Percentage gain and loss - Bogleheads (2024)

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.

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.

• With a loss of 10%, you need a gain of about 11% to recover. (A market correction)^[1]
• 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.

Figure 1. Percentage gain and loss

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.^[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 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.

(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.

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

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 XcNp change in a quantity following a −XcNp change returns that quantity to its original value. For example, if an investment return doubles, this corresponds to a 69.3cNp change (an increase). When it halves again, it is a −69.3cNp 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

See also

• 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)

References

External links

• Bogleheads forum topic: "[Wiki] - Percentage Gain and Loss (for new investors)"