1. Davis was paid more in real terms, since his salary is more than twenty times Ruth's salary and prices increased by only a factor of less than 10. More formally, real income is calculated by deflating nominal income by the price level:

Babe Ruth $80,000/15.2 = 5,263
Chili Davis: $1.75 million/147 = 11,905

   A 1993 salary X with the same purchasing power as Ruth's 1931 salary, X/147 = $80,000/15.2 implies X = $80.000(147/15.2) = $773,684.

2. The monthly payments X (that begin after six months) are determined by equating the unpaid balance to the present value of the 24 monthly payments:

   (The solution is X = $49.92.) The effective annual interest rate R equates the present value of the X = $49.92 payments to the initial loan (remembering that the first payment is not due for 7 months):

   The solution, by trial and error, is R = 11.93 percent.

3. For a 10-year $100,000 monthly amortized loan at 10% + 7% = 17%, the monthly payments are determined by this present value equation:

   The solution works out to be X = $1,737.98. The borrower only receives $100,000 - 7%($100,000) = $93,000 and $1,737.98 monthly payments on a $93,000 loan implies an effective annual interest rate of 19.03%, determined by this equation:

   With a 5-year loan, the initial 7% fees become more burdensome because they are spread over a shorter loan, raising the effective interest rate (to 20.41%).

4. An increase in the cash flow increases the price. Because the bond has a duration of two years, the addition of cash flow two years from now does not affect the duration, which measures the average wait until receiving the cash flow.

5. 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 1 year, because the monthly payments are lower so that the loan is not repaid as quickly. (The unpaid balance on the 30-year mortgage works out to be $148,975.25; the unpaid balance on the 15-year mortgage is $145,041.99.)
   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 9 percent required return (the interest rate on the loan) is equal to the amount borrowed: $150,000.

6. Security 2 offer the higher expected return at all possible standard deviations:

   According to CAPM, Security 2 has a higher beta coefficient than Security 1 (because it has the higher expected return).

7. If a bond's coupons are larger than its yield to maturity, then the bond sells for a premium over its face (or maturation) value. The generous coupons make up for the high price.

8. Using the constant-dividend-growth model's equation R = D/P + g, for two stocks to have comparable rates of return, the one with the higher anticipated dividend growth rate must be priced to have a lower dividend yield.

9. The market wouldn't be efficient if experts agreed that stock prices were soon going to rise or fall, because then stocks would clearly be mispriced.

10. This bank has more pound-denominated liabilities than assets. If the dollar appreciates relative to the pound, the dollar value of its pound-denominated assets declines by less than the dollar value of its pound-denominated liabilities, giving the bank capital gains. For example, if the value of the dollar increases by 50 percent, from $1.80 per pound to $0.90 per pound (and the mark also appreciates by 50 percent relative to the pound, leaving the dollar price of marks at $0.60), the revised balance sheet shows a $9 million increase in net worth:

   b. To reduce the exposure of its net worth to fluctuations in the value of the mark relative to the pound and the dollar, this bank needs a closer match between the size of its mark-denominated assets and liabilities. It should consequently swap some of its pound-denominated liabilities for mark-denominated liabilities.

11. a. The bank's degree of leverage is 12.5:1,

   b. A comparison of asset and liability durations shows that the present value of this bank's assets are somewhat more sensitive to interest rates than are this bank's liabilities. Therefore, a one-percentage-point increase in interest rates will reduce the present value of this bank's assets more than it reduces the present value of its liabilities, reducing the bank's net worth.
   We can confirm this reasoning by examining the bank's balance sheet after a one-percentage-point increase in interest rates. The value of the 6-year zeros on the asset side of the bank's balance sheet falls to $57.13/1.09^6 = $34.06, and the value of the bank's 2-year zeros liability falls to $35.96/1.07^2 = $31.40. Therefore, the bank's balance sheet after the one-percentage-point increase in interest rates is as follows:

   Net worth falls by $8.00 million - $6.66 million = $1.34 million.
   c. The bank can trade 6-year zeros for 1-year zeros in order to reduce its asset duration to match its liability duration better.

12. Alternatively, if the bank keeps its portfolio with a positive duration gap, its net worth will decline if interest rates interest unexpectedly. To hedge against this possibility, the bank can buy or sell financial options and futures that will yield capital gains if interest rates increase (and bond prices decline). Therefore, it should buy bond puts (which become more valuable if bond prices decline), sell bond calls (which become less valuable if bond prices decline), and sell bond futures (which become less valuable if bond prices decline).
   Another possibility is to use interest-rate swaps, swapping some of the interest the bank pays on its variable-rate deposits for longer-term fixed-rate interest payments.

13. When the term structure is upward sloping, low-coupon securities should be priced to have higher yields to maturity than high-coupon securities with the same maturity. If the spread described in the exercise is inexplicably large, then the 4-percent-coupon bonds have temptingly high yields and low prices. By purchasing 20-year Treasury bonds with 4 percent coupons and selling 20-year Treasury bonds with 8 percent coupons, the security dealer bets on the spread narrowing, regardless of whether the general level of interest rates goes up or down.

14. Let P be the price of Waste Management stock on the expiration date in January of 1994. The dollar profit from strategy (a) is 100(P - $35 5/8). For strategy (b), the Treasury zeros will be worth $3,562.50/(1 + 0.0485)^(17/12) = $3,839.91, for a profit of $3,839.91 - $3,562.50 = $277.41, no matter what the price of Waste Management stock.
   Each call option purchased in strategy (c) will be worth P - $40 if P is above $40, and $0 otherwise.Therefore the dollar profit from 100 call options purchased for $387.50 will be

   The Treasury notes bought in strategy (c) will be worth $3,175.00/(1 + 0.0485)^(17/12) = $3,422.24, for a dollar profit of $3,422.24 - $3,175.00 = $247.24. The total dollar profit from strategy (c) is consequently

   Here is a graph comparing the dollar profits from the three strategies:

   Buying stock is the most profitable strategy if the price of Waste Management stock on the expiration date is above the striking price of $40. The option strategy provides a cushion if the price falls drastically, but is less profitable than buying stock if the price of Waste Management stock on the expiration date is above $34.22:

   The option strategy is less profitable than the Treasury-note strategy if the price of Waste Management stock on the expiration date is less than $44.18:

15. This quotation makes it seem that the beta coefficient is the ratio of two variances. It is instead a measure of the amount by which the return on the stock tends to increase when the market return increases by 1 percentage point. (Formally, it is the ratio of the covariance between Ri and RM to the variance of RM.) A beta of 1 indicates that a 1 percentage point increase in the market return increases the expected value of the stock return by 1 percent.

16. Duration is an estimate of the percentage change in the price resulting from a one-percentage point change in the yield. The duration of a zero is equal to its maturity (here 40 years), implying a 40% price change. The actual percentage changes are:

17. a. Let PC be the probability that Bill Clinton becomes the Democratic nominee. On the day that the Democratic National Convention determines the party's nominee, the holder of a Clinton futures share purchased for 45.5 cents pays 45.5 cents and either receives $1 (a net gain of 54.5 cents) or receives nothing (a net loss of 45.5 cents). Thus the expected return is

   For the expected return to be positive, a person who pays 45.5 cents for a Clinton futures must believe that Clinton's probability of winning is larger than 0.455:

   b. The prices of stock and bond futures depend on the cost of carry because it is possible to arbitrage between the spot price and the futures price by buying the item and selling futures contracts for the item, or vice versa. The prices of presidential futures depend on expectations rather than the cost of carry, because (unlike stock or bond futures) there is nothing with a spot price that can be bought or sold to arbitrage against presidential futures.
   c. The Clinton calculations demonstrated that the purchase of a nomination futures contract has a zero expected return if the price of the futures is equal to the candidate's probability of winning the nomination. Because the six types of shares traded (including rest of field") take into account all of the possible nominees, the probabilities must add to one and it seems reasonable that the prices of these shares (which reflect the probabilities) will add to one dollar as well.
   In fact, arbitrage enforces this condition. Suppose that the prices were as follows, adding to more than $1:

18. Principal risk (or capital risk) refers to fluctuations in the market price of a bond as interest rates fluctuate. An increase in interest rates reduces bond prices. Income risk (or reinvestment risk) refers to the effects of fluctuating interest rates on the income earned when the coupons and maturation value are reinvested. An increase in interest rates increases the income from reinvested coupons and maturation values.
   Bonds with long maturities (actually long durations) have relatively more principal risk as their prices are more sensitive to interest rates. Bonds when short maturities have more income risk as their maturation values will soon need to be reinvested.

19. This analysis is very similar to the discussion of immunization in Chapter 5 of the textbook.
   a. For a 25-year horizon, the reinvestment risk is most important (the value of the bond is simply equal to its maturation value), and the terminal value is largest if interest rates rise to 10 percent.
   b. For a 1-year horizon, there has been almost no reinvestment of coupons and the capital risk is most important. The terminal value is largest if interest rates fall to 6 percent.
   c. The terminal value is (approximately) the same, regardless of whether interest rates go up or down (by small amounts) for a horizon that is equal to the duration of the bond. (The duration of a 25-year 8-percent bond is 11.2 years and, as expected, Bogle's graph shows the three terminal values to be equal at this point.)

20. a. U.S. Treasury bonds have the least quality risk because the federal government has the power to tax and the Federal Reserve has the power to print money and buy Treasury securities if necessary.
   b. Prepayment risk refers to the fact that bonds can be called and the mortgages that underlie GNMA and other pass-through mortgage securities can be prepaid. Bonds are more likely to be called and mortgages are more likely to be prepaid when interest rates decline and these borrowings can be refinanced at lower interest rates. Another way of seeing this is that when interest rates decline, the market value of these liabilities is larger than the cost of calling or prepaying them. Investors lose for the very same reasons: they must sell their assets for less than their market value and they must now reinvest at low interest rates.
   c. Currency risk refers to the fact that the dollar value of securities denominated in foreign currencies fluctuates with the foreign exchange value of the dollar. An investor's dollar return on foreign-denominated securities declines when the dollar appreciates (and foreign currencies depreciate relative to the dollar).
   d. Municipal bonds have the most favorable tax treatment, in that their interest is generally exempt from both federal income taxes and the appropriate state and local income taxes.