4. (Bond matrix ) The cash matching and other problems can be conveniently represented in matrix form. Suppose there are m bonds. We define for each bond j its associated yearly cash flow stream (column) vector cj, which is n-dimensional. The yearly obligations are likewise represented by the n-dimensional vector y. We can stack the c; vectors side by side to form the columns of a bond matrix C. Finally we let p and x be m-dimensional column vectors. The cash matching problem can be expressed as maximize px subject to Cx>y x>0. (a) Identify C,y,p, and x in Table 5.3. (b) Show that if all bonds are priced according to a common term structure of interest rates, there is a vector v satisfying c?v=p. What are the components of v? (c) Suppose b is a vector whose components represent obligations in each period. Show that a portfolio x meeting these obligations exactly satisfies Cx=b. (d) With x and v defined as before, show that the price of the portfolio x is v b. Interpret this result. TABLE 5.3 CASH MATCHING EXAMPLE Bonds Yr 1 2 3 4 5 6 7 8 9 Actual -Nm 7 7 6 6 10 10 110 7 107 10 10 10 10 10 110 5 5 5 105 8 8 108 7 7 7 7 107 7 7 7 107 8 8 8 108 6 6 106 4 5 6 10 Req'd 100 100 200 800 100 800 1,200 171.74 200.00 800.00 119.34 800.00 1,200.00 109 94.8 99.5 93.1 97.2 92.9 110 104 102 95.2 2,381.14 0 11.2 06.81 0 0 0 6.3 0.28 0 Cost A spreadsheet layout clearly shows the problem and its solution. In this example, the cash flow streams of 10 different bonds are shown, year by year, as 10 columns in the array. The current price of each bond is listed below the stream, and the amount to be included in a portfolio is listed below the price. Cash flows required to be generated by the portfolio are shown in the penultimate column, and those actually generated are show in the last column. 4. (Bond matrix ) The cash matching and other problems can be conveniently represented in matrix form. Suppose there are m bonds. We define for each bond j its associated yearly cash flow stream (column) vector cj, which is n-dimensional. The yearly obligations are likewise represented by the n-dimensional vector y. We can stack the c; vectors side by side to form the columns of a bond matrix C. Finally we let p and x be m-dimensional column vectors. The cash matching problem can be expressed as maximize px subject to Cx>y x>0. (a) Identify C,y,p, and x in Table 5.3. (b) Show that if all bonds are priced according to a common term structure of interest rates, there is a vector v satisfying c?v=p. What are the components of v? (c) Suppose b is a vector whose components represent obligations in each period. Show that a portfolio x meeting these obligations exactly satisfies Cx=b. (d) With x and v defined as before, show that the price of the portfolio x is v b. Interpret this result. TABLE 5.3 CASH MATCHING EXAMPLE Bonds Yr 1 2 3 4 5 6 7 8 9 Actual -Nm 7 7 6 6 10 10 110 7 107 10 10 10 10 10 110 5 5 5 105 8 8 108 7 7 7 7 107 7 7 7 107 8 8 8 108 6 6 106 4 5 6 10 Req'd 100 100 200 800 100 800 1,200 171.74 200.00 800.00 119.34 800.00 1,200.00 109 94.8 99.5 93.1 97.2 92.9 110 104 102 95.2 2,381.14 0 11.2 06.81 0 0 0 6.3 0.28 0 Cost A spreadsheet layout clearly shows the problem and its solution. In this example, the cash flow streams of 10 different bonds are shown, year by year, as 10 columns in the array. The current price of each bond is listed below the stream, and the amount to be included in a portfolio is listed below the price. Cash flows required to be generated by the portfolio are shown in the penultimate column, and those actually generated are show in the last column
