It is currently the beginning of 2010. A city (labeled C for convenience) is trying to sell municipal bonds to support improvements in recreational facilities and highways. The face values (in thousands of dollars) of the bonds and the due dates at which principal comes due are listed in the file S14_104.xlsx. (The due dates are the beginnings of the years listed.) An underwriting company (U) wants to underwrite C’s bonds. A proposal to C for underwriting this issue consists of the following:
(1) An interest rate, 3%, 4%, 5%, 6%, or 7%, for each bond, where coupons are paid annually, and
(2) An up-front premium paid by U to C. U has determined the set of fair prices (in thousands of dollars) for the bonds listed in the same file.
For example, if U underwrites bond 2 maturing in 2013 at 5%, it will charge C $444,000 for that bond.
U is constrained to use at most three different interest rates. U wants to make a profit of at least $46,000, where its profit is equal to the sale price of the bonds minus the face value of the bonds minus the premium U pays to C. To maximize the chance that U will get C’s business, U wants to minimize the total cost of the bond issue to C, which is equal to the total interest on the bonds minus the premium paid by U. For example, if the year 2012 bond (bond 1) is issued at a 4% rate, then C must pay two years of coupon interest: 2(0.04)($700,000) = $56,000. What assignment of interest rates to each bond and up-front premiums ensure that U will make the desired profit (assuming it gets the contract) and maximize the chance of U getting C’s business? To maximize this chance, you can assume that U minimizes the net cost to C, that is, the cost of its coupon payments minus the premium from U to C.