1. The annual rate of growth g works out to be 5.2 percent:

2. a. The price of the bond with the 4 percent coupon could be sufficiently low to make it more attractive.

   b. The yield to maturity y is given by

   The solution works out to be y = 0.0473 (4.73 percent).

3. a. Because the yield to maturity is equal to the coupon rate, it sold for par.

   b. The total return is the 7.5% coupon plus the capital gain (or loss):

7.5% + x% = –4.19% implies x = –11.69%
7.5% + x% = 23% implies x = +15.5%

  The duration is evidently between 11.69 and 15.5 years. (Because of the nonlinearity, the actual percentage changes are not exactly equal to each other or to the duration.) Halfway between is (11.69 + 15.5)/2 = 13.6. (The actual duration works out to be 13.8 years.) Alternatively, we can divide the difference in the percentage returns by the two percentage-point difference in the yield to maturity: (4.19 + 23)/2 = 13.6.

4. Long-term interest rates must have fallen. [Robert McGough and Jonathan Clements, "Zero-Coupon Returns Lead Bond Funds With Average 7.05% Gain in 1st Quarter," The Wall Street Journal, April 6, 1993.]

5. a. The Journal explains: "if cash dries up, companies can halt a repurchase program easily and more quietly than they can cut a dividend, which often results in much publicity and a big drop in stock price. What's more, dividends are taxable; a successful buyback program creates capital gains taxed at a lower rate." [I expected the second explanation on this test.]

   b. The former may have been abandoning insufficiently profitable current operations or planned investment (an important economic event if r < R), while the latter were dispersing surplus cash (a financial nonevent). For example, Sun Microsystems financed its stock repurchases with $1.1 billion cash that it had accumulated.

6. a. The monthly payments are lower on the 30-year mortgage, so that the loan is not paid off as quickly.

   b. The 30-year mortgage has the higher unpaid balance after 5 years, because the monthly payments are lower so that the loan is not repaid as quickly.

   c. The 30-year mortgage has the higher total payments because the funds are borrowed for a longer period of time and, consequently, more interest must be paid.

   d. For either loan, no matter when the mortgage is paid off, the present value of the payments at a 7 percent required return (the interest rate on the loan) is equal to the amount borrowed.

   e. In comparison with the 15-year mortgage, the 30-year mortgage has lower monthly payments and a higher unpaid balance after 5 years; it therefore has the longer duration. An increase in the required return will reduce the present value of the loan with the longer duration, here the 30-year mortgage. Thus the 15-year mortgage has the higher present value.

7. In the short run, the increase in interest rates gives the investor capital losses (answer A). As time passes, the extra income on the reinvested coupons boosts the return (answer B). For a horizon equal to the duration of the bond (here, about 7 1/2 years), the total return--reinvested income plus capital gain or loss--is (approximately ) unaffected by changes in interest rates.

8. If the term structure flattens, the yield spread will narrow so that the yield on 20-year zeros falls (and the price rises) relative to 1-year zeros. To profit from such a flattening of the term structure, the investment bank should buy 20-year zeros and sell (twenty times as many) 1-year zeros. By purchasing the 20-year zeros and selling the 1-year zeros, instead of just buying the 20-year zeros, the investment bank will profit from a flattening of the term structure even if the overall level of interest rates rises or falls.
   Suppose, for example, that the 20-year rate goes up 1 percentage point while the 1-year rate goes up 2 percentage points. The bank will lose money if had simply bought 20-year zeros, but make money if it bought 20-year zeros and sold 1-year zeros.

9. An investor who puts up $250 to purchase $1,000 worth of stock has 4:1 leverage:

10. The Lynch ratio is reasonable because a high growth rate, a high dividend yield, and a low P/E are all attractive to investors. The ratio is not compelling because the required rate of return is ignored and, depending on risk and the level of interest rates, the required return is sometimes much higher than at other times. Consider the constant-dividend-growth model and suppose, that either d = 1 or p = R, so that E/P = R. The Lynch ratio is 100(D/P + g)/(P/E) = 100R/(1/R) = 100R^2, which can be larger or smaller than 1, depending on shareholders' required return R. Consider this stock, which is fairly priced at either a 10% or 20% required returns, but has very different Lynch ratios: