A Bank Is Trying To Determine Where Its Assets Should Be
A Bank Is Attempting To Determine Where Its Assets Should Be Invested
A bank is attempting to determine where its assets should be invested during the current year. At present, $500,000 is available for investment in bonds, home loans, auto loans, and personal loans. The annual rate of return on each type of investment is known to be: bonds, 10%; home loans, 16%; auto loans, 13%; personal loans, 20%. To ensure that the bank's portfolio is not too risky, the bank's investment manager has placed the following three restrictions on the bank's portfolio: a) The amount invested in personal loans cannot exceed the amount invested in bonds. b) The amount invested in home loans cannot exceed the amount invested in auto loans. c) No more than 25% of the total amount invested may be in personal loans. The bank's objective is to maximize the annual return on its investment portfolio. Formulate an LP that will enable the bank to meet this goal.
Paper For Above instruction
Implementing an effective investment strategy is vital for banking institutions aiming to optimize returns while managing risks. Linear Programming (LP) provides a robust mathematical framework to model such decision-making processes, allowing banks to allocate assets efficiently under specified constraints. This paper formulates a linear programming model based on the scenario described, outlining key decision variables, the objective function, and the relevant constraints.
Decision Variables
Let:
- \( x_b \) = amount invested in bonds
- \( x_h \) = amount invested in home loans
- \( x_a \) = amount invested in auto loans
- \( x_p \) = amount invested in personal loans
All variables are constrained to be non-negative:
\[ x_b, x_h, x_a, x_p \geq 0 \]
Objective Function
The goal is to maximize the annual return on the entire investment portfolio. Given the rates of return:
- Bonds: 10%
- Home loans: 16%
- Auto loans: 13%
- Personal loans: 20%
The total return \( R \) can be expressed as:
\[
\text{Maximize } Z = 0.10x_b + 0.16x_h + 0.13x_a + 0.20x_p
\]
Constraints
1. Total investment constraint: The sum of investments in all asset classes should not exceed $500,000:
\[
x_b + x_h + x_a + x_p \leq 500,000
\]
2. Investment proportionality constraints:
a) The amount invested in personal loans cannot exceed the amount invested in bonds:
\[
x_p \leq x_b
\]
b) The amount invested in home loans cannot exceed the amount invested in auto loans:
\[
x_h \leq x_a
\]
3. Maximum allocation in personal loans: No more than 25% of total investment:
\[
x_p \leq 0.25 \times (x_b + x_h + x_a + x_p)
\]
Expanding this:
\[
x_p \leq 0.25x_b + 0.25x_h + 0.25x_a + 0.25x_p
\]
which simplifies to:
\[
0.75x_p \leq 0.25x_b + 0.25x_h + 0.25x_a
\]
or
\[
3x_p \leq x_b + x_h + x_a
\]
Summary of LP Model
Maximize:
\[
Z = 0.10x_b + 0.16x_h + 0.13x_a + 0.20x_p
\]
Subject to:
\[
x_b + x_h + x_a + x_p \leq 500,000
\]
\[
x_p \leq x_b
\]
\[
x_h \leq x_a
\]
\[
3x_p \leq x_b + x_h + x_a
\]
\[
x_b, x_h, x_a, x_p \geq 0
\]
This LP model ensures the investment allocations are optimized to achieve maximum return while adhering to the constraints designed to control risk levels and asset distribution proportions.
Conclusion
The formulated linear programming model provides a systematic approach for the bank's investment manager to determine the optimal asset distribution. By solving this LP, the bank can allocate its $500,000 efficiently across bonds, home loans, auto loans, and personal loans with an aim to maximize returns without exceeding risk thresholds. Implementing such models enhances decision-making accuracy, aligns investments with strategic risk appetite, and drives better financial outcomes in competitive banking environments.
References
Financial Institutions and Markets. John Wiley & Sons. Convex Optimization. Cambridge University Press. Journal of Financial Modeling, 32(4), 78-89. Practical Optimization. Academic Press. Financial Analysts Journal, 76(2), 45-60. Linear and Nonlinear Programming. Springer. Banking & Finance Review, 11(3), 124–138. Optimization Methods in Finance. Springer. Journal of Portfolio Management, 43(2), 22-36. International Journal of Financial Engineering, 8(1), 2150012.