Why do we use relative numbers for gains and losses?

By Greg Turnquist

Greg L. Turnquist worked on the Spring team for over thirteen years and is a senior staff technical content engineer at Cockroach Labs. He was the lead for Spring Data JPA and Spring Web Services. He wrote Packt's best-selling title, Learning Spring Boot 2.0 2nd Edition, and its 3rd Edition follow-up along many others.

March 26, 2012

I suppose you’re very familiar with hearing news like “The Dow is down 2.5% today,” or “The S&P is up 3%.” Mapping into basic mathematics, that would be -2.5% and +3%, relatively speaking.

But why are these figures couched in relative numbers? In my previous article about rebalancing my portfolio, I talked about a scenario where we suffered a -50% loss, followed by three +20% gains. Calculating the average annual gain was easy: +2.5%. But the ACTUAL gain was harder to figure out. Do you know why? Because everything we read and hear on the news is oriented towards calculating the arithmetic mean, which can give a real false positive on the situation.

I’m not alleging there is some huge controversy to cover up the real performance of the investment products. It’s more likely that the math to calculate the arithmetic mean is simpler and easier to understand than what we should be doing to properly evaluate something. What we SHOULD be using is the geometric mean.

  • Arithmetic mean is when we add up all four of numbers up above and divide by four.
  • Geometric mean is when we multiply all four numbers together and take the 4th root.

First of all, geometric mean doesn’t work with relative numbers. Can you imagine what -50%*20%*20%*20% ^ 1/4 would equal? If you studied advanced math, then you realize this would involve taking the square root of a negative number. Actually when something neither grows or declines, we multiply by 1. A 50% loss is represented as 0.5, or half of 1. A 20% gain is 1.2. You get the idea?

Our original scenario would be (0.5 * 1.2 * 1.2 * 1.2) ^ 1/4 = 0.964. Since this is below 1, it is a loss (as it should be!) This comes to roughly -3.6% annual loss (instead of the +2.5% we figured out earlier).

So there you have it: the way you calculate things can tell the difference between a loss or a gain. Now let’s check this out with a more realistic situation, like the performance of the S&P 500 index. That is something we always hear people citing.

The S&P showed the following average gains/losses (in %) from 2001-2010: -13, -23.4, 26.4, 9.0, 3.0, 13.6, 3.5, -38.5, 23.5, and 12.8.[1]

If we add up all 10 numbers and then divide by 10, we get => +1.69%. A paltry gain, but there were two market corrections, so not too bad, ehh? After all, the total gain (when we don’t divide by 10) was 16.9%, right?

Sorry, but that is wishful thinking. If we convert them to absolute numbers and then calculate the geometric mean by multiplying together and taking the 10th root, we get => 0.995, or a -0.5% loss. Total loss over that 10 year span (when we don’t take the 10th root) was really -4.7%.

Compared with our assumed total gains, this is more than a 20% gap! The differences between arithmetic and geometric mean can be devastating, especially if you were planning on retiring anywhere in the last decade.

The differences between what financial pundits say on TV shows and what is really happening can be huge!

Remember in that last post how I talked about an EIUL, or Equity Indexed Life Insurance policy, and how it could shield us from losses? Well, let’s run that scenarion and assume our money has been saved up in an EIUL. For this example, we’ll assume our EIUL has a minimum of 0% and maximum of 15%.

Revised numbers for 2001-2010: 0, 0, 15, 9, 3, 13.6, 3.5, 0, 15, and 12.8.

I know what you’re thinking. Doesn’t look good getting nipped on those really big gains, does it? Instead of the first 26.4%, we only get 15%. And trading in that 23.5% for 15%? Whew! But what is our REAL performance given all this? What if we calculate the geometric mean?

Result: 7.0% average annual growth. Wow! Compare that with our -0.5% loss. Total growth over 10 years was 96%. We almost doubled our money over that time span. Compared with basically standing still in the face of an average 2-4% inflation makes our EIUL a tremendous boon compared to some 500 index fund.

This begs the question: why do people use the arithmetic mean so much? For one thing, it’s easier. Your simple calculator is able to divide by 10, but you probably needed a scientific one to calculate the 10th root. It also may stem from certain financial calculations being valid with the arithmetic mean. For example, a financial instrument that is providing you with money without having it’s principle value being impacted. A key example here are dividends. If you hold a certain amount of stock and are taking cash dividends, then the capital value of your shares aren’t part of the equation. The dividend yield is what’s important. If you aren’t reinvesting the money, than simply averaging the numbers over a certain period of time will give you an average amount of dividend money you can expect to receive. But if you start investing the money back into the stock, your principle value will start to change, and this simple math tool is no longer valid.

So the next time you hear people give average rates of return on various products, think twice and double check how they are making their calculations. It can mean the world between gains and losses.

UPDATE: For the previous set of numbers, I used a spreadsheet, and found it clunky because there was no equivalent to SUM(list) like MULTIPLY(list). Instead, I wrote a short scala application. (Stay tuned for this getting published). After verifying the above numbers, I went on to add S&P 500 performance numbers all the way back to 1971. Here are the results:

Arithmetic mean of the S&P 500: 8.31%
Geometric mean of the S&P 500: 6.76%
Total growth factor: 13.7 x your original money

Arithmetic mean of our EIUL: 8.70%
Geometric mean of our EIUL: 8.50%
Total growth factor: 26.1 x your original money

A) If you hear people saying that the S&P performs close to 12%, well then, the last 40 years don’t reflect that. B) The differences when you remove losses is almost a 2% boost, which over 40 years can result in doubling your money.

[1] – http://www.forecast-chart.com/historical-sp-500.html

I am not a licensed financial advisor nor an insurance agent, and cannot give out financial advice. This is strictly wealth building opinion and should be treated as such.


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