Please assist with question 2 and all of its parts. Part 4 must be graphed. Please show work with the answers.
Business Finance: Fixed Income Homework 1 Treasury Bonds and Trades Due in class on January 26, 2023 The focus of this assignment is to understand how to construct portfolios of Treasuries and to hedge interest rate risk. You can work in a group of 1 – 4 people. Each group should hand in one copy of their answers. Show your work and provide explanations where relevant. However, you do not need to print out entire spreadsheets. Your answers should be summarized in a write-up with relevant details of calculations, tables and charts (if applicable), and explanations. Question 1: Trading on the Level of Interest Rates Suppose that you are given the following zero-curve (interest rates are annual, but com- pounded semi-annually): t 0.5 1 1.5 2 2.5 3 3.5 4 r 0.02 0.021 0.023 0.025 0.03 0.035 0.04 0.041 1. Calculate the value of a 4-year Treasury bond with a 10% coupon rate, coupons paid semi-annually, and a face value of $100. 2. Calculate the modified duration and convexity of the bond in (1). What is the approx- imate percentage change in the bond price if the zero-curve shifts up by one percent (i.e. dy = 1% or r2,0.5 = 0.03, r2,1 = 0.031, etc.) using the modified duration and convexity you just calculated? 3. Because you believe that the zero-curve will shift upward and decrease the bond price, you decide to use a reverse repo trade to short the bond. Let’s suppose that a repo dealer quotes you a special repo rate of 2% and you borrow 100 units of the Treasury bond (which you then sell on the market as part of your reverse repo transaction). (a) Draw a picture of the initial transactions in the repo trade. (b) If the zero-curve shifts up by one percent, what is the value of the Treasury bond in part 1? (c) What is your net payoff to unwinding the repo trade tomorrow? Recall that you need to buy the bond and sell it to the repo dealer at the prespecified price. Assume that the price tomorrow is the price that you calculated in (b). 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: 1 Maturity Yield 2 years 5% 5 years 5% 10 years 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? 2