Why a 10% Investment Loss Followed by a 10% Gain Does Not Restore Your Principal
An investment does not return to its original value simply because it gains 10% after losing 10%. Starting with USD 100, a 10% decline reduces the investment to USD 90. A subsequent 10% increase brings it to USD 99. That is USD 1 less than the initial investment, resulting in a final return of -1%. To get back to the original USD 100, the investment must rise by approximately 11.11% from its reduced value.
This difference occurs because the percentage decline and subsequent percentage increase are calculated using different starting amounts.
1. How USD 100 Falls to USD 90 and Then Rises to USD 99
A 10% decline and a 10% increase do not cancel each other out because the two percentages are applied to different amounts.
A 10% loss on USD 100 amounts to USD 10, while a 10% gain on USD 90 amounts to USD 9.
The following is a hypothetical example designed to explain the calculation, not an actual historical price record for a stock or cryptocurrency.
| Stage | Calculation | Investment Value |
|---|---|---|
| Initial investment | Starting amount | USD 100 |
| 10% decline | 100 × 0.90 | USD 90 |
| 10% increase | 90 × 1.10 | USD 99 |
| Final result | 99 - 100 | -USD 1 |
| Return relative to initial investment | (99 - 100) ÷ 100 × 100 | -1% |
Author's calculation
USD 100 × 0.90 × 1.10 = USD 99
Final return = (99 ÷ 100 - 1) × 100 = -1%
Although the initial investment was USD 100, only USD 99 remains after the two price changes.
The crucial distinction is not the percentage increase or decrease itself, but the starting amount to which each percentage applies.
2. Why Equal Loss and Gain Percentages Produce Different Investment Values
The percentage loss is calculated using the investment's value before the decline, while the subsequent percentage gain is calculated using its value after the decline.
As a result, even when both percentages are 10%, the amounts lost and recovered are different.
When USD 100 falls to USD 90, the investment loses USD 10.
By contrast, when USD 90 increases by 10%, the gain amounts to USD 9.
The difference between these two changes is USD 1.
When calculating investment returns over multiple periods, the correct approach is to multiply the successive return factors rather than simply add the percentage returns.
This mathematical relationship is also explained in the Finance & Quant Society's educational material on investment losses and recovery.
Successive returns can be calculated using the following relationship.
Final investment value = Initial investment × (1 + first-period return) × (1 + second-period return)
Entering the 10% decline and 10% increase as -0.10 and +0.10, respectively, gives the following calculation.
100 × (1 - 0.10) × (1 + 0.10) = 99
In other words, adding -10% and +10% and concluding that the return is 0% does not accurately describe the actual change in investment value.
3. How Much Must an Investment Rise to Fully Recover a 10% Loss?
Recovering from a 10% loss requires a gain of approximately 11.11%.
To return from USD 90 to the original USD 100, the investment must increase by USD 10. The required percentage gain is calculated using the current value, USD 90, as the starting point.
Author's calculation
Required gain (%) = (Target amount - current amount) ÷ current amount × 100
= (100 - 90) ÷ 90 × 100
= 11.111…%
Therefore, the precise percentage gain required to recover the initial investment is 11.111…%, or approximately 11.11% when rounded to two decimal places.
If the percentage loss is known, the gain required to recover the principal can be calculated regardless of the original investment amount.
Required gain to recover principal (%) = Loss (%) ÷ [100 - loss (%)] × 100
For example, if the loss is 10%, the calculation is as follows.
10 ÷ (100 - 10) × 100 = approximately 11.11%
This formula applies when no additional deposits or withdrawals are made and separate expenses, including fees and taxes, are excluded.
4. How Much Must an Investment Rise to Recover Losses of 20%, 30%, or 50%?
As losses grow, the percentage gain needed to recover the original investment increases faster than the percentage loss itself.
For example, if an investment falls by 50%, its remaining value is half the original investment. Returning to the original amount from that level requires a 100% increase.
The following table shows the remaining investment values and required recovery gains for different percentage losses, calculated directly using an initial investment of USD 100.
| Percentage Loss | Remaining Investment Value | Gain Required to Recover Principal |
|---|---|---|
| 5% | USD 95 | Approximately 5.26% |
| 10% | USD 90 | Approximately 11.11% |
| 20% | USD 80 | 25% |
| 30% | USD 70 | Approximately 42.86% |
| 40% | USD 60 | Approximately 66.67% |
| 50% | USD 50 | 100% |
Author's calculations. Required recovery gains have been rounded to two decimal places. Additional deposits, withdrawals, and transaction costs are excluded.
The Caibull Academy's educational material on the mathematics of recovering investment losses also explains that recovery rates must be calculated using the reduced asset value after a loss, and that recovering from a 20% loss requires a 25% gain.
The difference becomes particularly clear when examining the recovery gain required after a 30% loss.
If USD 100 falls to USD 70 and then rises by 30%, the investment reaches USD 91. Recovering the original USD 100 requires a gain of approximately 42.86%.
As the size of the loss increases, a rebound of the same percentage is not enough to restore the original investment value.
5. Does the Result Change If an Investment First Gains 10% and Then Loses 10%?
If an investment starts at USD 100, rises by 10%, and then falls by 10%, its final value is still USD 99.
Changing the order of the returns does not change the calculated result when the same two return factors are multiplied.
The following hypothetical examples illustrate the comparison.
| Order of Price Changes | Change in Investment Value | Final Return |
|---|---|---|
| 10% decline → 10% increase | 100 → 90 → USD 99 | -1% |
| 10% increase → 10% decline | 100 → 110 → USD 99 | -1% |
In the first scenario, the investment loses USD 10 and then recovers USD 9.
In the second scenario, the investment gains USD 10, but a subsequent 10% decline calculated from USD 110 reduces its value by USD 11.
In both scenarios, the final investment value is USD 99, which is USD 1 below the original USD 100.
However, these mathematical results do not mean that actual investment profits will be identical in every situation.
Additional purchases, sales, deposits, withdrawals, or transaction costs during the investment period can change an investor's final profit or loss.
6. What Should You Use as the Basis for Calculating Actual Investment Returns?
To determine whether an investment has recovered its original value, investors should examine both the reference amount and the current value rather than relying solely on the most recent percentage gain.
A recent 10% increase does not necessarily mean that an earlier 10% loss has been fully recovered.
When evaluating investment returns, it is useful to distinguish among three figures.
Initial investment: The starting amount used to compare gains and losses.
Current market value: The investment's current value after price changes.
Required gain to recover principal: The percentage increase needed to return from the current market value to the reference amount.
For example, if an initial investment of USD 100 declines to a market value of USD 90, the loss relative to the original investment is 10%.
However, the starting amount used to calculate the recovery gain is the current market value of USD 90.
It is also important to distinguish between a recovery in market price and an investor's actual break-even point.
Trading fees, taxes, exchange rate fluctuations, dividends or distributions, and additional deposits or withdrawals can all affect actual investment results. Price changes alone cannot fully account for these factors.
In particular, the U.S. dollar-denominated return on a foreign-currency asset may differ from its Korean won-denominated return because of exchange rate movements.
The calculations in this article are simplified hypothetical examples using a consistent monetary basis and excluding these additional factors.
7. Frequently Asked Questions
If a Stock Falls by 10% and Then Rises by 10%, What Is the Remaining Loss?
If a stock falls by 10% after the initial purchase and then rises by 10% from the reduced price, it remains 1% below its original price.
Without additional transactions or costs, the return relative to the initial investment is -1%.
If the Loss Is 10%, Does a 11.11% Gain Restore the Investment to Exactly Break Even?
The precise recovery gain is 11.111…%.
11.11% is an approximation rounded to two decimal places, so an extremely small mathematical difference may remain.
Does the Same Calculation Apply to Bitcoin and ETFs as Well as Stocks?
Yes. The same mathematical relationship applies to stocks, ETFs, cryptocurrencies, and other assets when successive percentage changes are applied to the same underlying investment value.
However, an investor's actual profit or loss may differ depending on the product's structure and trading conditions.
How Much Must an Investment Rise to Recover a 100% Loss?
If the investment's market value has fallen to exactly 0, no finite percentage increase applied to that amount can restore the original principal.
This is the extreme case of the mathematical relationship in which the required recovery gain grows as the percentage loss approaches 100%.
To Determine Whether an Investment Has Recovered, Check the Starting Amount Before the Percentage Gain
If USD 100 falls by 10%, it becomes USD 90. A subsequent 10% increase from that amount brings it to USD 99.
The final loss relative to the initial investment is 1%, while returning from USD 90 to USD 100 requires a gain of approximately 11.11%.
The reason the percentage loss and the required recovery gain differ is that each calculation uses a different starting amount.
When comparing investment performance, it is more accurate to calculate the final value after applying each period's return and compare that value with the original reference amount than to simply add the percentage gains and losses.
※ This article is provided for informational purposes to help readers understand market behavior. It is not a recommendation to invest in any particular security or asset. Readers are responsible for their own investment decisions and their consequences.
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