# Help me understand the oddity of percentage gains and losses?

If you start with \$100K, and you get a 50% gain, it's \$150K. Then, you get a 50% loss: now it's \$75K. Reverse the order (i.e. lose 50%, gain 50%), it's the same thing (a 25% loss). So, 50% up then 50% down, or 20% up then 20% down, you don't get back to "even"; it's a loss.

My question is: how can one make this notion work to their advantage, instead of to their disadvantage? I'm wondering if you have any strategies/thoughts/ideas on how to make this effect work for you, instead of against you. Or is my thinking on this as a strategy somehow incorrect?

The effect seems to diminish with lower swings in returns. E.g. the +20% / -20% case, results in a 4% loss. I.e. 100K -> 120K -> 96K. Not clear what the formula for it is, but it's not linear.

• The loss is the multiple of the two: 50% × 50% is 25%; 20% × 20% is 4%. – Nicole Jan 17 '12 at 17:14
• It's possible that you are worrying that an N% gain and an N% loss are equally likely. Generally they aren't, for exactly this reason. – keshlam Oct 15 '16 at 9:36

Other answers have pointed out the mathematics behind this "notion", as you put it – and I hope the explanations were helpful – while failing to answer with a potential strategy (save for duffbeer703). Here it is:

The way you take advantage of swings in the market (to generalize) is to:

• invest more cash in the market before the up moves, and
• remove cash from the market before the down moves.

As you've seen, if you start with \$100K fully invested, with the market dropping 50% first and then gaining 50% from there while remaining fully invested, then you're down \$25K. But what if you were to add \$100K when you are down? Then you are ahead of the game by \$25K after the 50% gain.

Next, consider the opposite ordering of the moves: the market gaining 50% first, then dropping 50%. Similarly, with the \$100K fully invested, you end up down \$25K. But what if you take half your stake out at the top? Then you are ahead of the game by \$12.5K. Or you could be ahead the full gain if you sell everything at the top.

Of course, this "buy low, sell high" or "market timing" strategy is easier to describe than actually perform, and the market isn't likely to deliver X% gains followed by X% losses (or vice versa) consistently .. it is more temperamental than that. In fact it often goes exactly where you didn't expect it: down when you are fully invested, and up when you aren't.

But, successful trading is the way to take advantage of moves in the market that would otherwise do damage to a fully-invested portfolio ... "successful" being the highly elusive secret ingredient.

This isn't an effect that works either for you or against you, it's just a notational coincidence or numerology-like observation.

A \$10 absolute gain and a \$10 absolute loss are even. The mistake here is to think there's any reason percent changes should match when you get back to even.

I grant you, people may psychologically connect a 50% loss with a 50% gain and therefore think the loss is easier to recover from than it really is. But this is a mistake (i.e. connecting the two just because the "50%" matches is incorrect).

This is because the percent change is from some base value, and the base value changes before and after the loss. Or in other words, relative changes are relative, and not comparable when they aren't relative to the same thing.

If you have two percent change values for adjacent time intervals, they compound rather than adding up. So 50% loss then 50% gain is base amount times 0.5 times 1.5. Which is NOT base amount minus 50 plus 50. But that isn't an "effect" it's just math.

• Exactly. One shouldn't confuse a dollar change with a ratio of dollars change. If you start with \$100k, gain \$50K, then lose \$75K, you have lost \$25K overall. Ratios (percentages) are tricky; they are always relative to something. In calling these two changes +50% and -50%, you are using two different somethings. If you instead use your starting investment amount as the consistent divisor, you have \$50k/\$100k = +50% and -\$75k/\$100k = -75%, and you can see you have lost money overall. – mgkrebbs Jan 1 '12 at 23:53

Ray - Algebra I tells us that (X+Y)(X-Y)= X^2 - Y^2 so a gain and loss of 10% is a net loss of 1%, 20%, -4%, and so on. I'm not clear on how you want to benefit from this aside from avoiding the losses.

• Joe - thanks for the formula. Avoiding the loss is definitely good. I guess I haven't seen a lot of "embrace" of this effect/problem, which it seems is necessary, before good solutions can be found, e.g. make this effect work on losses, instead of gains/investment. There may be other techniques, hence my question, or how does one execute applying it to losses? After I wrote the question, I thought, if one always takes X% off the table where one makes a gain of X%, what will that result in. That's simple, but ideally just a start to "study of this issue", something to solve I think. – Ray K Jan 1 '12 at 2:11
• Do me a favor, if you agree, and up-vote this question, so it doesn't have a negative vote. It's a very legitimate question, and I think a good/important investing knowledge. It looks like from the comment that perhaps someone was disappointed to learn, that a 50% gain and a 50% loss results in a 25% overall loss. Unfortunately, denial doesn't solve problems/create solutions. – Ray K Jan 1 '12 at 2:16

Ok, so, I still think my other answer is much more useful, but just to make @littleadv happy, here's how you can directly profit off of the percentage gain/loss phenominum:

`You short a double to triple short leveraged ETF`

The reason this works is because of the same thing that makes those 2x/3x ETFs suck:
Imagine you thought the S&P 500 was going to go down, and you were pretty sure about it. So, you bought an ETF that's double short that index. You figure the S&P 500 goes down, this ETF will go up by twice that amount.

Now, three months later, you check the markets and the S&P is indeed down 10%. Yay! You're ecstatic, you figure you made 20%! Right? Wrong. In fact, so wrong, that you could have actually lost money.

The reason is because of the phenomenon the poster asked about. Those double & triple short ETFs are double & triple the percentage loss or gain. So, if over the course of that three months the S&P had first gone up 20%, and then down 25% to end at 10% down, the double short ETF would have gone down 40% then up 50% to end at 10% down from the purchase price.

But, the numbers don't have to be as dramatic as that to make this work. The fact is each day the S&P will either go up or down a percentage, and the double short ETF will go down or up twice as much. This ends up having a ratcheting effect downward on that ETF's price over time.

So, if you think the index is going up, you can short the double short ETF, and put not only your long guess in your favor, but also this gain/loss ratcheting effect. You can end up being wrong about the index, and still making money.

I can't remember if this works on the 2x/3x long ETFs, but IIRC it does, just not quite as well as on the short ETFs.

I've never actually tried this myself.

• Wow, that sounds scary:) But I'm happy always, don't you worry about me. Upvoting both your answers, if only just to spread some happiness around :-) – littleadv Jan 3 '12 at 4:27
• I'm not clever enough to come up with the flaw for this strategy, but the very idea of shorting a 3x short leveraged ETF made me feel vaguely ill. Back to the index funds I go! – Fomite Jan 3 '12 at 21:46
• @EpiGrad I know, right? That's why I think my other answer is actually the better one. – Patches Jan 3 '12 at 21:59
• Thanks Patches, this is intersting. Is there a way to "simulate" what you have written, without shorting the 2x or 3x leveraged ETF? I know they have ETFs in both directions, e.g. SDS? On the other hand, if one had to short, I wonder what would happen return-wise if one shorted both directions, given that these do decline in value? Thoughts? – Ray K Jan 4 '12 at 2:10
• ... my concern about shorting was paying the interest to borrow the ETFs – Ray K Jan 4 '12 at 2:19

You can't beat math, so you manage your portfolio to reduce downside risk. Diversification allows you to avoid the big dips.

The math isn't actually working against you.

If you invest 100k in an investment for 2 years and it has a 50/50 chance of gaining or losing you 50%, the possible outcomes are like this:

25% Gain/Gain = +125k

25% Gain/Loss = -25k

25% Loss/Gain = -25K

25% Loss/Loss = -75k

So if you invest for 2 years four separate times with 100k, the statistical average is that you will get each of these once. If you get each of them once you will end up with the same amount you invested.

The reason this happens is because compound interest makes losing and gaining a net loss, but it also makes losing twice less of a loss than it should be intuitively (two 50% losses don't wipe out 100% your funds, they only wipe out 75%) and makes winning twice a bigger gain (two 50% gains is a 125% gain). So statistically you have a higher chance of losing, but you'll lose less than you "should" and gain more than you "should" when you do to balance it out.

The first rule of making money is to not lose money. So, when looking for, say, mutual funds, first look at, and spend most of your time analyzing, how much you could lose, not how much you could make.

So, that means:

• Don't be too impressed by 3, 5, and 10 year return averages. As your numbers show, they can give you a false impression. 50% up one year, and 50% down the next gives you a 2-year average of 0%... but you would have lost 25%.
• Instead, look at a table of returns for the last ten years. You want to see what each year's result actually was, and if there was a loss, consider what it took to break even. Consider whether or not that loss was acceptable considering the style of investing they're doing -- and if the gain in the positive years justifies it.
• Try to avoid funds with losses more than one year in a row. Multi-year losses just put compounding to work against you. This rule is particularly true for actively managed funds. After all, index funds are mechanically invested and so can't help it. But with actively managed funds you're paying them to not be stupid twice in a row.
• Wrap your head around mean reversion. Whether it's a sector, or investing style, or specific manager, if it's been doing better than the average for several years, going forward it's likely to under perform and revert to the mean. Of course, the key is when that will happen.
• Another one for active funds: don't discard one just because they've got alot sitting in cash. You want a manager with the guts to sell when everything is expensive (bubble) and have the bankroll to buy when everything is cheap (crash).
• Invest for a level of return, not a level of risk. Everyone knows that if you're young you're supposed to invest with high risk, right? Well, that's BS. Why would you take on more risk just because you're "supposed to"? You should target a level of return, then take on the least risk to get that return. If it's impossible to get that return with a level of risk you're willing to take, then reduce your expectations for the return.
• Do not underestimate the damage done by large losses. Over time, many of those high-risk/high-return funds can underperform the steady-eddie, more conservatively invested funds, precisely because of this phenomenon. The last 10 years has really exposed this fact. So, have a look at some of those moderate or conservative allocation funds, even if you're "not supposed to". Who knows, you may find something that gets you exactly what you need, but without the antacids.
• This might be a great answer... But not to the question asked. – littleadv Jan 2 '12 at 7:41
• @littleadv i still think this one is a better answer, but I just added another answer that answers the question more directly. Happy? :-) – Patches Jan 3 '12 at 3:10
• @keshlam ummm... that's my point exactly. – Patches Dec 7 '15 at 23:09

The reason for this: If you make a percent gain, followed by the same percent loss, you made a gain on the original amount, and a loss on the larger amount. So in percent, gain and loss were equal, but in dollars, the loss is bigger than the gain. For example, starting with \$100 and 20% gain and loss, you gain 20% of \$100, but you lose 20% of \$120.

If you make a percent loss, followed by the same percent gain, you made a loss on the original amount, and a gain on the smaller amount. So in percent, loss and gain were equal, but in dollars, the loss is bigger than the gain. For example, starting with \$100 and 20% loss and gain, you lose 20% of \$100, but you gain only 20% of \$80.

To make this work for you instead of against you: Make two gains. Two twenty percent gains for example give you a total of 44% gain, not 40%. The not recommended method: Make two losses. Two twenty percent losses for example give you a total loss of 36%, not 40%.

Mathematically, if you start with any fixed amount, and then have any sequence of percent gains and losses where the percentages add up to zero, you will have less money than the original amount.