# 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 Percentage gain and loss, 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 $1,000 investment is listed in Table 1 below.  If the value changes by Getting back to theinitial value requires a Percent Gain or Loss New Value Change of Gain or Loss -100% Loss$ 0.00 - - -90% Loss $100.00 900% Gain -80% Loss$200.00 400% Gain -70% Loss $300.00 233% Gain -60% Loss$400.00 150% Gain -50% Loss $500.00 100% Gain -40% Loss$600.00 67% Gain -30% Loss $700.00 43% Gain -20% Loss$800.00 25% Gain -10% Loss $900.00 11% Gain 0% No change$1,000.00 0% No change 10% Gain $1,100.00 -9% 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%, 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.[note 1]

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:[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'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.

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.