Question 2: Butterﬂy Strategy One of the more popular strategies in the Fixed Income market is the Butterﬂy Strategy which constructs a trade that is proﬁtable if the term structure of interest rates moves in parallel. In this problem, you will be asked to construct the right portfolio weights for this trade and to determine the proﬁtability of the trade. Start with the following table of zero-coupon bonds:Maturity: 2 Years, 5 Years, 10 YearsYield: 5% 5% 5%Yields are quoted in an annually compounded basis. The strategy is based on buying the 2-year and 10-year bonds and shorting the 5-year bond. It will end up having a positive convexity. The relative amounts to invest in each bond are given by the two constraints: (1) It should be (modiﬁed) duration neutral. That is, construct the strategy such that the dollar value of your short position times the modiﬁed duration of your short position is equal to the dollar value of your long position times the modiﬁed duration of your long position. Thus, it is hedged against small (level) changes in yields. (This is the constraint from a number of the examples given in class.) (2) It should be “cash neutral”. That is, the initial value of the short position should equal the initial value of the long position. 1. Calculate the modiﬁed duration of the 2-year, 5-year, and 10-year bonds. 2. Suppose you short $1,000 in face value of the 5-year bond. First, calculate the market value of the 5-year bond. Denote x to be the amount (market value) of the 2-year bond that you buy and z the amount (market value) of the 10-year bond you buy. Write down the two equations for the two constraints given above. 3. Solve the two equations for x and z. These are the market values of the 2-year and 10-year bonds that you buy in this strategy. What is the face value of the 2-year and 10-year bonds that you buy? 4. Suppose that yields move in parallel to 4.75%. What is the proﬁt/loss to your strategy? What about 4.5%? Do this calculation for each 0.25% increment from a yield of 0% to a yield of 10% and plot the proﬁt/loss against the yield. 5. Suppose that instead, the yields move to y2 = 3%, y5 = 5%, and y10 = 8%. What is the proﬁt/loss of your position?

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