60% in the financial markets. Thus, the weight of the debt used by the second expert in his DCF valuation (60%) is too low, resulting in a cost of capital which is too high and consequently a valuation which is too low. Following the iteration process, the true value of the non-rebalancing firm, based on the second expert's forecasted cash flows is found to be a higher $1.395 billion.Figure 1 demonstrated the value distortion discussed above by means of a graph. The figure describes the relationship between the cash flows estimated by both experts and the firm's resulting value assuming policies of both rebalancing and non-rebalancing. The dashed line represents the valuation of the firm under the standard DCF which assumes that the firm rebalances its debt. The solid line represents the true value of the firm assuming it does not rebalance its debt. For the non-rebalancing firm, the vertical difference between the two lines represents the distortion in the valuation when one uses the standard DCF analysis for the non-rebalancing firm. The crossing point of the lines is at $200 million on the x axis and $1.625 billion on the y axis and represents the annual cash flow implied by the market and the market valuation of the firm, respectively.Note that for cash flow estimates above $200 million, the line representing the rebalancing firm implies valuations greater than those implied by the line representing firms adhering to a policy of fixed debt. This results because under rebalancing, as cash flows increase, both the firm's value and its debt will increase, keeping the WACC constant. Therefore, the tax shield generated by the firm's increased debt also increases. Thus, under debt rebalancing, the increased value is generated by both the effects of increased cash flow and the benefits resulting from the increased tax shield. For firms with fixed debt, increased value is also generated as cash flow is increased, yet no additional value is generated from the benefit of the tax shield since no new debt is incurred. Thus, as value increases for cash flows greater than the crossing point ($200 million), the value of the rebalanced firm (with its associated tax shields) will exceed the value of the firm with a fixed debt policy (no additional tax shields).Similarly, as cash flow estimates fall below the crossing point, the value of the rebalancing firm will fall, and as a result of the rebalancing process the firm will reduce its debt. This reduction in debt will in turn result in a reduction in the firm's tax shield. However, for the firm with a policy of fixed debt, its debt will remain fixed as the firm's estimated cash flow falls. This firm will maintain its tax shield with a reduction in cash flow. As a result, the value of the firm adhering to a fixed-debt policy will exceed that of the firm that rebalances its debt.In terms of the numerical example, the first expert assumes cash flows of $250 million and calculates the value of the firm using standard DCF to be $2.031 billion. The value obtained using the DCF iteration process is determined to be $1.855 billion. The difference, $176 million or 9.5% of the value, represents distortion. Moreover, the market mispricing which represents the difference between the true value of the non-rebalancing firm and that implied by the market is $230 million ($1.855 — $1.625)/$1.855 = 12.4%). The second expert assumes cash flows of $150 million and calculates the value of the firm using standard DCF to be $1.219 billion. The value obtained upon performing the DCF iteration is $1.395 billion. The distortion in the valuation for the non-rebalancing firm is thus $176 million or 12.7% of the firm's true value. The market mispricing given the second expert's forecasts is $230 million ($1.625 — $1.395)/$1.395 = 16.5%).While corporate valuation litigation often results in the application of discounted cash flow analysis, an inherent tautology exists in its use. This paper analyzes that tautology. The paper demonstrates that the tautology can be explained by an inherent assumption in the DCF analysis, namely, that a firm continuously rebalances its debt. Although continuous rebalancing is the standard framework implicitly assumed in a DCF analysis, most firms only slowly, if at all, adjust their capital structure. This suggests that the true value of the typical firm lies somewhere between that determined under each of the extreme assumptions, continuous rebalancing, or no rebalancing at all. It is important to assess the policy of the firm being analyzed. For those firms that are expected to frequently rebalance, the standard DCF framework may be appropriate for valuation purposes. However, for those firms that are not expected to continuously rebalance their debt, this article provides an iteration process for modifying the standard DCF analysis. The economic expert should be aware that if a standard DCF analysis is performed to value a firm, there is likely to be distortion from the firm's true value unless the firm is continuously rebalancing its debt. Moreover, any calculation of damages based on the expert's determination of value is likely to be distorted as well.A. Analytic solution of the circularity problemIn the case in which cash flows are stable for all dates (i.e., cfi = cf), there exists an alternative straightforward approach to determine the DCF solution for the case in which the firm does not continuously rebalance. Specifically, the iteration process can be avoided by solving three equations: an equation defining the WACC, an equation defining firm value as a function of the WACC, and an equation defining the ratio of debt to value. These equations are: and The three equations (3), (4), and (5) with three unknowns, w, WACC, and VA can be easily solved: substituting equation (4) into (5) we obtain Then, substituting for the WACC into (3) we find and upon rearrangement Indeed, in our numerical example we have We then substitute this value of the weight w into (6) and rearrange to find: Substituting this WACC into (4) we obtain: Note that the values of w, WACC and VA calculated by solving the equations are consistent with the ending values of the iteration process (see Table 1).This straightforward method can also be used to evaluate firms whose cash flows are growing perpetuities. However, for valuations in which each of the cash flows is separately estimated, the iteration method must be used since each cash flow is different and must be discounted separately.B. Convergence of the Iteration ProcessThe iteration process starts with an assumption about the ratio w = D/V and then improves the estimation until convergence. Denote the initial value of w with w0. We will use subscript i = {0,1,2...} to indicate the iteration. For simplicity of exposition we will denote rd =rD (1 – τC). The iteration process starts with and and It then calculates and The relation between w1 and w0 is thus which can be rearranged to and or Similarly we can express w2 as a function of w1 as Upon substitution of w1 in terms of w0 w0 we have In the same manner we can recursively calculate that and In generalConvergence requires that the difference between wt and wt – 1 will converge to zero. That is Using (8) we can write It is enough to show convergence of the absolute value |wt – wt–1|. or The term in the brackets is constant and hence (9) converges to zero whenever For completeness, assuming (10), we can use (8) with T→∞ to write the true weight to which the series converges as which is what we found in (7) upon substitution of rD(1 – τC) for rd.To conclude, as long as condition (10) holds, convergence is warranted. For example, in our numerical example and hence the process is guaranteed to converge.