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 , the percentage gained will be different than the percentage lost.

Market performance, along with anything else related to investing, is stated in terms of percentages. It's important to understand the context of how this information is presented.

Overview
The formula is expressed as a change from the initial value to the final value.



\text{Percent change} = \frac{(\text{Final value} - \text{Initial value})}{\text{Initial value}} * 100% $$ The equation is plotted in Figure 1 below. To show gains and losses in percentages alone, the actual value of the investment is not needed. This allows comparison of investment performance.

The plot shows that you always need a larger percentage increase to come back to the same value. For example, a loss of 10% requires an 11% increase to recover its value, while a loss of 50% requires a 100% increase (double its value) to recover. Conversely, a gain of 100% (double its value) only needs to drop 50% to return to the initial value.

A common misconception, explained
A common error investors make is to assume that percentages can be added and subtracted. Instead, they are actually a multiplication operation.

Question: When is this equation true?


 * P + (10%) - (10%) = P.

Answer: It's never true.

And +10% is not the inverse gain of a 10% loss, either. First, the percentages must be converted to a decimal or fractional representation. You cannot write:


 * P * (+10%) * (-10%) = P.

Again, this is not true.


 * +10% is equivalent to a decimal multiplication of 1.1
 * -10% is equivalent to a decimal multiplication of 0.9

And then you can see how a gain of 10% followed by a loss of 10% does not get you back to even. Because the two returns compounded after one another is equivalent to writing:


 * P * (1.1) * (0.9) = P * 0.99

which is not back to even.

If you do it old school (in fractions), then a 10% gain can be written as 11/10, or any multiple such as 110/100. A 10% loss can be written as 9/10, or 90/100.

Now, we can write the equation as:


 * P * (110/100) * (90/100) = P * (9900/10000)

and you see what's happening. Multiplying by 90/100 is not the inverse operation of multiplying by 110/100. The inverse operation of multiplying by 90/100 is multiplying by 100/90 (which is +11.11%).

And the inverse to "* (110/100)", is "* (100/110)" (which is +9.09%). They are never exactly equal, no matter how small the number.

So, a 10% loss requires an 11.1% gain to recover, and a 10% gain would take a 9.09% loss to be even.

In dollar terms, you did nothing but either gained, or lost, 10 coins on a hundred and then they were reversed back to 100.

This is why percentage gains compared to losses can never be equal.

A related misconception would be to assume that a continual sequence of +10%, -10%, +10%, -10%... gains and losses would result in breaking even. In reality, the value of your portfolio is declining over time. This is a mistake in the math of what an "equivalent" gain or loss is.

Perspective is important
While it is always true that a percentage loss will require a greater percentage increase to return to the same value, keeping this in perspective of "the big picture" is important.

For example, stock market drops are often reported in the news with attention-grabbing headlines. Wait a week or so, and those attention-grabbing headlines have disappeared. Why? Because the market has recovered.

When the market takes a large negative swing, recovery will take a longer time because the market needs a larger percentage increase to get back to the same point.

The important point is to stay the course. If you believe that the market has an underlying upward trend, then spending time out of the market is dropping you below that trend.

In practical terms, if you pull out during the drop you may miss some of the recovery, and that may become an unrecoverable sequence of events unless your timing is so good it compensates--and odds are your timing will not be good but instead neutral or worse.

A detailed example
For example: A $125 investment drops by $25, then increases by $25 to recover the initial $125 investment.


 * Drop by $25 :
 * Initial Value: $125
 * Drops to: $100
 * Loss in $: -25 = $100 - $125
 * Percent change: -20% = ($100 - $125) / $125


 * Then, increase by $25 :
 * From: $100
 * Back to initial value: $125
 * Gain in $: +25 = $125 - $100
 * Percent change: +25% = ($125 - $100) / $100

The example showed a sequence containing a $25 loss, then a $25 gain. The same amount of money was lost as was gained.

In percentage, the sequence is described as a 20% loss, then a 25% gain. To say this another way, you will need to gain much more in percentage than you have lost to return to the initial value.

The impact of percentage changes on the value of the above example ($125) is listed in Table 1 below. A spreadsheet is available on Google Drive for Microsoft Excel or LibreOffice Calc.