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4. Risk Management, relaring the parallel shift assumption - 25 points Currently, t -0, a portfolio of 5 annually coupon-paying bonds, each with a face value of $100. Their characteristics are given by Bond A Bond B Bond C Bond D Bond E $98 Current Price $98 $98 3 Maturity Annual Coupon payment 5.50% 6.75% 7.00% 8.25% 6.00% In addition to the above bond prices, you also know that the current one-year con- tinuously compounded spot rate is R(0.1) 4%. Your objective here is construct a portfolio that hedges the changes in the 1-year, 2-year, and 3-year spot rates. You decide to leave your portfolio exposed to changes in the interest rates at the longer horizon (i.e., 4-, 5- and 6-year spot rates). This is because you are betting on the changes in the zero rates at the longer end of the maturity. However, you want to limit your risk of the short-term rate changes. (a) Construct a zero initial cost portfolio using Bond A to Bond E such that you are hedged against the changes in the 1-year, 2-year, and 3-year spot rates. This portfolio must have zero initial investment cost. Note, you only have to hedge against the first-order changes in these rates, see for example, equation (1). Hint: there are more than one way to construct your portfolio. Show me the possible solutions that are sensible (at least three). Use your common sense. Then finally choose the portfolio that is most appropriate for your objective. Use this optimal portfolio in your Part B and C (b) Assume that exactly a year has gone by, t 1, and you want to close out the portfolio that you constructed in (a). Notes, all these bonds are now a year closer to their maturities. The zero rates at time t-1 is flat at 5% across all the maturities. How much profit or loss has this trading strategy been for you? 4. Risk Management, relaring the parallel shift assumption - 25 points Currently, t -0, a portfolio of 5 annually coupon-paying bonds, each with a face value of $100. Their characteristics are given by Bond A Bond B Bond C Bond D Bond E $98 Current Price $98 $98 3 Maturity Annual Coupon payment 5.50% 6.75% 7.00% 8.25% 6.00% In addition to the above bond prices, you also know that the current one-year con- tinuously compounded spot rate is R(0.1) 4%. Your objective here is construct a portfolio that hedges the changes in the 1-year, 2-year, and 3-year spot rates. You decide to leave your portfolio exposed to changes in the interest rates at the longer horizon (i.e., 4-, 5- and 6-year spot rates). This is because you are betting on the changes in the zero rates at the longer end of the maturity. However, you want to limit your risk of the short-term rate changes. (a) Construct a zero initial cost portfolio using Bond A to Bond E such that you are hedged against the changes in the 1-year, 2-year, and 3-year spot rates. This portfolio must have zero initial investment cost. Note, you only have to hedge against the first-order changes in these rates, see for example, equation (1). Hint: there are more than one way to construct your portfolio. Show me the possible solutions that are sensible (at least three). Use your common sense. Then finally choose the portfolio that is most appropriate for your objective. Use this optimal portfolio in your Part B and C (b) Assume that exactly a year has gone by, t 1, and you want to close out the portfolio that you constructed in (a). Notes, all these bonds are now a year closer to their maturities. The zero rates at time t-1 is flat at 5% across all the maturities. How much profit or loss has this trading strategy been for you

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