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. But when expressed as a , 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.

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



\text{Percentage change} = \frac{(\text{Final value} - \text{Initial value})}{\text{Initial value}} * 100% $$

The impact of percentage changes on the value of a $1,000 investment is listed in Table 1 below.


 * 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.

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:
 * 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 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.

Summary
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.

Spreadsheet
There is a spreadsheet on Google Drive.

Spreadsheets are also available on Google Drive for Microsoft Excel and LibreOffice Calc. These versions contain the chart used in Figure 1.

Each spreadsheet contains a worksheet for calculating centinepers described in the Appendix below.

Appendix: Other units
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.

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:


 * $$ D_{cNp} = 100 \cdot \ln\frac{V_2}{V_1} \approx 100 \cdot \frac{V_2 - V_1}{V_1} = \text{Percentage change} \text{ when }\left | \frac{V_2 - V_1}{V_1} \right | << 1 $$

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.