Answer: 1.93%

## Explanation To calculate the time-weighted rate of return (TWRR), we need to compute the holding period returns for each period between cash flows, then link them geometrically. ### Step 1: Identify the periods and cash flows **Period 1: Beginning of Year 1 to End of Year 1** - Initial investment: 10 shares × €160 = €1,600 - End of period value: 10 shares × €168 = €1,680 - Dividend received: 10 shares × €3.00 = €30 - Cash flow at end of period: Purchase 5 shares × €168 = €840 (outflow) **Period 2: End of Year 1 to End of Year 2** - Beginning value: 15 shares × €168 = €2,520 - End of period value: 15 shares × €175 = €2,625 - Dividend received: 15 shares × €4.00 = €60 - No cash flows during this period **Period 3: End of Year 2 to End of Year 3** - Beginning value: 15 shares × €175 = €2,625 - End of period value: 15 shares × €165 = €2,475 - Dividend received: €0 - Cash flow at end: Sell 15 shares × €165 = €2,475 (inflow) ### Step 2: Calculate holding period returns (HPR) **HPR for Period 1:** \[ HPR_1 = \frac{Ending\ Value + Dividend - Cash\ Flow}{Beginning\ Value} - 1 \] \[ HPR_1 = \frac{1,680 + 30 - 840}{1,600} - 1 = \frac{870}{1,600} - 1 = 0.54375 - 1 = -0.45625 \] **HPR for Period 2:** \[ HPR_2 = \frac{Ending\ Value + Dividend}{Beginning\ Value} - 1 \] \[ HPR_2 = \frac{2,625 + 60}{2,520} - 1 = \frac{2,685}{2,520} - 1 = 1.065476 - 1 = 0.065476 \] **HPR for Period 3:** \[ HPR_3 = \frac{Ending\ Value + Dividend - Cash\ Flow}{Beginning\ Value} - 1 \] \[ HPR_3 = \frac{2,475 + 0 - 2,475}{2,625} - 1 = \frac{0}{2,625} - 1 = 0 - 1 = -1 \] ### Step 3: Calculate time-weighted rate of return \[ TWRR = (1 + HPR_1) \times (1 + HPR_2) \times (1 + HPR_3) - 1 \] \[ TWRR = (1 - 0.45625) \times (1 + 0.065476) \times (1 - 1) - 1 \] \[ TWRR = (0.54375) \times (1.065476) \times (0) - 1 = 0 - 1 = -1 \] Wait, this gives -100% which is incorrect. Let me recalculate properly. Actually, for Period 3, the cash flow (sale) should be treated differently. When we sell at the end, the ending value before cash flow is €2,475, and then we receive that amount as cash flow. The proper calculation is: **HPR for Period 3 (corrected):** \[ HPR_3 = \frac{Ending\ Value + Dividend}{Beginning\ Value} - 1 \] \[ HPR_3 = \frac{2,475 + 0}{2,625} - 1 = 0.942857 - 1 = -0.057143 \] Now recalculate TWRR: \[ TWRR = (1 - 0.45625) \times (1 + 0.065476) \times (1 - 0.057143) - 1 \] \[ TWRR = (0.54375) \times (1.065476) \times (0.942857) - 1 \] \[ TWRR = 0.54375 \times 1.065476 = 0.579355 \] \[ 0.579355 \times 0.942857 = 0.546429 \] \[ TWRR = 0.546429 - 1 = -0.453571 \] This still gives -45.36%, which is not matching any options. Let me reconsider the approach. Actually, for time-weighted return, we should calculate returns for subperiods between cash flows: **Subperiod 1: Beginning to End of Year 1 (before purchase)** - Beginning: 10 shares × €160 = €1,600 - Ending before purchase: 10 shares × €168 = €1,680 - Dividend: €30 - Return = (1,680 + 30)/1,600 - 1 = 1,710/1,600 - 1 = 0.06875 **Subperiod 2: After purchase to End of Year 2** - Beginning: 15 shares × €168 = €2,520 - Ending: 15 shares × €175 = €2,625 - Dividend: €60 - Return = (2,625 + 60)/2,520 - 1 = 2,685/2,520 - 1 = 0.065476 **Subperiod 3: End of Year 2 to End of Year 3** - Beginning: 15 shares × €175 = €2,625 - Ending: 15 shares × €165 = €2,475 - Dividend: €0 - Return = 2,475/2,625 - 1 = 0.942857 - 1 = -0.057143 **TWRR = (1.06875 × 1.065476 × 0.942857)^{1/3} - 1** \[ = (1.06875 × 1.065476) = 1.138393 \] \[ 1.138393 × 0.942857 = 1.073684 \] \[ TWRR = 1.073684^{1/3} - 1 = 1.02398 - 1 = 0.02398 \] **TWRR ≈ 2.40%** This matches option B (2.40%). **Final Answer: B. 2.40%**