Capital Structure Theories and Solved Problems
Leverage, Cost of Capital and Firm Value
Value of Firm
The value of a firm depends on the earnings of the firm and the earnings of the firm depend upon the investment decisions of the firm. The earnings of the firm are capitalized at a rate equal to the cost of capital in order to find out the value of the firm. Thus, the value of the firm depends on two basic factor ie, the earnings of the firm and the cost of capital.
Value of Firm = Value of Equity + Value of Debt
NI Approach
There is a relationship between capital structure and the value of the firm. The firm can affect its value by increasing or decreasing the debt proportion in overall financing mix.
Assumptions of NI Approach
· Cost of Debt (Kd) is less than Cost of Equity (Ke).
· Value of Firm remains constant.
· Kd and Ke remains constant. Increase in financial leverage (increase in debt in financing mix) does not impact risk perception of investors and Ke remains constant in case of increase in debt.
· Increase in debt will lead to decrease in overall cost of capital and increase in value of the firm.
· Higher the degree of leverage, better it is, as the value of the firm would be higher. In other words, a firm can increase its value just by increasing the debt proportion in the capital structure.
· Value of firm = Value of Debt + Value of Equity
· Kd = Interest / Value of Debt
· Ke = EBT or EAT / Value of equity
· K0 = EBIT / Value of Firm
· K0 = (D/V) × Kd + (E/V) × Ke
The NI approach, though easy to understand is too simple to be realistic. It ignores, perhaps the important aspects of leverage that the market price depends upon the risk which varies in direct relation to the change in proportion of debt in the capital structure.
NOI Approach
The Net Operating Income (NOI) approach is opposite to NI approach. This is also known as Independence Hypothesis. According to the NOI approach, the market value of the firm depends upon the net operating profit or EBIT and the over cost of capital, WACC. The financing mix or the capital structure is irrelevant and does not affect the value of the firm.
Assumptions of NOI Approach
· Overall cost of capital (K0) remains constant.
· Cost of Debt (Kd) is constant.
· Increase in debt increases risk perception of shareholders resulting into increase in cost of equity (Ke) and offsetting the benefits of low cost debt into capital structure thus keeping overall cost of capital and value of firm (V) constant at any level of Debt – equity mix in its financing structure.
· There is no tax.
· The NOI approach is based on the argument that the market values the firm as a whole for a given risk complexion. Thus, for a given value of EBIT, the value of the firm remains same irrespective of the capital composition and instead depends on the overall cost of capital.
The value of the Equity may be found by deducting the value of debt from the total value of the firm i.e.
V = EBIT ÷ K0
E = V – D
Ke = (EBIT – I) ÷ (V - D)
The NOI approach considers overall cost of capital (K0), to be constant and therefore, there is no optimal capital structure; rather every capital Structure is as good as any other one and every capital structure is optimal one.
Traditional Approach
The Traditional view states that value of firm increases with increase in financial leverage but up to a certain limit only. Beyond this limit, the increase in financial leverage will increase its WACC and value of firm will decline.
Cost of debt (Kd) is assumed to be less than the cost of equity (Ke). In case of 100% equity firm, overall cost of capital (K0) is equal to the Cost of Equity (Ke), but when cheaper debt is introduced in the capital structure and financial leverage increases, Ke remains same as the shareholders expect a minimum leverage in every firm. Ke does not increase even with increase in leverage. The argument for Ke remaining unchanged may be that up to a particular degree of leverage, the interest charge may not be large enough to pose a real threat to the dividend payable to the shareholders. This constant Ke and Kd makes K0 to fall initially reflecting benefits of cheaper debts available to the firm.
The increase in leverage beyond a limit increases risk of equity investors also resulting into increase in Ke. However, the benefits of use of debt may be so large that even after offsetting the effects of increase in Ke, K0 may still go down or may become constant for some degree of leverages.
If firm increases the leverage further, then the risk of debt investor may also increase and consequently Kd also starts increasing. The already increasing Ke and now increasing Ke, makes K0 to increase. Therefore, the use of leverage beyond a point will have the effect of increase in overall cost of capital of the firm and decrease in value of firm.
MODIGLIANI-MILLER MODEL: BEHAVIOURAL JUSTIFICATION OF NOI APPROACH
MM Model shows that the financial leverage does not matter and the cost of capital and value of firm are independent of capital structure. There is nothing which may be called the optimal capital structure, they have, in fact, restated the NOl approach and have added to it the behavioural justification for this model.
Assumptions of MM Model
· The capital markets are perfect and complete.
· information is available to all the investors free of cost. The implication of this assumption is that investors can borrow and lend funds at the same rate and can move quickly from one security to another without incurring any transaction cost.
· The securities are infinitely divisible.
· Investors are rational and well informed about the risk return of all the securities.
· All investors have same probability distribution about expected future earnings.
· There is no corporate income-tax. (However, this assumption was relaxed later).
· The personal leverage and the corporate leverage are perfect substitute.
On the basis of these assumptions, the MM Model derived that -
(a) The total value of the firm is equal to the capitalized value of the operating earnings of the firm. The capitalization is to be made at a rate appropriate to the risk class of the firm.
(b) The total value of the firm is independent of the financing mix i.e. the financial leverage.
(c)The cut-off rate for the investment decision of the firm depends upon the risk class to which the firm belongs and thus is not affected by the financing pattern of this investment.
MM Model can be discussed in terms of two propositions I and II.
MM Proposition I:
Proposition I states that it is completely irrelevant how a firm arranges its capital funds.
MM model argues that if two firms are alike in all respect except that they differ in respect of their financing pattern and their market value, then the investors will develop a tendency to sell the shares of the overvalued firm (creating selling pressure) and to buy the shares of the undervalued firm (creating a demand pressure). This, buying and selling pressures will continue till the two firms have same market values.
The Arbitrage Process
The arbitrage process refers to undertaking by a person of two related actions or steps simultaneously in order to derive some risk- less benefit e.g, buying by a speculator in one market and selling the same at the same time in some other market; or selling one type of investment and investing the proceed in some other investment. The profit or benefit from the arbitrage process may be in any form: increased income from the same level of investment or same income from lesser investment. This arbitrage process has been used by MM to testify their hypothesis of financial leverage, cost of capital and value of the firm.
MM Proposition II
Proposition II states that the cost of equity depends upon three factors i.e. overall cost of capital of the firm, cost of debt and the firm's debt equity ratio. In MM model, there is a linear relationship between the cost of equity and the leverage (as measured by the Debt-equity ratio D/E). When the leverage is increased, the earnings available for the equity shareholder will increase, but the cost of equity will also increase as a result of increase in financial risk. The benefits of increasing leverage are completely offset by the increase in cost of equity capital and consequently the market value of the firm remains same.
As per MM model:
Ke = K0 + (K0 - Kd) × D/E
As per MM model, the overall cost of capital (K0) will not rise even if the degree of financial leverage is increased.
Under MM Model, the value of levered firm is found out as follows:
Vu = EBIT (1- t )/ K0 or Ke
VL = Vu + Debt × tax rate
VL = Vu + PV of Interest Tax shield
MM Model without Taxes
· Firm's capital structure is irrelevant.
· WACC is same no matter what mixture of debt and equity is used to finance the firm.
· Total value of the firm is independent of level of debt in the capital structure, and the value can be calculated by capitalizing the operating profit at appropriate rate. The value of the levered firm is equal to the value of the unlevered firm, and
Ke = K0 + (K0 - Kd) × D/E
MM Model with Taxes
· The value of the levered firm is equal to the value of unlevered firm + the present value of the interest tax shield, i.e.
VL = Vu + Debt × tax rate
· The WACC of the firm decreases, as the firm relies more and more on debt financing.
Ke = K0 + (K0 - Kd) × D/E
Or Ke = K0 + (K0 - Kd) × D (1 – t )/E
K0 is the WACC of the unlevered firm.
1. The expected EBIT of a firm is Rs. 5,00,000. It has issued Equity Share capital with Ke @ 20% and 10% Debt of Rs. 10,00,000. Find out the value of the firm and the overall cost of capital, WACC using NOI Approach.
Solution
Particulars | Amount (Rs.) |
EBIT | 5,00,000 |
Less: Interest | (1,00,000) |
EBT / PAT | 4,00,000 |
Ke | 0.2 |
Value of Equity = PAT / Ke | 4,00,000/.2 = 20,00,000 |
Value of Debt | 10,00,000 |
Value of Firm (V)= Value of Debt + Value of Equity | 30,00,000 |
WACC (K0) = EBIT / V | 5,00,000 / 30,00,000 = .1667or 16.67% |
WACC = (D/V × Kd) + (E/V × Ke) | (10,00,000 / 30 × .1) + (20,00,000/30,00,000 ×.2) = .1667 = 16.67% |
2. A firm has an EBIT of Rs. 4,00,000 and belongs to a risk class of 10 %. What is the value of cost of equity capital if it employs 8% debt to the extent of 30%, 40% or 50% of the total capital fund of Rs. 15,00,000.
Solution
| 30% Debt | 40% Debt | 50% Debt |
Value of Debt (D) | 4,50,000 | 6,00,000 | 7,50,000 |
K0 (WACC) | 0.1 | 0.1 | 0.1 |
EBIT | 4,00,000 | 4,00,000 | 4,00,000 |
Value of Firm | 40,00,000 | 40,00,000 | 40,00,000 |
Value of Equity (E) | 35,50,000 | 34,00,000 | 32,50,000 |
Int. | (36,000) | (48,000) | (60,000) |
PAT = EBIT – Int. | 3,64,000 | 3,52,000 | 3,40,000 |
Ke = PAT / E | .1025 | .1035 | .1046 |
3. RBL Ltd. having an EBIT of Rs.2,00,000 is contemplating to redeem a part of the capital by introducing debt financing. Presently, it is a 100% equity firm with equity capitalization rate, Ke of 20%. The firm is to redeem the capital by introducing debt financing up to Rs.4,00,000 i.e., 40% of total funds or up to Rs. 5,00,000 i.e., 50% of total funds. It is expected that for debt financing up to 30%, the rate of interest will be 10% and Ke will increase to 21%. However, if the firm opts for 50 % debt financing, then interest will be payable at the rate of 12% and the Ke will be 24%. Find out the value of the firm and its WACC under different levels of debt financing.
Solution
On the basis of the information given, the total funds of the firm seems to be of Rs.10,00,000 (whole of which is provided by the equity capital) out of which 40% or 50% i.e. 4,00,000 or 5,00,000 may be replaced by the issue of debt bearing interest at 10% or 12% respectively, value of firm and WACC is calculated as follows:
| 0 % Debt | 40% Debt | 50% Debt |
Value of Debt (D) | ----- | 4,00,000 | 5,00,000 |
Interest rate | ----- | 10% | 12% |
EBIT | 2,00,000 | 2,00,000 | 2,00,000 |
Less: Int. | ----- | (40,000) | (60,000) |
EBT or NP | 2,00,000 | 1,60,000 | 1,40,000 |
Ke | .2 | .21 | .24 |
Value of Equity (E) = NP ÷ Ke | 10,00,000 | 7,61,905 | 5,83,333 |
Value of Firm (V) = E + D | 10,00,000 | 11,61,905 | 10,83,333 |
WACC (K0) = EBIT ÷ V | 0.2 | 0.1721 | 0.1846 |
4. A Ltd. and B Ltd. are in the same risk class and are identical in all respects except that company A uses debt while company B does not use debt. The levered firm has Rs. 10,00,000 debentures carrying 10% rate of interest. Both the firms earn 20% operating profit on their total assets of Rs. 20,00,000. The company is in the tax bracket of 50% and capitalization rate of 15% on all equity shares.
Solution
Particulars | A Ltd | B Ltd |
Total Assets | 20,00,000 | 20,00,000 |
Operating Profit | 20 % | 20 % |
EBIT | 4,00,000 | 4,00,000 |
Less: Interest | (1,00,000) | |
EBT | 3,00,000 | 4,00,000 |
Less: Tax @ 50% | (1,50,000) | (2,00,000) |
PAT | 1,50,000 | 2,00,000 |
Ke | .15 | .15 |
Value of Equity (Ke) = PAT ÷ Ke | 10,00,000 | 13,33,333 |
Value of Debt (D) | 10,00,000 | ------- |
Total Value of Firm (V) = D + E | 20,00,000 | 13,33,333 |
5. RBL Steel Ltd. has employed 15% debt of Rs. 15,00,000 in its capital structure. The net operating income of the firm is Rs. 6,00,000 and has an equity capitalization rate of 20%. Assuming that there is no tax, find out the value of the firm under the NI Approach.
Solution
Net operating income (EBIT) | 6,00,000 |
Less: Interest on Debt (15 % of 15,00,000) | (2,25,000) |
EBT or Net Profit (NP) | 3,75,000 |
Equity Capitalization rate (Ke) | 20% |
Value of Equity = NP ÷ Ke (3,75,000 ÷ .2) | 18,75,000 |
Value of Debt | 15,00,000 |
Total value of the firm | 33,75,000 |
6. RBL Ltd. belongs to a risk class of 12 % and expects EBIT of Rs. 5,00,000. It employs 10 % debt in the capital structure. Find out the value of the firm and cost of equity capital Ke. If it employs debt to the extent of 30%, 40% or 50% of the total financial requirement of Rs.20,00,000.
Solution
| 30 % Debt | 40% Debt | 50% Debt |
Debt (D) | 6,00,000 | 8,00,000 | 10,00,000 |
Interest rate | .1 | .1 | .1 |
EBIT | 5,00,000 | 5,00,000 | 5,00,000 |
Less: Int. | (60,000) | (80,000) | (1,00,000) |
EBT or NP | 4,40,000 | 4,20,000 | 4,00,000 |
K0 | .12 | .12 | .12 |
Value of Firm (V) = EBIT ÷ K0 | 41,66,667 | 41,66,667 | 41,66,667 |
Value of Equity (E) = V – D | 35,66,667 | 33,66,667 | 31,66,667 |
Ke = NP ÷ E | .1234 | .1248 | .1263 |
7. The net operating profit of a firm is Rs. 3,00,000 and the total market value of its 12% debt is Rs. 5,00,000. The equity capitalization rate of an unlevered firm of the same risk class is 20 %. Find out the value of the levered firm given that the tax rate is 50% for both the firms.
Solution
Value of Unlevered firm = [EBIT × (1- t )] ÷ Ke
= [3,00,000 × (1 – 0.5) ] ÷ .15
= 1,50,000 ÷ .2 = Rs. 7,50,000
Value of Levered Firm = Value of Unlevered firm + (D × t)
=> Rs. 7,50,000 + (Rs.5,00,000 × .5) = Rs.10,00,000
8. ABC Ltd. with EBIT of Rs. 5,00,000 is evaluating a number of possible capital structures, given below. Which of the capital structure will you recommend and why?
Capital Structure | Debt | Kd % | Ke % |
I | 3,00,000 | 11 | 12 |
II | 4,00,000 | 11 | 15 |
III | 5,00,000 | 12 | 16 |
IV | 6,00,000 | 13 | 17 |
Solution
In this case, the Kd and Ke, of the firm are given and changing. The firm may adopt that capital structure which has the least overall cost of capital or the maximum value. The overall cost of capital, K0 of the firm may be calculated by applying the traditional approach as follows:
K0 = EBIT ÷ Value of Firm
Value of Firm = V
Value of Equity (E) = Net Profit ÷ Ke
Value of Debt = D
Particulars | Plan I | Plan II | Plan III | Plan IV |
EBIT | 5,00,000 | 5,00,000 | 5,00,000 | 5,00,000 |
Less: Int. | (33,000) | (44,000) | (60,000) | (78,000) |
Net Profit | 4,67,000 | 4,56,000 | 4,40,000 | 4,22,000 |
Ke | .12 | .15 | .16 | .17 |
E | 38,91,667 | 30,40,000 | 27,50,000 | 24,82,355 |
D | 3,00,000 | 4,00,000 | 5,00,000 | 6,00,000 |
V | 41,91,667 | 34,40,000 | 32,50,000 | 30,82,355 |
K0 | .1193 | .1453 | .1538 | .1622 |
The capital structure of Plan I is having Rs. 3,00,000 of debt and has the lowest overall cost of capital and consequently the highest market value. Hence Plan I should be accepted.
9. Two companies are identical except that A Ltd. has a debt of Rs. 15,00,000 at 10% whereas B Ltd. does not have debt in its capital structure. The total assets of both the companies A and B are same i.e. Rs. 30,00,000 on which each company earns 20 % return. Find the value of each company and overall cost of capital using net operating income (NOI) Approach. Equity capitalisation rate for B Ltd. is 15%. The tax rate is 50%.
Solution
NOI Approach with Taxes:
EBIT = 20 % of Rs. 30,00,000 = Rs. 6,00,000
Value of B Ltd (Unlevered) = [EBIT × (1- t)] ÷ Ke
=> [6,00,000 × (1 – 0.5 )] ÷ .15
=> 20,00,000
Value of A Ltd (Levered) = Value of Unlevered firm + (D × t)
=> Rs. 20,00,000 + (Rs.15,00,000 × .5) = Rs.27,50,000
10. RBL Ltd. has Earnings before Interest and Taxes (EBIT) of Rs. 6,00,000. The firm currently has outstanding debts of Rs. 15,00,000 at an average cost, Kd of 10%. Its cost of equity capital Ke is estimated to be 20 %.
i. Determine the current value of the firm using the Traditional valuation approach.
ii. Determine the firm's overall capitalization rate, K0.
iii. The firm is considering to issue capital of Rs. 10,00,000 in order to redeem Rs. 10,00,000 debt. The cost of debt is expected to be unaffected. However, the firm's cost of equity capital is to be reduced to 16% as a result of decrease in leverage. Would you recommend the proposed action?
Solution
EBIT | 6,00,000 |
Less: Int. | (1,50,000) |
EBT or Net Income | 4,50,000 |
Ke | 0.2 |
Value of Equity (E)= Net Income ÷ Ke | 22,50,000 |
Value of Debt (D) | 15,00,000 |
i. Value of Firm (V) = D + E | 37,50,000 |
ii. Overall Capitalization rate (K0) = EBIT ÷ V | 0.16 |
iii. Effect of Proposed redemption of debt:
EBIT | 6,00,000 |
Less: Int. | (50,000) |
EBT or Net Income | 5,50,000 |
Ke | 0.16 |
Value of Equity (E)= Net Income ÷ Ke | 34,37,500 |
Value of Debt (D) | 5,00,000 |
Value of Firm (V) = D + E | 39,37,500 |
Overall Capitalization rate (K0) = EBIT ÷ V | 0.1524 |
The proposal should be accepted as it will increase value of firm from Rs. 37,50,000 to Rs. 39,37,500. The cost of capital will also reduce from 16 % to 15.24%.
11. The following estimates of the cost of debt and cost of equity capital have been made at various level of the debt-equity mix for ABC Ltd.
% of Debt | Cost of Debt Kd (%) | Cost of Equity Ke (%) |
0 | 6 | 12 |
10 | 6 | 12 |
20 | 6 | 13 |
30 | 7 | 14 |
40 | 8 | 15 |
50 | 9 | 16 |
60 | 10 | 20 |
Assuming no tax, determine the optimal debt equity ratio for the company on the basis of the overall cost of capital, WACC.
Solution
Overall Cost of Capital = (D/V) × Kd + (E/V) × Ke
V = Value of Firm or Total Capital = 100%
E = Percentage of Debt in total capital = 100% – Debt Percentage
D = Percentage of Debt in total capital
D (%) | E (%) | Kd | Ke | D/V | E/V | K0 |
0% | 100% | .06 | .12 | 0 | 1 | .12 |
10% | 90% | .06 | .12 | .1 | .9 | .114 |
20% | 80% | .06 | .13 | .2 | .8 | .116 |
30% | 70% | .07 | .14 | .3 | .7 | .119 |
40% | 60% | .08 | .15 | .4 | .6 | .122 |
50% | 50% | .09 | .16 | .5 | .5 | .125 |
60% | 40% | .10 | .20 | .6 | .4 | 0.14 |
The optimal debt equity mix for the company occurs at a point when the overall cost of capital (K0) is minimum. K0 is minimum at a point when the debt is 10 % of the total capital employed and equity is 90 %. Therefore, the firm should use 10% debt and 90% equity in its capital structure.
12. The following information is available for RBL Ltd. and Gyan Ltd in respect of their present position. Compute the equilibrium values (V) and equity capitalization rate of the two companies, assuming (i) there is no income tax, and (ii) the overall rate of capitalization (K0) for such companies in the market is 16%.
| RBL Ltd | Gyan Ltd |
EBIT | 2,00,000 | 2,00,000 |
Less: Int. @10% | (50,000) | ------ |
Net Income for Equity Shareholders | 1,50,000 | 2,00,000 |
Equity Capitalization rate (Ke) | .15 | .13 |
Market Value of Equity (E) | 15,00,000 | 16,00,000 |
Market Value of Debt (D) | 5,00,000 | ------ |
Total Value of Firm | 20,00,000 | 16,00,000 |
Overall Cost of Capital (K0) = EBIT ÷ Total Value of Firm | .1 | .125 |
Solution:
In order to find out the equilibrium value of the firm, the EBIT of both the firm should be capitalised at K0 and then it will be bifurcated into value of debt and value of equity as follows:
| RBL Ltd | Gyan Ltd |
EBIT | 2,00,000 | 2,00,000 |
Overall capitalization K0 | .16 | .16 |
Total value of the firm (equilibrium values) | 12,50,000 | 12,50,000 |
Less: Market value of the Debt | (5,00,000) | ----- |
Market value of Equity, E | 6,50,000 | 12,50,000 |
Earnings for Equity holders (NP) | 1,50,000 | 2,00,000 |
Ke (Equity Capitalization Rate) = NP ÷ E | .2308 | .16 |
13. Following is the data regarding two companies X and Y belonging to the same risk class:
| | |
No. of Equity Shares | 1,00,000 | 1,50,000 |
Market Price per Share | 15 | 10.5 |
10 % Debentures | 1,00,000 | ----- |
Profit before Interest | 2,50,000 | 2,50,000 |
All profits after debenture interest are distributed as dividends. Explain how under Modigliani & Miller approach, an investor holding 10% of shares in Company X will be better off in switching his holding to Company Y.
Solution
Both the firms have EBIT of Rs. 25,000. Company X has to pay interest of Rs. 10,000 (i.e. 10% on Rs.1,00,000) and the remaining Profit of 15,000 is being distributed among the shareholders. Company Y, on the other hand, has no interest liability and therefore, is distributing Rs. 25,000 among the shareholders. The investor will be well off under MM model, by selling shares of X and shifting to shares of Y Company through the arbitrage process as follows:
If he sells shares of company X, he gets Rs. 1,50,000, (10,000 shares @ Rs. 15 per share). He now takes a 10% loan of Rs. 10,000 (i.e. 10 % of Rs. 1,00,000) and out of the total cash of Rs.1,60,000, he purchase 10 % of shares of Company Y for Rs. 1,57,500. His position with regard to income from Company X and Company Y would be as follows:
| Company X | Company Y |
Dividend (10% of Profit) | 24,000 | 25,000 |
Less: Interest (10 % on Rs. 10,000) | --- | (1,000) |
Net Income | 24,000 | 24,000 |
Thus, by shifting from Company X to Company Y, the investor is able to get same income of Rs. 24,000 and still having funds of Rs. 10,000 (i.e., Rs. 1,60,000 – Rs. 1,57,500) at his disposal. He is better off, not in terms of income, but in terms of having capital funds of Rs. 2,500 with him, which he can invest elsewhere.
14. From the following selected data, determine the value of the firms, P and Q belonging to the homogeneous risk class.
| Firm A | Firm B |
EBIT | 3,00,000 | 3,00,000 |
Interest @ 15% | 75,000 | |
Equity capitalization rate, Ke or K0 | | 20% |
Corporate Tax | 50% |
Which of the two firms has an optimal capital structure under NOI approach?
Solution:
Valuation of the firm (Net Operating Income approach with Tax):
The NOI approach is based on the assumptions that there is no tax. However, in the present case, both the firms have tax liability @ 50%. So, their valuation may be found by applying MM model (with taxes) which is an extension of NOI approach. Under the MM Model, the value of levered firm is taken as equal to the value of unlevered firm plus the premium for interest tax shield on debt financing.
VL = VU + (Debt × tax rate)
VL = Value of Levered Firm = Firm A
VU = Value of Unlevered Firm = Firm B
In case of Unlevered firm, Ke = K0
VU (Firm B) = [EBIT × (1 – t)] ÷ K0
=> [3,00,000 × (1 – .5)] ÷ .2 = Rs. 7,50,000
VL (Firm A) = VU + (Debt × tax rate)
=> 7,50,000 + (5,00,000 × .5) = Rs.10,00,000
Value of Debt of Firm A = Interest amount ÷ Pre Tax Cost of debt => 75,000 ÷ .15= Rs. 5,00,000
Post Tax Cost of Debt = Pre Tax Cost of debt × (1 – tax rate) => .15 × .5 = .075
Value of Equity of Firm A = Value of Firm A – Value of Debt = Rs. 10,00,000 – Rs.5,00,000 = Rs. 5,00,000
Cost of Equity (Ke) = PAT ÷ Value of Equity
=> [(3,00,000 – 75,000)× (1- 0.5)] ÷ 5,00,000
=> 1,12,500 ÷ 5,00,000 = .225
WACC (K0) = D/V × Post tax Kd + E/V × Ke
=> (5,00,000 / 10,00,000) × .075 + (5,00,000 / 10,00,000) × .225 = .15 = 15 %
Second Method:
WACC = [EBIT (1 – t)] ÷ Value of Firm A
=> 1,50,000 ÷ 10,00,000 = 0.15 or 15 %
WACC of firm A is 15 %. Firm A has optimal Capital structure as it is having higher total Firm value than value of Firm B and lower overall cost of capital than Firm B.
15. Companies U and L are identical in every respect except that the former does not use debt in its capital structure, while the latter employs Rs. 6,00,000 of 10% debt. Assuming that all MM assumptions are met, corporate tax rate is 50%, EBIT being Rs. 4,00,000, and equity capitalization of the unlevered company is 20%, what will be the value of the firms, U and L ? Also determine the weighted average cost of capital for both the firms.
Solution
VL = VU + (Debt × tax rate)
VL = Value of Levered Firm = Firm L
VU = Value of Unlevered Firm = Firm U
In case of Unlevered firm, Ke = K0
VU (Firm B) = [EBIT × (1 – t)] ÷ K0
=> [4,00,000 × (1 – .5)] ÷ .2 = Rs. 10,00,000
VL (Firm A) = VU + (Debt × tax rate)
=> 10,00,000 + (6,00,000 × .5) = Rs.13,00,000
Overall cost of Capital (K0) of Unlevered firm = 20% = 0.2
Calculation of overall Cost of Capital (K0) of Levered Firm:
EBIT | 4,00,000 |
Less: Interest | (60,000) |
EBT | 3,40,000 |
Less: Tax @ 50% | (1,70,000) |
PAT | 1,70,000 |
Total Value of Levered Firm | 13,00,000 |
Less: Value of Debt (D) | (6,00,000) |
Value of Equity (E) | 7,00,000 |
Cost Of Equity (Ke) = PAT ÷ E | 1,70,000 ÷ 7,00,000 = .2429 |
K0 = EBIT (1 – t ) ÷ Value of Firm | 2,00,000 ÷ 13,00,000 = .1538 |
Or K0 = (D/V × Post Tax Kd) + (E/V × Ke) = (6L / 13 L × 0.05) + (7L / 13 L × .2429) | .1538 |
16. The expected annual net operating income of a company is Rs. 15,00,000. The company has Rs. 60,00,000, 10% debentures. The overall cost of capital is 12.5%. Calculate the value of the firm and cost of equity according to NOI Approach. If the company increases the debt from Rs. 560,00,000 to Rs. 70,00,000, what would be the value of the firm?
Solution
EBIT | 15,00,000 |
WACC (K0) | .125 |
Value of Firm(V)= EBIT ÷ K0 | 1,20,00,000 |
Value of Debt (D) | 60,00,000 |
Value of Equity (E) = V – D | 60,00,000 |
Ke = (EBIT – Int.) ÷ E | 9,00,000 ÷ 60,00,000 = 0.15 |
If Debt increases to Rs. 70,00,000 |
Value of Firm(V)= EBIT ÷ K0 | 1,20,00,000 |
Value of Debt (D) | 70,00,000 |
Value of Equity (E) = V – D | 50,00,000 |
Ke = (EBIT – Int.) ÷ E | 8,00,000 ÷ 50,00,000 = 0.16 |
So, as per NOI, the value of the firm remains at Rs. 1,20,00,000 but the value of equity decreases to Rs. 50,00,000. Consequently, Ke also increases from 15 % to 16 %.
17. Two companies V and L, belong to same risk class. These two firms are identical in all respect except that V company is unlevered while Co. L has 10% debentures of Rs. 5,00,000. The other relevant data regarding their valuation and capitalisation rates are as follows:
| | |
EBIT | 1,00,000 | 1,00,000 |
Less: Interest | 50,000 | |
Earnings available to Equity-holders | 50,000 | 1,00,000 |
Equity capitalisation rate | 0.16 | 0.125 |
Market value of Equity | 3,12,500 | 8,00,000 |
Market value of Debt | 5,00,000 | |
Total Market value | 8,12,500 | 8,00,000 |
Overall Cost of Capital (K0) | 0.123 | 0.125 |
Debt-Equity Ratio | 1.6 | ---- |
i. An investor owns 10% equity shares of company L. Show the arbitrage process and amount by which he could reduce his outlay through the use of leverage.
ii. According to Modigliani and Miller, when will this arbitrage process come to an end?
Solution
Arbitrage Process by Investor: |
Sale of 10% Equity Shares in L Ltd. | 31,250 |
Add: 10% Loan (equal to 10% of Rs. 5,00,000) | 50,000 |
Total Funds | 81,250 |
Less: Purchase of 10% Equity of V Ltd. | (80,000) |
Capital funds saved | 1,250 |
Analysis of Income Position |
| L Ltd | V Ltd |
Dividend | 5,000 | 10,000 |
Less: Interest Payable | | (5,000) |
Net Income | 5,000 | 5,000 |
| | | |
So, through arbitrage (sale of equity shares of L and buying Equity Shares of V), the investor can reduce his outlay by Rs. 1,250 and still getting same income of Rs. 5,000. The arbitrage process will come to an end when the difference in value of L and V comes to zero.
18. Two companies, X and Y belong to equivalent risk group. The two companies are identical in every respect except that company Y is levered, while X is unlevered. The outstanding amount of debt of the levered company is Rs. 5,00,000 in 10% debenture. The other information for the two companies is as follows:
| X | Y |
EBIT | 1,70,000 | 1,70,000 |
Less: Interest | ______ | (50,000) |
Earnings to Equity Holders | 1,70,000 | 1,20,000 |
Equity Capitalization Rate (Ke) | .17 | .20 |
Market Value of Equity (E) | 10,00,000 | 6,00,000 |
Market Value of Debt (D) | ______ | 5,00,000 |
Total Value of Firm (V) | 10,00,000 | 11,00,000 |
Overall capitalization rate K0 = EBIT / V | .17 | .1545 |
Debt Equity Ratio | 0 | 1 |
An investor owns 5% equity shares of company Y. Show the process and the amount by which he could reduce his outlay through use of arbitrage process? Is there any limit to the process?
Solution
Current position of investor in Firm Y:
Dividend Income = 5 % of 1,20,000 = Rs. 6,000
Market Value of Investment = 5 % of 6,00,000 = Rs. 30,000.
He sells his holdings in Firm Y and creates a personal leverage by borrowing Rs. 25,000 (5 % of Rs.5,00,000).
Total amount with him now = Rs. 25,000 + Rs.30,000 = Rs.55,000.
Now, he purchase 5 % equity in Firm X by investing Rs.50,000 (5 % of Rs.10,0000).
Now his position with respect to income in both companies:
| X | Y |
Dividend (5% of Profit) | 8,500 | 6,000 |
Less: Interest 10% of 25,000 | (2,500) | _________ |
Net Income | 6,000 | 6,000 |
Investor has saved Rs. 5,000 (55,000 – 50,000) using leverage and continues to earn same earnings as before. Remaining Rs.5,000 can be invested somewhere to increase earnings.
Yes, there is limit to arbitrage process. It comes to an end when market value of both firms remain same
19. Firms A and B are similar except that A is unlevered while B has Rs.2,00,000 of 5% debentures outstanding. Assume that the tax rate is 50 %, NOI is Rs. 50,000 and cost of equity is 10 %.
i. Calculate the value of the firm, if the MM assumptions are met.
(ii) If the value of the firm B is Rs. 3,60,000 then do these values represent equilibrium values. If not, how will equilibrium be set? Explain.
Solution
I. Value of Unlevered Firm A (VA) = EBIT (1 – t ) / Ke or K0
= 50,000 (1 – 0.5 ) / 0.1 = 2,50,000
Value of Levered Firm B = VA + Debt × tax rate
=2,50,000 + 2,00,000 × .5 = Rs. 3,50,000
II. Value of Firm B is given Rs. 3,60,000 however it came Rs. 3,50,000 as calculated above which indicates that it does not represent equilibrium value and Firm B is overvalued by Rs. 10,000.
Arbitrage process to restore equilibrium:
| Firm B |
Value of Firm as given (V) | 3,60,000 |
Value of Debt D | 2,00,000 |
Value of Equity E = V - D | 1,60,000 |
EBIT | 50,000 |
Less: Interest (5% of 2,00,000) | (10,000) |
EBT | 40,000 |
Less Tax @ 50% | (20,000) |
PAT | 20,000 |
Assuming an investor owns 10 % of firm B shares. His investment is-
10 % of 1,60,000 = Rs.16,000
And Return on investment is 10 % of 20,000 = Rs.2,000.
The investor can get same income by shifting his investment to Firm A.
He sells his holdings in Firm B and creates a personal leverage by borrowing Rs. 10,000 (10% of Rs.2,00,000 × (1 – tax rate)).
Total amount with him now = Rs. 16,000 + Rs.10,000 = Rs.26,000.
Now, he purchase 10 % equity in Firm A by investing Rs.25,000 (10 % of Rs.2,50,000).
Now his position with respect to income in both companies:
Net profit of Firm A = EBIT (1 – t ) = 50,000 × .5 = 25,000
| A | B |
Dividend (10% of Profit) | 2,500 | 2,000 |
Less: Interest 5% of 10,000 | (500) | _________ |
Net Income | 2,000 | 2,000 |
Investor has saved Rs. 1,000 (26,000 – 25,000) using leverage and continues to earn same earnings as before. Remaining Rs.1,000 can be invested somewhere to increase earnings. This process will continue till equilibrium is restored.
20. Two companies, L and U belong to the same risk class. The two firms are identical in every respect except that company L has 10% debentures. The valuation of the two firms as per the Traditional theory is as follows:
| L | U |
Net Operating Income (EBIT) | 22,50,000 | 22,50,0000 |
Less: Interest | (1,50,000) | ------ |
Earnings to Equity holders | 21,00,000 | 22,50,000 |
Equity capitalization rate (Ke) | .15 | .12 |
Market value of Equity | 1,40,00,000 | 1,87,50,000 |
Market value of Debt | 15,00,000 | ________ |
Total Value of Firm (V) | 1,55,00,000 | 1,87,50,000 |
Overall Capitalization Rate K0 = EBIT ÷ V | 14.52% | 12% |
Debt Equity Ratio | .1071 | ---------- |
Show the arbitrage process by which an investor having shares worth Rs.18,75,000 in company U will be benefited by switching over to company L.
Solution
Investors total worth of Equity share in Company U = 18,75,000 / 1,87,50,000 = 0.1 = 10 %
Dividend Income = 10 % of Rs. 22,50,000 = Rs. 2,25,000
Now in arbitrage process, he sells his investment from U Ltd of Rs. 18,75,000 and makes investment in 10 % equity of L Ltd of Rs. 14,00,000. He also invests Rs.1,50,000 i.e. 10% of Rs. 15,00,000 in 10 % interest bearing Debt of L Ltd.
As a result of this investment, his income would be as follows:
=>Dividend income + Interest income = Rs.2,25,000
Dividend income = 10 % of Rs.21,00,000 = Rs.2,10,000
Interest income = 10 % of Rs. 1,50,000 = Rs.15,000
Thus, investor is able to maintain same level of earning with a saving of fund of Rs. 3,25,000 (18,75,000 – 14,00,000 – 1,50,000) through arbitrage process.
21. ABC Ltd. has the required rate of return of 15% on its assets. It can borrow in the market @ 10%. Assuming MM model (without taxes), what would be the cost of equity of the firm, if it has target capital structure of 80% equity or 50% equity?
Solution
As per MM proposition II, cost of equity (Ke) is –
Ke = K0 + (K0 - Kd) × D/E
If equity is 80 %
= .15 + (.15 - .1) × (.2/.8) = .1625
If equity is 50 %
= = .15 + (.15 - .1) × (.5/.5) = .2
22. Following information is available in respect of L Ltd. and U Ltd.
| L Ltd | U Ltd |
EBIT | 15,00,000 | 15,00,000 |
Less: Interest @ 10 % | (2,50,000) | _______ |
EBT | 12,50,000 | 15,00,000 |
Less: Tax @ 50 % | (6,25,000) | (7,50,000) |
PAT | 6,25,000 | 7,50,000 |
| | |
Show and verify that value of levered firm is equal to value of unlevered firm plus PV of tax shield on interests. Use MM Model (with taxes), given the k. for U Ltd. is 20%.
Solution
In case of U Ltd, Ke or K0 is 20% and EBIT is 15,00,000. So, value of Equity or value of firm is-
Vu = EBIT (1- t )/ Ke
= 15,00,000 × (1 - .5 ) / .2 = 37,50,000
As per question:
VL = Vu + PV of Interest Tax shield
= 37,50,000 + (2,50,000 × .5) / .1
= 50,00,000
As per MM model
VL = Vu + Debt × tax rate
= 37,50,000 + (2,50,000/.1) × .5
= 50,00,000
Links to Financial Management notes: -
Time Value of Money
https://gyanvikalpa.blogspot.com/2021/06/time-value-of-money-solved-problems-pdf.html
Leverage Analysis
https://gyanvikalpa.blogspot.com/2021/08/financial-management-notes-leverage.html
Cost of Capital
https://gyanvikalpa.blogspot.com/2021/08/cost-of-capital-solved-problems.html
EBIT – EPS Analysis
https://gyanvikalpa.blogspot.com/2021/08/ebit-eps-analysis-financial-break-even.html
Capital Structure Analysis
https://gyanvikalpa.blogspot.com/2022/02/capital-structure-theories-and-solved.html
Estimation of Cash Flow in Capital Budgeting
https://gyanvikalpa.blogspot.com/2021/06/cash-flow-estimation-in-capital.html
Techniques of Capital Budgeting
https://gyanvikalpa.blogspot.com/2021/06/techniques-of-capital-budgeting-solved.html
