Final answer:
The student can use the $43,200 to purchase 981 shares of Sophie Shoes, borrowing $35,280 to meet the initial margin requirement and invest a total of $78,480 in stock.
Step-by-step explanation:
The student has $43,200 to invest in Sophie Shoes, with the stock priced at $80 per share. Given the initial margin requirement of 55 percent, the maximum purchase of stock can be computed. The margin requirement means that you can borrow up to 45% (100% - 55%) of the stock purchase price. Therefore, with the available investment cash, the student can buy shares to the point where the cost of the shares amounts to 55% of the combined value of the cash plus the loan amount.
First, we calculate the maximum total value of the stocks that can be purchased, including the borrowed amount:
Total Value = Investment Cash / Initial Margin Requirement = $43,200 / 0.55 = $78,545.45 (not rounded)
Next, we determine the number of shares that can be bought with this total value:
Number of Shares = Total Value / Price per Share = $78,545.45 / $80 ≈ 981.82 (not rounded)
Since a fraction of a share cannot be bought, the student can buy 981 full shares. The total investment in shares would therefore be:
Total Investment in Shares = Number of Shares * Price per Share = 981 * $80 = $78,480
The amount of money borrowed can then be calculated as:
Borrowed Amount = Total Investment in Shares - Investment Cash = $78,480 - $43,200 = $35,280