Capital Structure Decisions: Extensions

ANSWERS TO END-OF-CHAPTER QUESTIONS

17-1 a. MM Proposition I states the relationship between leverage and firm value. Proposition I without taxes is V = EBIT/rsU. Since both EBIT and rsU are constant, firm value is also constant and capital structure is irrelevant. With corporate taxes, Proposition I becomes V = Vu + TD. Thus, firm value increases with leverage and the optimal capital structure is virtually all debt.

b. MM Proposition II states the relationship between leverage and cost of equity. Without taxes, Proposition II is rsL = rsU + (rsU – rd)(D/S). Thus, rs increases in a precise way as leverage increases. In fact, this increase is just sufficient to offset the increased use of lower cost debt. When corporate taxes are added, Proposition II becomes rsL = rsU + (rsU – rd)(1 – T)(D/S). Here the increase in equity costs is less than the zero-tax case, and the increasing use of lower cost debt causes the firm’s cost of capital to decrease, and again, the optimal capital structure is virtually all debt.

c. The Miller model introduces personal taxes. The effect of personal taxes is, essentially, to reduce the advantage of corporate debt financing.

d. Financial distress costs are incurred when a leveraged firm facing a decline in earnings is forced to take actions to avoid bankruptcy. These costs may be the result of delays in the liquidation of assets, legal fees, the effects on product quality from cutting costs, and evasive actions by suppliers and customers.

e. Agency costs arise from lost efficiency and the expense of monitoring management to ensure that debtholders’ rights are protected.

f. The addition of financial distress and agency costs to either the MM tax model or the Miller model results in a trade-off model of capital structure. In this model, the optimal capital structure can be visualized as a trade-off between the benefit of debt (the interest tax shelter) and the costs of debt (financial distress and agency costs).

g. The value of the debt tax shield is the present value of the tax savings from the interest payments. In the MM model with taxes, this is just interest x tax rate / discount rate = iDT/r, and since i = r in the MM model, this is just TD. If a firm grows and the discount rate isn’t r, then the value of this growing tax shield is rdTDg/(1+rTS) where rd is the interest rate on the debt and rTS is the discount rate for the tax shield.

h. When a firm has debt outstanding it can choose to default if the firm is not worth more than the face value of the debt. This decision to default when the value of the firm is low is like the decision not to exercise a call option when the stock price is low. If management (and hence the stockholders) make the debt payment, they get to keep the company. This makes equity like an option on the underlying value of the entire firm, with a strike price equal to the face value of the debt. If D is the face value of debt maturing in one year and S is the value of the entire firm (the firm’s debt plus equity) then the payoff to the stockholder when the debt matures is: Payoff = max(S-D, 0). This is the same payoff as a call option on S with a strike, or exercise, price of D.

17-2 Modigliani and Miller show that the value of a leveraged firm must be equal to the value of an unleveraged firm. If this is not the case, investors in the leveraged firm will sell their shares (assume they owned 10%). They will then borrow an amount equal to 10% of the debt of the leveraged firm. Using these proceeds, they will purchase 10% of the stock of the unleveraged firm (which provides the same return as the leveraged firm) with a surplus left to be invested elsewhere. This arbitrage process will drive the price of the stock of the leveraged firm down and drive up the price of the stock of the unleveraged firm. This will continue until the value of both stocks are equal.

The assumptions of the MM model are:

Firms must be in a homogeneous business risk class. If the firms have varying degrees of risk, the market will value the firms at different rates. The earnings of the firms will be capitalized at different costs of capital.

Investors have homogeneous expectations about expected future EBIT. If investors have different expectations about future EBIT then individual investors will assign different values to the firms. Therefore, the arbitrage process will not be effective.

Stocks and bonds are traded in perfect capital markets. Therefore, (a) there are no brokerage costs and (b) individuals can borrow at the same rate as corporations. Brokerage fees and varying interest rates will, in effect, lower the surplus available for alternative investment.

Investors are rational. If by chance, investors were irrational, then they would not go through the entire arbitrage process in order to achieve a higher return. They would be satisfied with the return provided by the leveraged firm.

There are no corporate taxes. With the existence of corporate taxes the value of the leveraged firm (VL) must be equal to the value of the unleveraged firm (VU) plus the tax shield provided by debt (TD).

17-3 MM without taxes would support AT&T, although if AT&T really believed MM, they should not object to Gordon’s 50 percent debt ratio. MM with taxes would lead ultimately to 100 percent debt, which neither Gordon nor AT&T accepted. In effect, Gordon and AT&T seemed to be taking a “traditional” or perhaps a “compromise” view, but with different conclusions about the optimal debt ratio. We might note, in a postscript, that AT&T did raise its debt ratio, but not to the extent that Gordon recommended.

17-4 The value of a growing tax shield is greater than the value of a constant tax shield. This means that for a given initial level of debt a growing firm will have more value from the debt tax shield than a non-growing firm. Thus for a given face value of debt, D, and unlevered value of equity, U, a growing firm will have a smaller wD, a larger levered cost of equity, reL, and a larger WACC. So the MM model will underestimate the value of the levered firm and its cost of equity and WACC.

17-5 If equity is viewed as an option on the total value of the firm with a strike price equal to the face value of debt then the equity value should be affected by risk in the same way that an option is affected by risk. An option is worth more if the underlying asset is more risky, so a manager wanting to maximize the option value of the firm might want to switch investment decisions to make the firm more risky. Of course bondholders will not like this, since the increase in equity value comes at their expense. They will write covenants in to the bonds specifying how the proceeds can be used, and if management still manages to engage in this “bait and switch” tactic, the firm will find it difficult to raise capital through bond issues in the future.

SOLUTIONS TO END-OF-CHAPTER PROBLEMS

17-1 a. bL = bU[1 + (1 - T)(D/S)].

bU = = = = 1.125.

b. rsU = rRF + (rM - rRF)bU = 10% + (5%)1.125 = 10% + 5.625% = 15.625%.

c. $2 Million Debt: VL = VU + TD = $10 + 0.25($2) = $10.5 million.

rsL = rsU + (rsU - rRF)bU(1 - T)(D/S)

= 15.625% + (15.625% - 10%)(0.75)($2/$8.5)

= 15.625 + 5.625% (0.75)($2/$8.5) = 16.62%.

$4 Million Debt: VL = $10 + 0.25($4) = $11.0 million.

rsL = 15.625% + 5.625%(0.75)($4/$7) = 18.04%.

$6 Million Debt: VL = $10 + 0.25($6) = $11.5 million.

rsL = 15.625% + 5.625% (0.75)($6/$5.5) = 20.23%.

d. $6 Million Debt: VL = $8.0 + 0.40($6) = $10.4 million.

rsL = 15.625% + 5.625%(0.60)($6/$4.4) = 20.23%.

The mathematics of MM result in the required return, and, thus, the same financial risk premium. However, the market value debt ratio has increased from $6/$11.5 = 52% to $6/$10.4 = 58% at the higher tax rate. Hence, a higher tax rate reduces the financial risk premium at a given market value debt/equity ratio. This is because a higher tax rate increases the relative benefits of debt financing.

17-2 a. VU = = = $20 million.

b. rsU = 10.0%. (Given)

rsL = rsU + (rsU - rd)(D/S) = 10% + (10% - 5%)($10/$10) = 15.0%.

c. SL = = = $10 million.

SL + D = VL = VU + TD.

$10 + $10 = $20 = VL = $20 + (0)$10 = $20 million.

d. WACCU = rsU = 10%.

For Firm L, we know that WACC must equal rsU = 10% according to Proposition I. But, we can demonstrate this as follows:

WACCL = (D/V)rd + (S/V)rs = ($10/$20)5% + ($10/$20)15%

= 2.5% + 7.5% = 10.0%.

e. VL = $22 million is not an equilibrium value according to MM. Here’s why. Suppose you owned 10 percent of Firm L’s equity, worth 0.10($22 million - $10 million) = $1.2 million. You could (1) sell your stock, (2) borrow an amount (at 5%) equal to 10 percent of Firm L’s debt, or 0.10($10 million) = $1 million, and (3) end up with $1.2 million + $1 million = $2.2 million. You could spend $2 million to buy 10% of Firm U’s stock, and invest $200,000 in risk-free debt. Your cash stream would now be 10 percent of Firm U’s flow, or 0.10(EBITU) = 0.10($2 million) = $200,000, plus the return on the $200,000 of risk-free debt, minus the 0.05($1 million) = $50,000 interest expense for $150,000 plus the return on the extra $200,000. Before the arbitrage, your return was 10 percent of the $2 million - 0.05($10 million) = $1.5 million, or $150,000. Investors would do this arbitrage until VL = VU = $20 million.

17-3 a. VU = = = $12 million.

VL = VU + TD = $12 + (0.4)$10 = $16 million.

b. rsU = 0.10 = 10.0%.

rsL = rsU + (rsU - rd)(1 - T)(D/S)

= 10% + (10% - 5%)(0.6)($10/$6) = 10% + 5% = 15.0%.

c. SL = = = $6 million.

VL = SL + D = $6 + $10 = $16 million.

d. WACCU = rsU = 10.00%.

WACCL = (D/V)rd(1 - T) + (S/V)rs = ($10/$16)5%(0.6) + ($6/$16)15%

= 7.50%.

17-4 a. VU = = = = $12 million.

b. VL = VU + D = $12 + $10

= $12 + [1 - 0.67]$10 = $12 + 0.33($10)

= $15.33 million.

VL = $15.93 million. Gain from leverage = $3.33 million.

c. The gain from leverage under Miller is 0.33($10) = $3.33 million. The gain from leverage in Problem 17-3 is 0.4($10) = $4 million. Thus, the addition of personal tax rates reduced the value of the debt financing.

d. VU = VL = $20 million. Gain from leverage = $0.00.

e. VU = $12 million. VL = $16 million. Gain from leverage = $4 million.

f. VU = $12 million. VL = $16 million.

Gain from leverage = $4.0 million. Note that the gain from leverage is the same as in Part (e) and will be the same value, as long as Td = Ts.

17-5 a. VU = $500,000/(rsU – g) = $500,000/(0.13 - 0.09) = 12,500,000.

b. . So since

D = 5, S = 16 – 5 = $11.0 million.

= 15.7%

c. Under MM, VL = VU + TD = $12.5 million + (0.40)(5 million)

= $14.5 million. S = $14.5 – 5 = $9.5 million. rsL = 0.13+(0.13-0.07)(1-.40)(5/9.5) = 14.9%

d. VL is greater under the extension that incorporates growth than under MM because MM assumes 0 growth. A positive growth rate gives a larger value to the tax shield. In this case the value of the tax shield under MM is 2.0 million and is $3.5 million if growth is included. The cost of capital when growth is included is higher because the relative weight of equity is higher and the relative weight of debt is lower than when growth is ignored.

17-6 a. VU = SU = = = $14,545,455.

VL = VU = $14,545,455.

b. At D = $0:

rs = 11.0%; WACC = 11.0%

At D = $6 million:

rsL = rsU + (rsU – rd)(D/S)

= 11% + (11% - 6%) = 11% + 3.51% = 14.51%.

WACC = (D/V)rd + (S/V)rs

= 6% + 14.51%

= 11.0%.

At D = $10 million:

rsL = 11% + 5% = 22.00%.

WACC = 6% + 22%

= 11.0%.

Leverage has no effect on firm value, which is a constant $14,545,455 since WACC is a constant 11%. This is because the cost of equity is increasing with leverage, and this increase exactly offsets the advantage of using lower cost debt financing.

c. VU = [(EBIT - I)(1 - T)]/rsU = [($1,600,000 - 0)(0.6)]/0.11 = $8,727,273.

VL = VU + TD = $8,727,273 + 0.4($6,000,000) = $11,127,273

d. At D = $0:

rs = 11.0%. WACC = 11.0%.

At D = $6 million:

VL = VU + TD = $8,727,273 + 0.4($6,000,000) = $11,127,273.

rsL = rsU + (rsU - rd)(1 - T)(D/S)

= 11% + (11% - 6%)(0.6)($6,000,000/$5,127,273) = 14.51%.

WACC = (D/V)rd(1 - T) + (S/V)rs

=($6,000,000/$11,127,273)(6%)(0.6) + ($5,127,273/$11,127,273)(14.51%)

= 8.63%.

At D = $10 million:

VL = $8,727,273 + 0.4($10,000,000) = $12,727,273.

rsL = 11% + 5%(0.6)($10,000,000/$2,727,273) = 22.00%.

WACC = ($10,000,000/$12,727,273)(6%)(0.6) + ($2,727,273/$12,727,273)(22%)

=7.54%.

Summary: (in millions)

D V D/V rs WACC

$ 0 $ 8.73 0% 11.0% 11.0%

6 11.13 53.9 14.5 8.6

10 12.73 78.6 22.0 7.5

e. The maximum amount of debt financing is 100 percent. At this level

D = V, and hence

VL = VU + TD = D

$8,727,273 + 0.4D = D

D - 0.4D = $8,727,273

0.6D = $8,727,273

D = $8,727,273/0.6 = $14,545,455 = V.

Since the bondholders are bearing the same risk as the equity holders of the unleveraged firm, rd is now 11 percent. Now, the total interest payment is $14,545,455(0.11) = $1.6 million, and the entire $1.6 million of EBIT would be paid out as interest. Thus, the investors (bondholders) would get $1.6 million per year, and it would be capitalized at 11 percent:

VL = = $14,545,455.

f. (1) Rising interest rates would cause rd and hence rd(1 - T) to increase, pulling up WACC. These changes would cause V to rise less steeply, or even to decline.

(2) Increased riskiness causes rs to rise faster than predicted by MM. Thus, WACC would increase and V would decrease.

a.The inputs to the Black and Scholes option pricing model are P = 5, X = 2, rRF = 6%,

s = 50%, and t = 2 years. Given these inputs, the value of a call option is calculated as:

=

=

Using Excel’s Normsdist function N(d1) = 0.9656, and N(d2) = 0.8669. This gives a value of the call option equal to:

= .

b. The debt must therefore be worth 5-3.29 = $1.71 million. Its yield is .

c. At a volatility of 30% d1 = 2.566 and N(d1) = 0.996. d2 = 2.230 and N(d2) = 0.987. This gives an option value of $3.32 million. The debt value is then 5.0 – 3.23 = $1.77 million. Its yield is 6.8%. The value of the stock goes down and the value of the debt goes up because with lower risk, Fethe has less of a chance of a “home run.”

SOLUTION TO SPREADSHEET PROBLEM

17-8 The detailed solution for the problem is available both on the instructor’s resource CD-ROM (in the file Solution for FM11 Ch 17 P8 Build a Model.xls) and on the instructor’s side of the web site, http://brigham.swcollege.com.

MINI CASE

David Lyons, CEO of Lyons Solar Technologies, is concerned about his firm’s level of debt financing. The company uses short-term debt to finance its temporary working capital needs, but it does not use any permanent (long-term) debt. Other solar technology companies average about 30 percent debt, and Mr. Lyons wonders why they use so much more debt, and what its effects are on stock prices. To gain some insights into the matter, he poses the following questions to you, his recently hired assistant:

a. Business Week recently ran an article on companies’ debt policies, and the names Modigliani and Miller (MM) were mentioned several times as leading researchers on the theory of capital structure. Briefly, who are MM, and what assumptions are embedded in the MM and Miller models?

Answer: Modigliani and Miller (MM) published their first paper on capital structure (which assumed zero taxes) in 1958, and they added corporate taxes in their 1963 paper. Modigliani won the Nobel Prize in economics in part because of this work, and most subsequent work on capital structure theory stems from MM. Here are their assumptions:

Firms’ business risk can be measured by ?EBIT, and firms with the same degree of risk can be grouped into homogeneous business risk classes.

All investors have identical (homogeneous) expectations about all firms’ future earnings.

There are no transactions (brokerage) costs, either to individuals or to firms.

All debt is riskless, and both individuals and corporations can borrow unlimited amounts of money at the same risk-free rate.

All cash flows are perpetuities. This implies that firms and individuals issue perpetual debt, and also that firms pay out all earnings as dividends, hence have zero growth. Additionally, this implies that expected EBIT is constant over time, although realized EBIT may turn out to be higher or lower than was expected.

In their first paper (1958), MM also assumed that there are no corporate or personal taxes.

These assumptions--all of them--were necessary in order for MM to use the arbitrage argument to develop and prove their equations. If the assumptions are unrealistic, then the results of the model are not guaranteed to hold in the real world.

b. Assume that firms U and L are in the same risk class, and that both have EBIT = $500,000. Firm U uses no debt financing, and its cost of equity is rsU = 14%. Firm L has $1 million of debt outstanding at a cost of rd = 8%. There are no taxes. Assume that the MM assumptions hold, and then:

1. Find v, s, rs, and WACC for firms U and L.

Answer: First, we find Vu and VL:

VU = = = $3,571,429.

VL = VU = $3,571,429.

To find rsL, it is necessary first to find the market values of firm L’s debt and equity. The value of its debt is stated to be $1,000,000. Therefore, we can find s as follows:

D + SL = VL

SL = VL - D = $3,571,429 - $1,000,000 = $2,571,429.

Now we can find L’s cost of equity, rsL:

rsL = rsU + (rsU - rd)(D/S)

= 14.0% + (14.0% - 8.0%)($1,000,000/$2,571,429)

= 14.0% + 2.33% = 16.33%.

We know from Proposition I that the WACC must be WACC = rsU = 14.0% for all firms in this risk class, regardless of leverage, but this can be verified using the WACC formula:

WACC = wdrd + wcers = (D/V)rd + (S/V)rs

= ($1,000/$3,571)(8.0%) + ($2,571/$3,571)(16.33%)

= 2.24% + 11.76% = 14.0%.

b. 2. Graph (a) the relationships between capital costs and leverage as measured by D/V, and (b) the relationship between value and D.

Answer: Figure 1 plots capital costs against leverage as measured by the debt/value ratio. Note that, under the MM no-tax assumption, rd is a constant 8 percent, but rs increases with leverage. Further, the increase in rs is exactly sufficient to keep the WACC constant--the more debt the firm adds to its capital structure, the riskier the equity and thus the higher its cost. Figure 2 plots the firm’s value against leverage (debt). With zero taxes, MM argue that value is unaffected by leverage, and thus the plot is a horizontal line. (Note that we should not really extend the graphs to D/V = 100% or D = $2.5 million, because at this amount of leverage the debtholders become the firm’s owners, and thus a discontinuity exists.)

Figure 1

c. Using the data given in part B, but now assuming that firms L and U are both subject to a 40 percent corporate tax rate, repeat the analysis called for in B(1) and B(2) under the MM with-tax model.

Answer: With corporate taxes added, the MM propositions become:

Proposition I: VL = VU + TD.

Proposition II: rsL = rsU + (rsU – rd)(1 - T)(D/S).

There are two very important differences between these propositions and the zero-tax propositions: (1) when corporate taxes are added, VL does not equal VU; rather, VL increases as debt is added to the capital structure, and the greater the debt usage, the higher the value of the firm. (2) rsL increases less rapidly when corporate taxes are considered. This is seen by noting that the Proposition II slope coefficient changes from (rsU – rd) to (rsU – rd)(1 – t), so at any positive T, the slope coefficient is smaller.

Note also that with corporate taxes considered, VU changes to

VU = = = $2,142,857 versus $3,571,429.

This represents a 40% decline in value, and it is logical, because the 40% tax rate takes away 40% of the income and hence 40% of the firm’s value.

Looking at VL, we see that:

VL = VU + TD = $2,142,857 + 0.4($1,000,000)

VL = $2,142,857 + $400,000 - $2,542,857 versus $2,142,857 for VU.

Thus, the use of $1,000,000 of debt financing increases firm value by T(D) = $400,000 over its leverage-free value.

To find rsL, it is first necessary to find the market value of the equity:

D + SL = VL

$1,000,000 + SL = $2,542,857

SL = $1,542,857.

now,

rsL = rsU + (rsU - rd)(1 - T)(D/S)

= 14.0% + (14.0% - 8.0%)(0.6)($1,000/$1,543)

= 14.0% + 2.33% = 16.33%.

Firm L’s WACC is 11.8 percent:

WACCL = (D/V)rd(1 - T) + (S/V)rs

= ($1,000/$2,543)(8%)(0.6) + (1,543/$2,543)(16.33%)

= 1.89% + 9.91% = 11.8%.

The WACC is lower for the leveraged firm than for the unleveraged firm when corporate taxes are considered.

Figure 3 below plots capital costs at different D/V ratios under the MM model with corporate taxes. Here the WACC declines continuously as the firm uses more and more debt, whereas the WACC was constant in the without-tax model. This result occurs because of the tax deductibility of debt financing (interest payments), which impacts the graph in two ways: (1) the cost of debt is lowered by (1 - T), and (2) the cost of equity increases at a slower rate when corporate taxes are considered because of the (1 - T) term in Proposition II. The combined effect produces the downward-sloping WACC curve.

Figure 4 shows that, when corporate taxes are considered, the firm’s value increases continuously as more and more debt is used.

Figure 3

d. Now suppose investors are subject to the following tax rates:

TD = 30% and TS = 12%.

1. What is the gain from leverage according to the miller model?

Answer: To begin, note that Miller’s Proposition I is stated as follows:

VL = VU + D.

Here the bracketed term replaces T in the earlier MM tax model, and

Tc = corporate tax rate, Td = personal tax rate on debt income, and

Ts = personal tax rate on stock income.

If there are no personal or corporate taxes, then Tc = Ts = Td = 0, and Miller’s model simplifies to

VL = VU,

Which is the same as in MM’s 1958 model, which assumed zero taxes.

If there are corporate taxes, but no personal taxes, then Ts = Td = 0, and Miller’s model simplifies to

VL = VU + TCD,

Which is the same as MM obtained in their 1963 article, which considered only corporate taxes.

We can now analyze the firm’s value numerically, using Miller’s model: if Tc = 40%, Td = 30%, and Ts = 12%, then Miller’s model becomes

d. 2. How does this gain compare to the gain in the MM model with corporate taxes?

Answer: If only corporate taxes were considered, then

VL = VU + TCD = VU + 0.40D.

The net effect depends on the relative effective tax rates on income from stocks and bonds, and on corporate tax rates. The tax rate on stock income is reduced vis-a-vis the tax rate on debt income if the company retains more of its income and thus provides more capital gains. If Ts declines, while Tc and Td remain constant, the slope coefficient, which shows the benefit of debt in a graph like figure 4, is increased. Thus, a company with a low payout ratio gets greater benefits under the miller model than a company with a high payout.

Note that the effects of leverage as computed by Miller’s model were much more important before 1987, because in earlier years capital gains were taxed at only 40 percent of the rate imposed on dividends (Ts . 20% and Td . 50%). Now the advantages of capital gains are (1) the fact that taxes on them are deferred, and (2) individuals in the higher tax brackets obtain an advantage because the tax rate imposed on long-term capital gains is 20 percent.

d. 3. What does the Miller model imply about the effect of corporate debt on the value of the firm, that is, how do personal taxes affect the situation?

Answer: The addition of personal taxes lowers the value of debt financing to the firm. The underlying rationale can be explained as follows: the U.S. corporate tax laws favor debt financing over equity financing, because interest expense is tax deductible while dividends are not. This provides an incentive for firms to use debt financing, and this was the message of the mm 1963 paper. At the same time, though, the U.S. personal tax laws favor investment in equity securities over debt securities, because equity income is effectively taxed at a lower rate. Thus, investors require higher risk-adjusted before-tax returns on debt to be induced to buy debt rather than equity, and this reduces the advantage to issuing debt.

The bottom line conclusion we reach from an analysis of the Miller model is that personal taxes lower, but do not eliminate, the value of debt financing.

e. What capital structure policy recommendations do the three theories (MM without taxes, MM with corporate taxes, and Miller) suggest to financial managers? Empirically, do firms appear to follow any one of these guidelines?

Answer: In a zero tax world, MM theory says that capital structure is irrelevant--it has no impact on firm value. Thus, one capital structure is as good as another. With corporate but not personal taxes considered, the MM model states that firm value increases continuously with financial leverage, and hence firms should use (almost) 100 percent debt financing. Miller added personal taxes to the analysis, and the value of debt financing is seen to be reduced but not eliminated, so again firms should use (almost) 100 percent debt financing.

The Miller model is the most realistic of the three, but if it were really correct, we would expect to see firms using almost all debt financing. However, on average, firms use only about 40 percent debt. Note, though, that debt ratios increased all during the 1980s, so companies were moving toward the miller position. However, in the 1990s we see firms reducing their debt.

f. How is the analysis in part C different if firms U and L are growing? Assume that both firms are growing at a rate of 7 percent and that the investment in net operating assets required to support this growth is 10 percent of EBIT.

Answer: If a firm is growing, the assumptions that MM made are violated. The extension to the MM model shows how growth affects the value of the debt tax shield and the cost of capital. The first difference in this situation is that the appropriate discount rate for the debt tax shield is the unlevered cost of equity, not the cost of debt. The second difference is that a growing debt tax shield is more valuable than a constant debt tax shield.

First, calculate expected free cash flow:

NOPAT = EBIT X (1-T) = 500,000 X (1 – 0.40) = $300,000

Investment In Net Operating Assets = 0.10 X EBIT = $50,000

Free Cash Flow = NOPAT – Investment In Net Operating Assets

= $300,000 - $50,000 = $250,000

(Note that this is an expected value for the coming year since EBIT is an expected value for the coming year.)

Next, note that WACC = unlevered cost of equity if there is no debt so

WACC = rsU = 14%

The Value Of U = Expected FCF/(WACC – g)

= 250,000/(0.14 – 0.07) = $3,571,429

Which is greater than in part C because the firm is growing.

If there is $1,000,000 in debt then:

The value of l = the value of U + value of debt tax shield

The value of the (growing) debt tax shield = rdTD/(rsU – g)

= 0.08(0.40)(1,000,000)/(0.14 – 0.07)

= $457,143

Therefore, the value of the firm = $3,571,429 + $457,143 = $4,028,571.

The value of the equity is the value of the firm less the value of the

debt = $4,028,571 - $1,000,000 = $3,028,571.

In this case the increase in the firm’s value due to the debt tax shield as a percent of its zero debt value is $457,143/$3,571,429 = 12.80%

This is less than the increase in the non-growing firm’s value as calculated using the MM model: $400,000/$2,142,857 = 18.7%.

To calculate the new levered cost of equity:

rsL = rsU + (rsU – rd)(D/S)

= 14% + (14% - 8%)(1,000,000/3,028,571)

= 15.98%

And the new levered WACC:

WACCL = (D/V)rd(1 - T) + (S/V)rs

= (1,000,000/4,028,571)8%(1-.40)

+ ($3,028,571/4,028,571)15.98%

= 13.2%.

g. What if L’s debt is risky? For the purpose of this example, assume that the value of L’s operations is $4 million—which is the value of its debt plus equity. Assume also that its debt consists of 1-year zero coupon bonds with a face value of $2 million. Finally, assume that L’s volatility is 0.60 (s = 0.60) and that the risk free rate is 6 percent.

Answer: L’s equity can be considered as a call option on the total value of l with an exercise price of $2 million, and an expiration date in one year. If the value of L’s operations is less than $2 million in a year, then L’s management will not be able to make its required payment on the debt, and the firm will be bankrupt. The debtholders will take over the firm and the equity holders will receive nothing. If L’s value is greater than $2 million in one year, then management will repay the debt and the stockholders will keep the company.

This option can be valued with the Black-Scholes Option Pricing Model:

V = PN(D1) – Xe-RTN(D2)

where

D1 = [ln(P/X) + (r + 0.5s2)T]/[sT0.5]

D2 = D1 - sT0.5

And n() is the cumulative normal distribution function, from either appendix a in the back of the text, or the NORMSDIST() function in excel.

in this case, P = $4

X = $2

s = 0.60

T = 1.0

R = 0.06

and calculating,

D1 = 1.552

D2 = 0.9552

N(D1) = 0.9491

N(D2) = 0.8303

and V = $2.1964 million.

This leaves debt value of $4 million - $2.1964 million = $1.8036 million.

The yield on this debt is calculated as

Price = (Face Value)/(1+Yield)N

so that

Yield = [Face Value/Price]1/N – 1.0

= [2.0/1.8036] – 1.0

= 10.89%

In this case, the value of the debt must be $1.8036 million, and it is yielding 10.89%. The value of the equity is $2.1964 million.

h. What is the value of L’s stock for volatilites between 0.20 and 0.95? What in-centives might the manager of L have if she understands this relationship? What might debtholders do in response?

Answer: The mini case model shows the calculations for the table below.

Value of Stock and Debt for Different Volatilities

Volatility

Equity

Debt

0.20

2.12

1.88

0.25

2.12

1.88

0.30

2.12

1.88

0.35

2.12

1.88

0.40

2.13

1.87

0.45

2.14

1.86

0.50

2.16

1.84

0.55

2.17

1.83

0.60

2.20

1.80

0.65

2.22

1.78

0.70

2.25

1.75

0.75

2.28

1.72

0.80

2.31

1.69

0.85

2.34

1.66

0.90

2.38

1.62

0.95

2.41

1.59

The value of the equity increases as the volatility increases—and the value of the debt decreases as well. A manager who knows this may choose to invest the proceeds from borrowing in assets that are riskier than usual. This is called “bait and switch.” This action decreases the value of the debt, because now its claim is riskier. It increases the value of equity because the worse the stockholders can do is default on the bonds, but the best they can do is potentially unlimited.

Bondholders who face this possibility will write covenants into their bond contracts limiting management’s ability to invest in assets other than originally planned. If this isn’t possible, then bondholders will demand a higher rate of return in order to compensate them for the possibility that management will switch investments.