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.

Percentages can be misleading if not combined correctly. For example, will a market loss of 10% followed by a gain of 10% get you back to the same point? This article explains why the answer is "No". |

## 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.

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)
^{[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.

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

This section is intended for those familiar with logarithms and is not necessary for understanding the concepts presented in the previous sections. |

See also: Neper (Applications)

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 *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 69.3cNp 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

- ↑ It is also true that a percentage gain will require a smaller percentage decrease to return to the same value.
- ↑ Multiplication of the terms "(1 + P1) * (1 + P2)" is known as
*compounding*, meaning that you are reinvesting the proceeds of your investment. No money is added to or withdrawn from your investment. See: Compounding Interest: Formulas and Examples, on Investopedia, viewed August 25, 2023.

For example, "Compound interest" is the term used for the investment return of a bank CD. The interest paid every year is added to the value of the CD. All of the reinvested interest is paid to you when the CD matures. - ↑ The LibreOffice Calc version corrects a compatibility issue with the Microsoft Excel chart. The chart will not display in Google Drive, but is present in the downloaded file.

## 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

- ↑ Bogleheads forum post: "Re: [Wiki] - Percentage Gain and Loss (for new investors)", by forum member Peter Foley.
- ↑ Bogleheads forum post: "Re: [Wiki] - Percentage Gain and Loss (for new investors)", by forum member TD2626.
- ↑ Bogleheads forum post: "Re: [Wiki] - Percentage Gain and Loss (for new investors)", by forum member livesoft.
- ↑ Bogleheads forum post: "Re: [Wiki] - Percentage Gain and Loss (for new investors)", follow-up post by forum member livesoft.
- ↑ "Compound Interest". mathisfun.com. Retrieved August 25, 2023.
- ↑ Bogleheads forum post: "Re: [Wiki] - Percentage Gain and Loss (for new investors)", based on tables supplied by forum member #Cruncher.
- ↑
^{7.0}^{7.1}^{7.2}"Relative change and difference". Wikipedia. Retrieved August 25, 2023. - ↑ Robert F. Nau. "The logarithm transformation". Duke University: The Fuqua School of Business. Retrieved August 25, 2023.
- ↑ "Use of logarithms in economics". Econbrowser. Retrieved August 25, 2023.

## External links

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

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