A 12% gain on a stock and a 12% gain on a leveraged real-estate position are not the same return. Same number, different meaning. This lesson teaches you how to read return figures correctly and how to decompose required returns into their building blocks.
The single-period return on an asset is the holding period return (HPR). Buy a share at $50, collect a $2 dividend, sell at $54: HPR = ($54 + $2 − $50) / $50 = 12%.
where:
- is the price at the start of the holding period
- is the price at the end of the holding period
- is the income (dividend or interest) received during the period
To chain multiple HPRs, compound them. Three years of 10%, −5%, and 15% give total HPR = (1.10)(0.95)(1.15) − 1 = 20.18%.
DECISION: Use arithmetic mean to estimate next period's expected return. Use geometric mean to describe realized compound growth across past periods.
Common mistakes
- Adding nominal and real instead of compounding. The approximation nominal ≈ real + inflation works only for small numbers. For exam-quality precision, use the Fisher equation: (1 + nominal) = (1 + real)(1 + inflation). Trap value: writing 7% − 3% = 4% real instead of 1.07/1.03 − 1 = 3.88%.
- Confusing arithmetic and geometric. Arithmetic mean overstates compound growth. If the question asks "what did the portfolio actually compound at," use geometric. If it asks "what is the expected single-period return," use arithmetic.
- Multiplying instead of compounding when annualizing. A 3% quarterly return is NOT 12%. It is (1.03)^4 − 1 = 12.55%.
Bottom line
- Holding period return (HPR) = (Ending + Income) / Beginning − 1. Compound HPRs to chain multi-period returns
- Geometric mean ≤ arithmetic mean, gap widening as volatility rises. Geometric measures realized compound growth; arithmetic measures expected single-period return
- Money-weighted return = IRR (sensitive to cash flow timing). Time-weighted return = compounded HPRs (manager-agnostic to flows)
- Annualize by compounding, not multiplying: . A 3% quarterly return is 12.55%, not 12%
Exam shortcut
For nominal-to-real: if numbers are small (under 5%), subtraction approximates well; if larger, use exact Fisher. For arithmetic vs geometric, the verb tells you. "Expected" points to arithmetic, "compounded" or "realized" points to geometric. For money-weighted vs time-weighted: a large cash flow before a strong period flatters money-weighted, while a flow before a weak period penalizes it. Time-weighted stays clean either way.
The full lesson (about 1,876 words, 13 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.
Learning objectives
- returns of financial assets and instruments
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