Chapter 04

# and the linear programming problem is minimize

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Unformatted text preview: ment times of the obligation y • The decision variable is the amount xi ≥ 0 of bond i to be purchased (long only!), and the linear programming problem is minimize subject to m ￿ i=1 m ￿ i=1 pi xi cij xi ≥ yj , xi ≥ 0, Kay Giesecke j = 1, 2, . . . , n i = 1, 2, . . . , m Finance 14 Cash matching An example (Example 5.3 in Luenberger’s “Investment Science”) • We consider m = 10 bonds over n = 6 years with 6 × 10 cash ﬂow matrix C , price vector p = (p1 , . . . , p10 )T and obligation vector y = (y1 , . . . , y6 )T • The linear programming problem is min pT · x subject to x C · x ≥ y, • Solution: simplex method for example Kay Giesecke x≥0 Finance 15 Cash matching Matlab formulation using linprog p = [109; 94.8; 99.5; 93.1; 97.2; 92.9; C = [10 7 8 6 7 5 10 7 8 6 7 5 10 7 8 6 7 5 10 7 8 6 7 105 10 7 8 106 107 0 110 107 108 0 0 0 y = [100; 200; 800; 100; 800; 1200] lb = [0; 0; 0; 0; 0; 0; 0; 0; 0; 0] [x,fval] = linprog(p,-C,-y,,,lb) Kay Giesecke 110; 10 10 110 0 0 0 104; 8 8 108 0 0 0 102; 7 107 0 0 0 0 95.2] 100; 0; 0; 0; 0; 0] Finance 16 Cash matching Matlab output x= 0.0000 11.2150 0.0000 6.8066 0.0000 0.0000 0.0000 6.3024 0.2826 0.0000 fval = 2381 Kay Giesecke Finance 17 Cash matching Surpluses • Our formulation did not incorporate the surpluses generated by the bond portfolio; these can be computed in Matlab as s = C*x-y 71.7411 0.0000 0.0000 19.3440 0.0000 0.0000 Kay Giesecke Finance 18 Cash matching Allowing bounded short selling p = [109; 94.8; 99.5; 93.1; 97.2; 92.9; 110; 104; 102; 95.2] C = [10 7 8 6 7 5 10 8 7 100; 10 7 8 6 7 5 10 8 107 0; 10 7 8 6 7 5 110 108 0 0; 10 7 8 6 7 105 0 0 0 0; 10 7 8 106 107 0 0 0 0 0; 110 107 108 0 0 0 0 0 0 0] y = [100; 200; 800; 100; 800; 1200] lb = [-10; -10; -10; -10; -10; -10; -10; -10; -10; -10] [x,fval] = linprog(p,-C,-y,,,lb) Kay Giesecke Finance 19 Cash matching We r...
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## This document was uploaded on 02/10/2014.

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