Investment problems
Investment Problems
Investment problems allocate capital among financial alternatives.
The objective is usually to maximize expected return subject to budget, category, and risk constraints.
Basic Structure
An investment LP has:
- investment options
- cost or price per unit
- expected return
- total budget
- category constraints
- nonnegative investment quantities
Decision Variables
There are two common choices.
Unit Variables
If the problem gives purchase cost per unit, define:
\[x_j=\text{number of units purchased of fund }j\]Then money invested in fund $j$ is:
\[p_jx_j\]where $p_j$ is the unit price.
Euro Variables
If the problem is easier in money amounts, define:
\[y_j=\text{euros invested in fund }j\]Then the budget constraint is usually:
\[\sum_j y_j\le B\]Both approaches are valid if used consistently.
Objective with Unit Variables
If fund $j$ costs $p_j$ per unit and has annual return rate $r_j$, then expected return per unit is:
\[r_jp_j\]So the objective is:
\[\max \sum_j r_jp_jx_j\]Objective with Euro Variables
If $y_j$ is euros invested in fund $j$, then expected return is:
\[r_jy_j\]So the objective is:
\[\max \sum_j r_jy_j\]Budget Constraint
With unit variables:
\[\sum_j p_jx_j\le B\]With euro variables:
\[\sum_j y_j\le B\]If all capital must be invested, use equality:
\[\sum_j p_jx_j=B\]or:
\[\sum_j y_j=B\]Category Constraints
Investment options often belong to categories such as:
- bond
- balanced
- equity
Minimum investment in a category uses $\ge$.
Maximum investment in a category uses $\le$.
Example:
\[\sum_{j\in \text{Bond}} p_jx_j\ge 15000\]means at least $15000$ euros must be invested in bond funds.
Example: Mutual Funds
An investor has $50000$ euros and can buy units of $8$ funds.
Let:
\[x_A,\ldots,x_H\]be the number of units purchased of funds $A,\ldots,H$.
Suppose the unit costs are:
| Fund | Type | Cost | Return Rate |
|---|---|---|---|
| A | Bond | 4.5 | 7% |
| B | Bond | 4 | 8% |
| C | Bond | 2.5 | 6% |
| D | Balanced | 3 | 6% |
| E | Balanced | 4.5 | 9% |
| F | Balanced | 5 | 9% |
| G | Equity | 6 | 10% |
| H | Equity | 5.5 | 12% |
The objective is:
\[\max 0.07(4.5)x_A +0.08(4)x_B +0.06(2.5)x_C +0.06(3)x_D +0.09(4.5)x_E +0.09(5)x_F +0.10(6)x_G +0.12(5.5)x_H\]The budget constraint is:
\[4.5x_A+4x_B+2.5x_C+3x_D+4.5x_E+5x_F+6x_G+5.5x_H\le 50000\]Bond requirement:
\[4.5x_A+4x_B+2.5x_C\ge 15000\]Balanced requirement:
\[3x_D+4.5x_E+5x_F\ge 20000\]Equity limit:
\[6x_G+5.5x_H\le 5000\]Nonnegativity:
\[x_A,\ldots,x_H\ge 0\]Alternative with Euro Variables
Let:
\[y_A,\ldots,y_H\]be euros invested in each fund.
Then the model is simpler:
\[\max 0.07y_A+0.08y_B+0.06y_C+0.06y_D+0.09y_E+0.09y_F+0.10y_G+0.12y_H\]subject to:
\[y_A+\cdots+y_H\le 50000\] \[y_A+y_B+y_C\ge 15000\] \[y_D+y_E+y_F\ge 20000\] \[y_G+y_H\le 5000\] \[y_A,\ldots,y_H\ge 0\]This is valid if fractional euro allocation is allowed.
Liquidity or Treasury Bonds
If excess liquidity can be invested in treasury bonds, introduce another variable:
\[y_0=\text{euros invested in treasury bonds}\]If treasury bonds return $4\%$, include:
\[0.04y_0\]in the objective.
Then the budget equality can be:
\[y_0+y_1+\cdots+y_n=B\]Common Mistakes
Avoid these mistakes:
- multiplying return rate by units but forgetting unit cost
- using percentages as $7$ instead of $0.07$
- confusing units purchased with euros invested
- forgetting category constraints
- using $\le$ for “at least”
- omitting nonnegativity
- forcing all capital to be invested when the problem says only “wants to invest up to”
Key Takeaway
An investment LP allocates capital to maximize expected return while satisfying budget and portfolio-composition rules.
Exam checkpoint
For formulation questions, show variables, objective, constraints, sign restrictions, and units. Do not solve unless the question asks for a solution.