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.