Explanation

Time-Weighted Return Calculation

Year 1:

• Initial investment: $200.00
• Dividend received: $5.00
• Ending value: $225.00 + $5.00 = $230.00 (price at t=1 plus dividend)
• Return for Year 1: (230/200) - 1 = 1.15 - 1 = 15%

Year 2:

• Beginning value: $225.00 (price of second share) + $225.00 (value of first share) = $450.00
• Dividends received: $5.00 × 2 shares = $10.00
• Ending value: $235.00 × 2 shares = $470.00 + $10.00 = $480.00
• Return for Year 2: (480/450) - 1 = 1.06667 - 1 = 6.667%

Time-weighted return: TWR = (1.15 × 1.06667)^{1/2} - 1 = (1.22667)^{0.5} - 1 = 1.1075 - 1 = 10.75% ≈ 10.8%

Money-Weighted Return Calculation

Cash flows:

• t=0: -$200.00 (initial purchase)
• t=1: -$225.00 (additional purchase) + $5.00 (dividend from first share) = -$220.00
• t=2: +$470.00 (sale of 2 shares) + $10.00 (dividends from 2 shares) = +$480.00

Solve for IRR (Money-weighted return): -200 - 220/(1+r) + 480/(1+r)^2 = 0

Let x = 1/(1+r): -200 - 220x + 480x^2 = 0 480x^2 - 220x - 200 = 0

Using quadratic formula: x = [220 ± √(220^2 - 4×480×(-200))] / (2×480) x = [220 ± √(48400 + 384000)] / 960 x = [220 ± √432400] / 960 x = [220 ± 657.57] / 960

Taking positive root: x = (220 + 657.57) / 960 = 877.57/960 = 0.9141

Since x = 1/(1+r): 1/(1+r) = 0.9141 1+r = 1/0.9141 = 1.094 r = 0.094 = 9.4%

Therefore:

• Time-weighted return: 10.8%
• Money-weighted return: 9.4%

Why the difference? The money-weighted return is lower because the investor invested more money ($225) when the stock price was higher, which reduces the overall return compared to the time-weighted return that simply measures the performance of the investment itself regardless of cash flow timing.