please answer question 2

Problem 1 (16 points): Assume that: Forward rates for 6 month and 1 year are r(0.5)=7% and r(1)=8%; Spot rates for 18 months and 2 years are f(1.5) =7.8% and f(2)=8.2% The price of a zero-coupon bond maturing 2.5 years from now is $81.50 a) (3 points) Find the 1-year spot rate (1). b) (3 points) Find the 18-month forward rate r(1.5) c) (3 points) Find the price of a 10% coupon bond maturing 2 years from now d) (3 points) Find 2.5 years forward rate r(2.5) e) (4 points) Assume 2-year 5% coupon bond and 2-year 7% coupon bond are priced correctly while 2-year zero coupon bond is incorrectly priced at $85. You want to make an arbitrage by trading only these 3 bonds. Find an arbitrage strategy (i.e., state how many of each bond you want to buy or sell) Problem 2 (5 points): Assume that forward rates for the next year are given by r(0.5)=7% and r(1)-6% and consider a 5% coupon bond maturing 1 year from now. a) (1 point) Find the bond price. b) (2 points) Assume at t=0.5 the half-year spot rate will be exactly 6%. Find the price of the bond at t=0.5. c) (2 points) Assume at t=0.5 the half-year spot rate can be either 5% or 7% with equal probabilities so that the expected spot rate is 6%. Find the expected price of the bond at t=0,5. Is it higher, lower, or the same as the price you found in part (b)? Problem 3 (9 points): The price of a risky 10-year zero-coupon bond depends on two parameters: risk-free 10-year spot rate f(10) (to simplify notations, denote (10) by r), and the $1,000,000 risk factors and it is given by Pr,s) = Today r=0.05 and s=0.12, so, (1+0.5r+2rs 20 Pr.s) = $483,531.61. You want to estimate the change in the bond (1+0.5*0.05+2*0.05-0.12)20 price using first-order Taylor series approximation, i.e., write the change in price AP as a linear function of the change in interest rate Ar and risk factor As as AP=A*Ar+B*As, where A and B are some constants. $1,000,000 a) (5 points) Find A and B. Round to the nearest integer number. $1,000,000 b) (2 points) Using the original price function, Pr.s) = if the risk factor increases (1+0.5r+2rs)20 by 0.01, by how much the interest rate should change to keep the price unchanged? c) (2 points) Using the Taylor series approximation, if the risk factor increases by 0.01, how much the interest rate should change to keep the price unchanged? Specific assignment instructions: All interest rates are annual interest rates with semi-annual compounding. All coupon rates are annual rates paid semi-annually. All bonds have a $100 face value. Keep at least 6 decimal digits in all your calculation and answers unless specified otherwise. Show your work!!! Write down the equations you are using and then write down the same equations with appropriate numbers plugged in. If your work is clear but you make an arithmetie mistake, you will receive generous partial credit; however, if the work is unclear or missing, a significant penalty may be applied. Problem 1 (16 points): Assume that: Forward rates for 6 month and 1 year are r(0.5)=7% and r(1)=8%; Spot rates for 18 months and 2 years are f(1.5) =7.8% and f(2)=8.2% The price of a zero-coupon bond maturing 2.5 years from now is $81.50 a) (3 points) Find the 1-year spot rate (1). b) (3 points) Find the 18-month forward rate r(1.5) c) (3 points) Find the price of a 10% coupon bond maturing 2 years from now d) (3 points) Find 2.5 years forward rate r(2.5) e) (4 points) Assume 2-year 5% coupon bond and 2-year 7% coupon bond are priced correctly while 2-year zero coupon bond is incorrectly priced at $85. You want to make an arbitrage by trading only these 3 bonds. Find an arbitrage strategy (i.e., state how many of each bond you want to buy or sell) Problem 2 (5 points): Assume that forward rates for the next year are given by r(0.5)=7% and r(1)-6% and consider a 5% coupon bond maturing 1 year from now. a) (1 point) Find the bond price. b) (2 points) Assume at t=0.5 the half-year spot rate will be exactly 6%. Find the price of the bond at t=0.5. c) (2 points) Assume at t=0.5 the half-year spot rate can be either 5% or 7% with equal probabilities so that the expected spot rate is 6%. Find the expected price of the bond at t=0,5. Is it higher, lower, or the same as the price you found in part (b)? Problem 3 (9 points): The price of a risky 10-year zero-coupon bond depends on two parameters: risk-free 10-year spot rate f(10) (to simplify notations, denote (10) by r), and the $1,000,000 risk factors and it is given by Pr,s) = Today r=0.05 and s=0.12, so, (1+0.5r+2rs 20 Pr.s) = $483,531.61. You want to estimate the change in the bond (1+0.5*0.05+2*0.05-0.12)20 price using first-order Taylor series approximation, i.e., write the change in price AP as a linear function of the change in interest rate Ar and risk factor As as AP=A*Ar+B*As, where A and B are some constants. $1,000,000 a) (5 points) Find A and B. Round to the nearest integer number. $1,000,000 b) (2 points) Using the original price function, Pr.s) = if the risk factor increases (1+0.5r+2rs)20 by 0.01, by how much the interest rate should change to keep the price unchanged? c) (2 points) Using the Taylor series approximation, if the risk factor increases by 0.01, how much the interest rate should change to keep the price unchanged? Specific assignment instructions: All interest rates are annual interest rates with semi-annual compounding. All coupon rates are annual rates paid semi-annually. All bonds have a $100 face value. Keep at least 6 decimal digits in all your calculation and answers unless specified otherwise. Show your work!!! Write down the equations you are using and then write down the same equations with appropriate numbers plugged in. If your work is clear but you make an arithmetie mistake, you will receive generous partial credit; however, if the work is unclear or missing, a significant penalty may be applied