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M08_MCDO8122_01_ISM_C08 - Chapter 8 Swaps Question 8.1 We...

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Chapter 8 Swaps Question 8.1 We first solve for the present value of the cost per two barrels: 2 $22 $23 41.033. 1.06 (1.065) + = We then obtain the swap price per barrel by solving: 2 41.033 (1.065) 1.06 22.483, x x x + = = which was to be shown. Question 8.2 1. We first solve for the present value of the cost per three barrels, based on the forward prices: 2 3 $20 $21 $22 55.3413. 1.06 (1.065) (1.07) + + = Hence we could spend $55.3413 today to receive 1 barrel in each of the next three years. We then obtain the swap price per barrel by solving: 2 3 55.3413 1.06 (1.065) (1.07) 20.9519 x x x x + + = = 2. We first solve for the present value of the cost per two barrels (Year 2 and Year 3): 2 3 $21 $22 36.473. (1.065) (1.07) + = Hence we could spend $36.473 today and receive 1 barrel of oil in Year 2 and Year 3. We obtain the swap price per barrel by finding two equal payments we would make in Years 2 and 3 that have the same present value: 2 3 36.473 (1.065) (1.07) 21.481 x x x + = =
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94 McDonald • Fundamentals of Derivatives Markets Question 8.3 Since the dealer is paying fixed and receiving floating, each year she has a cash flow $20.9519. T S She can hedge this risk by selling 1 barrel forward (i.e., short one forward) in each of the three years. Her payoffs from the swap, the short forward contracts, and the net are summarized in the following table: Year Net Swap Payment Short Forwards Net Position 1 1 $20.9519 S 1 $20 S 0.9519 2 2 $20.9519 S 2 $21 S + 0.0481 3 3 $20.9519 S 3 $22 S + 1.0481 We need to discount the net cash flows to year zero. We have: 2 3 0.9519 0.0481 1.0481 PV(net CF) 0. 1.06 (1.065) (1.07) = + + = Indeed, the present value of the net cash flow is zero. Question 8.4 The fair swap rate was determined to be $20.9519. Therefore, compared to the forward curve price of $20 in one year, we are overpaying $0.9519. In year two, with interest, this overpayment increases to $0.9519 1.070024 $1.01853, × = where we used the appropriate forward rate to calculate the interest payment. In year two, we underpay by $0.0481, so that our total accumulative underpayment is $1.01856 $0.0481 = $0.97042. In year three, using the appropriate 1-year forward rate of 8.007%, this net overpayment increases to $0.97046 1.08007 $1.0481. × = However, in year three, we receive a fixed payment of 20.9519, which underpays relative to the forward curve price of $22 by $22 $20.9519 $1.0481. = Therefore, our cumulative balance is indeed zero, which was to be shown. Question 8.5 Question 8.3 showed that, at the initial yield curve, the swap has a zero present value. To look at how the yield curve affects the value of the dealer’s swap position, we repeat the hedge and then look at the present value of the hedged cash flows under the new yield curve.
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