What is the minimum cost portfolio, consisting of up to 6 bonds, that provides enough cash flow to cover liabilities in each period? Interest Rate 7% Characteristics of bonds Bond 1 Bond 2 Bond 3 Bond 4 Bond 5 Face Value \$1,000 \$1,000 \$1,000 \$1,000 \$1,000 Coupon Payment \$100 \$125 \$150 \$175 \$75 Years to Maturity 3 5 6 4 6 Price \$1,078.73 \$1,225.51 \$1,381.32 \$1,355.66 \$1,023.83 Bond 1 Bond 2 Bond 3 Bond 4 Bond 5 Cost Number Purchased 50 50 50 50 50 \$303,253 Cash Flow Bond 1 Bond 2 Bond 3 Bond 4 Bond 5 Total w/Int Liability Year 1 \$5,000 \$6,250 \$7,500 \$8,750 \$3,750 \$31,250 \$32,000 Year 2 \$5,000 \$6,250 \$7,500 \$8,750 \$3,750 \$30,448 \$25,000 Year 3 \$5,000 \$6,250 \$7,500 \$8,750 \$3,750 \$37,079 \$22,000 Year 4 \$6,250 \$7,500 \$8,750 \$3,750 \$42,384 \$28,000 Year 5 \$6,250 \$7,500 \$3,750 \$32,891 \$25,000 Year 6 \$7,500 \$3,750 \$19,694 \$20,000 Problem An investor wants to put together a portfolio consisting of up to 6 different bonds. He has certain cash-flow requirements in the future that the coupons of the bonds should cover. (For example, a pension fund must meet requirements for future pension payments.) These payments are independent of interest rate changes. Excess payments in a period can be reinvested, to be available in the next period, at a certain interest rate. How should the investor choose his portfolio to minimize the cost of the bonds, while making sure that the payments cover his future cash-flow requirements? Solution 1) The variables are the number of each bond to include in the portfolio. In worksheet BOND3 these are given the name Purchased_bonds. 2) The constraints are very simple. First we have the logical constraints: Purchased_bonds >= 0 via the Assume Non-Negative option Purchased_bonds = integer (We can not buy fractions of a bond) Then there is the constraint to make sure that the cash-flow requirements are met: Cash_flow >= Liabilities 3) The objective is to minimize the portfolio cost. This is given the name Total_cost. Remarks The solution is similar to the one in BOND3. The difference is that the cash-flow takes into account the reinvestment of excess funds in one period for the next period. Remember that the original idea behind exact matching was to minimize the influence of interest rate changes. In this model, however, we are again more dependent on the interest rate, since a shift in the future rate would affect the solution to the model. Thus, the market value of the portfolio may fluctuate to a greater extent than if we ignored reinvestment opportunities.