Quantitative risk management

12. Consider a fixed income product that pays a coupon of $25 at the end of each year for the next 5 years,and at the end of 5 years, pays the principal of $1000. Suppose the current yield on the bond is 7%.Then find the following quantities (I would use the formulas in the notes, and then confirm calculationsin the spreadsheet provided along with the class notes):(a) The current price of the bond, and the price if the yield rises to 7.3%.(b) The approximate duration based price change if the yield was to increase to 7.3%.(c) The approximate duration and convexity based price change if the yield was to increase to 7.3%.(d) If the annual volatility of yield changes is 0.06, then find the monthly VaR(0.05) using the durationbased as well as the duration-convexity based approximation. You may assume that the monthlymean change in yields is zero.(e) Explain in words and with a diagram why the two VaRs calculated differ.
13. Consider the case of a bank with assets of $400 and liabilities of $300. Suppose the modified durationof the assets is 10 years, and the modified duration of the liabilities is 7 years. Also let the convexity ofthe assets be 600 and the convexity of the liabilities be 700. Suppose all bonds have a 5% yield. Then,(a) Suppose the bank wishes to hedge its equity position by going long/short in a 25-year discountbond. Find the duration based variance minimizing hedge ratio of the bank and the dollar positionin the discount bond.(b) Write the value of the hedged bank one year from today as a function of the interest rate a yearfrom today. What would be the value of the bank if all rates rise 200 basis points?(c) Suppose the bank wishes to hedge its duration and convexity by buying/selling 25-year and 10-year discount bonds. Find the dollar position in each of the bonds that will nullify both theduration and the convexity of its equity.(d) Explain in words why two bonds were used to hedge the bank’s position.(e) If the annual volatility of yield changes is 3%, then calculate the 1-day VaR(0.05) based onduration only of the bank’s equity assuming that the mean yield change is zero.