1. Model 3: A Linear Model
By Evan Nixon

2. The problem:
We have been given $1.5 million to invest over three years
4 Different rates are available:
Our goal is to maximize the value of the investment over three years
Year 1
Year 2
Year 3
Investment A 7%
Investment B 5% 8%
Investment C
20 % (three years)
Investment D
N/A
16% (two years)

3. How to solve the problem
A linear model can be created to solve this problem
Excel offers a useful ‘solver’ add-in

4. Assumptions
We may invest any available amount of money in any investment
Each investment has its interest compounded at the end of the investment period
Any different combination of investments is allowed
Each investment must be ≥ $0

5. Decision Variables
Let Iij be the amount invested in choice i (A,B,C, or D) in year j (1, 2, or 3)

6. Additional Variables
Let Rij be the annual return of investment choice i in year j
Years that an investment is not available will have 0 return
Rij can be visualized in this table:
j=1
j=2
j=3
i=A
1.07
i=B
1.05
1.08
i=C
1.20
i=D
1.16

7. Additional Variables Let Xij = Ri(j-n+1) * Ii(j-n+1)
Xij is the amount that is returned after investing Iij in choice i in year j
n is the number of years the investment last

8. Explanation Xij = Ri(j-n+1) * Ii(j-n+1) Xij = Rij* Iij when n = 1
An investment in year j returns when j = j+n-1
Simple arithmetic leads to this equation:
Xi(j+n-1) = Rij* Iij Xij = Ri(j-n+1) * Ii(j-n+1)

9. Constraints Each investment must be ≥ $0
∑ Iij ≤ ∑ Xi(j-1) (the investments made in year j must be less than or equal to the returns in year j-1)
Xij = initial investment when j=0

10. Objective Function
Maximize

11. The Excel solver

12. Rij

13. Iij

14. Xij

15. Results
The optimized solution invested all available money in investment A for year one and invested in B for years two and three
Optimized investments return $1,872,072
This is a 24.8% increase from $1,500,000 over three years

16. Sensitivity
In order for investment C to have the highest returns for all three years, it must yield higher than %
In order for investment D to have the highest returns for the last two years, it must yield higher than 16.64%