Question

What is wrong with the following example: Assume a Modigliani Miller (without taxes) world. An unlevered firm is worth$1000. Another firm with $500 debt has the same value, $1000. The risk-free rate is 5%. The income of both firms is $100.

According to Modigliani and Miller, capital structure does not matter in this case, hence there should be no arbitrage opportunities under such circumstances.

However, here is one: You own 10% of the all equity firm, and obviously, your investment entitles you to 10% of the profit, i.e. $10. You sell your share in the unlevered company and have $100 to invest. You now buy 20% of the equity of the levered firm ($500 equity, $500 debt). That firm's income is:$100 -0.05 x $500(interest) = $75. Since you own 20% of the firm, you receive -0.2 x $75 =$15. (hint and request -it is a long question, but the answer should be short. You need no more than two sentences).

Answer 1

Modigliani and Miller Theorem's Proposition 1 talks about the Value of firm remaining same irrespective of Capital structure. This means Value of all equity financed firm will be same as a Levered firm.

In our case, we are talking about difference in equity cashflows from an all equity financed and a levered firm. If we simply look at the cashflows to the investors, including equity holders and debtholders, the firm's value will be same as both all equity financed and a levered firm would yield $20 cashflow to the firm for 20% of the firm. In case of all equity financed, the whole cashflow goes to equityholders, whereas, in case of Levered firm, the cashflow for equityholders, for same amount of investment, is higher because of additional risk that comes with leverage which is reflected in the company's cost of equity.

If we look at Modigliani Miller's Proposition 2, the Cost of equity of the firm increases with as Leverage of the firm increases.

For example: If Cost of equity of all-equity financed firm is 10%, then value of $10 cashflow for the investor is $100.

However, with Debt to equity of 1 and cost of debt of 5%, Cost of equity of a Levered firm would be 10% + 1*(10% - 5%) = 15%, due to probability of default that gets added to firm with leverage which keeps the value of $15 cashflow for the equityholder at $100.

Even though, it may seem like an Arbitrage opportunity, however, the Increased risk reflected in increased cost of equity of a Levered firm keeps the value of its cashflow same for equityholders as it were in case of an all equity financed firm.