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Unformatted text preview: the college cost is: PV of college = $23,165.50 + 21,252.76 + 19,497.94 + 17,888.02 PV of college = $81,804.22 Now, we can set up the following table to calculate the liability’s duration. The relative value of each payment is the present value of the payment divided by the present value of the entire liability. The contribution of each payment to the duration of the entire liability is the relative value of the payment multiplied by the amount of time (in years) until the payment occurs. Year 7 8 9 10 PV of college PV of payment $23,165.50 21,252.76 19,497.94 17,888.02 $81,804.22 Relative value 0.28318 0.25980 0.23835 0.21867 Payment weight 0.84955 1.03920 1.19174 1.31201 4.39250 Duration = 12. 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. We need to find the present value of the bond’s payments at the market rate. 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. Since this bond has semiannual coupons, the years will include halfyears. So, the duration of the bond is: Year 0.5 1.0 1.5 2.0 Price of bond PV of payment $28.71 27.47 26.29 863.72 $946.19 Relative value 0.03034 0.02903 0.02778 0.91284 Duration = Payment weight 0.01517 0.02903 0.04168 1.82568 1.91156 13. Let R equal the interest rate change between the initiation of the contract and the delivery of the asset. Cash flows from Strategy 1: Today Purchase silver Borrow Total cash flow Cash flows from Strategy 2: Purchase silver Total cash flow Today 0 0 1 Year –F –F –S0 +S0 0 1 Year 0 –S0(1 + R) –S0(1 + R) 496 Notice that each strategy results in the ownership of silver in one year for no cash outflow today. Since the payoffs from both the strategies are identical, the two strategies must cost the same in order to preclude arbitrage. The forward price (F) of a contract on an asset with no carrying costs or convenience value equals the current spot price of the asset (S0) multiplied by 1 plus the appropriate interest rate change between the initiation of the contract and the delivery date of the asset. 14. a. The forward price of an asset with no carrying costs or convenience value is: Forward price = S0(1 + R) Since you will receive the bond’s face value of $1,000 in 11 years and the 11 year spot interest rate is currently 9 percent, the current price of the bond is: Current bond price = $1,000 / (1.09)11 Current bond price = $387.53 Since the forward contract defers delivery of the bond for one year, the appropriate interest rate to use in the forward pricing equation is the oneyear spot interest rate of 6 percent: Forward price = $387.53(1.06) Forward price = $410.78 b. If both the 1year and 11year spot interest rates unexpectedly shift downward by 2 percent, the appropriate interest rates to use when pricing the bond is 7 percent, and the appropriate interest rate to use in the forward pricing equation is 4 percent. Given these changes, the new price of the bond will be: New bond price = $1,000 / (1.07)11 New bond price = $475.09 And the new forward price of the contract is: Forward price = $475.09(1.04) Forward price = $494.10 15. a. The forward price of an asset with no carrying costs or convenience value is: Forward price = S0(1 + R) Since you will receive the bond’s face value of $1,000 in 18 months, we can find the price of the bond today, which will be: Current bond price = $1,000 / (1.0473)3/2 Current bond price = $933.03 497 Since the forward contract defers delivery of the bond for six months, the appropriate interest rate to use in the forward pricing equation is the six month EAR, so the forward price will be: Forward price = $933.03(1.0361)1/2 Forward price = $949.72 b. It is important to remember that 100 basis points equals 1 percent and one basis point equals 0.01%. Therefore, if all rates increase by 30 basis points, each rate increases by 0.003. So, the new price of the bond today will be: New bond price = $1,000 / (1 + .0473 + .003)3/2 New bond price = $929.03 Since the forward contract defers delivery of the bond for six months, the appropriate interest rate to use in the forward pricing equation is the six month EAR, increased by the interest rate change. So the new forward price will be: Forward price = $929.03(1 + .0361 + .003)1/2 Forward price = $947.02 Challenge 16. The financial engineer can replicate the payoffs of owning a put option by selling a forward contract and buying a call. For example, suppose the forward contract has a settle price of $50 and the exercise price of the call is also $50. The payoffs below show that the position is the same as owning a put with an exercise price of $50: Price of coal: Value of call option position: Value of forward position: Total value: Value of put position: $40 0 10 $1...
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 Spring '10
 eshmalwi
 Finance, Corporate Finance

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