Model 3: A Linear Model
By Evan Nixon
The problem:
• We have been given $1.5 million to invest over
three years
• 4 Different rates are available:
Year 1
Year 2
Year 3
Investment A
7%
7%
7%
Investment B
5%
8%
8%
Investment C
Investment D
20 % (three years)
N/A
16% (two years)
• Our goal is to maximize the value of the
investment over three years
How to solve the problem
• A linear model can be created to solve this
problem
• Excel offers a useful ‘solver’ add-in
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
Decision Variables
• Let Iij be the amount invested in choice i (A,B,C,
or D) in year j (1, 2, or 3)
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
1.07
1.07
i=B
1.05
1.08
1.08
i=C
1.20
0
0
i=D
0
1.16
0
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
Explanation
Xij = Ri(j-n+1) * Ii(j-n+1)
Xij = Rij* Iij when n = 1
Xij = Ri(j-n+1) * Ii(j-n+1)
Xij = Ri(j-1)* Ii(j-1)when n = 2
Xij = Ri(j-n+1) * Ii(j-n+1)
Xij = Ri(j-2)* Ii(j-2)when n = 3
• 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)
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
Objective Function
• Maximize
The Excel solver
Rij
Iij
Xij
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
Sensitivity
• In order for investment C to have the highest
returns for all three years, it must yield higher
than 24.8048%
• In order for investment D to have the highest
returns for the last two years, it must yield
higher than 16.64%