Geometric Mean Return

The geometric mean return is the answer to: what steady annual rate would have produced exactly the same ending value?

Instead of adding returns, it multiplies growth factors. A year of +15.5% is a factor of 1.155; a year of −6.9% is 0.931. Multiply them all, take the root for the number of years, subtract 1.

Because it follows the money through each year, it depends only on the starting and ending values, not on the path in between. That is why the workbook calls it "the only way to compare long-term accumulations".

The workbook's method, step by step:

1. Convert each period return into a wealth relative — add 1 to the return in decimal form (−5% becomes 0.95).
2. Multiply the wealth relatives to get the cumulative wealth relative — what ₹1 grew to.
3. Raise it to the power 1/n, where n is the number of years.
4. Subtract 1.

On the workbook's five-year series (−5.00%, −15.20%, 8.10%, 30.75%, 17.65%):

0.95 × 0.848 × 1.081 × 1.3075 × 1.1765 = 1.3396
1.3396^(1/5) − 1 = 6.02%

The workbook states that TWRR is the same as geometric return, and that CAGR and annualised return use the same calculation.

GMR = [(1 + R₁) × (1 + R₂) × … × (1 + Rₙ)]^(1/n) − 1

CAGR = (Ending value ÷ Beginning value)^(1/t) − 1

The workbook's annualising example, applied to an illustrative ₹10 lakh PMS account. Returns: 2017 +15.5%, 2018 +9.5%, 2019 −6.9%.

Cumulative wealth relative = 1.1550 × 1.0950 × 0.9310 = 1.1775
GMR = 1.1775^(1/3) − 1 = 5.56%

Check: ₹10 lakh × 1.0556³ ≈ ₹11.77 lakh. ✔

The arithmetic average of the same three returns is (15.5 + 9.5 − 6.9) ÷ 3 = 6.03%. Compounding ₹10 lakh at 6.03% for three years would suggest about ₹11.92 lakh — roughly ₹14,600 more than the account actually holds.

Chapter 20 (Performance Measurement and Evaluation of Portfolio Managers) uses the geometric method four times: for TWRR (section 20.2.2), for the geometric mean return (20.2.3), for CAGR (20.2.6) and for annualising (20.2.7). A question can hand you any series of yearly returns and ask for the annualised figure, so the chain-linking steps are worth doing until they are automatic.

• Multiply wealth relatives; never multiply raw percentages. 15.5% × 9.5% is meaningless.
• Take the root for the number of years, not the number of data points plus one.
• Do not annualise a return for less than a year. The workbook says it is inadvisable, and the GIPS advertising rules forbid it.
• The workbook's CAGR table has a typo. Section 20.2.6 computes CAGR as 6.02%, but the follow-up table is headed "growing at the rate 6.0622%". The formula and the TWRR on the same returns (6.021%) both give 6.02%.
• GMR equals TWRR for a series of period returns; it is not the same as MWRR, which depends on cash flow timing.