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$8\mathrm{.}7.$ A stockbroker, Richard Smith, has just received a call from his most important client, Ann Hardy. Ann has $$50,000$

Page $326$

to invest and wants to use it to purchase two stocks. Stock $1$ is a solid blue$-$chip security with a respectable growth

potential and little risk involved. Stock $2$ is much more speculative. It is being touted in two investment newsletters as having

outstanding growth potential, but also is considered very risky. Ann would like a large return on her investment, but also has

considerable aversion to risk. Therefore, she has instructed Richard to analyze what mix of investments in the two stocks would be

appropriate for her. She also informs him that her plan is to hold the stock being purchased now for three years before selling it$.$

After doing some research on the historical performances of the two stocks and on the current prospects for the companies

involved, Richard is able to make the following estimates. If the entire $$50,000$ were to be invested in Stock $1$ now, the profit when

sold in three years would have an expected value of $$12,500$ and a standard deviation of $$5,000.$ If the entire $$50,000$ were to be

invested in Stock $2$ now, the profit when sold in three years would have an expected value of $$20,000$ and a standard deviation of

$$30,000.$ The two stocks behave independently in different sectors of the market so Richard's calculation from historical data is that

the covariance of the profits from the two stocks is $0.$

Richard now is ready to use a spreadsheet model to determine how to allocate the $$50,000$ to the two stocks so as to minimize

Ann's risk while providing an expected profit that is at least as large as her minimum acceptable value. He asks Ann to decide what

her minimum acceptable value is$.$

a$.$ Without yet assigning a specific numerical value to the minimum acceptable expected profit, formulate a quadratic

programming model in algebraic form for this problem.

b$.$ Display this model on a spreadsheet.

c$.$ Solve this model for four cases: Minimum acceptable expected profit $=$$$13,000,$$$15,000,$$$17,000,$ and $$19,000.$

d$.$ Ann was a statistics major in college and so understands well that the expected return and risk in this model represent

estimates of the mean and standard deviation of the probability distribution of the profit from the corresponding

portfolio. Ann uses the notation $$ and $$ for the mean and standard deviation. She recalls that, for typical probability

distributions, the probability is fairly high $($about $0\mathrm{.}8$ or $0\mathrm{.}9)$ that the return will exceed $-,$ and the probability is

extremely high $($often close to $0\mathrm{.}999)$ that the profit will exceed $-3.$ Calculate $-$ and $-3$ for the four

portfolios obtained in part $c.$ Which portfolio will give Ann the highest $$ among those that also give $-0?$