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					                              Chapter 17
                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)(1 – T)(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 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.




                                                                    Answers and Solutions: 17 - 1
       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.

Answers and Solutions: 17 - 2
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.




                                                                 Answers and Solutions: 17 - 3
              SOLUTIONS TO END-OF-CHAPTER PROBLEMS


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

                         bL                     1 .8             1 .8
          bU =                       =                         =      = 1.125.
                 1  (1  T )( D / S) 1  (1  0.4)( 0.5 / 0.5) 1.6

       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.




Answers and Solutions: 17 - 4
                 EBIT   $2 million
17-2   a. VU =        =            = $20 million.
                  rsU      0.10

       b. rsU = 10.0%. (Given)

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

                 EBIT  rd D   $2  0.05($10)
       c. SL =               =                = $10 million.
                     rsL            0.15

          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.


                 EBIT (1  T )   $2(1  0.4)
17-3   a. VU =                 =             = $12 million.
                     rsU            0.10

          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%.




                                                                  Answers and Solutions: 17 - 5
                 ( EBIT  rd D)(1  T )   [$2  0.05($10)]0.6
       c. SL =                          =                     = $6 million.
                         rsL                      0.15

            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%.


                  EBIT (1  TC )(1  Ts )   EBIT (1  TC ) $2(0.6)
17-4   a. VU =                            =               =        = $12 million.
                      rsU (1  Ts )             rsU         0.10

                          (1  TC )(1  Ts ) 
       b.      VL = VU + 1                   D
                                 (1  Td )    
                              (0.6)(0.8) 
                    = $12 + 1               $10
                                   (0.72) 
                    = $12 + [1 - 0.67]$10 = $12 + 0.33($10)
                    = $15.93 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.




Answers and Solutions: 17 - 6
17-5   a. VU = $500,000/(rsU – g) = $500,000/(0.13 - 0.09) = 12,500,000.

                                0.07 x 0.40 x 5 million 
       b. VL  $12.5 million                             $16.0 million . So since
                                      0.13 - 0.09       

          D = 5, S = 16 – 5 = $11.0 million.

                                       5
           rsL  0.13 (0.13 0.07)      = 15.7%
                                      11



       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.




                                                                     Answers and Solutions: 17 - 7
                       EBIT   $1,600,000
17-6   a. VU = SU =         =            = $14,545,455.
                        rsU      0.11

          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)
                                      $6,000,000 
              = 11% + (11% - 6%)                  = 11% + 3.51% = 14.51%.
                                      $8,545,455 


          WACC = (D/V)rd + (S/V)rs
              $6,000,000         $8,545,455 
           =               6% +               14.51%
              $14,545,455        $14,545,455 
           = 11.0%.


At D = $10 million:

                          $10,000,000 
          rsL = 11% + 5%               = 22.00%.
                          $4,545,455 

                  $10,000,000        $4,545,455 
          WACC =               6% +               22%
                  $14,545,455        $14,545,455 
           = 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




Answers and Solutions: 17 - 8
       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
            Value (Millions of Dollars)

             15

             14

             13

             12

             11

             10

             9

             8




                                 25            50           75            100
                                                                        D/V (%)


                                                                    Answers and Solutions: 17 - 9
       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:

                                              $1,600,000
                                       VL =              = $14,545,455.
                                                 0.11



                 Cost of Capital (%)

                    25                                                      ks
                                                                            rS

                    20



                    15



                    10

                                                                            WACC
                     5

                                                                            r (1-T)
                                                                            kd(1-T)
                                                                             d


                                       25             50           75                 100
                                                                          D/V (%)




       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.


Answers and Solutions: 17 - 10
17-7   a.The inputs to the Black and Scholes option pricing model are P = 5, X = 2, rRF = 6%,
         = 50%, and t = 2 years. Given these inputs, the value of a call option is calculated as:

                  ln( P / X)  [rRF   2 / 2]t       ln( 5 / 2)  [0.06  0.52 / 2]2
           d1                                    =                                      1.819
                               t                                0.5 2

           d 2  d1   t = 1.819  0.5 2  1.112


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

           V  P[N(d 1)]  Xe-rRF t [N(d2 )] = 5[1.819] 2e-0.06(2) [1.112]  3.29.

       b. The debt must therefore be worth 5-3.29 = $1.71 million. Its yield is
           2.0 / 1.71  1  0.81  8.1% .

       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.‖




                                                                             Answers and Solutions: 17 - 11
                   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.swlearning.com.




Solution to Spreadsheet: 17 - 12
                                        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 σEBIT, 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.


                                                                               Mini Case: 17 - 13
             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:

                                        EBIT   $500,000
                                 VU =        =          = $3,571,429.
                                         rsU     0.14

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




Mini Case: 17 - 14
           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.)




                                                                              Mini Case: 17 - 15
               Figure 1


                                   25%                   Without Taxes


                                   20%
                 Cost of Capital
                                   15%
                                                                                                              rs
                                   10%                                                                        WACC
                                                                                                              rd
                                       5%

                                       0%
                                            0%            20%                      40%                    60%
                                                          Debt/Value Ratio
                                                               Figure 2
              Value of Firm, V
                                   ($)


                            4

                         VU                                                                              VL

                            3

                                                                       Firm Value ($3.6 Million)

                            2




                            1




                                   0             0.5     1.0                 1.5     2.0           2.5
                                                                                             Debt ($)
                                                           (Millions of $)




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).

Mini Case: 17 - 16
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, V L
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

               EBIT (1  T )   $500,000(0.6)
        VU =                 =               = $2,142,857 versus $3,571,429.
                   rsU             0.14

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 + S L = 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%.




                                                                       Mini Case: 17 - 17
          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


                                                                                rs
                                                  With Taxes
                                                                                WACC
                              50%
                              45%                                               rd x (1-T)
                              40%
            Cost of Capital




                              35%
                              30%
                              25%
                              20%
                              15%
                              10%
                               5%
                               0%
                                 0%     20%     40%      60%      80%    100%
                                               Debt/Value Ratio




Mini Case: 17 - 18
                                                              Figure 4
                        Value of Firm, V
                                 ($)

                             4




                                                                                                      VL
                             3


                                                                                       TD

                             2                                                                        VU




                             1




                                 0           0.5        1.0                1.5   2.0            2.5
                                                                                            Debt ($)
                                                         (Millions of $)




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:

                                                       (1  TC )(1  TS ) 
                                            VL = VU + 1                   D.
                                                           (1  TD )      

           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,



                                                                                                           Mini Case: 17 - 19
          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

                           (1  TC )(1  TS ) 
                VL  VU  1                  D
                                (1  TD       
                             (1  0.40)(1  0.12 
                      VU  1                    D  VU  (1  0.75) D  VU  0.25D.
                                  (1  0.30)     


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-à-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                    d
          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.




Mini Case: 17 - 20
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.




                                                                              Mini Case: 17 - 21
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.


Mini Case: 17 - 22
          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 ( = 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.




                                                                              Mini Case: 17 - 23
          This option can be valued with the Black-Scholes Option Pricing Model:

                                   V = PN(D1) – Xe-RTN(D2)
          where
                               D1 = [ln(P/X) + (r + 0.52)T]/[T0.5]
                                          D2 = D1 - T0.5
          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
                              = 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.




Mini Case: 17 - 24
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.




                                                                              Mini Case: 17 - 25

				
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