Problem 1 (Issuance and Repurchase in a Modigliani-Miller World)

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1 Problem 1 (Issuance and Repurchase in a Modigliani-Miller World) A rm has outstanding debt with a market value of $100 million. The rm also has 15 million shares outstanding with a market value of $10 per share. The CEO then announces that the rm will issue a further $60 million of debt and that proceeds will be used to buy back common stock. Debtholders, seeing the extra risk, mark the value of the existing debt down to $70 million. The M&M irrelevance theorem holds in this market. (a) How is the market value of the stock a ected by the announcement? (b) How many shares can the company buy back with the $60 million of new debt that it issues? (c) What is the market value of the rm (equity plus debt) after the change in capital structure? (d) What is the debt ratio after the change in structure? (e) Who (if anyone) gains or loses? Let s work through the rm s balance sheet as it rst issues new debt and then uses the proceeds to buy back common stock. Note though that we will be working with market instead of book values. Initially, the rm s balance sheet is Assets Liabilities Assets in Place 50 Debt 100 Equity 150 Total 50 Total 50 It has some assets in place, worth $50 million, and these are split between existing debt and equity as speci ed by the problem. Once the rm issues new debt but before it buys back common stock, its balance sheet becomes Assets Liabilities Assets in Place 50 Old Debt 70 Cash 60 New Debt 60 Equity 180 Total 310 Total 310 In addition to the existing assets in place, the rm now has $60 million in cash on the asset side of its balance sheet. Overall rm value is now $310 million. Note that this is not a violation of the Modigliani-Miller theorem, because the theorem says that it is the value of assets in place that is not a ected when the rm issues new debt. On the liabilities side we have $60 million in new debt. Old debt is now worth $70 million. As the problem states old debt is worth less because of its increased riskiness. Finally, since total liabilities have to equal total assets, equity must be worth $180 million. Another way to see this is that since new debt 1

2 is issued at a fair price, the value of shareholder equity has to increase by the same amount that the value of existing debt decreases. Since there are still 15 million shares outstanding after the new debt is issued, price per share is P = = $1: $60 million The company can buy back $1 = 5 million shares. Once it buys back 5 million share, the rm s balance sheet is Assets Liabilities Assets in Place 50 Old Debt 70 New Debt 60 Equity 10 Total 50 Total 50 Having used all of its new cash, the rm is now back to having only its existing assets on the asset side of its balance sheet. The values of old and new debt are same as before, and the value of equity is now $10 million, 10 million shares at $1 per share. The debt ratio is now Debt = = 0:5 Debt + Equity 50 Overall, existing creditors lose $30 million, while shareholders gain $30 million or $30 15 = $ per share. Problem (Agency Problems and Debt) There are two states of nature that occur with probability 50% each. The cash ows (F ) of the rm are $80 and $40 in each state. Assume that there are not taxes and that this is a risk neutral economy with interest rate equal to 0%. The rm lasts only for one period. Because of ine ciency or theft, the manager throws away half of the cash ows after paying the outstanding debt (D). This loss function (L) can be written as L = max 0; F D The cash ows that are left are distributed as dividends. (a) If the rm is nanced 100% with equity, what is the value of the rm today? If there is only equity, losses in the two states (call the states G and B) are and the value of the rm is L G = 40 L B = 0 V = 0:5 (80 L G ) + 0:5 (40 L B ) = 30

3 We can summarize this in our usual payo table, modi ed to show losses and actual cash ows, B G PV (b) What is the value of the rm if it has outstanding debt of $50 (promised amount)? The new payo table is B G PV Losses are now L G = 15 L B = 0 and the values of the rm (V ) ; its debt (V D ) and equity (V E ) are respectively V = 0:5 (80 L G ) + 0:5 (40 L B ) = 5:5 V D = 0: :5 40 = 45 V E = 0: :5 0 = 7:5 (c) Assume now that a new project is made available to the rm. The project reports cash ows of $0 in each state and requires an initial investment of $9. If the rm only considers equity nancing for this project, would the project be taken if the rm currently has debt for $50? The new payo table is B G PV Losses are now L G = max ; 0 = L B = max ; 0 = 5 3

4 and rm, debt, and equity values are V = 0:5 (100 L G ) + 0:5 (60 L B ) = 65 V D = 0: :5 50 = 50 V E = 0: :5 5 = 15 Since the rm value increases by (65 5:5) = 1: 5 while the project costs 9; NPV from the perspective of the rm is positive, and if the manager were to maximize rm value he would take the project. However, the NPV to equityholders is (15 7:5) 9 = 1:5 negative because 5 are e ectively transferred to creditors, whose debt becomes risk-free. manager maximizes equityholder s value, he will not accept the project. So, if the (d) Assume now that the rm has debt of $30 instead of $50. What is the value of the rm without the project mentioned in (c)? Would the rm take this project if it has the opportunity to invest in it? Before the project is undertaken, rm payo s are as follows B G PV With the new project B G PV Thus once again from the perspective of the rm overall, this is a positive NPV project since rm value increases by (55 45) = 10 while the project costs 9: If the manager were to maximize rm value he would take the project. However, now the project has positive NPV from the perspective of equityholders (5 15) 9 = 1 So, if the manager maximizes equityholder s value, he will accept the project. (e) What mechanisms described by the capital-structure literature are represented in this simple example? The problem shows two concepts related to agency problems. The rst one is the free cash problem: if it is easier for the manager to use cash for his own bene t as long as it is in the rm, the rm will gain by increasing the amount of debt to remove the free cash from the rm. 4

5 However, increasing debt causes a debt overhang problem, which discourages some positive NPV investments since the equityholders do not reap the full bene t of those projects (which are instead captured by the debtholders). Problem 3 (Equity Issues and Information) Consider a market with rms of high and low value of assets in place. All rms are nanced 100% with equity. The owners of each rm know the value of the assets in place, but the market is not sure. At t = 0, the market assigns a probability p to assets in place being of high value (V H = 100), and a probability 1 - p to assets in place being of low value (V L = 40). In the following period t = 1, a new investment opportunity appears and everyone knows about it. This new project yields a cash ow of $1 and requires an initial investment of $0.5. Therefore, and assuming that this is a risk-neutral economy with an interest rate of 0%, the NPV of the project is positive and equal to $0.5. In period t = 1 rms can issue equity to raise funds for the new project. (a) For what range of the probability p the rm with high value of assets in place will decide to issue equity and invest in the new project? Let s start with the pooling equilibrium. The value of equity in the pooling equilibrium is V = 100p + 40(1 p) + 1 The new equityholders are injecting 0:5 of capital into the rm, which means they are going to get a share of the rm equal to 0:5 = 100p + 40(1 p) + 1 so that the existing shareholders will get a share 1 = 1 0:5 100p + 40(1 p) + 1 The high-value rm is now worth So, the high-value shareholders, who know the true value of their claims is 100 without the project, will proceed with the investment only if (1 0:5 ) p + 40(1 p) + 1 ) p :83% (b) Explain why if p = 5% the rm with low value of assets in place decides to issue equity and invest in the new project. Since p = 5% < 15:83% there can be no pooling equilibrium as the high-value rms would not issue equity. Therefore, only the low-value rms will issue, be recognized as such, and invest in the positive NPV project. The reason the low-value rms want to issue and invest is that the market will eventually recognize them to be low-value. Since in the meantime there is no way for these rms to bene t from their temporary overvaluation, they might as well be recognized as low-value sooner rather than later, issue fairly priced equity and invest in the positive NPV project. Thus, existing owners of low-value rms get 40:5 if they issue and invest versus 40 if they don t issue. 5

6 (c) If p = 10%, what is the change in value between t = 0 and t = 1 for the rm that issues new equity? If p = 10%, again high-value rms do not issue equity. At t = 0 low-value rms were not recognized as such, and their value was: V LOW;0 = 100p + 40(1 p) = 46 Now, V LOW;1 = 41 Therefore their values declines by 5: Once again note that even though their value declines by 5 once they issue, the low-value rms do want to issue. Otherwise their value would have eventually declined by (46 40) = 6: Thus they are better o issuing and investing. Problem 4 (Payout Policy in a Modigliani-Miller World) A rm has 1 million shares outstanding with total market value of $0 million. The rm is expected to pay $1 million in dividends next year and thereafter the amount paid out is expected to grow by 5% per year in perpetuity. Thus, the expected dividend is $1:05 million in year, $1:105 million in year 3 and so on. However, the company has heard that the value of a share depends on the ow of dividends and announces that next year s dividend will be increased to $ million, and that the extra cash to pay the dividend will be raised by a simultaneous secondary equity o ering. After that, the total amount paid out each year will be as previously forecasted ( $1:05 million in year and increasing by 5% in each subsequent year). All of the Modigliani-Miller assumptions hold in this market. (a) At what price will the new shares be issued in year 1? At t = 0 each share is worth $0. This value is based on the expected stream of dividends: $1 at t = 1, and increasing by 5% in each subsequent year. Thus we can nd the appropriate discount rate for this company as follows P 0 = D 1 r g ) r = D 1 + g = 1 + 0:05 = 0:10 P 0 0 Beginning at t =, each share in the company will enjoy a perpetual stream of growing dividends: $1:05 at t =, and increasing by 5% in each subsequent year. Thus, the total value of the shares at t = 1 after the t = 1 dividend is paid and after N new shares have been issued is given by 1:05 V 1 = 0:1 0:05 = 1m If P 1 is the price per share at t = 1 then V 1 = P 1 (1m + N) = 1m 6

7 But since the new N shares were issued to raise $1 million, we know Therefore P 1 = $0 and N = 50; 000 shares. (b) How many shares will the rm need to issue? P 1 N = 1 With P 1 = $0 and $1 million to raise, the rm will sell 50; 000 new shares. (c) What will be the expected dividend payments on these new shares, and what will be paid out to the old shareholders after year 1? The expected dividends paid at t = are $1:05 million, increasing by 5% in each subsequent year. With 1:05 million shares outstanding, dividends per share are: $1 at t =, increasing by 5% in each subsequent year. Thus, total dividends paid to old shareholders are: $1 million at t =, increasing by 5% in each subsequent year. (d) Show that the present value of the cash ows to the current shareholders remains $0 million. Current shareholders receive the rst dividend of $ million at t = 1 and then from t = they get their $1 million that grows at 5% per year. Therefore, the present value is P V 0 = m 1: :1 1m = $0 million (0:1 0:05) 7

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