Solution
We first use the random numbers to simulate the states of the economy Δ, followed by the simulation of the states of the three different industries Δ
_{1}, Δ
_{2}, and Δ
_{3}. This allows us to compute the stock prices. All of these computations are shown in Table
14.16.
Table 14.16 Simulation of stock prices

Strategy 1: Leaving the entire amount of $1,000,000 in the shortterm account for 10 weeks will net us $1,001,000.45, or a 0.1 % gain.

Strategy 2: In week 1, none of the stocks has increased by at least 2 %, so that we keep our entire amount in the shortterm account. By the end of week 2, we have $1,000,200.01. Since Stock 2 increases by 3 % in week 2, we purchase it with half of the available money, i.e., $500,100. At the price of $61.38 a share, we obtain 8,147.6051 shares. The first time we can realize a gain is at the end of week 8, at which point we sell the shares at $61.99 a share for a total of $505,070.04. This money is kept in the shortterm account for 2 weeks, resulting in a payoff at the end of week 10 in $505,171.06. The remaining $500,100 that were not invested in week 2 will remain in the shortterm account, resulting in $504,114.83 for a total of $1,009,285.89 or an increase of 0.929 %.

Strategy 3: By the end of week 1, Stock 3 has declined in value, which leads the investor to invest half of the available money in that stock. The $1,000,000 has appreciated due to its investment in the shortterm account for 1 week, so that 1,000,100 are available, half of which ($500,050) are invested in Stock 3. Each share costs $101.97, so that 4,903.8933 shares are purchased. Since the shares will never exceed that value again during the 10 weeks, they will not be sold.
The remaining $500,050 are left in the shortterm account for 2 weeks until the end of Week 3, when they have appreciated to $500,150.02. As Stocks 1 and 2 decreased in value in Week 3, half of the available amount is invested in each. (Note, by the way, that Stock 3 decreased in Week 2, but since we still hold shares of that stock, we do not invest in it again). The sum of $250,075.01 is invested in Stock 1, which costs $16.66 per share, so that we obtain 15,010.5048 shares. We hold them until the end of Week 8, when their price increases to $16.83, which gives us $252,626.80.
Back to the end of Week 3, we invested $250,075.01 in Stock 2 at $60.77 a share, so that we obtain 4,115.1063 shares. We sell these shares at the end of week 5 for $61.38 each, resulting in $252,585.22. We hold this money in the shortterm account until the end of Week 8, when the investment in Stock 1 is liquidated. By that time, we have $505,212.02. Since none of the stocks declined in Week 8, we hold the amount in the shortterm account for a week, resulting in $505,262.54. We are now at the end of Week 9. During Week 9, we observed all stocks declining. As we still hold Stock 3, we cannot invest in it, so that we invest the entire remaining money in Stocks 1 and 2 in equal parts. For the 252,631.27 invested in Stock 1, we obtain 15,163.9418 shares, while for the same amount invested in Stock 2, we obtain 4,116.5271 shares of Stock 2.
Since none of the stock prices increases during Week 10, the account by the end of the planning period consists of 15,163.9418 shares of Stock 1, 4,116.5271 shares of Stock 2, and 4,903.8933 shares of Stock 3. The total value of the portfolio is thus $982,989.70, for a loss of 1.7 %.
Comparing the three strategies, it appears that the second investment strategy is best.