An Investor Wants to Invest $50,000 in Two Mutual Funds,A

question 57

Essay

An investor wants to invest $50,000 in two mutual funds,A and B.The rates of return,risks and minimum investment requirements for each fund are:
$\begin{array}{lll}\text { Fund }&\text { Rate of return }&\text { Risk }&\text { Minimum investment }\\\hline \text { A } & 12 \% & 0.5&\$20,000 \\\text { B } & 9 \% & 0.3&\$10,000\end{array}$
Note that a low Risk rating means a less risky investment.The investor wants to maximize the expected rate of return while minimizing his risk.Any money beyond the minimum investment requirements can be invested in either fund.The investor has found that the maximum possible expected rate of return is 11.4% and the minimum possible risk is 0.32.
The following Excel spreadsheet has been created to solve a goal programming problem with a MINIMAX objective based on the following goal programming formulation with MINIMAX objective and corresponding solution.
MINIMIZE Q
Subject to: ^X₁ ^{+ X}₂ ^{= 50000}
X₁ ≥ 20000
X₂ ≥ 10000
$w _ { 2 } \left( \frac { \left( \frac { 0.5 X _ { 1 } + 0.3 X _ { 2 } } { 50000 } \right) - 0.32 } { 0.32 } \right) \leq Q$
$w _ { \mathrm { i } } \left( \frac { \left( 0.114 - \frac { 0.12 X _ { 1 } + 0.09 X _ { 2 } } { 50000 } \right) } { 0.114 } \right) \leq Q$
$w_{\mathrm{i}}\left(\frac{\left(0.114-\frac{0.12 X_{1}+0.09 X_{2}}{50000}\right)}{0.114}\right) \leq Q$
$w_{2}\left(\frac{\left(\frac{0.5 x_{i}+0.3 X_{2}}{50000}\right)-0.32}{0.32}\right) \leq Q$
X_i ≥ 0 for all i,Q ≥ 0
with solution X₁,X₂)= 15,370,34,630).
What values should go in cells B2:D14 of the spreadsheet?
$\begin{array}{|c|l|c|c|c|c|}\hline &{\text { A }} & \text { B } & \text { C } & \text { D } & \text { E } \\\hline 1 & \text { Problem data } & \text { A } & \text { B } & & \\\hline 2 & \text { Expected retum } & & & & \\\hline 3 & \text { Risk rating } & & & & \\\hline 4 & & & & \\\hline 5 & \text { Variables } & \text { A } & \text { B } & \text { Total } \\\hline 6 & \text { Amount invested } & & & \\\hline 7 & \text { Minimum required } & & & \\\hline 8 & & & & \\\hline 9 & & & & & \text { Weighted } \\\hline 10 & \text { Goals } & \text { Actual } & \text { Target } & \text { Weights } & \% \text { Deviation } \\\hline 11 & \text { Average return } & & & 1 & \\\hline 12 & \text { Average risk } & & & 1 & \\\hline 13 & & & & & \\\hline 14 & \text { Objective: } & & & &\\\hline\end{array}$

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Definitions:

Borda Count

A voting system used to rank candidates in which voters rank options, and rankings are then converted into scores to determine the winner.

Arrow's Impossibility Theorem

A theorem stating that no rank-order voting system can simultaneously fulfill all of a certain set of reasonable fairness criteria.

Pairwise Voting

A voting system in which choices are compared one pair at a time, with participants selecting their preference in each matchup.

Pairwise Voting

A voting system where participants compare options in pairs, and the option with the most preferences wins.