Returns across time, costs and leverage

Learning goals

• Choose arithmetic, geometric or continuously compounded returns for the question.
• Distinguish annualization, real returns, net returns and equity leverage.

Draft · approximately 60 minutes, excluding further revision.

Average does not mean compound growth

The arithmetic mean estimates the average of separate one-period observations. Wealth over consecutive periods instead grows by the product of their gross returns. The geometric mean is that product raised to 1/n, minus one. With variable returns it does not exceed the arithmetic mean. A +20% year followed by −20% leaves 96% of the original wealth, not 100%.

A log return is ln(1 + R); log returns add over time when the wealth measure is consistent. Convert back with exp(sum of log returns) − 1. It is undefined for a simple return at or below −100%. Annualizing a short sample describes a rate convention, not a forecast.

The investor receives a net, purchasing-power result

For an effective annual return R and annual inflation π, real return is (1 + R)/(1 + π) − 1. Tax and fee treatment must follow the stated cash-flow convention: a fee of 1% of ending gross assets makes net gross wealth (1 + R)(1 − 0.01), whereas a charge on beginning capital is subtracted differently. Do not silently treat every fee as a percentage-point subtraction.

If assets A are funded by debt D and equity E with A = D + E, a simple one-period equity return is (A × asset return − D × debt rate)/E, ignoring taxes and other costs. This amplifies losses as well as gains; collateral calls and insolvency are additional risks outside that arithmetic.

Volatility drag, annualization and leverage in one example

Start with 100, earn 20%, then lose 20%. Wealth becomes 120 then 96. Arithmetic mean return is 0%, but two-period total return is −4% and geometric mean is sqrt(0.96)−1≈−2.0204% per period. Averages answer different questions: an arithmetic one-period estimate does not reconstruct compounded wealth. Annualize only after identifying the length and reinvestment assumptions of each period.

Now finance a 100 asset with 50 equity and 50 borrowing. If the asset gains 10% and interest is 5% of debt, equity ends at 110−50−2.5=57.5: return 15%. If the asset loses 10%, equity ends at 37.5: return −25%. Leverage magnifies outcomes after funding cost; it does not create a free improvement. Margin calls and changing funding rates are excluded from this simple example.

Formula and assumptions

R_geo = [∏(1 + Rₜ)]^(1/n) − 1; r_log = ln(1 + R); R_real = (1 + R)/(1 + π) − 1

Compound, annualize, then deflate

1. Two yearly returns of +25% and −20% give 1.25 × 0.80 = 1.00: geometric mean 0%, arithmetic mean 2.5%.
2. A 3% effective quarterly return annualizes to 1.03⁴ − 1 = 12.5509%, assuming repeatable reinvestment at that rate.
3. At 4% annual inflation, its corresponding real annual rate is 1.12550881/1.04 − 1 = 8.2220%.

Calculation lab

Errors to catch

• Using arithmetic means to reconstruct cumulative wealth.
• Applying leverage to gains while ignoring interest and losses.

Original knowledge check

Answer every question before checking. Review explanations after submitting; retry starts a new attempt.

Sources and coverage status

These original lessons are drafts. Public outlines are used to check scope; the official learning-outcome text and proprietary question banks are not reproduced here. Follow the source for exam-year requirements and updates.

• Reference · www.cfainstitute.org