# 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 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.

${\text{Percentage change}}={\frac {({\text{Final value}}-{\text{Initial value}})}{\text{Initial value}}}*100\%$ The impact of percentage changes on the value of a $1,000 investment is listed in Table 1 below. Table 1. Percentage of gain or loss$1,000 initial investment
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 $0,000.00 - - -90% Loss$0,100.00 900% Gain
-80% Loss $0,200.00 400% Gain -70% Loss$0,300.00 233% Gain
-60% Loss $0,400.00 150% Gain -50% Loss$0,500.00 100% Gain
-40% Loss $0,600.00 067% Gain -30% Loss$0,700.00 043% Gain
-20% Loss $0,800.00 025% Gain -10% Loss$0,900.00 011% Gain
00% No change $1,000.00 000% No change 10% Gain$1,100.00 -09% 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%, 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.[note 1] 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:[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.

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.

## 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.