Abstract
This paper identifies three sources of risk for tax shields (TS): two of them associated with debt risk and one associated to operating risk. A set of conditions for defining risky debt associated to cash flow and not to accounting earnings is presented. The paper shows that realization of tax shields for finite cash flows in any period of time t are correlated with Earnings before Interest and Taxes and are not with interest expenses at time t.
The paper questions the validity of Miles and Ezzell’s proposal.
With the results of a Monte Carlo Simulation (MCS) the behavior of tax shields, Cash Flow to Debt (CFD) and Earnings before Interest and Taxes plus Other income OI, (EBITO) is examined.
As a conclusion, the paper suggests that it is not reasonable to define the risk of TS as the measure of a single risk, but it is a mix of debt risk and operational risk.
Introduction
In current financial literature the volatility of tax shields (TS) from debt is usually associated with debt risk. This paper suggests that the volatility of tax shields is also associated with operating risk. The existence of Earnings before Interest and Taxes (EBIT) plus Other Income, (OI) (EBITO, in short) is what makes it possible for the firm to earn tax shields. Interest expenses are the origin of debt tax shields; however, the realization of TS depends on EBITO. This is relevant because TS plays a crucial role when defining cash flows and cost of capital for valuation purposes.
This work is organized as follows: In Section One relevant literature is reviewed. In Section Two Cash Flow to Debt, (CFD) and in particular the interest expenses and tax shields (TS) are studied. In this Section sources of TS risk are identified and a critique to the Miles and Ezzell, (1985)’s proposal is presented. In Section Three with the results of a Monte Carlo Simulation, (MCS) the behavior and relationships of tax shields, CFD and EBITO are examined. Section Four summarizes the findings. There are four appendixes that explain the financial planning model used, the descriptive statistics resulting from the MCS, the derivation of conditions for default and the derivation of an algorithm to calculate TS.
Section One. Literature review
There are several approaches to discounting tax shields. One of them establishes that tax shields have to be discounted at the cost of debt (more precisely at the risk free cost of debt), see for example, Modigliani and Miller (1958, 1963), Myers (1974), Luherman
(1997), Brealey and Myers (2003), and Damodaran (2002, 2005). Another school of thought says that the discount rate for tax shields should be the unlevered cost of equity, see for instance, Harris and Pringle (1985), Ruback (2002), Tham and VélezPareja (2001, 2004). Finally, Miles and Ezzell (1985) and Arzac and Glosten (2005), propose to discount tax shields at the cost of debt for year t and at the cost of unlevered equity for all subsequent years from t+1.
Regarding the behavior of tax savings, Cordes and Sheffrin, (1983), Dammon and Senbet (1998), Graham (2000) and Grabowski (2009) report that some firms do not fully use their tax shields in the current period but receive them in the future when losses carried forward are allowed.
New bould, Chatfield and Anderson (1992) and Kaplan and Ruback (1995), Fama and French (1998), Graham (2000), and Kemsley and Nissim (2002) agree that tax shields are relevant and might share a good part of firm value.
Section Two. Understanding Cash Flow to Debt and Tax Shields
This section examines cash flow to debt, tax shields and the sources of risk for tax shields and also makes a digression on the Miles and Ezzel (1985) proposal.
Cash Flow to Debt
Cash Flow to Debt (CFD) is what debt holders lend to the firm and/or receive back from the firm when loans are paid. The CFD has two components:
1. Interest payments
2. Net principal payments/loans inflows
Interest payments in t are calculated on the amount of debt Dt1. This means that interest charges are fully known at t1, assuming constant cost of debt. Interest payments are
interest charge, Kd is the cost of debt and D is debt.
Alternatively, net principal payments/loans inflows are an item defined in time t. In case of deficit, a financial planning model will define when and how much debt should be acquired and when to pay the principal. In other words, this part of CFD is based on situations that occur and/or decisions made at time t. Deficits depends on sales revenues, expenses, capital investments and the like. That is, it depends on the operating and investment activities of the firm.
Proper Calculation of Tax Shields
Wrightsman (1978) and VélezPareja (2009 and 2010) propose an algorithm for calculating tax shields. It is of interest to analysts when forecasting financial statements and cash flows to estimate values of firm and equity. While interest charges are fully known at any t1 assuming constant cost of debt, tax shields are not. Tax shields at t depend on EBITO and are a piecewisedefined function of it, as follows:
TS is tax shields, FE is financial expenses, T is corporate tax rate and EBITO is EBIT + OI. This piecewisedefined function is depicted in exhibit 1. See derivation in Appendix D.
Exhibit 1. Behavior of tax shields (TS) as a function of EBITO
The right to earn tax shields occurs when interest is subtracted and deduced in the Income Statement, not when interest is paid. In other words, a firm could never pay the accrued interest and yet, obtain the right to the tax shield. However, the firm actually receives the tax shields when taxes are paid. Tax shields reduce the amount of taxes paid, compared with an unlevered firm.
For a better understanding of these ideas, table 1 shows the conditions for interest, tax shields and taxes for actually earning the TS.
Table 1. Conditions of accrual and payment for earning and accruing TS
Taxesand interestpaid att 
Interestpaidint and Taxespaidatt+1 
Taxesand interestpaidat t+1 
Taxespaidatt and interestpaidatt+1 

Year 
t 
t 
t+1 
t 
t+1 
t 
t+1 
Interest 
Accrual/Cash 
Accrual/Cash 
Accrual 
Cash 
Accrual 
Cash 

Taxshields 
Accrual/Cash 
Accrual 
Cash 
Accrual 
Cash 
Accrual/Cash 

Taxes 
Accrual/Cash 
Accrual 
Cash 
Accrual 
Cash 
Accrual/Cash 
Accrual means that the item is listed in the Income Statement as an accrual. Cash means that the item is received/paid.
Previous table indicates when each item is accrued and/or is paid/received. Observe that in all cases the right to earn tax shields occurs when interest and taxes are accrued, no matter if interest is paid or not. Note as well that TS are received only when taxes are paid.
Another way to understand the idea of the right to earn tax shields when interest/taxes are accrued is to think on what happens when losses carried forward (LCF) are allowed. The firm that has financial expenses(interest charges) has the right to earn the tax shields (T times financial expenses) but when EBITO is zero or negative (the extreme case) the tax shields are apparently lost because taxes are zero. However, when LC Fare allowed, those tax shields from the year when EBITO was negative can be recovered when losses from previous years are carried forward to a future year where the firm has enough Earnings before Taxes (EBT) to offset previous losses.
Sources of Tax Shields Risk
Wrightsman (1978) defines risky debt “in terms of the possibility that EBIT may turn out not to fully cover interest expense. If some possible EBIT outcomes from a given probability distribution of EBIT fall short of interest, then interest can fall into jeopardy, and debt is risky”. (p. 652).
On the other hand, Talmor, Haugen and Barnea (1985) consider the conditions for risky debt are associated to cash flow, not to EBIT. They however, suggest that the tax deduction and tax shields are obtained when the firm pays the interest charges and not when the firm accrues those financial expenses and pays taxes. Generally Accepted Accounting Principles (GAAP) regulations require companies construct their financial statements according to the accrual method, hence, the tax deduction (and the tax shields) is obtained on an accrual and not on a cash basis.
Tax shields have two sources of risk: debt risk and operational or realization risk. Debt risk is associated with cash flows (from Cash Budget or Cash Flow Statement) and operational risk is associated with EBITO, from the Income Statement.
Debt risk has two components: Market or systematic risk and default risk. Creditors estimate the level of default risk (probably looking at leverage and other items) and add it to the risk free rate to obtain the cost of debt. Market or systematic risk is associated to the volatility of Kd, the cost of debt. Insolvency occurs if there is not cash availability and this is measured with the Cash Budget or Cash Flow Statement, not with the Income Statement. In fact, the firm will pay interest either from operating income or from other sources in order to avoid insolvency. Interest can be paid out from internally generated funds such as depreciation,from other debt or from new equity investment.
Assuming that losses carried forward are not allowed, and given the residual characteristic of cash flow to equity, CFE, the firm will default when the following general conditions are fulfilled, from the point of view of cash flow:
where Dep is depreciation charges, CAPEX is capital expenditures, dWC is change in working capital, dCS is the change in capital stock and other variables have been defined above. See Appendix C.
However, the firm will default under four scenarios:
derivation of (3e) is presented in Appendix C.
This idea complements the proposal from Talmor, Haugen and Barnea (1985). From these conditions for default it can be seen that the criteria for defining risky debt comparing EBIT and financial expenses, is not appropriate. Insolvency does not come out because EBIT or EBITO falls short of interest.
The other source of tax shields risk is when EBITO outcomes from a given probability distribution are less than interest. This is exactly what equation (2) is showing. Even with a risk free debt, tax shields might be risky due to EBITO. A firm might have a risk free debt (from a cash availability point of view) and yet, tax shields are risky (from the Income Statement point of view). Put differently, EBITO could be greater than financial expenses and yet, the firm could default and hence, debt is risky. EBITO is calculated on an accrual basis. Insolvency is related to cash flow, to cash availability.
The fact that D is known at t1 and that interest is fully known at that time assuming constant cost of debt, does not mean that tax shields are certain or riskless, nor debt is riskless. Tax shields depend on EBITO which is a random variable that depends in turn several other variables such as prices, inflation, increase in units sold, expenses, etc.
In summary, tax shields risk depends on EBITO and this risk is independent of debt risk in the sense explained above: the firm could have a risk free debt and yet have a risky tax shields. Tax shields are associated to the accrual of interest. On the contrary, tax shields are received when taxes are paid, not when interest are accrued nor when interest is paid.
Said this, three sources of risk are identified and associated to TS risk:
 Risk of default in debt. There exists a possibility that there is not enough cash for paying interest and/or principal. If the default is such that the firm does not pay the accrued taxes (default in taxes), it will not earn the TS (on a cash flow basis). This means risky debt and risky TS. If the firm defaults in debt and not in taxes, it means risky debt but it might effectively earn TS.
 Market cost of debt risk. Variable Kd is subject to systematic risk. The market rate could change and that is a source of risk for the TS. However, what creates TS is not the market rate but the contractual rate.Unless there is a link between the contractual rate and the market rate the risk of Kd (market rate) does not affect Kd contractual. There is one way to link that rate to market rate: when the contractual rate is indexed to inflation. This means risky debt and risky TS.
 Operational or realization risk of TS is associated with EBITO and the financial expenses. This is a clear dependence of TS from EBITO. This is the piecewise function of TS as a function of EBITO (see eq (2)). This means risky TS. This can be seen in Exhibit 2.
From exhibit 2, it can be said:
 A debt is risk free if the installments are certain. If debt is at perpetuity, then the installments are interest payments. A debt is risky if it is not risk free.
 The TS is risky if the stream of TS is not known with certainty.
Then
 A risk free debt does not imply a risk free TS.
The proof is evident: if the debt is riskfree, then FE is fixed, so TS is a function of EBITO (see piecewise function, above). But (EBITO) is a risky variable, which implies that TS is risky as well. See equation (2) above.
In other words, risky market cost of debt implies risky TS, non risky debt does not imply risk free TS. And vice versa: risky TS do not imply risky debt. Risk free TS imply EBITO greater than financial expenses, risk free cost of debt and risk free debt.
The M&E Proposal
Miles and Ezzel (1985) assume that as D is known in t, interest charges at t are fully defined and hence tax shields at t+1 are risk free and tax shields from t+1 will not be risk free and will be as risky as the FCF afterwards.
As in t1 it is not known in advance on which interval in (2) EBITO will occur in t, the conclusion is that TS are risky although interest charges are fully known. This risk has to be taken into account when defining the discount rate for TS. Lewellen and Douglas (1986) have the same understanding: the risk of tax shields comes from the certainty or uncertainty of the interest charges. Interest charges might be fully known and that does not make tax shields risk free. What makes tax shields risk free given a risk free debt, is the certainty that EBITO is greater than or equal to financial expenses. If EBITO is not known in advance (at time t1) then tax shield are risky and the assumption of Miles and Ezzel has limited application
Next section examines the behavior of tax shields compared with EBITO and with CFD.
Section Three. A Monte Carlo Simulation
A selected a sample for non financial firms (for year 2009) from Superintendencia de Sociedades (Governmental office for supervising non traded firms) was examined Table
2 depicts the sample disaggregated considering EBITO and compared with Other expenses, (OE). Due to lack of disaggregated data, the assumption is that OE are 100% financial expenses. Financial expenses include interest expenses, bank commissions and losses in exchange rate, and other expenses.
Previous table shows that 8.98% of firms loss the debt tax shields, 7.90% earns less than full tax shields, and only 83.12% earn full tax shields. This means that a significant amount (almost 17%) of firms loses or at least postpones part or all of the benefits of tax shields from debt payment. This is in line with the findings of Cordes and Sheffrin, (1983), Dammon and Senbet (1998), Graham (2000) and Grabowski (2009).
As per eq (2) the third section of TS is calculated as T FE and it is strictly related to debt risk. The proportion of this section with the other two (related to operating risk) is 4.92 (83.12%/16.88%).
Now the relationship between EBITO, CFD and TS, the items that are affected by the sources of TS volatility, are explored. The purpose of examining this behavior is to shed light upon the risk associated with TS. This exploration is done simulating CFD, EBITO and TS.
As mentioned in Section One there is a debate on which discount rate should be used to discount TS. As can be seen in their general formulas, cost of equity and average cost of capital depend on this discount rate. See Taggart (1991) and Tham and VélezPareja, (2004) as follows.
Where Ke is the cost of levered equity, Ku is the cost of unlevered equity, Kd is the cost of debt, D is market value of debt, E is market value of equity, ψ is the discount rate for the tax shields (TS), VTS is the present value of TS at ψ and WACC is the weighted average cost of capital.
Observe that equations (4) and (5) are quite different from the traditional and widely used textbook formulas for cost of capital. This means that the greater the debt TS lost or postponed, the larger the cost of capital and the greater the risk of overestimating the valuation of cash flows if the appropriate formula for the cost of capital and cash flows are not used.
With the results of table 2 a Monte Carlo Simulation scenario was designed. Based upon a hypothetical financial planning model 1,000 simulations were made with growth in units as input variable and TS, CFD and EBITO as output variables. Two critical points were estimated: one point is the growth rate, G, where EBITO is zero and the other is where EBITO matches Other expenses. These two points were identified using reverse engineering (Goal seek in a spreadsheet). With this information a typical profile in terms of EBITO and Other expenses was constructed in order to examine their behavior. These two critical points define the intervals in the piecewise equation (2). The intervals are
1. Interval 1 is defined from a minimum G to a second level of G associated to EBITO equal to zero (first critical point, above). Referring to table 2 it is 8.98% of total interval.
2. Interval 2 is defined from the growth rate associated to EBITO equal to zero up to the growth associated to EBITO equal to Other expenses (second critical point, above). The size of this interval is the difference of the two critical points above. Referring to table 2 it is 7.90% of total interval. Known the size of this interval and its proportion on the total interval length, the minimum and maximum G can be calculated[2].
3. Interval 3 is defined from the growth rate associated to EBITO equal to Other expenses to a maximum growth rate. Referring to table 2 it is 83.12% of total interval.
The financial planning model can be downloaded from http://cashflow88.com/decisiones/cige_ME_web.xlsx. The planning model has some characteristics listed in Appendix A.
The probability distribution of the input variable growth in units, (G) was assumed as a uniform distribution. Appendix B shows the minimum and maximum values of real growth for running the simulation.
As has been discussed, TS risk has two sources: debt and operating results. For that reason Table 3 shows the correlation coefficients among TS, CFD and EBITO. CFD includes interest payment that is the basis of TS calculation and EBITO includes the operating earnings (EBIT).
Table 3. Correlation Coefficients between TS and Selected Outputs
Outputs 
TS 
CFD 
0.37 
EBITO 
0.60 
If the relationship among TS, CFD and EBITO were linear, then it can be expressed as
Tables 4a and 4 b show the Ordinary Least Squares (OLS) regression for eq. (6).
Table 4a OLS with TS, CFD and EBITO (with intercept)
Multiplecorrelationcoefficient 
0.96394224 
F 
Fcriticalvalue 
R2 
0.929184643 
6540.9335 
0 
AdjustedR2 
0.929042586 
Observations 
1000 
Coefficients 
tStatistic 
pvalue 

Intercept(0) 
9.752142103 
–71.41685 
0 
CFD(1) 
0.612510629 
89.140581 
0 
EBITO(2) 
0.096948197 
105.48618 
0 
Table 4b OLS with TS, CFD and EBITO (without intercept)
Multiplecorrelationcoefficient 
0.975292445 
F 
Fcriticalvalue 
R2 
0.951195354 
9725.4363 
0 
AdjustedR2 
0.950144447 
Observations 
1000 
Coefficients 
tStatistic 
pvalue 

CFD(1) 
0.125739194 
58.434605 
0 
EBITO(2) 
0.032505212 
75.378033 
0 
Results from OLS suggest, as expected, from Table 2 and eq. (2), that the discount rate for TS is a mixture of debt risk and operating risk. As it has been suggested, the sources of TS risk are debt risk and operational risk. The non intercept model in table 4b depicts the proper relationship among TS, CFD and EBITO. In this case, the contribution of debt risk to TS is 3.87 (0.125739194/0.032505212) times the contribution of operational risk.
A naïve approximation to the elusive discount rate for TS is to weight the risk of debt (default, for instance) and the risk associated with EBITO (2) as in (7)
Table 5. Firms in Chapter 11 versus total Employer firms in the US
Year 
# of employerfirms[3] 
FirmsinChapter11 
Proportion 
2008 e 
6,145,500 
9,272 
0.15% 
2007 
6,049,655 
5,736 
0.09% 
2006 
6,022,127 
4,643 
0.08% 
2005 
5,983,546 
5,923 
0.10% 
2004 
5,885,784 
9,186 
0.16% 
2003 
5,767,127 
8,474 
0.15% 
2002 
5,697,759 
10,286 
0.18% 
2001 
5,657,774 
10,641 
0.19% 
Average 
0.14% 
Source: U.S. Small Business Administration, Office of Advocacy: http://www.sba.gov/advo/research/us88_07.pdf. US.
Courts: www.rib.uscourts.gov/Docs/Stats/table_f2.pdf (visited on August 7, 2010).
The average of 0.14% could be considered as the probability for a firm to be included in Chapter 11 for defaulting. With this probability one could figure out the effect of the risk of debt for default in the naïve approach.
The information regarding the number of firms in Colombia is volatile depending on the inclusion or not of micro enterprises or not. According to information from The World Bank, in Colombia exist near 500,000formalized firms. (See http://data.worldbank.org/indicator/IC.BUS.TOTL visited on Nov. 2, 2010). The latest data is shown in table 6.
Table 6. Total businesses registered in Colombia
2005 
2006 
2007 

Numberoffirms 
461,061 
477,742 
497,778 
Number of firms with financial stress using the rate of firms in Chapter 11 in the US. 
645 
669 
697 
Source: http://data.worldbank.org/indicator/IC.BUS.TOTL and calculations by author.
From public information (see http://www.portafolio.com.co/economia/economiahoy/200905
11/ARTICULOWEBNOTA_INTERIOR_PORTA5171194.html and http://www.eltiempo.com/archivo/documento/MAM1591980, visited Nov 2, 2010) it is known that from January 2000 to July 2004, there were 964 firmswhich filed for bankruptcy or agreement with creditors (210 firms per year). Only in 2000 there were 360 firms that declared financial distress. This means that even 0.14% the rate of firms in Chapter 11 in the US, is anoverestimation of what occurs in Colombia in terms of bankruptcy rate. If the average of firms per year (210) is doubled to assess the bankruptcy rate from 2005 to 2007, that rate is 0.09%.
Three practical approaches are suggested for estimating Wrd and WEBITO in eq. (7)
1. According to the intervals as shown in table 2. In this case, Wrd = 83.12% and WEBITO = 16.88%.
2. According to the regression coefficients from table 4b. In this case Wdr = 79.46% (0.125739194/0.158244406) and WEBITO = 20.54% (0.032505212/0.158244406).
3. According to an estimation of the probability of default. In this case, Wdr =0.09% and WEBITO = 99.91%.
Other sophisticated approaches could be considered. For instance, one possible approach is assuming TS as a contingent cash flow or a real option and working with risk neutral probabilities or considering the TS as a portfolio with different risks and weight and estimate a composite beta and applying the Capital Asset Pricing Model. See Díaz and VélezPareja (2010). All this is beyond the scope of this exploratory work.
In a work in process, Kolari and VélezPareja (2010) have identified a flaw in the traditional Modigliani and Miller’s model for valuation. This problem is solved when the discount rate for the TS is Ke, the cost of levered equity. In addition, this approach has the property that Ke captures part of the bankruptcy costs and defines an optimal capital structure. Tham and VélezPareja (2010) developed the formula for Ke when Ke is the discount rate for TS.
As seen, there is no clear and specific answer to which is the discount rate for TS.
Section Four. Concluding Remarks
From this work it is possible to draw some conclusions regarding the TS, TS risk, and TS realization as follows:
1. Different sources of risk for the tax savings have been identified: risk of debt default, risk of volatility of cost of debt and volatility of EBITO. Conditions for risky debt relate CFD to the available cash and not to earnings (EBIT) that include accruals and in fact do not show the complete cash availability.
2. The initial reasoning and critique to Wrightsman, 1978 is supported by the Monte Carlo Simulation results: risk of tax shields includes operating risk and debt risk: operating income (EBITO) and CFD.
3. Miles and Ezzell, 1985, proposal is questioned because knowing the interest charge in t does not mean a risk free TS in t+1. Hence, their proposal is limited to very specific scenarios where EBITO is greater than financial expenses and when the assumption of EBITO > FE means no risky debt
4. There is no clear cut answer to the question about the discount rate for TS.
5. There is evidence that in Colombia near 17% firms in 2009 lose or postpone tax shields due to low operational earnings (EBITO). This means that if TS are not defined properly their cost of capital estimates might be undervalued (see (4) and (5)). As a consequence, they are exposed to make sub optimal decisions.
6. Available data suggest the rate of bankruptcy filings in Colombia is much less than the same rate in the U.S.
7. Using a naïve approach as in (7) and with available data, results are contradictory. Depending on how to assess the weights of debt risk and operational risk, the former could overweigh the later. This is, TS risk would be closer to debt risk or to operational risk.
8. The analysis of the relation among TS, CFD and EBITO should be extended to more years. This paper only explored the behavior as per 2009.
9. Other approaches to defining the discount of TS have to be developed.
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U.S. Small Business Administration, Office of Advocacy, (2010). http://www.sba.gov/advo/research/us88_07.pdf visited on August7, 2010. VélezPareja, I., (2010). Calculating Tax Shields from Financial Expenses withLosses
Carried Forward (May). Available at SSRN: http://ssrn.com/abstract=1604082
VélezPareja, I., (2009). Return to Basics: Are You Properly Calculating Tax Shields? (July). Available at SSRN: http://ssrn.com/abstract=1306043 World Bank, (2010). http://data.worldbank.org/indicator/IC.BUS.TOTL visited on Nov. 2, 2010.
Wrightsman, D., (1978). Tax Shield Valuation and the Capital Structure Decision. The Journal of Finance, Vol. 33, No. 2 (May), pp. 650656.
Appendix A. Financial Planning Model Characteristics Some features and assumptions of the model are:
1. The financial planning model includes three financial statements: Cash Budget or Cash Flow Statement, Income Statement and Balance Sheet.
2. No plugs for solving the balancing problem.
3. No circularity in calculating interest because the model assumes end of period convention.
4. Input data are defined with a high degree of disaggregation. For instance, input data include inflation rate, real rates for interest and increases in prices, growth in units, and policies for different items such asAccounts Receivable, Accounts Payable, inventory, dividends payment, minimum cash, etcetera.
5. Nominal increases and interest rates are synthesized from inflation and real rates using the Fisher equation and a risk premium is included for interest rates.
6. A new simple manufacturing firm (starting from zero) with only one product.
7. The model has three types of inventory: Raw material, Inprocess and finished goods.
8. Taxes are paid the same year as accrued.
9. Losses carried forward are not allowed.
10. All expenses are paid on a cash basis. The only credit is from the supplier and has to be paid according to commercial terms (Accounts Payable).
11. Dividends are not greater than the Net Income of previous year. The payout ratio is a percent of previous year Net Income.
12. Dividends are paid the next year after the Net Income is generated.
13. Input prices are fixed and do not depend on volume purchased.
14. It is expected to invest in new assets (Capital expenditures, CAPEX) every year. Yearly CAPEX investment equals the amount of depreciation charge.
15. Deficit in the operating module (Module 1 in the Cash Budget) should be covered with shortterm loans.
16. Any long term deficit is covered by new debt and new equity investment. In the example, a percent of long term deficit is covered by longterm debt, the remainder by equity. Deficit in the after investment in fixed assets module (Module 2 in the Cash Budget) should be covered with longterm loans or new equity investment and not with shortterm loans.
17. Shortterm loans will be repaid the following year.
18. Longterm loans are repaid in 10 years.
19. Any cash excess is invested in marketable securities.
20. Inventory valuation is done under the First in, First Out (FIFO) policy.
21. A deficit can arise in any year when a dynamic approach is used (input data can be changed and the results reflect that change).
22. Shortterm portion of longterm debt is not considered in the current liabilities.
Appendix B
Descriptive Statistics and Results for the Monte Carlo Simulation
Table B1. Designed Intervals for growth rate, G in Monte Carlo Simulation
Minimum G 
34.29% 
G for EBITO = 0 
21.81% 
G for EBITO = Financial expenses 
10.83% 
Maximum G 
104.70% 
TableB2.Originalandsimulateddistributionof G for intervals.
Interval 
Original 
Simulated 
EBITO < 0 
8.98% 
8.60% 
0 < EBITO < Interest 
7.90% 
7.80% 
Interest > EBITO 
83.12% 
83.60% 
TableB3.DescriptiveStatisticsfor simulatedresults
G 
TS 
CFD 
EBITO 

Max 
104.65% 
3.82 
20.09 
125.49 
Min 
34.25% 
0.00 
2.44 
12.35 
Mean 
34.12% 
3.33 
12.58 
55.50 
Stdev 
39.71% 
1.19 
5.27 
39.40 
Appendix D
Derivation of piecewisedefined equation (2)
Assume two firms with identical EBITO. The difference is that one has debt and the other is unlevered.
Table D1. Case 1 EBITO >FE
No debt 
Debt 
TaxShields=differenceintaxes 
EBITO 
EBITO 

0 
FE 

EBT=EBITO 
EBT=EBITOFE 

Tax=TEBITO 
Tax=T(EBITO– FE) 
TS=TFE 
InthepreviouscasewesaythattheTSisequaltothefinancialexpense,FE,times thetaxrate,T.Thisisthecasewherethetraditionaltextbookformulafor WACC works.
TableD2. Case2 0<EBITO<FE
No debt 
Debt 
TaxShields=differenceintaxes 
EBITO 
EBITO 

0 
FE 

EBT=EBITO 
EBT=EBITO– FE<0 

Tax=T(EBITO) 
Tax=0 
TS=T(EBITO) 
In thissecondcaseTSISNOT Ttimesthefinancialexpenses,butTtimesEBITO.
TableD3. Case3 EBITO<0
No debt 
Debt 
TaxShields=differenceintaxes 
EBITO 
EBITO 

0 
FE 

EBT=EBITO<0 
EBT=EBITO– FE<0 

Tax=0 
Tax=0 
TS=0 
In cases 2 and 3 traditional textbook formula for WACC is not correct.
This can be summarized in the piecewisedefined equation
Equation (A1) is equation (2) in the body of the paper.
[1] I am grateful to Rauf Ibragimov from Graduate School of Financial Management, Moscow, Carlo Alberto Magni from University of Modena, Italy and Gonzalo Díaz from Universidad Tecnológica de Bolívar, Cartagena for their comments on this work. The responsibility of what is written in this work is entirely mine.
[3] “Employer firm means that has more than one employee. “A nonemployer firm is defined as one that has no paid employees, has annual business receipts of $1,000 or more ($1 or more in the construction industries), and is subject to federal income taxes” (http://www.sba.gov/advo/research/data.html, visited on Nov. 2, 2010).
Previously published by the Universidad Tecnológica de Bolívar, November 2010
Key Words
Weighted Average Cost of Capital, WACC, firm valuation, tax shields, cash flows, Monte Carlo Simulation, discount rate for tax shields, risky debt, risky tax shields.
JEL codes
M21, M40, M46, M41, G12, G31, J33