Selecting an Investment Portfolio:

An investment manager wants to determine an optimal portfolio for a wealthy client. The fund

has $$2\mathrm{.}5$ million to invest, and its objective is to maximize total dollar return from both growth

and dividends over the course of the coming year. The client has researched eight high$-$tech

companies and wants the portfolio to consist of shares in these firms only. Three of the firms

$($S$1-$S$3)$ are primarily software companies, three $(H1-H3)$ are primarily hardware companies, and

two $($C$1-$C$2)$ are internet consulting companies. The client has stipulated that no more than $40$

percent of the investment be allocated to any one of these three sectors$.$ To ensure

diversification, at least $$100,000$ must be invested in each of the eight stocks. Moreover, the

number of shares invested in any stock must be a multiple of $1000.$

The table below gives estimates from the investment company's database relating to these

stocks. These estimates include price per share, the projected annual growth rate in the share

price, and the anticipated annual dividend payment per share.

a$.$ Determine the maximum return on the portfolio. What is the optimal number of shares

to buy for each of the stocks? What is the corresponding amount invested in each

stock?

b$.$ If there were no integer restriction on the number of the shares invested, how would

your solution $($in part a$)$ change?