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### Practice Problems

A bank makes four kinds of loans to its personal customers

3. Financial planning: A bank makes four kinds of loans to its personal customers. These loans 10 Marks yield the following annual interest rates to the bank:

• First mortgage 15%

• Second mortgage 20%

• Home improvement 20%

• Personal overdraft 10%

The bank has a maximum foreseeable lending capability of \$350 million and is further constrained by the following policies:

1. ﬁrst mortgages must be at least 55% of all mortgages issued and at least 25% of all loans issued (in \$ terms)

2. second mortgages cannot exceed 25% of all loans issued (in \$ terms)

3. to avoid public displeasure and the introduction of a new windfall tax the average interest rate on all loans must not exceed 16%.

Formulate the bank’s loan problem as an LP problem so as to maximize interest income while satisfying the policy limitations. Note here that these policy conditions, while potentially limiting the proﬁt that the bank can make, also limit its exposure to risk in a particular area. It is a fundamental principle of risk reduction that risk is reduced by spreading money (appropriately) across diﬀerent areas.

Hint
Mathematicsobjective -> max : 15%x1 + 20%x2 + 20%x3 + 10%x4constraints -> x1 + x2 + x3 + x4 <= 350 x1 >= 0.55(x1 + x2) x1 >= 0.25(x1 + x2 + x3 + x4) x2 <= 0.25(x1 + x2 + x3 + x4) 0.14x1 + 0.20x2 + 0.20x3 + 0.10x4 <= 0.15(x1 + x2 + x3 + x4) x1 >= 0 x2 >= 0 x3 >= 0 x4 >= 0 ...

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