## [Wiki] - Percentage Gain and Loss (for new investors)

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### [Wiki] - Percentage Gain and Loss (for new investors)

Question: If your investment drops by 20%, what percentage increase is needed to get back to the original value? Is it:

(a) Less than 20%
(b) 20%
(c) More than 20%
(d) I don't know

Investors can lose sight of the fact that percentages, which is a ratio, is not the same as working with the actual dollar value.

If an investment loses \$25, you get back to the same value by gaining \$25. When stated as a percentage, the amount of the percentage the investment drops is not the same as the amount needed to get back to the original value.

There is a new wiki article to put this in perspective: Percentage gain and loss

The answer is in the article. Hint: It's not (d).

This article was written because there's quite a bit of attention focused on Risk tolerance, which is how much of a drop you can take and not be overly concerned (What will let you sleep well at night?).

What isn't usually mentioned is how much is needed to get back to the original investment. It may be more than you think - when working in percentages. However, putting this into a dollar perspective may help.*

Also not mentioned is the impact of percentages on gains. The wiki article attempts to address this perspective as well.

What do you think? If there are any errors, please post here.

Comments / questions / concerns are welcome. Wiki editors can edit the page directly.

* Or, the currency of your choice.
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dbr
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

I think this article needs more discussion. The proper math to do this is not percentage changes in asset values but ratios. That is because successive returns in time are compounded. In other words if your investment is cut in half, the value needs to double to return to the original value. The question then comes to what is the distribution of price changes and why we should think that stocks can't double in price as often as they halve in price.

In any case, what is the point of the demonstration. The obvious implication is that the stock market can only be a losing proposition, but the opposite is demonstrated by history, so there is a paradox here. I am not in favor of creating paradoxes by messing around with arithmetic.

#Cruncher
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

LadyGeek in original post wrote:Investors can lose sight of the fact that percentages, which is a ratio, is not the same as working with the actual dollar value.
The Wiki article is correct. But it may create the false impression that what it describes refers specifically to investing and that it results from expressing changes in percentages. But neither is so.

Consider the case of a person who weighs 125 pounds; loses 25 pounds; and then regains it. We can express these two changes as percents. But the question is "percents of what?". If we choose the initial weight (125), then both percents are equal at 20%. Or if we choose the lower weight (100), again they are equal, this time at 25%. But if we choose to express them as percents of two different amounts, the results obviously will differ.

Or expressed algebraically:

Code: Select all

``````Let V0 = value at time 0 (initial value)
and V1 = value at time 1
and V2 = value at time 2
Then if  V2 = V0,
1)      (V1 - V0)      =  -(V2 - V1)
2) and  (V1 - V0) / V0 =  -(V2 - V1) / V0
3) or   (V1 - V0) / V1 =  -(V2 - V1) / V1
4) but  (V1 - V0) / V0 <> -(V2 - V1) / V1 when V1 <> V0``````
What the article describes is simply an example of the obvious arithmetic fact: When we divide equal amounts by amounts that are unequal, the results are also unequal. In addition the article is confusing because it uses the term "initial value" two different ways: In the example it is the value at the start of each period. But in the graph it is the value at the start of the first period.

NiceUnparticularMan
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

Just thinking big picture:

If you have a geometric random walk model for stocks, you are assuming normally-distributed random variation around some upward underlying exponential trend, which is represented by your expected compounding annualized return.

It is true a percentage loss from a certain value will require a greater percentage increase from the new lower value to get back to the same value. But in a way it is even worse than that given that model for stocks, because to get back to where you expected to be will require getting back to some higher value after some time has passed. And that expected value is getting higher all the time if the recovery is not immediate. This is one of the major reasons a true recovery from a bad stock market event can take much longer than some people might have internalized after our last rather rapid recovery.

Still, on the plus side, you do have that upward underlying trend working for you. And one very big question is whether when there is a negative stock market event, does that upward underlying trend also decrease in proportion (basically meaning you just slipped backward on the exponential curve), or stay the same (basically meaning you stay at the same slope of the curve despite the random walk away from it), or something in between? It is misleading to call the latter two possibilities regression to the mean, but it has a similar sort of effect, and it at least seems to fit the historical data a little better (except when it doesn't, which gets us into tail risk, which is a whole other subject).

Finally, all this is really pointing not to short-term losses being worse than we think (and the last thing we need is more commentary persuading people to exhibit even more myopic loss aversion). Rather, it is pointing to the importance of being invested in stocks all along. If you believe that underlying exponential trend is there, then spending time out of the market is dropping you below that trend. In practical terms, if you pull out during the drop you may miss some of the recovery, and that may become an unrecoverable sequence of events unless your timing is so good it compensates--and odds are your timing will not be good but instead neutral or worse.

So from my point of view, this is a somewhat dangerous article without all that context. Specifically, it would seem to be encouraging more myopic loss aversion, whereas I would suggest we should be doing the opposite, meaning giving people the conceptual tools (and math) they need to combat myopic loss aversion.

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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

First, I have isolated the article to keep it out of the main wiki area. IOW, it's bounced back to "draft" status so we can discuss.

See: Percentage gain and loss

Everyone talks about market swings, but there is never any mention of what you need to do to get back to your starting point. I was thinking that new investors may not understand how the math works for percentages - hence the reason for the page.

I honestly thought this would be a simple explanation of a math formula, which is why I was being bold and skipped the draft / discussion part. OK, lesson learned and moving on...
#Cruncher wrote:
LadyGeek in original post wrote:Investors can lose sight of the fact that percentages, which is a ratio, is not the same as working with the actual dollar value.
The Wiki article is correct. But it may create the false impression that what it describes refers specifically to investing and that it results from expressing changes in percentages. But neither is so.
My intention was percent change. I now see where the interpretation of ratio is not correct. "initial/final" is a ratio, but the "percent change" formula is not.
#Cruncher wrote:What the article describes is simply an example of the obvious arithmetic fact: When we divide equal amounts by amounts that are unequal, the results are also unequal. In addition the article is confusing because it uses the term "initial value" two different ways: In the example it is the value at the start of each period. But in the graph it is the value at the start of the first period.
I was trying to describe an "obvious" arithmetic fact to a reader where it might not be so obvious. I'll take a look at the terminology.
dbr wrote:In any case, what is the point of the demonstration. The obvious implication is that the stock market can only be a losing proposition, but the opposite is demonstrated by history, so there is a paradox here. I am not in favor of creating paradoxes by messing around with arithmetic.
NiceUnparticularMan wrote:...So from my point of view, this is a somewhat dangerous article without all that context. Specifically, it would seem to be encouraging more myopic loss aversion, whereas I would suggest we should be doing the opposite, meaning giving people the conceptual tools (and math) they need to combat myopic loss aversion.
Context is definitely needed.

I'll see what I can do to update the article. If anyone has a suggestion, post here. Wiki editors can update the article directly.
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TD2626
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

The chart is very eye opening. A large loss in something (80-90% or more) seems very hard to recover from. This sort of loss is possible in some asset classes during the worst crashes (Great depression / Japan type scenarios). It is especially possible in individual stocks (buy-and-hold Citi or Bank of America investors who invested in 2006 or 2007 will need to wait a really long time to recover).

Some discussion of or links to differences between arithmetic mean and geometric mean may help. People can be deceived by an "average" return that is really the wrong average - it is importantthat this not happen.

A reminder that deep drops are often followed by rapid rises (and vice versa) may help as well - reversion to the mean is important to understand.

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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

The article has been revised: Percentage gain and loss I think it's a lot closer than the last version.
#Cruncher wrote:What the article describes is simply an example of the obvious arithmetic fact: When we divide equal amounts by amounts that are unequal, the results are also unequal. In addition the article is confusing because it uses the term "initial value" two different ways: In the example it is the value at the start of each period. But in the graph it is the value at the start of the first period.
Could you please point out what, exactly, should be fixed? I honestly don't see how the graph implies the start of the first period. The graph is the same as the example - from the starting value to the final value. It doesn't matter how you got there.

NiceUnparticularMan - I have incorporated your context, but paraphrased the wording. How's it look?

TD2626 - I'm not sure if arithmetic vs. geometric mean fits within the context of this article (percentage is different than averaging methods). If you have any suggested text, please feel free to supply.

Comments / questions / suggestions are welcome.
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The Wizard
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

I agree with #cruncher and others that the Wiki article is correct as regards the math.
I've encountered this before and my comment has to do with its relevance to the stock market. It's pretty much non relevant in that the market doesn't care about percentages. It's just a feeble analysis tool we apply in retrospect or maybe in prediction for the s&p500 contest.

If the market oscillated each month up 10% then down 10%, then it would gradually decline, year after year. But it doesn't.

Some sort of caveat might be needed for newbies on this percentage business...
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The Wizard
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

Using logarithms eliminates the apparent discrepancy. After a 20% drop, log (0.8) = -0.09691
Then after a 25% gain, log(1.25) = +0.09691
So equal logarithmic changes cancel out.

Good luck explaining this to the general public...
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pkcrafter
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

What isn't usually mentioned is how much is needed to get back to the original investment. It may be more than you think - when working in percentages. However, putting this into a dollar perspective may help.*

Also not mentioned is the impact of percentages on gains. The wiki article attempts to address this perspective as well.
I agree that this is not well known by most newer investors, and I don't see anything wrong with adding it to the Wiki. Looking at tables of losses/required gains in percent can create a different perspective and may sober up a newly enthused investor and make this investor think a little more carefully about AA and risk.

The table in this link goes to a 75% loss with a needed 300% gain to recover, but if 80% loss were shown, the needed gain goes to 400%.

Paul
When times are good, investors tend to forget about risk and focus on opportunity. When times are bad, investors tend to forget about opportunity and focus on risk.

dbr
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

What isn't usually mentioned is how much is needed to get back to the original investment. It may be more than you think - when working in percentages. However, putting this into a dollar perspective may help.*

Also not mentioned is the impact of percentages on gains. The wiki article attempts to address this perspective as well.
I agree that this is not well known by most newer investors, and I don't see anything wrong with adding it to the Wiki. Looking at tables of losses/required gains in percent can create a different perspective and may sober up a newly enthused investor and make this investor think a little more carefully about AA and risk.

The table in this link goes to a 75% loss with a needed 300% gain to recover, but if 80% loss were shown, the needed gain goes to 400%.

Paul
More likely it would make new investors who are typically fearful and too conservative afraid to invest most effectively. Of course, what I just wrote there is a presumption, but it is better to get presumptions out in the open rather than hide them behind correct but possibly (probably - certainly) irrelevant mathematical trivia. You are correct and the proposed posting is correct in recognizing that variability of return is not a subject that is intuitive and that the topic is critical to investors. I am not sure I could think of a simple and quick to grasp presentation of the subject.

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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

What isn't usually mentioned is how much is needed to get back to the original investment. It may be more than you think - when working in percentages. However, putting this into a dollar perspective may help.*

Also not mentioned is the impact of percentages on gains. The wiki article attempts to address this perspective as well.
I agree that this is not well known by most newer investors, and I don't see anything wrong with adding it to the Wiki. Looking at tables of losses/required gains in percent can create a different perspective and may sober up a newly enthused investor and make this investor think a little more carefully about AA and risk.

The table in this link goes to a 75% loss with a needed 300% gain to recover, but if 80% loss were shown, the needed gain goes to 400%.

Paul
The table was already in the wiki, but in a collapsed format so it wouldn't detract from the graph. The table is now fully visible: Percentage gain and loss
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Fallible
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

Not sure how you plan to run this, but it could flow nicely from the key question about loss asked in the "Important" section on the Risk Tolerance page and also the Asset Allocation page, beginning with a brief reference to it to let readers know it's available and then linking to the page.

Readers are mainly new investors or experienced ones brushing up on tolerance, but I think many of them would appreciate this feature at some point. However, I think dbr has raised a good point that the the idea may spook, or further spook, some new investors who might then lower risk too much. Also, new investors already have a lot to take in at first, although they can do the figuring at any time rather than doing it as they are just beginning to learn about investing overall. Possibly this can be conveyed to them.

Fallible

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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

In reference to this earlier quote:
dbr wrote:I think this article needs more discussion. The proper math to do this is not percentage changes in asset values but ratios. That is because successive returns in time are compounded. In other words if your investment is cut in half, the value needs to double to return to the original value. The question then comes to what is the distribution of price changes and why we should think that stocks can't double in price as often as they halve in price.

In any case, what is the point of the demonstration. The obvious implication is that the stock market can only be a losing proposition, but the opposite is demonstrated by history, so there is a paradox here. I am not in favor of creating paradoxes by messing around with arithmetic.
and
The Wizard wrote:Using logarithms eliminates the apparent discrepancy. After a 20% drop, log (0.8) = -0.09691
Then after a 25% gain, log(1.25) = +0.09691
So equal logarithmic changes cancel out.

Good luck explaining this to the general public...
...brings the distribution of the stock market into the discussion. I don't know if we want to go in that direction, as discussing long-term historical performance (50+ years) might detract from investors. My concern is that looking too far out might increase sequence of return risk. For example, going for a long-term bond fund, e.g. 10 year duration, when your need is within the next 5 years. The volatility 5 years out might have a negative return.

Additionally, the statistical distribution over time is not normal. It's probably lognormal, here's one perspective: Stock Return Distributions (from Gummy-stuff - finiki, the Canadian financial wiki)

=================
A useful property of logarithms is that they can flatten curves (turn into a straight line). Here's an interesting article why natural logarithms are used in economics: Use of logarithms in economics . Note what was done with the S&P 500. I don't think this type of explanation would be helpful in the context of the wiki article.
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

LadyGeek in [url=https://www.bogleheads.org/forum/viewtopic.php?p=3403743#p3403743]this post[/url] wrote:Could you please point out what, exactly, should be fixed? ... The graph is the same as the example ...
No, the graph and the example aren't the same in the use of the term "Initial Value". Here is the graph:

Now here is the example:
Drop by \$25:

Initial Value: \$125

Final Value: \$100

Loss in \$: -25 = Final value - Initial value = \$100 - \$125

Percent change: -20% = (\$125 - \$100)/\$125 * 100%

Then, increase by \$25:

Initial Value: \$100

Final Value: \$125

Gain in \$: +25 = Final value - Initial value = \$125 - \$100

Percent change: +25% = (\$125 - \$100)/\$100 * 100%
To use the term "Initial Value" consistently, I'd leave the graph alone, but change the example to read:

Code: Select all

``````Drop by \$25:
Initial value:         \$125
Drops to:              \$100
Loss in \$:              -25  =  \$100 - \$125
Percent change:         -20% = (\$100 - \$125) / \$125

Then, increase by \$25:
From:                  \$100
Back to initial value: \$125
Gain in \$:              +25  =  \$125 - \$100
Percent change:         +25% = (\$125 - \$100) / \$100``````
This also fixes two other things in the example:
• Fixes the sign: -20% = (100 - 125) / 125, not (125 - 100) / 125
• Eliminates superfluous term: "* 100%".

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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

^^^ Thank you, your suggestion was not clear until now. The changes have been incorporated, which includes your catch of a math formula error: Percentage gain and loss

As for the "superflous" term: "* 100%", I'm a big fan of specifying units. In engineering (what I do), it's a necessity and did not seem superflous to me. (I did remove the term, but wanted to explain why it was there in the first place.)

===========

Not sure how you plan to run this, but it could flow nicely from the key question about loss asked in the "Important" section on the Risk Tolerance page and also the Asset Allocation page, beginning with a brief reference to it to let readers know it's available and then linking to the page.

Readers are mainly new investors or experienced ones brushing up on tolerance, but I think many of them would appreciate this feature at some point. However, I think dbr has raised a good point that the the idea may spook, or further spook, some new investors who might then lower risk too much. Also, new investors already have a lot to take in at first, although they can do the figuring at any time rather than doing it as they are just beginning to learn about investing overall. Possibly this can be conveyed to them.

Fallible
I have included the Risk tolerance page as a reference, as I thought there should be some way to explain the relevance.

Do you have any suggestions for the Risk tolerance - Why is risk tolerance important? section? We can create a new draft page to discuss the changes.

I agree on dbr's points. We don't want to overload new investors, as any hint of a suggestion is taken as gospel.

==========
Anyone is welcome to provide suggested text.

To the new investors reading this discussion - what do you think?
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TD2626
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

TD2626 - I'm not sure if arithmetic vs. geometric mean fits within the context of this article (percentage is different than averaging methods). If you have any suggested text, please feel free to supply.
Just some ideas that could be incorporated in:

A portfolio gains 10% 1 year and then loses 10% the next. It has 99% of of its starting value.

A different portfolio loses 10% in a year and then gains 10% in the next. It is also at 99% of its starting value.

This nonlinear effect of percentage moves is shown in the article. I think there are two useful implications of this (besides the eye opening chart):

The first is that investors should not take the arithmetic average return (0% per year in my example) - that number is too high and is dangerous to use for investment planning. Too many investors think they can spend the average return. Instead, one must use geometric mean / CAGR to find that the investors in the example above had -0.5% per year returns.

The second implication is that volatility drag is an important consideration when selecting a risky asset.

You could also note that the fact that the chart goes to infinity at a 100% loss suggests the need for diversification - individual stocks can easily lose 100%, but it's far harder for this to happen in mutual funds (every company in the fund would need to be bankrupt).

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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

I have revised the section order to put the introduction and overview first, as I think this frames the context better: Percentage gain and loss Any of this text can be changed. We could also add more information after the math section.
TD2626 wrote:The second implication is that volatility drag is an important consideration when selecting a risky asset.
I think introducing two new words together in the same sentence - volatility and drag - will scare new investors away. Perhaps risky assets are best left for "experienced" investors who understand the risks when deviating from a total market approach?

The related wiki article: Variance drain does discuss "Arithmetic versus geometric returns". Perhaps something can be extracted from here?

Update: Perhaps that "eye opening chart" should go at the top of the article.
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Fallible
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

Not sure how you plan to run this, but it could flow nicely from the key question about loss asked in the "Important" section on the Risk Tolerance page and also the Asset Allocation page, beginning with a brief reference to it to let readers know it's available and then linking to the page.

Readers are mainly new investors or experienced ones brushing up on tolerance, but I think many of them would appreciate this feature at some point. However, I think dbr has raised a good point that the the idea may spook, or further spook, some new investors who might then lower risk too much. Also, new investors already have a lot to take in at first, although they can do the figuring at any time rather than doing it as they are just beginning to learn about investing overall. Possibly this can be conveyed to them.

Fallible
I have included the Risk tolerance page as a reference, as I thought there should be some way to explain the relevance.

Do you have any suggestions for the Risk tolerance - Why is risk tolerance important? section? We can create a new draft page to discuss the changes.

I agree on dbr's points. We don't want to overload new investors, as any hint of a suggestion is taken as gospel.
...
Yes, it is a bit tricky because of this reader.

My thought is that, because you have worked up the page and understand it best and also already understand the target audience for the tolerance page, you'd be the best to write the draft. The pickup could be from the Zweig quote re loss, possibly paraphrasing it to begin the transition to the gain/loss page (but leaving the main quote as is). It probably should be brief and let the new investors know that it's not urgent, but why it could be helpful if they'd like to try it at some point.

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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

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TD2626
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

Just a few more thoughts:

I agree that voltatility drag/variance drain may not be the best topic to include in the article, though it could be linked in a "see also" type section.

The table covers -100% to +100%. It may be worth noting that investors are limited in their losses to -100% (assuming no margin), while gains are potentially unlimited. Staying the course for decades will very likely get gains of far over 100%. The table shows that a 100% gain requires a 50% drop to get back to the initial value. Could one put 150%, 200%, 300%, etc in the table as well? Indeed, these are likely more common than the -70%, -80%, and -90% that are in the table.

Also, the 50% or more drop required after a 100%+ gain becomes fairly historically uncommon (though it's possible). Could one provide examples in the article of drops that have occurred historically in the article (Great Depression, Japan, 1987, 2008) in percentages and include them in a table?

MIpreRetirey
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

How's this? Percentages can be misleading if not combined correctly.

Maybe a common quick error or fallacy people make is making the assumption that percentages can be added and subtracted. But they are actually a multiplication operation.
When is this equation true? P +10% -10% = P. When would you expect it to be true. It's not true. And +10% is not the inverse gain of a 10% loss, either.

1st, the percentages must be converted to a decimal or fractional representation.
You, also, can't just write P * +10% * -10% = P. Not true, again.

+10% is equivalent to a decimal multiplication of 1.1
-10% is equivalent to a decimal multiplication of 0.9

And then you can see how a gain of 10% followed by a loss of 10% does not get you back to even. Because the two returns compounded after one another is equivalent to writing P * 1.1 * .9 = P * .99, which is not back to even.

If you do it old school, in fractions, then a 10% gain can be written 11/10 , or any multiple such as 110/100, and
a 10% loss can be written as 9/10, or 90/100.
Now the equation can be written P * 110/100 * 90/100 = P * 9900/10000, and you see what's happening (like in the graph and chart)?, multiplying by
90/100 is not the inverse operation of multiplying by 110/100. The inverse operation of multiplying by 90/100 is multiplying by 100/90 (which is +11.11...% These would be called geometrically inverse operations (as in multiplication and division.)
And the inverse to * 110/100, is * 100/110. (which is +9.09...%) As in the chart/graph. They are never exactly equal, no matter how small the number.
So, a 10% loss requires an 11.1% gain to recover, and a 10% gain would take a 9.09% loss to be even.

In dollar terms, you did nothing but either gained, or lost, 10 coins on a hundred and then they were reversed back to 100.

Edit.

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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

When the market takes a large negative swing, recovery will take a longer time because the market needs a larger percentage increase to get back to the same point.

Ron
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snarlyjack
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

Here is another consideration for you?

I believe you are talking about a constant value portfolio.
IE: I have a \$1000. portfolio. It takes a 50% hit, how long would
it take to get back up to my original \$1000. in time or the %
gain or loss I would need to make up.

Where it get's complicated in my mind is (DCA) dollar cost averaging
into the bear market and dividends buying more shares. I'll use Japan
as a example. If a person who took a large loss in their portfolio in the
1980's were to dollar cost average the whole way + reinvest dividends
I believe they would get back to their original portfolio + some, if they
were in the accumulation stage rather than the constant value stage
a lot quicker + the % gain & loss would be different.

Should dollar cost averaging into a bear market with reinvesting dividends
all along the way be considered in your analysis? (Think Japan).

Thanks...

The Wizard
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

Oicuryy wrote:
When the market takes a large negative swing, recovery will take a longer time because the market needs a larger percentage increase to get back to the same point.

Ron
As Ron alludes to, if The Market fundamentally worked as some sort of percentage driven machine, like CDs perhaps, then Oicuryy's statement would have major significance.
But stocks in particular do not care about percentages. Attempts to analyze market performance, especially for the future, with percentages or with more esoteric concepts will generally fail...
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MIpreRetirey
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

The Wizard wrote:
Oicuryy wrote:
When the market takes a large negative swing, recovery will take a longer time because the market needs a larger percentage increase to get back to the same point.

Ron
As Ron alludes to, if The Market fundamentally worked as some sort of percentage driven machine, like CDs perhaps, then Oicuryy's statement would have major significance.
But stocks in particular do not care about percentages. Attempts to analyze market performance, especially for the future, with percentages or with more esoteric concepts will generally fail...
This would be exactly half my beef.
The fair value, and a market priced-in trading value of an asset have nothing to do with some of the math statements sometimes construed in meaning. And, no, the market does not oscillate back and forth equally over time to get back to an average P/E. The math is just the math.

But we hear, eg., 2 things a lot:
You need a 100% gain to get back a 50% loss, and it's implied it always takes far longer for the gain than the loss.
But, this is usually true because 'the market tends to decline like an elevator and climb like the stairs.' Can this be the nature of risk aversion in risky assets? In any account, it's not the inherent math at all that describes this market phenominom, it is the market at work valuing the asset.

If you get a continual sequence of +10%, -10%,..., for a period of time, your portfolio is just steadily declining. Well, this is no majical math formula, it is a mistake in the math of what an equivalent gain or loss is.

MIpreRetirey
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

In other words, is a 50% needing a 100% gain, and a continual sequence of + and - 10% change, a math concern, or an asset valuation concern?
Sure, it's looking at the riskiness here, but that is not a math concern, it's an asset valuation/risk/return concern.
If an asset has a certain earnings and earnings growth, and then the economy sputters or almost stops, and then the economy recovers, and that asset resumes the same earnings and earnings growth, wouldn't it be expected to get back to a similar trading value, all other economic considerations the same?

The Wizard
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

MIpreRetirey wrote: ...But, this is usually true because 'the market tends to decline like an elevator and climb like the stairs.' Can this be the nature of risk aversion in risky assets? In any account, it's not the inherent math at all that describes this market phenominom, it is the market at work valuing the asset...
Loss Aversion, you mean.
Apparently studies show that many investors fear a 20% loss much more than they get overjoyed by an equivalent gain.
So yes, I would expect that panic selling would be more pronounced than exuberant buying. It should be simple to find the dozen biggest one-day drops in the S&P 500 over the last decade along with the dozen biggest gains, to verify that we are right...
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

MIpreRetirey, Thank You! Your tutorial was exactly what I was looking for. The basic concepts are explained and put in the proper context.

See: Percentage gain and loss

snarlyjack - This may not be an appropriate article to bring Dollar cost averaging into the mix.

More work is needed, but this is a big step in the right direction.
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

The wiki article says "The plot shows that you always need a larger percentage increase to come back to the same value." But this is purely an artifact of how the comparison is made. It leads to the fallacy that the gain needed to recover from a loss is bigger than the loss when expressed in percentage form.

It is instructive to look at the percentages in another way. Say your holding varies in value
\$100 on Monday
\$80 on Tuesday
\$100 on Wednesday

On Tuesday you can say, yesterday my holding was 25% greater than today.
On Wednesday you can say, yesterday my holding was 20% less than today.

Expressed in percentages this way, we get the opposite phenomenon. We can express the changes in the form of percentages such that we get the opposite fallacy, namely, that the gain needed to recover from a loss is SMALLER than the loss when expressed in percentage form in this manner.

I think there needs to be some thought as to what exactly is trying to be conveyed here.

itstoomuch
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

I still like my explanation
YPMV & YMMV
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dbr
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

The wiki article says "The plot shows that you always need a larger percentage increase to come back to the same value." But this is purely an artifact of how the comparison is made. It leads to the fallacy that the gain needed to recover from a loss is bigger than the loss when expressed in percentage form.

It is instructive to look at the percentages in another way. Say your holding varies in value
\$100 on Monday
\$80 on Tuesday
\$100 on Wednesday

On Tuesday you can say, yesterday my holding was 25% greater than today.
On Wednesday you can say, yesterday my holding was 20% less than today.

Expressed in percentages this way, we get the opposite phenomenon. We can express the changes in the form of percentages such that we get the opposite fallacy, namely, that the gain needed to recover from a loss is SMALLER than the loss when expressed in percentage form in this manner.

I think there needs to be some thought as to what exactly is trying to be conveyed here.
Nice. I would suggest the entire article is not needed in the Wiki because it is not useful to investors. It does not shed any particular light on how the stock market behaves. It tends to be misleading, if anything. The math, as pointed out by a number of people, is arbitrary even though it is correct on its own terms. If it is retained, the article might be a good item to point to when wanting to illustrate how mathematical confabulation can easily be constructed. There should probably be a companion article with the computations that show that dividends account for 70% of the returns of the stock market and capital gains account for the other 70% of the gains of the stock market. That fallacy also arises from some similar confabulation about how to account for changes in asset values and is commonly quoted here and there.

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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

itstoomuch wrote:I still like my explanation
YPMV & YMMV
Mind expanding that description a bit?

================
dbr wrote:Nice. I would suggest the entire article is not needed in the Wiki because it is not useful to investors. It does not shed any particular light on how the stock market behaves. It tends to be misleading, if anything. The math, as pointed out by a number of people, is arbitrary even though it is correct on its own terms. If it is retained, the article might be a good item to point to when wanting to illustrate how mathematical confabulation can easily be constructed. There should probably be a companion article with the computations that show that dividends account for 70% of the returns of the stock market and capital gains account for the other 70% of the gains of the stock market. That fallacy also arises from some similar confabulation about how to account for changes in asset values and is commonly quoted here and there.
Would it be better to remove everything except the basic math? Here's the first version: Percentage gain and loss (Revision as of 20:36, 5 June 2017)

Consider that we have other "basic math" articles which haven't generated as much criticism, such as: Comparing investments. How is this article different than calculating percentages?

Should we have "companion" articles as dbr suggests? The wiki aleady has Investing FAQ for the Bogleheads® forum. Would a similar compendium of "mathematical confabulation" be worthwhile?
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itstoomuch
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

It's all numbers. What is more importamt is the presentation of the numbers.
Also got you to repeat the terms and me to reinforce the concept.
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MIpreRetirey
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

But I'll ask on one smallish addition to it, if it could be changed.
There is no direct mention using the phrase 'percentages are a ratio', which is mentioned a lot in threads.
My idea on this is "can this phrase be create confusion over and over by what number belongs in the denominator? When calling it a multiplication of 1.1 or 0.9, and 11/10 or 9/10; and then you can leave out totally the math, in fractions and of reversing the operation(why is that necesssary?), which is a sidetrack from the pure subject of just about percentage representation/definition? I mean, to make it small and concise, and not uneededly an extensive amount of print and explanation?

Peter Foley
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

As the Wizard stated (paraphrasing) how do you make this understandable to the general public. I would only add, understandable without injecting too much emotion into the topic.

I would keep it very simple and just provide a table. Use \$1000 as your initial investment so many of us can do some of the math in our heads.

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.

The end.

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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

Peter Foley wrote:As the Wizard stated (paraphrasing) how do you make this understandable to the general public. I would only add, understandable without injecting too much emotion into the topic.

I would keep it very simple and just provide a table. Use \$1000 as your initial investment so many of us can do some of the math in our heads.

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.

The end.
If yesterday was 11% more than today, then you need today to be 10% less than tomorrow to recover.
If yesterday was 25% more than today, then you need today to be 20% less than tomorrow to recover.
If yesterday was 43% more than today, then you need today to be 30% less than tomorrow to recover.
If yesterday was 67% more than today, then you need today to be 40% less than tomorrow to recover.
If yesterday was 100% more than today, then you need today to be 50% less than tomorrow to recover. (That's right, if you have lost an amount equalling all of the money you now have left, then you'll only need to gain an amount equalling half of the money you'll end up with, to get back to even.)

If yesterday was 9999~~~9900% more than today, then you need today to be 99.99~~~99% less than tomorrow to recover (where 9999~~~99 denotes a string of N 9s for any given number N).

I hope I am succeeding in reinforcing the math and demolishing the myth.

itstoomuch
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

YMMV
{YourMathMayVary}
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Oicuryy
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

Here is the math.

(a/b)*(b/a) = 1

Now the wiki page just needs to explain how knowing this can help one be a better investor.

Ron
Money is fungible | Abbreviations and Acronyms

MIpreRetirey
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

post from the 'volatility drag' thread:
The arithmetic mean is the arithmetic average of multiplications. .
See wiki on variance drain:

viewtopic.php?f=10&t=199092&p=3410275#p3410275

The Wizard
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

Oicuryy wrote:Here is the math.

(a/b)*(b/a) = 1

Now the wiki page just needs to explain how knowing this can help one be a better investor.

Ron
I'm not sure this particular page can help much in that regard.
My main complaint in that simplistic math like this is implying that it's somehow "harder" for a broad index fund to go up than go down, given that a 25% increase is needed to return from a 20% decrease. That's not how the stock market works.

If I was writing a book about this, I would supplement the simple math with actual historical data: total return of \$10,000 invested in the S&P 500 over 20 years with dividends reinvested.
I would have a data point at the end of each calendar quarter, so 80 points in all.
I would have multiple graphs:
1) the value of the original \$10,000, updated each quarter.
2) the dollar change of that portfolio, compared to the previous quarter.
3) the percentage change of that portfolio, compared to the previous quarter.

I would probably also expound on why reinvested dividends matter, with graphs comparing the percentage change in the S&P 500's market value to the aforementioned total return...
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celia
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

I didn't read the entire thread or the whole wiki article, but some things in the article don't seem correct.
Question: When is this equation true?

P + (10%) - (10%) = P.

The 10% can't "float" by itself. It refers to 10% of what?

If you say:
P + (10% of P) - (10% of P) = P, that indeed is true.

In Table 1, I think it would be easier to understand if the starting value were to be 100 or 1,000 since most people in the U.S. grew up with the decimal system and learned per cents as "fractions" of 100. Some entries in the table are wrong, too. If you lose 60%, then you need to gain 250%. (Losing 60% is the same as keeping 40%, so you need to more than double before you have the original amount.)

MIpreRetirey
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

celia wrote:I didn't read the entire thread or the whole wiki article, but some things in the article don't seem correct.
Question: When is this equation true?

P + (10%) - (10%) = P.

The 10% can't "float" by itself. It refers to 10% of what?

If you say:
P + (10% of P) - (10% of P) = P, that indeed is true.

In Table 1, I think it would be easier to understand if the starting value were to be 100 or 1,000 since most people in the U.S. grew up with the decimal system and learned per cents as "fractions" of 100. Some entries in the table are wrong, too. If you lose 60%, then you need to gain 250%. (Losing 60% is the same as keeping 40%, so you need to more than double before you have the original amount.)
I know. That deserves an edit, or edit-out. Syntax error?

dbr
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

The basic mathematical picture would address something along the following lines:

1. What is return, meaning the simple return an investment delivers in a period of time, a year by convention.
2. How do we discuss the variability or uncertainty of that return. This introduces return conceived of as a statistical variable.
3. How does return compound year after year. How do we express that as a compound average growth rate, the number that is actually reported in the usual stats. At this point the difference between the arithmetic average return and the geometric average return probably can't be avoided. This is the context in which the canard about -50% takes +100% to recover is mentioned by those who mention it. I think that bit of mathematical fact is irrelevant and should be ignored. It is more important to focus on a more general and complete description of the time evolution of returns and resulting portfolio values stated in statistical language.
4. How do we discuss the variability of the compounded result.
5. Now we go back and look at the return and risk of portfolios rather than single assets. You have to do this in terms of annual return and introduce the efficient frontier. Then you have to look at how the return and risk compound.
6. One can extend the discussion to how all of this evolves in the real world examples of continuous contribution or continuous withdrawal. One might end up with the various models that have been developed to analyze retirement savings and withdrawal.

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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

celia wrote:Some entries in the table are wrong, too. If you lose 60%, then you need to gain 250%. (Losing 60% is the same as keeping 40%, so you need to more than double before you have the original amount.)
(a/b)*(b/a)=1

If a/b = 0.4 then b/a = 2.5

To convert the ratio a/b to percentage subtract 1 then multiply by 100.

(0.4 - 1)*100 = -60%
(2.5-1)*100 = 150%

But what does this math tell us about how much time it takes for the value of a portfolio to go from a to b?

Ron
Money is fungible | Abbreviations and Acronyms

dbr
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

Oicuryy wrote:
But what does this math tell us about how much time it takes for the value of a portfolio to go from a to b?

Ron
A time sequence of investment returns and investment values is much more complex than isolated events of "losing" or "gaining" wealth. The idea that we should look at our investments in terms of what did I lose today (or this year) or what did I gain today (or this year) might be something we should regard as pernicious as it leads us into bad psychology about investing.

The puzzle at the heart of this is how to be comfortable with an an enterprise, the results of which are important, but the nature of which is to be plagued by uncertainty. I am not sure I know of a solution to that other than to at least start with a comprehensive and global sense of what is involved and not get side-tracked by meaningless arithmetic sensations about short term results.

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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

celia wrote:I didn't read the entire thread or the whole wiki article, but some things in the article don't seem correct.
Question: When is this equation true?

P + (10%) - (10%) = P.

The 10% can't "float" by itself. It refers to 10% of what?
RUBBISH!

It's ALWAYS true!

P+X-X=P is ALWAYS true, for any X,
in particular, when X=10%=0.1!

itstoomuch
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

No one cares about the percentages. All they really care about is seeing the Money.
The sad commentary on the state of education, critical thinking, and wealth perceptions.
YMMV, and all the variations of Y_MV.
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

If my \$100 stock fund balance drops \$10, the balance must increase \$10 to return to \$100. The fact that this represents a 10% drop, and an 11.1% gain (90*1.11=100) to occur seems pretty irrelevant.

If my \$100 drops to \$50, and I ponder for a minute that 50% drop requires a 100% gain to return to \$100, and I recall folks regard a 10% annual gain a great return, I could be leery of investing in that fund. But I know stock funds are volatile, and that getting out low and missing the top couple day's gains can result in serious losses.

I think the 10%/11.1%, or 50%/100% loss/gain percents are "just the math." In reality, the balance dropped \$10, then gained the \$10 back. Or it dropped \$50 and gained the \$50 back. I'm not sure what to do with that.
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### Re: [Wiki] - Percentage Gain and Loss (for new investors)

Peter Foley wrote:As the Wizard stated (paraphrasing) how do you make this understandable to the general public. I would only add, understandable without injecting too much emotion into the topic.

I would keep it very simple and just provide a table. Use \$1000 as your initial investment so many of us can do some of the math in our heads.

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.

The end.
^^^ I agree, and have created a second draft page: User:LadyGeek/Percentage gain and loss

All of the commentary related to "emotion" and possibly incorrect context have been removed. Members can link to this article in forum discussions, thereby framing the issue in the proper context and provide appropriate guidance.

Otherwise, I don't think we can reach a consensus.

Here's the first version: Percentage gain and loss

Both versions have been revised to include a spreadsheet in Google Sheets format.
To some, the glass is half full. To others, the glass is half empty. To an engineer, it's twice the size it needs to be.