SKILL.md
Return Calculations
Core Concepts
Simple (Holding Period) Return
$$R = \frac{V_{end} - V_{begin} + D}{V_{begin}}$$
where D = distributions (dividends, interest) received during the period. If V_end already reflects reinvested distributions, do not add D again.
Mean and Log Return Conventions
- Arithmetic mean
R_a = (1/n) * sum(R_i)— unbiased estimate of the expected single-period return (use for forward-looking inputs, e.g., mean-variance optimization). Always >= geometric mean; overstates realized compound growth. - Geometric mean
R_g = [prod(1 + R_i)]^(1/n) - 1— the correct measure of realized multi-period compound growth. The gap below the arithmetic mean approximatessigma^2 / 2(volatility drag). - Log return
r = ln(V_end / V_begin)— time-additive (r_total = r_1 + ... + r_n), so preferred for statistical modeling and multi-period aggregation. Convert withR_simple = e^r - 1andr = ln(1 + R_simple). Log returns are additive across time but NOT across assets.
CAGR (Compound Annual Growth Rate)
$$CAGR = \left(\frac{V_{end}}{V_{begin}}\right)^{1/n} - 1$$