Hypothetical worked example and primary educational references; not observed market data.
Successive changes act on changing bases
A positive percentage change and an equal negative percentage change do not generally cancel. Each change applies to the balance present at its own date. Investor.gov's compounding framework follows this repeated-base mechanism; the illustration here allows a negative factor change to teach mathematical literacy. It does not describe a normal savings account's interest behavior or recommend a volatile asset. The initial increase expands the base, so the later equal-percentage decrease removes more currency than the first increase added. Reversing the order produces the same final product for an untouched lump sum. The mechanism is multiplication of factors, which an arithmetic average of the percentage labels does not preserve.
Supporting references: 1. SEC Investor.gov: Compound Interest Calculator
Calculate the path before summarizing it
The example begins with USD 5,000. One path rises ten percent and then falls ten percent; another remains unchanged in both periods. The arithmetic average change is zero in both paths, but their final balances differ. Multiply 1.10 by 0.90 for the first path, then apply that product to principal. Additions of the two percentage labels cannot reconstruct the result. The table reports ending currency amounts so the base effect is visible. The scenario is intentionally designed, not an observed sequence of returns. It provides a clean demonstration of why a zero average percentage can coexist with a lower endpoint without invoking fees, taxes, or any particular investment product.
Hypothetical worked example
Two-period paths with the same arithmetic-average change
Model setup
Read these assumptions before comparing the scenarios.
- All monetary amounts in this example are hypothetical U.S. dollars (USD), not current prices, quoted account terms or observed market data. Percentages, counts, time periods and index levels retain their stated units.
- Start with USD 5,000; no contributions or withdrawals.
- Path A changes +10% and then -10%.
- Path B stays unchanged in both periods; this is a mathematical stress example, not ordinary deposit behavior.
Calculated result (USD)
- 1. +10%, then -10%
- 4,950.00
- 2. 0%, then 0%
- 5,000.00
Expanded comparison scale, not zero-based. Range: 4,940.00 to 5,010.00, in the units above. Case numbers match the labels; no probability or time trend is implied.
Hypothetical model, not observed market data or a forecast. Assumptions above; original values and calculations below.
| Hypothetical scenario | Calculated result (USD) |
|---|---|
| +10%, then -10% | 4,950.00 |
| 0%, then 0% | 5,000.00 |
Calculation details
- +10%, then -10%
- 4,950.00 USD
((5000 * 1.1000000000000001) * 0.90000000000000002) - 0%, then 0%
- 5,000.00 USD
(5000 * (1 ^ 2))
A geometric average reproduces the endpoint
For two equal periods, take the square root of the cumulative factor and subtract one to obtain the constant effective change per period that would reproduce the same endpoint. For more periods, use the corresponding root. This geometric average is the appropriate fixed-lump-sum growth summary when no external cash flows occur. It differs from the arithmetic mean because it preserves the multiplication. An unchanged two-period path has a geometric average of zero; the up-and-down path has a negative one. The summary still compresses the interim movement. Someone needing money at the first-period date would care about that intermediate balance, not only the geometric rate calculated from the final endpoint.
The general equal-change identity explains the loss
If a balance rises by a fraction a and then falls by the same fraction, the combined factor is one plus a multiplied by one minus a. The product is one minus a squared. For any nonzero a smaller than one, the final factor is below one. This is an algebraic statement, not an empirical rule about particular markets. It also reveals why the size of the opposing changes matters: the loss depends on the square of their magnitude. A calculator showing the final balance unchanged for such a path may be adding percentages instead of multiplying factors. The identity provides a direct check that is independent of the chosen starting currency amount.
Keep growth summaries distinct from advice
FINRA's discussion of return emphasizes the relationship between gain and money committed. The endpoint calculation here uses one untouched initial amount; it does not describe the experience of deposits arriving between periods. New money would have different exposures, and its timing could make rate order matter. The article also excludes every contractual detail and protection question. Its purpose is to teach percentages, not suggest that savings should be placed in a volatile instrument. A transparent explanation states the factors, dates, cash-flow boundary, and summary method. That allows a reader to detect a misleading average while remembering that a correct geometric calculation says nothing by itself about suitability, future outcomes, or the reliability of the inputs.
Supporting references: 2. FINRA: Return and rate of return
Test another situationA midpoint withdrawal exposes what the endpoint average omits
A separate hypothetical case. Its inputs differ from the main worked example above.
A midpoint withdrawal exposes what the endpoint average omits
Assume hypothetical USD 2,000 first rises 25% and then falls 20%, with no fees. An untouched balance returns exactly to USD 2,000 because the factors multiply to one. Now remove USD 1,000 between periods and retain it without earnings elsewhere. The remaining account ends at USD 1,200 and combined resources are USD 2,200. Reversing the factor order with the same withdrawal leaves USD 750 in the account and USD 1,750 combined. The USD 450 difference reflects the withdrawn money's exposure. These are mathematical value stresses, not ordinary deposit interest. The trade-off is interim cash removal versus continued participation in the subsequent positive or negative change.
An equivalent rate summarizes only the declared boundary
FINRA ties return to the money committed. For an untouched lump sum, a geometric equivalent preserves the endpoint, but inserting external flows changes the question. A single endpoint divided by starting principal then mixes performance with those flows. Preserve the original factor path and date each contribution or withdrawal before interpreting household outcomes. A constant equivalent can conceal a low interim balance even if it reproduces the final one. That matters for a payment due before the horizon ends. The calculation demonstrates why a mathematically correct summary need not answer the separate question of how much usable money exists on an intermediate date.
Supporting references: 2. FINRA: Return and rate of return
Check this case’s calculations
- Untouched compound endpoint
- 2,000.00 USD
((2000 * 1.25) * 0.80000000000000004) - Up-first account after withdrawal
- 1,200.00 USD
(((2000 * 1.25) - 1000) * 0.80000000000000004) - Up-first combined resources
- 2,200.00 USD
(1200 + 1000) - Down-first account after withdrawal
- 750.00 USD
(((2000 * 0.80000000000000004) - 1000) * 1.25) - Down-first combined resources
- 1,750.00 USD
(750 + 1000) - Sequence difference with withdrawal
- 450.00 USD
(2200 - 1750)
Limits of this additional case
- The value changes are hypothetical stresses, not claimed insured-deposit behavior; retained withdrawals earn zero.
Limits of this analysis
- Negative balance changes are not asserted as typical insured-deposit interest.
- Rates may describe different periods and require matching before comparison.
- External cash flows invalidate a simple endpoint growth summary.
Sources and scope
- SEC Investor.gov: Compound Interest Calculator
Growth illustrations distinguish time, contributions, estimated rate, and compounding frequency.
Scope: US educational calculator; assumptions are not forecasts.. Reference accessed 2026-10-06.
- FINRA: Return and rate of return
Percentage return relates gain to the amount committed; inflation and costs affect interpretation.
Scope: US investor education; no portfolio or product recommendation.. Reference accessed 2026-10-06.
Illustrative calculations are not forecasts or personal recommendations. Editorial standards
