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Unformatted text preview: 6 Sensitivity Analysis In this section we study general questions involving the sensitivity of the solution to an LP under changes to its input data. As it turns out LP solutions can be extremely sensitive to such changes and this has very important practical consequences for the use of LP technology in applications. Let us now look at a simple example to illustrate this fact. Consider the scenario where we be believe the federal reserve board is set to decrease the prime rate at its meeting the following morning. If this happens then bond yields will go up. In this environment, you have calculated that for every dollar that you invest today in bonds will give a return of a half percent tomorrow so, as a bond trader, you decide to invest in lots of bonds today. But to do this you will need to borrow money on margin. For the 24 hours that you intend to borrow the money you will need to place a reserve with the exchange that is un-invested, and then you can borrow up to 100 times this reserve. Regardless of how much you borrow, the exchange requires that you pay them back 1.1 times your reserve tomorrow. To add an extra margin of safety you will limit the sum of your reserve and one hundreth of what you borrow to be less than 200,000 dollars. Model the problem of determining how much money should be put on reserve and how much money should be borrowed to maximize your return on this 24 hour bond investment. To model this problem, let R denote your reserve in $10,000 units and let B denote the amount you borrow in the same units. Due to the way you must pay for the loan (i.e. it depends on the reserve, not what you borrow), your goal is to maximize 0 . 005 B- . 1 R . Your borrowing constraint is B 100 R , and your safety constraint is B 100 + R 20 . The full LP model is maximize . 005 B- . 1 R subject to B- 100 R . 01 B + R 20 B, R . We conjecture that the solution occurs at the intersection of the two nontrivial constraint lines. We check this by applying the geometric duality theorem, i.e., we solve the system parenleftbigg . 005- . 1 parenrightbigg = y 1 parenleftbigg 1- 100 parenrightbigg + y 2 parenleftbigg . 01 1 parenrightbigg which gives ( y 1 , y 2 ) = (0 . 003 , . 2). Since the solution is non-negative, the solution does occur at the intersection of the two nontrivial constraint lines giving ( B, R ) = (1000 , 10) 73 with dual solution ( y 1 , y 2 ) = (0 . 003 , . 2) and optimal value 4, or equivalently a profit of $40,000 on a $100,000 investment (the cost of the reserve). But suppose that somehow your projections are wrong, and the Fed left rates alone and bond yields dropped by half a percent rather than increase by half a percent. In this scenario you would have lost $60 , 000 on the $100,000 investment. That is, the difference between a rise of the interest rate by half a percent to a drop in the interest rate by half a percent is one hundred thousand dollars. Clearly, this is a very risky investment opportunity. Inis one hundred thousand dollars....
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- Spring '10