PROBLEM 3.1. We want to construct a hedge for a coupon-paying bond in which we have a long position. The bond has face value equal to SEK 1000 , and pays out a 2% interest coupon at the end of each year. The bond has maturity in 4 years; the first interest coupon is due in one year from today. The Y'TM on every bond that we consider in this question is equal to 3%. 3.1.1. How can we use a standard, coupon-paying bond with face value equal to SEK 1000 , which pays out a 3% interest coupon at the end of each year and has maturity in three years, to construct the hedge? 3.1.2. Calculate the present value of the hedged bond portfolio constructed in 3.1.1. Also, calculate the present value that the same portfolio would have in one year from today, immediately after the payment of the interest coupons, if the Y'TM remained at its initial level of 3%. (In the latter case, you should consider the coupon paid as a part of the value of each bond, as we are tracking the full value of the portfolio.) 3.1.3. A correct answer to 3.1.2 should allow you to show (do try it!) that with an unchanged Y'TM, the value of the hedged portfolio constructed in 3.1.1 would change (decrease) by SEK 9.5961 between time 0 and time 1. The reason is that, as it is usually the case, the present value of the bond in the portfolio and the hedge constructed by imposing slope-matching only are different. A hedge that were meant to be in place for a relatively long period of time would thus be an autonomous driver of changes in the value of the hedged portfolio. To remedy this problem, we can construct a hedge that matches both the slope and the present value of the bond already in our portfolio. Construct a hedge that satisfies both requirements using the same bond used in 3.1.1 and a zero coupon-bond with maturity in five years from today. To this purpose, you should construct and solve a system with one equation imposing slopematching and one equation imposing PV-matching between the bond already in the portfolio and the hedge. 3.1.4. Using our notation, the expression for the duration of a generic portfolio with present value v(r) and slope s(r) can be written as d(r)=v(r)(1+r)s(r). Verify that the hedge constructed in 3.1.3 has the same duration as the bond to be hedged