return-calculations

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 approximates sigma^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 with R_simple = e^r - 1 and r = ln(1 + R_simple). Log returns are additive across time but NOT across assets.

CAGR (Compound Annual Growth Rate)

$$CAGR = \left(\frac{V_{end}}{V_{begin}}\right)^{1/n} - 1$$

where n is measured in years. The annualized geometric growth rate between two valuations with no intermediate cash flows.

Time-Weighted Return (TWR)

Chain-links sub-period returns calculated between each external cash flow, removing the effect of cash flow timing. TWR measures the manager's investment skill independent of investor deposit/withdrawal decisions, and is the GIPS standard for manager performance.

$$1 + R_{TWR} = \prod_{i=1}^{n}(1 + R_i), \qquad R_i = \frac{V_{end,i}}{V_{begin,i} + CF_i} - 1$$

Exact TWR requires a portfolio valuation on every cash flow date.

Modified Dietz Return

When valuations on each cash flow date are unavailable, Modified Dietz approximates the period return by day-weighting each external cash flow within the period:

$$R_{MD} = \frac{V_{end} - V_{begin} - CF_{net}}{V_{begin} + \sum_i CF_i \times w_i}, \qquad w_i = \frac{CD - D_i}{CD}$$