When an investment changes value, the dollar amount needed to return to its initial (starting) value is the same as the dollar amount of the change - but opposite in sign. But when expressed as a Percentage gain and loss, the percentage gained will be different from the percentage lost. This is because the same dollar amount is expressed as a percentage of two different starting amounts.
Percentages can be misleading if not combined correctly. For example, will a market loss of 10% followed by a gain of 10% get you back to the same point? This article explains why the answer is "No".
The formula is expressed as a change from the initial value to the final value.
The impact of percentage changes on the value of a $1,000 investment is listed in Table 1 below.
|If the value changes by||Getting back to the|
initial value requires a
|Percent||Gain or loss||New value||Change of||Gain or loss|
|0%||No change||$1,000.00||0%||No change|
- With a loss of 10%, you need a gain of about 11% to recover. (A market correction)
- With a loss of 20%, you need a gain of 25% to recover. (A bear market)
- With a loss of 30%, you need a gain of about 43% to recover.
- With a loss of 40%, you need a gain of about 67% to recover.
- With a loss of 50%, you need a gain of 100% to recover. (That is, if you lose half your money you need to double what you have left to get back to even.)
- With a loss of 100%, you are starting over from zero. And remember, anything multiplied by zero is still zero.
Here is the same equation shown as a graph. Showing gains and losses in percentages alone does not need the actual value of the investment.
After a percentage loss, the plot shows that you always need a larger percentage increase to come back to the same value.[note 1]
A simple example shows this.
- $1,000 = starting value
- $ 900 = $1,000 - (10% of $1,000), a drop of 10%
- $ 990 = $ 900 + (10% of $900), followed by a gain of 10%
The ending value of $990 is less than the starting value of $1,000.
A different perspective
Here is another way to express the same idea. You have an initial investment of $1,000. At the end of the first year, your investment goes down by 10%. Your investment then grows by 10% at the end of the second year.
- Starting value = $1,000
- First year return = -10% = -0.10
- Second year return = +10% = +0.10
At the end of the first year, you will have:
- $900 = $1,000 + ($1,000 * (-0.10)) = Starting value + (investment return)
We rearrange the formula to look like this:
- $900 = ($1,000 * 1) + ($1,000 * (-0.10))
- $900 = $1,000 * (1 + (-0.10))
The value at the end of the second year is calculated in the same way:
- $990 = (Starting value at the end of year 1) * (1 + 0.10)
- $990 = $1,000 * (1 + (-0.10)) * (1 + 0.10)
If we only wanted to know the percentage change from the initial investment to the end of the second year, the equation would look like this:[note 2]
- Starting value * (1 + P3) = Starting value * (1 + P1) * (1 + P2)
- P1 is the first year return
- P2 is the second year return
- P3 is the return over the 2 year period
We want to find P3. Since the starting value is common to both sides, it can be dropped.
- (1 + P3) = (1 + P1) * (1 + P2)
- P3 = ((1 + P1) * (1 + P2)) - 1
In this example:
- P3 = ((1 + P1) * (1 + P2)) - 1
- -0.01 = ((1 + (-.10)) * (1 + 0.10)) - 1
To say this another way, your investment returned -0.01 (a loss of 1%) over 2 years.
This means that you have ended up with 1% less than what you have started with. This is the same result as shown in Table 1 above. A 10% loss requires an 11% gain to break even.
Adding a 10% loss followed by 10% gain results in no change (breaking even, or 0% = -10% + 10%), which is not correct. This is why percentages cannot be added.
There are three key points:
- Percentages are a ratio, which can only use multiplication (or division)
- The period of time over which you measure performance matters.
- When measuring performance, you do not need the actual value of the investment. This allows an "apples-to-apples" comparison of different investments.
There is a spreadsheet on Google Drive.
(View Google Spreadsheet in browser, then File --> Download as to download the file.)
Note: If the spreadsheet is blank, select a different sheet, then back to that sheet. The image will be refreshed.
Appendix: Other units
This section is intended for those familiar with logarithms and is not necessary for understanding the concepts presented in the previous sections.
Change in a quantity can also be expressed logarithmically. Multiplication and division operations (ratios) become addition and subtraction of logarithms.
Normalization with a factor of 100, as done for percent, yields the derived unit centineper (cNp), which aligns with the definition for percentage change for very small changes:
An X cNp change in a quantity following a −X cNp change returns that quantity to its original value. For example, if an investment return doubles, this corresponds to a 69.3 cNp change (an increase). When it halves again, it is a −69.3 cNp change (a decrease).
Logarithms are also used for compounding (an investment's return) and to display economic data directly as percentage change.
- It is also true that a percentage gain will require a smaller percentage decrease to return to the same value.
- Multiplication of the terms "(1 + P1) * (1 + P2)" is known as compounding, meaning that you are reinvesting the proceeds of your investment. No money is added to or withdrawn from your investment. See: Compounding Interest: Formulas and Examples, on Investopedia, viewed August 25, 2023.
For example, "Compound interest" is the term used for the investment return of a bank CD. The interest paid every year is added to the value of the CD. All of the reinvested interest is paid to you when the CD matures.
- The LibreOffice Calc version corrects a compatibility issue with the Microsoft Excel chart. The chart will not display in Google Drive, but is present in the downloaded file.
- Risk tolerance
- Comparing investments
- Rate of return (How to calculate return when money is added to or withdrawn from an investment)
- Variance drain (For experienced investors)
- Bogleheads forum post: , by forum member Peter Foley.
- Bogleheads forum post: , by forum member TD2626.
- Bogleheads forum post: , by forum member livesoft.
- Bogleheads forum post: , follow-up post by forum member livesoft.
- "Compound Interest". mathisfun.com. Retrieved August 25, 2023.
- Bogleheads forum post: , based on tables supplied by forum member #Cruncher.
- "Relative change and difference". Wikipedia. Retrieved August 25, 2023.
- Robert F. Nau. "The logarithm transformation". Duke University: The Fuqua School of Business. Retrieved August 25, 2023.
- "Use of logarithms in economics". Econbrowser. Retrieved August 25, 2023.
- Bogleheads forum topic: