Let's break down the problem and explain the solution step by step. ### **Given:** - Security 1 (S1) price now: **$94.10** - S1 shares bought: **1,450** - Security 2 (S2) price now: **$82.53** - S2 shares bought: **1,000** - Initial Margin Requirement (IMR): **50%** - Maintenance Margin (MM): **25%** - Borrowed funds interest: **5% APR, monthly compounding** - No brokerage fees, no interest on cash deposits - All accounts aggregated - Security prices in 3 months: - S1: **$91.83** - S2: **$87.20** - Round answer to 4 decimals --- ## **Step 1: Calculate Initial Investment and Borrowed Amount** ### **Total Cost of Securities:** - S1: 1,450 × $94.10 = **$136,445.00** - S2: 1,000 × $82.53 = **$82,530.00** - **Total purchase = $136,445.00 + $82,530.00 = $218,975.00** ### **Initial Margin Requirement (IMR):** - IMR = 50% - **Your own money = 50% × $218,975.00 = $109,487.50** - **Borrowed amount = $218,975.00 - $109,487.50 = $109,487.50** --- ## **Step 2: Value of Portfolio After 3 Months** ### **New Values:** - S1: 1,450 × $91.83 = **$133,153.50** - S2: 1,000 × $87.20 = **$87,200.00** - **Total value = $133,153.50 + $87,200.00 = $220,353.50** --- ## **Step 3: Borrowed Amount After 3 Months (Interest Compounded Monthly)** ### **Monthly Interest Rate:** - APR = 5%, so monthly rate = 5% / 12 = **0.4167%** ### **Compounding for 3 Months:** - Future Value = Principal × (1 + i)^n - i = 0.004167, n = 3 \[ \text{Borrowed amount after 3 months} = 109,487.50 \times (1 + 0.004167)^3 \] \[ = 109,487.50 \times 1.012552 \] \[ = 110,861.4342 \] --- ## **Step 4: Compute Equity in the Account** \[ \text{Equity} = \text{Total portfolio value} - \text{Borrowed amount with interest} \] \[ = 220,353.50 - 110,861.4342 = 109,492.0658 \] --- ## **Step 5: Round to 4 Decimal Places** \[ \boxed{109,492.0658} \] --- ### **Summary Table** | Step | Amount | |----------------------------|---------------| | Your own money (IMR) | $109,487.50 | | Borrowed amount (initial) | $109,487.50 | | Portfolio value (3 months) | $220,353.50 | | Borrowed + interest | $110,861.4342 | | **Equity (rounded)** | **$109,492.0658** | --- **The correct answer, rounded to 4 decimals, should be:** \[ \boxed{109,492.0658} \]

Introduction: In the context of margin trading, investors borrow funds to purchase securities, leveraging their initial investment to potentially increase returns. Understanding the key concepts involved—such as initial margin requirement (IMR), maintenance margin (MM), and the impact of interest on borrowed funds—is essential for analyzing how the value of an account evolves over time. The initial margin specifies the proportion of the purchase price that must be funded with the investor’s own capital, while the remaining amount is borrowed. The maintenance margin represents the minimum equity percentage that must be maintained in the account to avoid margin calls. Additionally, interest charges on borrowed funds accrue over time, affecting the total liabilities. Calculating the account's equity after a specified period involves assessing the change in securities' value, the interest accrued on borrowed funds, and the initial leverage used, all of which are fundamental to effective margin trading management and risk assessment. Explanation: This introduction establishes the foundational concepts necessary for understanding the mechanics of margin accounts in investment scenarios. Recognizing how initial margin and maintenance margin requirements dictate the amount of leverage and the importance of interest calculations on borrowed funds provides clarity on the factors influencing account equity over time. These concepts underpin the subsequent calculations, enabling an accurate assessment of the account's value after three months, considering both market changes and financing costs. --- Presentation of Relevant Formulas Required To Solve The Question: 1. **Initial Purchase Cost:** \[ \text{Total Cost} = \text{Number of Shares} \times \text{Purchase Price per Share} \] *This formula calculates the total amount spent to acquire the securities at the initial time.* 2. **Initial Equity (Own Funds):** \[ \text{Initial Equity} = \text{Initial Margin Requirement} \times \text{Total Cost} \] *Represents the investor’s own capital invested initially, based on the margin requirement.* 3. **Borrowed Funds:** \[ \text{Borrowed Amount} = \text{Total Cost} - \text{Initial Equity} \] *The amount borrowed from the broker or lender.* 4. **Future Value of Borrowed Funds (Interest Accrued):** \[ \text{Future Borrowed Amount} = \text{Borrowed Amount} \times (1 + i)^n \] *Where \(i\) is the monthly interest rate, and \(n\) is the number of months. This accounts for interest accumulation over the period.* 5. **Portfolio Value after Period:** \[ \text{Market Value} = \sum (\text{Shares} \times \text{New Price per Share}) \] *Calculates the total value of securities at the end of the period based on new market prices.* 6. **Account Equity after Period:** \[ \text{Equity} = \text{Market Value of Securities} - \text{Future Borrowed Amount} \] *Determines the net value of the investor’s account after accounting for borrowed funds and accrued interest.* **Explanation:** These formulas are essential because they systematically relate initial investments, leverage, interest costs, and market valuation to derive the final account equity. They enable translating initial conditions and market changes into a quantifiable measure of account value, which is critical for risk management and decision-making in margin trading. --- Detailed Step-by-Step Solution: **Step 1: Calculate Initial Investment Costs** - *Market value of Security 1 (S1):* \(1,450 \times \$94.10 = \$136,445.00\) - *Market value of Security 2 (S2):* \(1,000 \times \$82.53 = \$82,530.00\) - *Total initial purchase cost:* \(\$136,445.00 + \$82,530.00 = \$218,975.00\) *Explanation:* The total cost reflects the combined market value of all purchased securities at the initial prices, forming the basis for margin calculations. **Step 2: Determine Initial Equity and Borrowed Funds** - *Initial Margin Requirement:* 50% - *Your own funds (initial equity):* \(50\% \times \$218,975.00 = \$109,487.50\) - *Borrowed amount:* \( \$218,975.00 - \$109,487.50 = \$109,487.50 \) *Explanation:* The initial equity is the investor’s own capital, while the borrowed amount is the leverage used, both critical for understanding risk and potential gains. **Step 3: Compute the Future Value of Borrowed Funds After 3 Months** - *Monthly interest rate:* \( 5\% / 12 = 0.0041667 \) - *Interest over 3 months:* \( (1 + 0.0041667)^3 = 1.012552 \) - *Interest accrued on borrowed funds:* \[ \$109,487.50 \times 1.012552 = \$110,861.4342 \] *Explanation:* Since interest is compounded monthly, the borrowed amount grows over the period, increasing the amount owed at the end of three months. **Step 4: Calculate Portfolio Value After 3 Months** - *New prices:* - S1: \( \$91.83 \) - S2: \( \$87.20 \) - *Value of S1 holdings:* \( 1,450 \times \$91.83 = \$133,153.50 \) - *Value of S2 holdings:* \( 1,000 \times \$87.20 = \$87,200.00 \) - *Total portfolio value:* \[ \$133,153.50 + \$87,200.00 = \$220,353.50 \] *Explanation:* The change in market prices affects the total value of securities, which is essential for evaluating the current account status. **Step 5: Determine the Equity in the Account After 3 Months** \[ \text{Equity} = \text{Market Value of Securities} - \text{Future Borrowed Amount} \] \[ = \$220,353.50 - \$110,861.4342 = \$109,492.0658 \] *Explanation:* Subtracting the accrued debt from the current market value yields the investor’s net equity, indicating the amount attributable to the investor after debt service. **Step 6: Final Result and Rounding** - Rounded to four decimal places: **109,492.0658** --- **Conclusion:** The equity in the account after three months, considering changes in security prices and interest on borrowed funds, totals approximately **\$109,492.0658**. This figure reflects the leverage effects, market fluctuations, and financing costs inherent in margin trading, illustrating the importance of understanding these variables in portfolio management and risk assessment.