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Unformatted text preview: Problem 1) Part 1) You have to set money aside for two payments, one at the end of the first and one at the end of the second year. For the first year's payment you need the present value of $20,000 discounted seminannually at 8%: $18,491.12 This is just the present value of $20,000 discounted at 8% semiannually. For the second payment we use: 20,000 / (1.04)^4 = $17,096.08 Hence, the total amount you need is: $35,587.21 Part 2 -- Invest in 4 year Zero Coupon Bonds Since the value today is $35,587.21 then at 8% we need to buy a par amount equal to 35,587.21*1.04^8 = $48,703.55 We sell $20,000 worth of bonds at the end of the first year and make a second payment at the end of the second year. Hence, at the end of year 1 and after making the payment the amount left in the bond account is: Year 1=$48,703.55/(1+yield/2)^6-$20,000 The question assumes only a one-off change in yield, hence the value of the portfolio at the end of year 2 is (Year1)*((1+yield/2))^2. Deducting the 20,000 fee payment gives: Rate Year 1 Year 2 Year 4 7% 19,620.37 1,017.83 $48,703.55 Rates went down, we get extra cash 8% 18,491.12 0.00 $48,703.55 Rates stay constant, amount is right. 9% 17,399.25-999.58 $48,703.55 Rates went up, we need more cash Your bonds are too long, hence you are subject to price risk. Hence, if interest rates increase by 1%, you are $999.59 short, otherwise you have $1017.83 left over. You could also do the computation another way. Work out the par value of the bonds you need to sell at the end of year 1, then work out the remaining par value. Finally, discount this at the prevailing yield: Rate 7% 24,585.11 24,118.45 21,017.83 1,017.83 8% 25,306.38 23,397.17 20,000.00 0.00 9% 26,045.20 22,658.35 19,000.42-999.58 Discussion: Think of a time line with 8 periods: The intuition is that you need to save the present value of your future cash outlays: 20,000 in year 1 (period 2) and 20,000 in year 2 (period 4). When we invest in the 4 year bond we can compute the future value of the bond using the interest rate that we have locked in (8%). We own the bond today (time 0). One year from today we're going to need to sell a fraction of our bond holding to make the first payment, to figure out the price of our bond in year one, we must discount the final cash flow to the bond (48,703.55) back to year one (discount 6 periods). After paying out 20,000, we need to future value what's left over, one year (2 periods) ahead to the next payment of 20,000 in year 2. The interest rate change does not affect the future value of our 4 year bond. We have locked in 8% for four years by purchasing the bond. What the interest rate change does affect is the rate at which we discount the future cash flow to the bond. When the interest rate falls, we're happy because we have locked in a higher rate with the bond. So, we'll end up with a bit extra after our expenses. The exact opposite occurs when the interest rate increases becauuse Par value sold Par value remaining Value at the end of year 2 Balance after 2nd payment 0 1 2 3 4 5 6 7 8 now we have locked in at a rate that is lower than the prevailing market rate and we wind up with...
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- Fall '09