Variable Savings Rates: Multiply the Path Instead of Averaging It

The sequence is a chain of factors

A variable savings yield requires a path rather than a single unexplained average. Each period takes the ending balance of the previous period as its starting point. Investor.gov includes a rate-variance input for illustrations; here the model instead assigns an explicit factor to each year. Begin with one plus each period's effective rate, then multiply those factors together. This method respects the base on which interest is earned. Averaging the rates before compounding changes the calculation because averaging and multiplying are different operations. A useful record keeps the dates and growth factors visible so a reader can distinguish an assumed scenario from a historical account reconstruction or a projection based on unknown future changes.

Supporting references: 1. SEC Investor.gov: Compound Interest Calculator

Arithmetic means describe rates, not cumulative wealth

The average of two and six percent is four percent. That statement correctly describes the arithmetic mean of two numbers. It does not establish that two years of four percent growth produce the same balance as one year at two percent and one at six percent. The worked example applies both approaches to USD 9,000. The difference follows from the cross-product terms in successive compounding. This is a mathematical consequence, not evidence that variable-rate accounts systematically underperform another product. The inputs were selected to illustrate an averaging error. A comparison involving actual accounts would also require consistent horizons, fees, balance methods, and information about what each quoted percentage represents.

Use a geometric equivalent for a single summary

When no external cash flows occur, a constant equivalent annual rate can be obtained from the cumulative growth factor. For two years, take its square root and subtract one. For n equal periods, take its nth root. The resulting geometric rate reproduces the same final balance if applied in every period. It is a summary of that particular chain, not the rate paid in either individual year. This distinction matters when communicating performance: someone can accurately report an equivalent rate while obscuring substantial variation along the way. For a savings goal, that variation may affect interim balances even when the terminal summary appears reasonable. Keep the original path available for any decision involving an earlier spending date.

Order changes exposure when money moves

For an untouched starting balance, multiplying a two-percent factor and a six-percent factor gives the same product in either order. Introduce a deposit between years and that symmetry disappears for the new money. The starting balance still experiences both rates, but the additional deposit experiences only the second year's rate. Withdrawals create the same timing issue in reverse. Therefore, a rate summary that works for a fixed lump sum cannot automatically describe a household's realized gains. Map each contribution to the factors it actually experiences. That method separates the return on existing funds from the effect of adding money just before a higher or lower growth interval.

Explain the measurement boundary

FINRA's discussion of return connects a gain with the amount committed. When the amount committed changes over time, the boundary of a return measure becomes particularly important. This article's rows deliberately exclude cash flows to isolate the arithmetic-versus-geometric distinction. A complete savings forecast should add deposits and withdrawals on their dates and apply the period factors sequentially. If future rates are unknown, use several explicitly hypothetical paths rather than treating the arithmetic average as a dependable substitute. The model should identify whether it is summarizing completed intervals or constructing possible future ones. That simple label prevents a technically correct cumulative-growth calculation from being mistaken for a promise about what a variable account will pay next.

Supporting references: 2. FINRA: Return and rate of return

Limits of this analysis

• The rates are invented rather than observed or predicted.
• Changing deposits make rate order relevant to household cash flows.
• Each input must be an effective rate for its stated period.