A finance manager is responsible for investing in a portfolio of stocks to maximize

returns while minimizing risk. The manager has identified five potential investment

options with the following expected returns and standard deviations:

$$ Stock A: Expected return $=8\%,$ Standard deviation $=12\%$

$$ Stock B: Expected return $=10\%,$ Standard deviation $=15\%$

$$ Stock C: Expected return $=12\%,$ Standard deviation $=18\%$

$$ Stock D: Expected return $=9\%,$ Standard deviation $=14\%$

$$ Stock E: Expected return $=11\%,$ Standard deviation $=16\%$

The manager can allocate investments in any proportion to these stocks, but the total

investment must sum up to $$1$ million. Considering a risk aversion coefficient of $0\mathrm{.}05,$

formulate a linear programming model to determine the optimal allocation of

investments to maximize the expected return while minimizing the risk.