Investment problem: mean, variance, etc.

David wants to invest his $4 million inheritance. He walks into
Stock-o-rama and examines the information about four particular stocks.
David will sell his stocks after exactly one year.

Each of the stocks will pay out a particular amount in one year, times
the amount invested. According to the experts at Stock-o-rama, the
following vectors represent the mean return per dollar invested for each
stock, and the standard deviations of each stock's return.

Stock 1 Stock 2 Stock 3 Stock 4

Mean 1.04 1.09 1.03 1.07

Standard Deviation 0.03 0.04 0.01 0.05



So, if David invests $1 million in Stock 1, then the mean amount of
money he will get in one year is $1.04 million, and the standard
deviation of the amount of money is $0.03 million.

The Stock-o-rama experts also claim that the returns of the four stocks
are relevant to each other. The following matrix represents the
correlation of each stock with each other.

Stock 1 Stock 2 Stock 3 Stock 4

Stock 1 1 -0.06 0.44 0.71

Stock 2 -0.06 1 0.09 0.60

Stock 3 0.44 0.09 1 -0.81

Stock 4 0.71 0.60 -0.81 1



David will invest his $4 million in these four stocks, and cares about
the amount of money he will get in one year. His portfolio will be worth
the sum of the worth of the stocks he buys.

represents the variance of that amount of money, then David wants to
maximize



where t = $5 million.(Here, t is a measure of David's fear of high
variance, known as David's risk tolerance) Which of the following
portfolios will David prefer?

a) $4 million invested in Stock 2

b) $2 million each in Stock 2 and Stock 4

c) $2 million each in Stock 3 and Stock 4

d) $1 million in each stock

And calculate the mean and variance of each of the four portfolios a, b,
c, and d.