Your client's portfolio gained 25% last year and lost 20% this year. She says her average return is 2.5%. She is wrong. She broke even. The difference between arithmetic and geometric mean is exactly the trap the exam sets.
HPR captures the total dollar gain on an investment (price change plus any income received) over whatever length of time you actually held it. Because it does not annualize, it is period-agnostic, which is why every annualized measure starts from HPR as its building block. HPR = (ending value - beginning value + income) / beginning value. A stock bought at $10,000, paying $400 in dividends, now worth $11,200: HPR = ($11,200 - $10,000 + $400) / $10,000 = 16%. HPR does not annualize.
Simple average of periodic returns. Returns of 20%, -10%, and 15% over three years: (20 + (-10) + 15) / 3 = 8.33%. The arithmetic mean is the best unbiased estimate of the expected return for any single future period. It always overstates compound growth, and the overstatement increases with volatility.
Common mistakes
- Confusing arithmetic and geometric mean. +25% then -20%: arithmetic = 2.5%, geometric = 0%. The exam loves this trap. Check whether the question asks for "average annual return" (arithmetic) or "compound annual growth" (geometric). Trap: 2.5% when the question asks about actual wealth growth.
- Using Treynor on an undiversified portfolio. If R-squared is low, beta does not capture the full risk picture. A portfolio with beta 0.8 and R-squared 0.40 has massive unsystematic risk that Treynor ignores. Use Sharpe instead. Trap: an impressive Treynor ratio from a portfolio with R-squared of 0.35.
- Calculating alpha as actual minus market return. Alpha = actual minus CAPM expected. 11% minus 9.10% = 1.90%, not 11% minus 10% = 1.00%. The beta adjustment is what makes alpha risk-adjusted. Trap: 1.00%.
Bottom line
- Arithmetic mean forecasts single periods; geometric mean measures actual compound growth. Geometric is always less than or equal to arithmetic.
- TWR = manager performance (strips cash flows); DWRR/IRR = investor experience (includes cash flows).
- R-squared > 0.70 = Treynor (beta); R-squared < 0.70 = Sharpe (std dev). This single check answers a disproportionate number of questions.
- Jensen's alpha = actual return minus CAPM expected return, not actual minus market return.
Exam shortcut
When a question gives beta, standard deviation, R-squared, risk-free rate, and portfolio return all at once, look at R-squared first. Above 0.70 = Treynor. Below 0.70 = Sharpe. Then check alpha for context. This decision framework answers a disproportionate number of D.30 questions. "Geo is always grounded", geometric ≤ arithmetic; the gap widens with volatility.
The full lesson (about 2,928 words, 20 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.
Learning objectives
- D.30
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