Compound returns and external cash flows

Compounding describes a linked capital base

For simple periodic returns, cumulative return is the product of one plus each return, minus one. PerformanceAnalytics documents this geometric chaining explicitly. Each period's return acts on the capital remaining from the previous period, so adding simple returns generally gives a different answer. The input must already represent an investment return rather than an unadjusted change in account balance. Deposits and withdrawals alter the amount invested without themselves being gains or losses. A sequence containing those balance changes can be multiplied perfectly and still answer the wrong economic question. Correct compounding begins with a correctly defined return series.

Arithmetic mean and growth are different summaries

The arithmetic mean averages individual period returns; the geometric mean is the constant per-period return that reproduces their cumulative wealth factor. For equal-length periods with positive gross returns, the geometric mean cannot exceed the arithmetic mean. This is a mathematical relationship, not evidence about a strategy. Returns must also be distinguished from log returns: summing log returns corresponds to multiplying positive gross simple returns, but the summed log value is not itself the simple cumulative return. A return of -1 represents complete loss in the ordinary funded account convention; subsequent percentage returns cannot reconstruct wealth without new capital.

External flows separate balance changes from performance

The account boundary determines what counts as an external flow. A client contribution enters the account from outside; a security sale that leaves proceeds inside the account is not the same event. Exact time-weighted measurement separates the investment history at external-flow boundaries and chains the resulting subperiod returns. The GIPS handbook explains this distinction and the need for valuations around those boundaries. Without those valuations, cash-flow-adjusted approximations introduce timing assumptions. Merely subtracting total contributions from the ending balance identifies a monetary gain or loss, but does not determine a rate of return on capital that was invested for different lengths of time.

One timeline produces different correct answers

The table is an original synthetic hypothetical in arbitrary currency units. An account begins with 100, earns 0.20 over year one and reaches 120. A contribution of 100 arrives exactly at that year-end boundary, before year two begins. The resulting 220 earns -0.10 and ends at 198. Time-weighted cumulative return is 1.20 times 0.90 minus 1, or 0.08. The arithmetic mean of the two annual returns is 0.05, while their geometric annual mean is 0.03923048. Ending balance grew by 98, but total contributed capital was 200, leaving a monetary investment loss of 2.

Money weighting answers the investor's timing question

For the same hypothetical timeline, the annual money-weighted internal rate of return solves 100 times (1 + q) squared plus 100 times (1 + q) = 198. The economically admissible solution with positive gross return is q = -0.00668155. It is negative because more capital experienced the losing year. This does not contradict the positive time-weighted result: the two measures weight investment experience differently. Cash-flow sign conventions, dates and terminal valuation determine the IRR problem. More complicated patterns of contributions and distributions can produce multiple roots or no suitable root, so an IRR is not automatically a uniquely defined answer.

Annualization does not repair missing information

For n equally spaced return periods and k periods per year, the mathematical annualized growth rate is the cumulative wealth factor raised to k divided by n, minus one, when that operation is defined. Here the two periods are full years, so the exponent is one half. Irregular dates require an explicit elapsed-time convention instead of an assumed row frequency. Money-weighted returns additionally require the timing of each flow. A calculator accepting only a return vector cannot infer deposits, reconstruct missing valuations or solve an unspecified cash-flow problem. No annualization rule supplies those absent inputs or turns a short history into a forecast.

What a reconciliation can establish

A reconciliation separates opening capital, external flows, investment gains or losses and closing capital before comparing percentage summaries. Return histories also need consistent treatment of dividends, fees, taxes and currency translation. A total-return sequence assumes income is included according to its stated convention; a price sequence may omit it. The synthetic account demonstrates that balance growth, monetary profit, arithmetic average, compound growth and money-weighted return can legitimately disagree. It does not certify an actual account or claim compliance with a reporting standard. GIPS is cited for measurement concepts, while these calculations remain educational examples without an assertion of implemented cash-flow accounting.

Primary references

These sources support methods and conventions. The example values are original constructions, not observations extracted from these documents.

• PerformanceAnalytics documentation: Return.cumulative
• GIPS Standards Handbook for Firms: Input data and calculation methodology

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