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What an account earned: money-weighted (XIRR), time-weighted and Modified Dietz returns when a large deposit comes before a fall

Tool	Personal Rate of Return
Source	CFA Institute (2010). GIPS Guidance Statement on Calculation Methodology. CFA Institute, Charlottesville, VA (sub-period returns, geometric linking and the Modified Dietz formula); CFA Institute (2019). Global Investment Performance Standards (GIPS) for Firms 2020 (definitions of money-weighted and time-weighted return). The case is illustrative (made-up account values), not a printed example.
Level	Undergraduate

Problem

An account is worth 10,000 on 31 December 2023. The owner deposits 2,000 on 28 June 2024 (account at 11,200 just before), deposits 30,000 on 31 December 2024 (account at 14,520) and withdraws 5,000 on 30 June 2025 (account at 38,733). On 31 December 2025 it is worth 36,094. Find the gain in money and the money-weighted, time-weighted and Modified Dietz returns, and say why they differ.

Inputs to Enter

Opening 2023-12-31, value 10,000; closing 2025-12-31, value 36,094; T = 731 days = 731 / 365 = 2.0027 years (actual / 365)
Rows (date, flow, value before): 2024-06-28, +2,000, 11,200; 2024-12-31, +30,000, 14,520; 2025-06-30, −5,000, 38,733
Advanced: benchmark 7 % a year; fees paid from outside the account 150 on 2024-12-31 and 400 on 2025-12-31

Open the Personal Rate of Return tool (loaded with this example)

Answers

Net deposits 2,000 + 30,000 − 5,000 = 27,000; gain = 36,094 − 10,000 − 27,000 = −906.00
Time-weighted: sub-periods 11,200 / 10,000 − 1 = +12.00 %, 14,520 / 13,200 − 1 = +10.00 %, 38,733 / 44,520 − 1 = −13.00 %, 36,094 / 33,733 − 1 = +7.00 %; linked 1.12 × 1.10 × 0.87 × 1.07 = 1.1469, +14.69 % over the period, 1.1469^(365/731) − 1 = +7.08 % a year
Modified Dietz: weights 551/731 = 0.7538, 365/731 = 0.4993, 184/731 = 0.2517; weighted flows 15,228.45; −906 / (10,000 + 15,228.45) = −3.59 % over the period, −1.81 % a year
Money-weighted (XIRR): trial −1.809552 % gives a closing value of 36,089.45; the next trial −1.800477 % gives 36,094.00; r = −1.80 % a year, −3.57 % over the period
The money-weighted return is 8.88 points a year below the time-weighted return: the account held the most money, 44,520, in the one sub-period that returned −13.00 %
Advanced: the same cash flows at 7 % a year reach 11,451.12 + 2,215.07 + 32,100.00 − 5,173.48 = 40,592.71, 4,498.71 above the account; with the outside fees the gain is −1,456.00 and the money-weighted return −2.89 % a year

Notes

The tool's default inputs are this case; press Submit. Year fractions are actual days / 365; a row's value is the account value just before that day's cash flow. Every figure is checked against an independent scipy / numpy implementation (brentq on the present-value equation; test_personal_rate_of_return.php). A calculator for education and analysis, not investment advice.