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. This is because the same dollar amount is being 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%, one needs a gain of about 11% to recover. (A market correction)
 * 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.

The concept can be shown with a simple example.
 * $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'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 "apples-to-apples" comparison of different investments.

Spreadsheet
A spreadsheet is available 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.