Explanation

To calculate the money-weighted return (MWR), we need to find the internal rate of return (IRR) that equates the present value of cash flows to zero.

Step 1: Determine cash flows

Year 1:

• Beginning investment: +$2,500 (cash outflow from investor's perspective)
• End-of-year value after -20% return: $2,500 × (1 - 0.20) = $2,000
• Withdrawal: $0
• Net cash flow at end of Year 1: $0

Year 2:

• Beginning investment: +$1,500 (additional cash outflow)
• Starting value: $2,000 + $1,500 = $3,500
• End-of-year value after 65% return: $3,500 × (1 + 0.65) = $5,775
• Withdrawal: -$500 (cash inflow to investor)
• Net cash flow at end of Year 2: +$500 (inflow)

Year 3:

• Beginning investment: +$1,000 (additional cash outflow)
• Starting value: $5,775 - $500 + $1,000 = $6,275
• End-of-year value after -25% return: $6,275 × (1 - 0.25) = $4,706.25
• Withdrawal: -$500 (cash inflow to investor)
• Net cash flow at end of Year 3: +$500 (inflow)

Year 4:

• Beginning investment: $0
• Starting value: $4,706.25 - $500 = $4,206.25
• End-of-year value after 10% return: $4,206.25 × (1 + 0.10) = $4,626.875
• Withdrawal: $0
• Final value: $4,626.875 (cash inflow at end of Year 4)

Step 2: Set up cash flow timeline

From investor's perspective:

• t=0 (beginning Year 1): -$2,500
• t=1 (beginning Year 2): -$1,500
• t=2 (end Year 2): +$500
• t=3 (beginning Year 3): -$1,000
• t=3 (end Year 3): +$500
• t=4 (end Year 4): +$4,626.875

Combining cash flows at same time points:

• t=0: -$2,500
• t=1: -$1,500
• t=2: +$500
• t=3: -$1,000 + $500 = -$500
• t=4: +$4,626.875

Step 3: Solve for IRR

We need to find r such that:

-2,500 + (-1,500)/(1+r) + 500/(1+r)² + (-500)/(1+r)³ + 4,626.875/(1+r)⁴ = 0

Testing the options:

• Option A (2.15%): NPV = -2,500 - 1,500/1.0215 + 500/1.0215² - 500/1.0215³ + 4,626.875/1.0215⁴ ≈ -$2,500 - $1,468.43 + $479.08 - $469.02 + $4,253.15 ≈ $294.78

• Option B (3.96%): NPV = -2,500 - 1,500/1.0396 + 500/1.0396² - 500/1.0396³ + 4,626.875/1.0396⁴ ≈ -$2,500 - $1,442.86 + $462.38 - $444.74 + $3,985.22 ≈ $61.00

• Option C (7.50%): NPV = -2,500 - 1,500/1.075 + 500/1.075² - 500/1.075³ + 4,626.875/1.075⁴ ≈ -$2,500 - $1,395.35 + $432.90 - $402.70 + $3,472.43 ≈ -$392.72

The IRR is the rate that makes NPV = 0. At 3.96%, NPV is closest to zero ($61), while at 2.15% it's $294.78 and at 7.50% it's -$392.72.

Step 4: Verification

Using financial calculator or Excel: CF0 = -2500 CF1 = -1500 CF2 = 500 CF3 = -500 CF4 = 4626.875 IRR ≈ 3.96%

Therefore, the money-weighted return is closest to 3.96%.