The issue of percentages comes up almost every time we talk about any kind of numerical data. This is especially true in the investment world where people generally tend to talk in terms of percentage gains and losses - rather than the absolute values. That’s simply because it’s easier to interpret financial changes on a common scale of 100. However, there are certain pitfalls that one should be aware of when dealing with percentage values. Here’s a quick (and elementary) look at some of these traps.

#### Absence of a time frame

When it comes to investments, simply stating percentage gain/loss is not nearly enough. There is something else that MUST go along with the percentage data: it is the **duration or time frame**. Without the time frame, most financial percentage figures are effectively meaningless. For example, simply saying “** Get a 100% return on your investment**” doesn’t make any sense -100% return in how much time? 69 years 8 months (corresponding to 1% annual return)? or 3 years 10 months (corresponding to 20% annual return)?

So, whenever you judge the performance of a stock (or compare two stocks), make sure that you are aware of the time frame before you start comprehending numbers like “300% increase”! *cough* penny stocks emails *cough*

#### Arithmetic mean against geometric mean

Let’s consider this statement “** Our picks have yielded 25% average annual returns over the past two years**“. Sounds genuine - after all it mentions the time frame clearly. But let’s run through an example and see how this can be pretty deceptive.

- Initial investment: $100

- Value of investment after one year: $200 (100% return this year)

- Value of investment after second year: $100 (50% loss this year)

- Net gain/loss amount over the two years: $0

Now, when you calculate “average annual return” using arithmetic mean, you are basically taking the average of **100%** and **-50%**, giving you a **25% average annual return**. But obviously, since your net gain amount is $0 (or 0% over two years), the simple arithmetic mean is not really telling you the entire story in this case.

This is where geometric mean comes into picture. Click here to learn how to calculate geometric mean of a given set of percentage values (including negative percentages). If you apply the geometric mean for the above example, you will end up with 0% average rate of return - which is correct. This rate of return - calculated using geometric mean - is known as “**annualized return**“.

Another way to look at it is that geometric mean takes into account the effect of compounding, whereas arithmetic mean does not.

I won’t be surprised to find a lot of proponents of volatile stocks using arithmetic mean to make their case of higher returns. So make sure you are aware about the difference between “average return” or arithmetic mean and “annualized return” or geometric mean.

#### Irrelevant time frames

Sometimes, data is presented with reference to a time frame (perhaps with careful consideration to the geometric mean) but this time frame may be largely misleading. In fact, I still don’t have a good explanation of what makes an “appropriate” time frame, but I am working on it.

For example, let us consider the share price of the S&P 500 exchange traded fund (SPY) over various time spans, going backwards from yesterday:

- 1 year return: 10.99%

- 5 year return: 69.69% [11.16% annualized (geometric) return; average (arithmetic) annual return of 13.94%]

- 10 year return: 62.83% [5% annualized (geometric) return, average (arithmetic) return of 6.28%]

Now, it is obvious that the annualized return differs if you choose different time windows over the history of the share. Depending on how you want to make your case, you could choose one of these time windows and go crazy with it. For example, you could pick the 5% annualized return for the last 10 years and blast the ETF’s performance, or you could pick the 11.16% return for the last 5 years and try to make a generalized statement like “*… gives almost a 12% return on an average*“.

Additionally, in all probability, people who want to downplay the 10 year performance will use geometric mean to make their case (geometric mean is always lower than the arithmetic mean), and people who want to glorify the 5-year performance, will use the arithmetic mean.

Readers should be wary of such pick-and-choose explanations. Try to find data that is most relevant to * your* investment time frame.

#### The disconnect between percentages and amounts

These types of examples are rarely seen in the investment world (I haven’t seen any), but are often observed in blog monetization circles. Here is an example:

My advertising income increased by 300%

Yeah right! It was 5 cents yesterday and today it’s 20 cents.

Obviously, in this case, the percentage value creates a greater boasting impact than the actual dollar value - so it might be used as a good marketing ploy to sell ideas/products.

It helps to have this scaling effect in your mind when you compare your income/net worth with others. For someone with $1,000,000 in hand, 10% means $100,000; whereas, for someone with $1,000 in hand, it’s just $100 (think in terms of buying power).

This becomes very obvious when you think in terms of percentage discounts on clearance products. What’s more attractive, a “**50% off**” sale on a $2 item you need or a “**5% off**” sale on a $100 item you need?

So, there you have it. Pay more attention to percentages in future.

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Steve Austin10.31.07 at 7:14 amStrong post. One other % snag (not really a trap) is the Percentage Inequality Illusion. I just made that name up, but I think the illusion is prevalent. E.g. say you invest $100, and at the end of the year the investment has appreciated to $125. That’s a 25% gain. The following year the investment sinks back to $100. That’s a 25% loss, right? Negative; it’s a 20% loss. $25 is 20% of the $125 start-of-year market value. I’ve read financial media that portray this as some sort of magical accounting fact.

25% gain and 20% loss somehow neutralize each other. But it’s nothing more than a notational illusion. Using fractions in lieu of % quickly illustrates that the two relative changes are visibly canceling: 25% gain is 5/4 in fractional notation; 20% loss is 4/5.

5/4 * 4/5 == 1

Not so mysterious now, is it? Unity is obvious as a product comparison, but deceptive when one compares each factor of the product to 1.

I suppose it’s dangerous to ignore unity. Even putting fractions aside (many people despise dealing with them), one can still bring unity back into the figures for clarity.

Instead of a 25% gain, start thinking of it as simply 1.25. Instead of a 20% loss, start thinking of it as 0.80. Inverting either shows the other, and confirms that they are indeed canceling factors.

The Dividend Guy10.31.07 at 8:39 amThis is a good post - it speaks to how the mutual fund industry can pull marketing tricks on us by representing results in different ways. I wonder what type of situations are better represented by each type of percentage?

J at Home Finance Freedom10.31.07 at 2:15 pmMMMM, Hello. Good summary.

Dividend Guy,

Geometric means are for multiplicative units (growth, rates, compounding).

Two more tricks are percentages of percentages and leveraged ROI calculations (click name link if interested).

Super Saver11.01.07 at 8:09 amGolbguru,

Great post. Shows why using absolute numbers and time frame is also important.

I recall reading a story about a car parts recycling dealer who became a millionaire. On average, he would buy parts for a dollar and sell them for three dollars. He once commented that it’s amazing how much money could be made in buying things and selling them for “1 to 2% more.”

MoneyNing11.01.07 at 9:08 amGreat post! Sometimes though these percentages help me stay optimistic and happy. For example, even my overall stock portfolio is doing bad, I can say that I doubled my money with stock X

It all depends on how you look at it!

Curtis11.02.07 at 4:36 amVery cool post. It’s fun to see others messing with numbers too. I teach economics and quanitative analysis at the graduate level in addition to my regular job. These are the kind of things I make sure to point out to students who’ve heard the saying that you can make statistics say what you want.

It’s a very true statement which means it requires us as consumers to inform ourselves on what the numbers mean rather than taking them at face value.

Laura11.02.07 at 5:55 amWonderful post. I loved how you showed that what they say isn’t exactly what is meant. I loved the disconnect between percents and amounts.

Pinyo11.02.07 at 11:22 amGreat post. I wrote a few times about percentage issues — i.e., gain/loss inequality, and geometric vs. arithmetic, etc. But this post definitely covers the whole gamut.

kitty11.02.07 at 6:47 pmVery good post. Interestingly the disconnect between percentage and amounts or in other words difference between relative and absolute is not limited to finances.

There are so many times you read in the media “this reduces your risk of X by Y%” without any regard of what your actual risk of X within some reasonable time frame like 10 years is. Sometimes they even say “this reduces your risk by a whopping 50%”. 50% of what? If your absolute risk is only 1%, 50% of 1% is still only .5 percentage points; nothing “whopping” about it. And if to get this 50% relative risk reduction you need to do something that carries a small risk as well, it’s kind of useful to know how this small risk is really smaller than the probability you’ll personally benefit. When I hear this misuse of meaningless relative numbers and want to scream “do you know the difference between absolute risk and relative risk?”.

Sorry for being a little off topic.

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