Question

Cardinal Capital is a company that makes relatively high risk/high return investments in promising small businesses across multiple markets. The company is in the process of developing a strategy/policy for its 2017 investment portfolio, which will have a maximum of $15 Million with which to work. The table below provides data gathered from analysis of historical investments and future projections.

Type of Business	Median Rate of Return	Loss Rate
Software	23%	15%
Restaurant/Food Production	10%	6%
Retail	7%	5%
Manufacturing	12%	7%
Tourism	14%	10%

The median rate of return is what the company has decided to use as a tool to inform decisions about the type of businesses in which to invest. The loss rate represents the historical likelihood that the company will lose all or part of its investment. For calculation purposes it is assumed that the entire investment is lost.

To meet its competition, business, and community goals, Cardinal has several entering goals:

-Retail and Restaurant/Food Production must be at least 40% of the dollar value of all investments

-Software must be no more than 30% of the dollar value of all investments

-Manufacturing must be at least 10% of the dollar value of all investments

Please provide the investment strategy that maximizes projected profit and meets the above constraints. Additionally please discuss some other key questions that may be asked by Cardinal to further refine its strategy.

Answer 1

Let

X1 = Dollar value (in million) invested in Software
X2 = Dollar value (in million) invested in Restaurant/Food Production
X3 = Dollar value (in million) invested in Retail
X4 = Dollar value (in million) invested in Manufacturing
X5 = Dollar value (in million) invested in Tourism

Expected return in Type-j asset = Prob of loss * Xj + (1 - Prob of loss) * % of return * Xj

So,

Max Z = overall expected return

= 0.15*X1 + (1 - 0.15)*0.23*X1 + 0.06*X2 + (1 - 0.06)*0.10*X2 + 0.05*X3 + (1 - 0.05)*0.07*X3 + 0.07*X4 + (1 - 0.07)*0.12*X4 + 0.10*X5 + (1 - 0.01)*0.14*X5

= 0.3455*X1 + 0.154*X2 + 0.1165*X3 + 0.1816*X4 + 0.226*X5

Subject to,

X1 + X2 + X3 + X4 + X5 <= 15

X2 + X3 >= 0.40*(X1 + X2 + X3 + X4 + X5)
or, -0.40*X1 + 0.60*X2 + 0.60*X3 - 0.40*X4 - 0.40*X5 <= 0

X1 <= 0.30*(X1 + X2 + X3 + X4 + X5)
or, 0.70*X1 - 0.30*X2 - 0.30*X3 - 0.30*X4 - 0.30*X5 <= 0

X4 >= 0.10*(X1 + X2 + X3 + X4 + X5)
or, 0.10*X1 + 0.10*X2 + 0.10*X3 - 0.90*X4 + 0.10*X5 <= 0

X1, X2, X3, X4, X5 >= 0

Excel formulation:

Solver inputs:

Solution:

Objective Cell (Max)
	Cell	Name	Original Value	Final Value
	$H$6	Expected return	0	3.86115
Variable Cells
	Cell	Name	Original Value	Final Value
	$C$3	Invested in Software	0	4.5	million
	$D$3	Invested in Restaurant/Food Production	0	0	million
	$E$3	Invested in Retail	0	0	million
	$F$3	Invested in Manufacturing	0	1.5	million
	$G$3	Invested in Tourism	0	9	million