Geometric mean return (time-weighted return) is the most appropriate method for performance measurement as it does not consider additions to or withdrawals from the account. This is particularly important for a buy-and-hold investor making regular deposits because:

1. Time-weighted return (geometric mean) eliminates the impact of cash flows (deposits/withdrawals), allowing for fair comparison of investment manager performance.
2. Money-weighted return (internal rate of return) is affected by the timing and size of cash flows, making it inappropriate for evaluating portfolio performance when the investor controls the timing of deposits.
3. Arithmetic mean return is a simple average that doesn't account for compounding effects and is less suitable for multi-period performance evaluation.

For a buy-and-hold strategy with regular deposits, the geometric mean provides the most accurate measure of the portfolio's compound growth rate over time, independent of the investor's deposit schedule.