The current treasury market can he summarized as follows, assuming that all rates/yields are quoted as annualized continuous compounding, and all zero- coupon bonds have face value $1: overnight federal fund rate 0.75%, 6-inonlh bill yield 0.67%, 2-year zero-coupon bond price $0.976: 5-year zero-coupon bond price $0.908: 10-year zero-coupon bond price $0.78: 30-year zero-coupon bond price $0.395. The zero-coupon bond price Z(0, T) and the yield y(0, T) are related via Z(0, T) = e^-y(0, T)T Here 0 in the first argument refers to the time-0 price or the time-0 yield. The function y(0, T) is called the current yield curve and it is the starting point for any quantitative work in the interest rate market. Use the market information provided above and a piecewise linear interpolation to build the current yield curve y(0, T) for 0 lessthanorequalto T lessthanorequalto 30. Based on this function y(0. T), derive the zero-coupon bond price Z(0. T). Compare this function with a direct piecewise linear interpolation of the bond prices, and explain why the interpolation based on y is to be preferred. The current continuously compounding forward rate for the period (T_1, T_2) is defined as f(0, T_1, T_2) = 1/T_2 - T_1 (log z(0, T_2)) Evaluate the zero-coupon bond price Z(0, T) generated above at T_j = 0.25j, j = 0, 1, 2, ..., 120, and calculate a set of 3-month forward rates f_j, j = 0, 1, ... 119. Plot the forward rates f_j = f(0, T, T_j + 1) and compare them with the yields y(0, T_j), j = 0, 1, ..., 119. Suppose you observe on the market, in addition to those prices listed in Problem 1, that there is a 7-year zero-coupon bond with price Z(0, 7) = 0.815. Is there any arbitrage opportunity? Suggest a bond portfolio to your investor if there is and explain to him/her how it is going to work