12.7 Constructing a portfolio of bonds to approrimate a sequence of liabilities. A business has a (pre- dicted) sequence of monthly liabilities (payments it must make) over 10 years, given by the 120- vector l. The business will purchase 20 different bonds, with quantities given by the 20-vector q. Bond i is associated with a sequence of monthly coupon payments given by the 120-vector C. (There are many different types of bonds, with different coupon payments. Some pay every month; others pay every 6 or 12 months. Each bond has a maturity date, after which the coupon payments are zero. But you don't need to know this.) Let p denote the 120-vector of total coupon payments from the bonds. It is given by p = q1c1 + ... +220C20. The bond quantities will be chosen by minimizing ||2 p||2, which means we are trying to match the payments to the liabilities. Explain how to set this up as the problem of minimizing || Aq b||2, where A is a matrix and b is a vector. You must say what A and b are, and give their dimensions. Remark. The liabilities are nonnegative, are the coupon payments, but we ignore that here. 12.7 Constructing a portfolio of bonds to approrimate a sequence of liabilities. A business has a (pre- dicted) sequence of monthly liabilities (payments it must make) over 10 years, given by the 120- vector l. The business will purchase 20 different bonds, with quantities given by the 20-vector q. Bond i is associated with a sequence of monthly coupon payments given by the 120-vector C. (There are many different types of bonds, with different coupon payments. Some pay every month; others pay every 6 or 12 months. Each bond has a maturity date, after which the coupon payments are zero. But you don't need to know this.) Let p denote the 120-vector of total coupon payments from the bonds. It is given by p = q1c1 + ... +220C20. The bond quantities will be chosen by minimizing ||2 p||2, which means we are trying to match the payments to the liabilities. Explain how to set this up as the problem of minimizing || Aq b||2, where A is a matrix and b is a vector. You must say what A and b are, and give their dimensions. Remark. The liabilities are nonnegative, are the coupon payments, but we ignore that here