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I set up the IP problem in Excel and solved it with Solver. The Excel spreadsheet is attached. You can see in this solution the annual return of $824.09 is gotten by investing $3500 in the 1-month investment in January, $3090 in the 7-month investment in February, etc. as read off the light grey cells on the right-hand side. The sum in column D is what Solver is ...
For the Solution, I set up the IP problem in Excel and solve it with Solver. The Excel spreadsheet is attached.