# Brealy Chapter 23

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1. Downward sloping. This is because high coupon bonds provide a greater proportion of their cash flows in the early years. In essence, a high coupon bond is a ‘shorter’ bond than a low coupon bond of the same maturity.

2. The key here is to find a combination of these two bonds (i.e., a portfolio of bonds) that has a cash flow only at t = 6. Then, knowing the price of the portfolio and the cash flow at t = 6, we can calculate the 6-year spot rate.
We begin by specifying the cash flows of each bond and using these and their yields to calculate their current prices:
Investment Yield C1 . . . C5 C6 Price
6% bond 12% 60 . . . 60 1,060 \$753.32
10% bond 8% 100 . . . 100 1,100 \$1,092.46 From the cash flows in years one through five, it is clear that the required portfolio consists of one 6% bond minus 60% of one 10% bond, i.e., we should buy the equivalent of one 6% bond and sell the equivalent of 60% of one 10% bond. This portfolio costs:
\$753.32 – (0.6  \$1,092.46) = \$97.84
The cash flow for this portfolio is equal to zero for years one through five and, for year 6, is equal to: \$1,060 – (0.6  1,100) = \$400
Thus:
\$97.84  (1 + r6)6 = 400 r6 = 0.2645 = 26.45%

3. Using the general relationship between spot and forward rates, we have:
(1 + r2)2 = (1 + r1)  (1 + f2) = 1.0600  1.0640  r2 = 0.0620 = 6.20%
(1 + r3)3 = (1 + r2)2  (1 + f3) = (1.0620)2  1.0710  r3 = 0.0650 = 6.50%
(1 + r4)4 = (1 + r3)3  (1 + f4) = (1.0650)3  1.0730  r4 = 0.0670 = 6.70%
(1 + r5)5 = (1 + r4)4  (1 + f5) = (1.0670)4  1.0820  r5 = 0.0700 = 7.00% If the expectations hypothesis holds, we can infer—from the fact that the forward rates are increasing—that spot interest rates are expected to increase in the future.

4. In order to lock in the currently existing forward rate for year five (f5), the firm should:
…...

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