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Unformatted text preview: price of the bond. The contribution of each payment to the duration of the bond is the relative value of the payment multiplied by the amount of time (in years) until the payment occurs. So, the duration of the bond is: Year 1 2 3 Price of bond PV of payment $74.07 68.59 857.34 $1,000 Relative value 0.07407 0.06859 0.85734 Duration = Payment weight 0.07407 0.13717 2.57202 2.78326 7. The duration of a bond is the average time to payment of the bond’s cash flows, weighted by the ratio of the present value of each payment to the price of the bond. Since the bond is selling at par, the market interest rate must equal 8 percent, the annual coupon rate on the bond. The price of a bond selling at par is equal to its face value. Therefore, the price of this bond is $1,000. The relative value of each payment is the present value of the payment divided by the price of the bond. The contribution of each payment to the duration of the bond is the relative value of the payment multiplied by the amount of time (in years) until the payment occurs. So, the duration of the bond is: Year 1 2 3 4 Price of bond PV of payment $74.07 68.59 63.51 793.83 $1,000 Relative value 0.07407 0.06859 0.06351 0.79383 Duration = Payment weight 0.07407 0.13717 0.19052 3.17533 3.57710 8. The duration of a portfolio of assets or liabilities is the weighted average of the duration of the portfolio’s individual items, weighted by their relative market values. a. The total market value of assets in millions is: Market value of assets = $31 + 630 + 390 + 98 + 346 Market value of assets = $1,495 493 So, the market value weight of each asset is: Federal funds deposits = $31 / $1,495 = 0.021 Accounts receivable = $630 / $1,495 = 0.421 Short-term loans = $390 / $1,495 = 0.261 Long-term loans = $98 / $1,495 = 0.066 Mortgages = $346 / $1,495 = 0.231 Since the duration of a group of assets is the weighted average of the durations of each individual asset in the group, the duration of assets is: Duration of assets = 0.021(0) + 0.421(0.20) + 0.261(0.65) + 0.066(5.25) + 0.231(12.85) Duration of assets = 3.57 years b. The total market value of liabilities in millions is: Market value of liabilities = $585 + 310 + 305 Market value of liabilities = $1,200 Note that equity is not included in this calculation since it is not a liability. So, the market value weight of each asset is: Checking and savings deposits = $585 / $1,200 = 0.488 Certificates of deposit = $310 / $1,200 = 0.258 Long-term financing = $305 / $1,200 = 0.254 Since the duration of a group of liabilities is the weighted average of the durations of each individual asset in the group, the duration of liabilities is: Duration of liabilities = 0.488(0) + 0.258(1.60) + 0.254(9.80) Duration of liabilities = 2.90 years c. Since the duration of assets does not equal the duration of its liabilities, the bank is not immune from interest rate risk. Intermediate 9. a. You’re concerned about a rise in corn prices, so you would buy May contracts. Since each contract is for 5,000 bushels, the number of contracts you would need to buy is: Number of contracts to buy = 140,000/5,000 = 28 By doing so, you’re effectively locking in the settle price in May, 2009 of $3.76 per bushel of corn, or: Total price for 140,000 bushels = 28($3.76)(5,000) = $526,400 494 b. If the price of corn at expiration is $3.92 per bushel, the value of you futures position is: Value of futures position = ($3.92 per bu.)(5,000 bu. per contract)(28 contracts) = $548,800 Ignoring any transaction costs, your gain on the futures position will be: Gain = $548,800 – 526,400 = $22,400 While the price of the corn your firm needs has become $22,400 more expensive since February, your profit from the futures position has netted out this higher cost. 10. a. XYZ has a comparative advantage relative to ABC in borrowing at fixed interest rates, while ABC has a comparative advantage relative to XYZ in borrowing at floating interest rates. Since the spread between ABC and XYZ’s fixed rate costs is only 1%, while their differential is 2% in floating rate markets, there is an opportunity for a 3% total gain by entering into a fixed for floating rate swap agreement. If the swap dealer must capture 2% of the available gain, there is 1% left for ABC and XYZ. Any division of that gain is feasible; in an actual swap deal, the divisions would probably be negotiated by the dealer. One possible combination is ½% for ABC and ½% for XYZ:
10.5% ABC LIBOR +1% LIBOR +1% Debt Market Dealer LIBOR +2.5% +2.5% 10.0% XYZ b. 10% Debt Market 11. The duration of a liability is the average time to payment of the cash flows required to retire the liability, weighted by the ratio of the present value of each payment to the present value of all payments related to the liability. In order to determine the duration of a liability, first calculate the present value of all the payments required to retire it. Since the cost is $30,000 at the beginning of each year for four years, we can find the present value of each payment using the PV equation: PV = FV / (1 + R)t So, the PV each year of college is: Year 1 PV = $30,000 / (1.09)3 = $23,165.50 Year 2 PV = $30,000 / (1.09)4 = $21,252.76 Year 3 PV = $30,000 / (1.09)5 = $19,497.94 Year 4 PV = $30,000 / (1.09)6 = $17,888.02 495 So, the total PV of...
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