Let's solve the IRR (Internal Rate of Return) for Smith's transaction, **removing all commission expenses** as you requested. We will express the answer in terms of \( i^{(2)} \) (the effective semiannual rate). --- ### **Step 1: Timeline and Cash Flows** Let's break down each event and calculate the cash flows. #### **Time 0 (Purchase 1,000 shares at $5.00)** - Shares bought: 1,000 - Price per share: \$5.00 - **Total cash outflow:** \$5,000 - **No commission** (since we're ignoring commissions). **CF at t = 0:** -\$5,000 --- #### **Time 0.5 years (6 months) (Dividend and Reinvestment)** - Dividend: \$0.20 per share × 1,000 = \$200 - Smith immediately reinvests the \$200 in new shares at \$4.00 each. - Shares bought: \$200 / \$4.00 = 50 shares - **Total shares after this:** 1,050 shares **CF at t = 0.5:** \$0 (dividend is immediately reinvested, so no cash flow out of the system) --- #### **Time 1 year (12 months) (Buys 500 shares at \$4.50)** - Shares bought: 500 - Price per share: \$4.50 - **Total cash outflow:** \$2,250 - **No commission.** - **Total shares after purchase:** 1,050 + 500 = 1,550 shares **CF at t = 1:** -\$2,250 --- #### **Time 1.5 years (18 months) (Dividend and Sale)** - Dividend: \$0.25 per share × 1,550 = \$387.50 - Smith sells all shares at \$5.00 per share. - Proceeds from sale: 1,550 × \$5.00 = \$7,750 - **Total inflow at this time:** \$7,750 (sale) + \$387.50 (dividend) = \$8,137.50 **CF at t = 1.5:** +\$8,137.50 --- ### **Step 2: Summarize the Cash Flows** | Time (years) | Cash Flow (\$) | |--------------|---------------| | 0 | -5,000 | | 0.5 | 0 | | 1.0 | -2,250 | | 1.5 | +8,137.50 | --- ### **Step 3: Set up IRR Equation** Let \( i^{(2)} \) be the **effective semiannual rate**. Let \( v = 1 / (1 + i^{(2)}) \). The IRR equation is: \[ -5,000 + 0 \cdot v^{1} + (-2,250) \cdot v^{2} + 8,137.50 \cdot v^{3} = 0 \] or: \[ -5,000 - 2,250 \cdot v^{2} + 8,137.5 \cdot v^{3} = 0 \] --- ### **Step 4: Solve for \( v \) (and thus \( i^{(2)} \))** Let’s denote \( v = 1 / (1 + i^{(2)}) \). So, \[ -5,000 - 2,250 v^{2} + 8,137.5 v^{3} = 0 \] Let's solve numerically. #### **Guess and check method (or use a financial calculator / spreadsheet):** Let’s try \( i^{(2)} = 10\% \) (so \( v = 1/1.10 = 0.90909 \)): - \( v^2 = 0.82645 \) - \( v^3 = 0.75131 \) Plug in: \[ -5,000 - 2,250 \times 0.82645 + 8,137.5 \times 0.75131 \\ = -5,000 - 1,860 + 6,115.69 = -744.31 \] Not zero, but close. Try \( i^{(2)} = 8\% \) (\( v = 1/1.08 = 0.92593 \)) - \( v^2 = 0.85714 \) - \( v^3 = 0.79365 \) \[ -5,000 - 2,250 \times 0.85714 + 8,137.5 \times 0.79365 \\ = -5,000 - 1,928.57 + 6,457.23 = -471.34 \] Still negative, but closer. Try \( i^{(2)} = 5\% \) (\( v = 1/1.05 = 0.95238 \)) - \( v^2 = 0.90703 \) - \( v^3 = 0.86104 \) \[ -5,000 - 2,250 \times 0.90703 + 8,137.5 \times 0.86104 \\ = -5,000 - 2,040.82 + 7,010.43 = -30.39 \] We're very close to zero now. Try \( i^{(2)} = 4.8\% \) (\( v = 1/1.048 = 0.95418 \)) - \( v^2 = 0.91046 \) - \( v^3 = 0.86910 \) \[ -5,000 - 2,250 \times 0.91046 + 8,137.5 \times 0.86910 \\ = -5,000 - 2,048.54 + 7,077.48 = +28.94 \] So, for \( i^{(2)} = 4.8\% \), the function is +28.94 For \( i^{(2)} = 5.0\% \), the function is -30.39 #### **Linear Interpolation:** \[ IRR = 4.8\% + \frac{28.94}{28.94 + 30.39} \times (5.0\% - 4.8\%) \] \[ = 4.8\% + \frac{28.94}{59.33} \times 0.2\% \] \[ = 4.8\% + 0.0975\% \] \[ \approx 4.90\% \] --- ### **Step 5: Conclusion** **The IRR (semiannual effective rate) for Smith's transaction, ignoring all commissions, is approximately:** \[ \boxed{i^{(2)} \approx 4.9\%} \] ----- ## **Summary Table** | Time | Cash Flow (\$) | Description | |------|------------------|-------------------------------------------------------| | 0 | -5,000 | Buy 1,000 shares at \$5.00 | | 0.5 | 0 | Reinvest dividend (\$200) into 50 shares at \$4.00 | | 1.0 | -2,250 | Buy 500 shares at \$4.50 | | 1.5 | +8,137.50 | Sell all 1,550 shares at \$5.00 and receive dividend | **Conclusion:** The effective semiannual rate of return (IRR) on Smith's transactions, with all commissions removed, is **approximately 4.9% per half-year**. If you wish to find the annual equivalent, you can use \( (1 + i^{(2)})^2 - 1 \). For \( 4.9\% \), this is about 10.05% annual effective return.