“The stock market returns an average of 10% per year.” How often have you heard that saying? Do you really understand what it means? Just why is Queen Latifah famous, anyway? Sorry…I got distracted.

Please carefully review the following numeric series: -20%, +20%, -20%, +20%, -20%, +20%. If those were your yearly investment results for the past six years, what was your average yearly rate of return? If you said “-2.02% per year”, you are correct. If you said “0% per year”, you have mistaken arithmetic return for geometric return. If you said “yellow”, you may have attended a public school in South Carolina, and for that I am truly sorry.

Let’s back up and assume that you invested $100,000 and earned those aforementioned returns. After one year, you would have $80,000$100,000 minus 20%. After two years, you would be worth $96,000The first year’s $80,000 plus 20%. Notice a pattern here? Even though your first two returns (-20% and +20%) averaged out to zero, you still lost money overall. Follow that trend for six years and you’d be left with just shy of $88,474, or barely enough to buy Apple’s newest iFad.

A frequent investing mistake when calculating yearly returns is to add the returns for each year, and divide by the number of years. While this is the common definition of “average”, this actually denotes an arithmetic average, which provides incorrect results when reviewing financial data.

In fact, the correct method is to convert to a decimal and multiply by your starting capital. In the case above, $100,000 x 1.2 x 0.8 x 1.2 x 0.8 x 1.2 x 0.8 = $88,473.60. And ($88,473 / $100,000) raised to .1666661 / number of years, which is six equals 0.979796, or a loss of about 2.02% per year. Let’s try it again with a zero “average” return but larger deviations: $100,000 x 1.4 x 0.6 x 1.4 x 0.6 x 1.4 x 0.6 = $59,270.40, or -8.35% per year. Oh no!

^ Not relevant to this topic, but a good Brad Pitt “Oh No” pic

The overriding lesson here is that the geometric return is always less than or equal to the arithmetic return, and will be drastically lower in times of high volatility. Consider if your financial adviser handed you the following results for four years of investing: +100%, +100%, +100%, -100%. Arithmetic average: 50% per year. Geometric (real) average: 0%. You lost all your money. Go back to the starting line. Do not pass Go. Do not collect $200. Do not adopt a Cambodian baby with Angelina Jolie.

To follow up with my starting paragraph, historically the United States stock market has returned approximately 10%equity appreciation plus dividends each year. This is a compound (geometric) return, which is tantamount to multiplying your money by 1.1 per annum. By now you should realize that this means the arithmetic average was well over 10% per year, and this also means the next time you hear your pretentious neighbor brag about last year’s 60% return on his volatile hedge fund after a rough starting stretch, you can rest easy knowing that the law of geometric averages is a high barrier to hurdle.

I’m just kidding, you should let your dog out to go pee in his yard.

Quick: I’ve handed you a quarter to flip. Heads, I’ll give you $50 million; tails, you get nothing. Oh, there’s another option – skip the flip and I’ll give you $20 million right now. Which will it be?

The conundrum here is that most people would pick the guaranteed $20 million over the $25 million expected valueExpected value, in economic terms, is the sum of the valuation of all possible outcomes of a situation multiplied by the probability that they will happen. In this case, there’s a 50% chance you get $50 million and a 50% chance you will get nothing. Hence, (0.5 x $50,000,000) + (0.5 x $0) = $25,000,000 expected value. you’d get from the coin flip. Some economics texts would call that irrational behavior. I couldn’t disagree more.

Let’s attack this using a different mental exercise. You’ve just won a few million from Carlos Mencia in a class action suit for pain and distress caused by extraordinarily awful comedic material. Being of sound financial judgement, you’ve decided to buy an Audi R8.

After plunking down $140,000 for this chunk of Viagra on wheels, the dealer offers you a second model (in green) for the same price. Do you take it?

Well, probably not.

Do you remember anything from Econ 101? There’s a term called “marginal utility“, which can be described as the utility gained from an additional unit of a good or service. If you’ve ever stuffed yourself with five pieces of pizza and declined a free sixth, you’ll know that the marginal utility of that sixth piece of pizza was zero.(or negative: you’d pay not to eat it)

Back to the car. The first car was worth more to you than $140,000 in cash, which is why you exchanged the latter for the former. The salient point here is the second car has less “marginal value” to you and, from your viewpoint, is actually worth less than the first car, although it’s exactly the same car.

You’re smart. If not, you’d be listening to Nickelback and stalking your exes on Facebook instead of reading a financial blog. The fact that a second Audi is less valuable to you than the first is no earth-shaking revelation. However, sometimes we forget to apply this concept to money itself. There’s a saying which goes, “the first million is the hardest”. That can, without altering the meaning of the phrase, be restated as “the first million is the most valuable.”

Study after financial study has shown that after essentials such as food, water, and shelter are taken care of, happiness increases very little with additional amounts of income or net worth. The first dollar is more valuable than the second.For the mathematically-inclined among you, this is tied to a phenomenon known as Benford’s Law, which gives the surprising result that among most real-world financial data, the first digit is a “1″ about 30% of the time, declining to the point where “9″ as a first digit occurs less than 5% of the time. Benford’s Law is used by auditors to sniff out cooked books, and is admissible in court as evidence of fraudulent data. All Benjamins are not created equal.

There are a few exceptions to the law of diminishing utility of money, namely, when the individual is trying to reach a certain monetary benchmark. In the beginning of the 1994 film Maverick, Bret Maverick, played by a pre-crazy Mel Gibson, is $3,000 short of a poker tournament entry fee of $25,000. While his efforts to make the needed $3,000 are entertaining, they also underscore the point that the final $3,000 is highly valued by Maverick. If you waited until right before the tournament started and asked him to choose between a guaranteed $2,000 or a coin flip for $3,000, he would pick the chance to win $3,000 ($1,500 expected value) over the risk-free $2,000.Note that this is the opposite result of the coin-flip choice from our first paragraph. Interesting.

Also, we get to see Jodie Foster:

In most cases, example above notwithstanding, the diminishing marginal utility of money is a reliable phenomenon.

I’m not telling you not to shoot for the stars. I’m not telling you to disregard your dream where Microsoft buys your social-networking-for-dogs website for 73 bajillion dollars and you live out your days in Tahiti wearing small shorts and drinking big drinks. Just keep in mind: the first million, well, that will be the sweetest.