1. He has to wait 27 years for his $1 million and the present value, at a 10 percent required return, is $76,277:

2. If you pay $25 for a bond that will be worth $50 at maturity, the sooner you get your $50 the higher is your annual rate of return. In order to reduce the interest rate on a bond that sells for a fixed price and pays a fixed dollar amount, the maturity must be lengthened. The interest rates were reduced because market interest rates had declined. The federal government no longer had to pay a 7.5 percent interest rate to find buyers.

3. The calculation ignores the time value of money. At a 5% interest rate, the present value of the two options are the same; at a higher interest rate, the prepayment plan is inferior. In addition, the student may not stay at Rensselaer or four years.

4a. If a company uses half its assets to buy back half its shares, it is unlikely that its earnings will be unaffected. One good reason for not liquidating a company's assets is that these assets are sufficiently profitable (profit rate greater than shareholders' required return) so that shareholders prefer the company investing on their behalf to receiving funds that they can invest on their own.

4b. The conservation-of-value principle implies that a 1-for-2 stock split will, by halving the number of shares, double earnings per share and double the market value of each share; but since each shareholder will have half as many of these twice-as-valuable shares, they are not made any better (or worse off) by this purely cosmetic financial transaction.

5. This strategy is intended to take advantage of the steep term structure of interest rates that then prevailed. The expectations hypothesis explanation of an upward sloping term structure is that interest rates are expected to increase. If interest rates do increase by more than the implicit forward rates embedded in the term structure, the U.S. Treasury will be forced to roll over its short-term debt at sharply higher interest rates, and its borrowing costs will be even higher than had it issued long-term bonds.

6. Speculation. There is little or no income from owning works of art, and purchases "made with Japanese clients in mind" sounds very much like the greater fool theory.

7. Because a corporation's interest on its debt is tax-deductible, but its dividends are not, it has an incentive to distribute more of its profits in the form of interest; i.e., using debt as a tax shield. Leveraged buyouts are often motivated by this tax-shield advantage of debt. By treating interest and dividends comparably, this incentive is removed.

8. The after-tax cost (analogous to an after-tax return) is X = (1 - t)R = (1 - 0.31)8% = 5.52%. We can simply look at not having to pay x% interest on this loan as effectively saving (and thereby earning) an x% return. The general principle that one cannot make money borrowing at an after-tax interest rate of x% to invest at an after-tax return of less than x% also implies that The Wall Street Journal is correct.

9. As with long-term Treasury bonds, there is no cash-flow risk from paying down a fixed-rate mortgage: you can be absolutely certain about how much you save each month by not having to make the monthly payments. The monthly savings are analytically equivalent to receiving fixed monthly income from Treasury securities. As with the purchase of long-term Treasury securities, the repayment of a fixed-rate mortgage will turn out to be a financial mistake if interest rates rise. (The repayment of variable-rate mortgage is equivalent to rolling over Treasury bills, and involves the very same reinvestment risk; you will regret paying off the mortgage if interest rates decline.)

10. Investors certainly may want an alternative to 4 1/2 percent CDs, but there is no guarantee that stocks will give a better return. As explained in the answer to the preceding exercise, stocks are a long-duration investment with considerable capital risk and should be compared with long-term bonds. The 8.5 percent yields on long-term corporate bonds are more relevant than the 4.5 percent returns on 3-month Treasury bills.
   The constant-dividend-growth model can help us judge whether stocks with 3 percent dividend yields are inexpensive compared to long-term corporate bonds with 8.5 percent yields to maturity. Equation 9.2 from the textbook gives the present value P of a stock with constantly growing dividends as

   where D1 is next period's dividend, g is the rate of growth of dividends, and R is the shareholder's required rate of return. We can rearrange this equation as

   For shareholders to anticipate receiving their required rate of return R, stocks should be priced so that the dividend yield D1/P plus the rate of growth of dividends g is equal to R. In November of 1992, dividend yields averaged 3 percent and yields on high-grade corporate bonds averaged 8.5 percent. Suppose that investors required a 5 percent risk premium on corporate stocks relative to high-grade corporate bonds, so that their required rate of return on corporate stock was

   For a more realistic 5 percent long-run growth rate of dividends, shareholders must have only an 8.0 percent required return on stocks to be satisfied holding stocks with a 3 percent dividend yield:

     An 8.0 percent required returns on stocks seems implausible when investors could have obtained an 8.5 percent yield on long-term high-grade corporate bonds and a 7.0 percent yield on tax-free high-grade municipal bonds. By fundamental analysis, stocks were not cheap in November of 1991.