## B.A. 130 Fall 1996 Midterm 2: Sketches of Answers to Parts B and C

Part B (25 minutes): Values. Show your work.

1. Suppose the risk-free rate r*f is 2% per year and the market portfolio expected return r*m is 8% per year. Suppose the standard deviation of the market portfolio's annual return is 30%, and the covariance of the annual return of security i with the market portfolio's annual return is 0.18. Suppose, further, that the expected cash flow from security i is \$14 each year, forever. What is the current market price of security i?

• Dividing the covariance by the variance of the market portfolio tells you that security i has a beta of two, and a required rate of return of 14% per year. Dividing the \$14 per year cash flow by 14% produces a current market price of \$100.

2. Suppose that it costs \$100,000,000 to construct and equip and \$30,000,000 each year to run a factory that makes 100,000 virtual reality home entertainment systems. If a factory (once built) is expected to last forever, if the appropriate discount rate is ten percent per year, and if the price of these virtual reality home entertainment systems is expected to remain constant each year, at what price for the product (the virtual reality home entertainment system) will building a factory be a zero net-present-value investment opportunity? If the price is higher, would it be profitable to build such a factory? If the price is lower, would it be profitable to build such a factory?

• The full annual equivalent cost of the factory is \$10,000,000 a year return owed on capital plus \$30,000,000 a year operating cost. So the factory is break-even in present value terms if its revenues are \$40,000,000 a year--or \$400 per VR system. If the price is higher, it is worth building. If the price is lower, it is not profitable.

3. Suppose that the standard deviation of the annual return to the Gamma Corporation's assets is 15% per year, that the covariance of the Gamma Corporation's annual return with the market portfolio is .00, and that the standard deviation of the return on the market portfolio is 30% per year. Suppose, further, that half of the market value of the Gamma Corporation is in the form of bonds; half is held in the form of common stocks; and that the Gamma Corporation's bonds have a beta of 0.0.

What is the beta of the Gamma Corporation's common stock?

• Because the covariance is zero, Gamma has a beta of zero (it has lots of idiosyncratic risk, but no systematic risk associated with it. With the company's assets having a beta of zero, and its bonds having a beta of zero, its stock must have a beta of zero as well.

4. Suppose that the High-Tech Startup Company has come to you, a venture capitalist, asking for start-up funding. They project--and you believe them: their internal estimates agree with yours--that there is one chance in a hundred that in one year they will be able to make an initial public offering [IPO] at a price that will value their entire company at \$525,000,000; that whether that one chance in a hundred will pay off is unconnected with the overall state of the market; and that once the IPO is completed the company will have a beta of two.

If this one in a hundred chance does not come off, in a year all investments in the company will be completely worthless. Furthermore, the risk-free rate is 5% per year, and that the market rate is 10% per year.

The founders of the High-Tech Startup Company offer to sell you ownership of half the company for \$2,480,000. This is their only and final offer. Should you accept the deal? How much profit do you expect to make on this transaction?

• "Unconnected with the overall state of the market" means that the beta associated with the risk of whether this company will pan out is zero. Hence you should use the risk-free rate to discount the one-in-a-hundred chance of success--giving the firm a present value today of \$5,000,000. The founders are offering you half the firm for \$2,480,000; this seems to be a good deal--a positive net present value of \$20,000.

Part C (25 minutes): Capital Structure. Show your work.

1. What is the present value of the interest tax shield generated by:

• A \$10,000,000, one-year loan at 6% per year?
• The tax shield saves you takes of \$150,000 next year, so divide \$150,000 by 1.06 to get 141502.
• A \$100,000,000 perpetual loan at 8% per year?
• The tax shield saves you \$2,000,000 a year in taxes--for a present value total of \$25 million (note that this is equal to the tax rate times the value of the loan).

Consider corporate income taxes only, and assume that the marginal tax rate is 25%.

2. What is the tax advantage of corporate debt if (a) the corporate tax rate is 35% and the personal tax rate is 31%, and (b) the average investor (like me) holds all securities in tax-favored 401(k), 403(b)(7), and other such accounts, and thus anticipates paying personal taxes at a marginal rate of not 31% but 5%?

• (a) 35% x 69% = 24%--the difference between sending money out of the corporation through the dividend channel, where it gets hit twice, and the interest channel. An answer saying that the edge is less because investors get to choose when to realize capital gains is fine.
• (b) If the marginal tax rate is 5% for all asset earnings, then the tax advantage of corporate debt is some 33%--the extra money you get by not exposing your operating earnings ot the corporate tax system.

3 [counts double]. The accepted theory of firm capital structure holds that businesses try (a) to save as much money as possible by reducing their taxes by using debt to generate interest tax shields, while balancing this off against (b) the possible costs of financial distress that may arise if the firm finds itself without the money to meet its debt payments. What are these costs of financial distress? What does this accepted theory predict about which kinds of businesses should have high debt-equity ratios and which kinds of businesses should have low debt-equity ratios? Does the theory fit the facts about which corporations borrow?

• Costs of financial distress: bankruptcy costs, lawyer costs, distracted managerial attention as financial maneuvering becomes more important, agency costs; businesses with secure operating earnings that face little risk of seeing their cash flow dry up (or businesses that can quickly sell their assets to raise cash) should have relatively high debt-equity ratios; businesses with insecure operating earnings should have relatively low debt-equity ratios. I think the theory--while not perfect--does pretty well.
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