Percentage gain and loss
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. Expressed as a Percentage gain and loss, the percentage gained will be different than the percentage lost. This is because the same dollar amount is being 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". 
Contents
Overview
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 
100%  Loss  $ 0.00     
90%  Loss  $100.00  900%  Gain 
80%  Loss  $200.00  400%  Gain 
70%  Loss  $300.00  233%  Gain 
60%  Loss  $400.00  150%  Gain 
50%  Loss  $500.00  100%  Gain 
40%  Loss  $600.00  67%  Gain 
30%  Loss  $700.00  43%  Gain 
20%  Loss  $800.00  25%  Gain 
10%  Loss  $900.00  11%  Gain 
0%  No change  $1,000.00  0%  No change 
10%  Gain  $1,100.00  9%  Loss 
20%  Gain  $1,200.00  17%  Loss 
30%  Gain  $1,300.00  23%  Loss 
40%  Gain  $1,400.00  29%  Loss 
50%  Gain  $1,500.00  33%  Loss 
60%  Gain  $1,600.00  38%  Loss 
70%  Gain  $1,700.00  41%  Loss 
80%  Gain  $1,800.00  44%  Loss 
90%  Gain  $1,900.00  47%  Loss 
100%  Gain  $2,000.00  50%  Loss 
 With a loss of 10%, one needs a gain of about 11% to recover. (A market correction)^{[1]}
 With a loss of 20%, one needs a gain of 25% to recover. (A bear market)
 With a loss of 30%, one needs a gain of about 43% to recover.
 With a loss of 40%, one needs a gain of about 67% to recover.
 With a loss of 50%, one needs a gain of 100% to recover. (That's right, 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. To show gains and losses in percentages alone, the actual value of the investment is not needed.
After a percentage loss, the plot shows that you always need a larger percentage increase to come back to the same value.^{[note 1]}
The concept can be shown with a simple example.^{[2]}
 $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.^{[3]}^{[4]} 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:^{[5]}
 $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)
where:
 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've 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.
Summary
There are three key points:
 Percentages are a ratio, which can only use multiplication (or division)
 The period of time over which the performance is measured matters.
 When measuring performance, the actual value of the investment is not needed. This allows an "applestoapples" comparison of different investments.
Spreadsheet
A spreadsheet is available on Google Drive.
(View Google Spreadsheet in browser, then File > Download as to download the file.)
Spreadsheets are also available on Google Drive for Microsoft Excel and LibreOffice Calc.^{[note 3]} These versions contain the chart used in Figure 1.
Each spreadsheet contains a worksheet for calculating centinepers described in the Appendix below.^{[6]}
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.
The neper (Np) is a unit of logarithmic change. One property of the natural logarithm is that small changes in value very closely approximate percentage change.^{[7]}^{[8]}
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:^{[7]}
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).^{[7]}
Logarithms are also used for compounding (an investment's return) and to display economic data directly as percentage change.^{[9]}
Notes
 ↑ 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 addedto or withdrawnfrom your investment. See: The Effect Of Compounding, on Investopedia, viewed June18, 2017.
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.
See also
 Risk tolerance
 Comparing investments
 Rate of return (How to calculate return when money is addedto or withdrawnfrom an investment)
 Variance drain (For experienced investors)
References
 ↑ Bogleheads® forum post: Re: [Wiki]  Percentage Gain and Loss (for new investors), by forum member Peter Foley.
 ↑ Bogleheads® forum post: Re: [Wiki]  Percentage Gain and Loss (for new investors), by forum member TD2626.}}
 ↑ Bogleheads® forum post: Re: [Wiki]  Percentage Gain and Loss (for new investors), by forum member livesoft.
 ↑ Bogleheads® forum post: Re: [Wiki]  Percentage Gain and Loss (for new investors), followup post by forum member livesoft.
 ↑ Compound Interest, mathisfun.com, viewed June 17, 2017.
 ↑ Bogleheads® forum post: Re: [Wiki]  Percentage Gain and Loss (for new investors), based on tables supplied by forum member #Cruncher.
 ↑ ^{7.0} ^{7.1} ^{7.2} Relative change and difference, Wikipedia, viewed June 19, 2017.
 ↑ The logarithm transformation, Robert F. Nau, Duke University: The Fuqua School of Business, viewed June 19, 2017.
 ↑ Use of logarithms in economics, Econbrowser, viewed June 19, 2017.
External links
 Bogleheads® forum topic: [Wiki]  Percentage Gain and Loss (for new investors)

