*Edit: There’ve been some excellent comments on this post by people who understand more about the way compound interest is presented than I do. So perhaps the most important thing about this is that when people talk about averages, they may mean the arithmatic mean, or they may be talking about an average determined by the end result. Which can be two different things. Or you might just want to skip this post. I promise there are better ones. 🙂*

We all know that when playing the market, even in safer categories like bonds and index funds and savings accounts, there’s some element of chance. The market fluctuates, and an average of 9.9% APY across 20 years might mean a sure thing or might mean wild jumps from 4% APY to 17% APY.

In the end, it’s ok because we weathered the storm and got our money, right? Well yes. Sort of. Being the spreadsheet geek I am and bored at work, I decided to put together a sheet which would illustrate the different possibilities.

What I came up with was one example of wild fluctuation, one of no fluctuation, one of mild fluctuation (less than 2% each way), and one of tiny fluctuation (0.5% each way). In each case, I put together some numbers rather randomly and then made changes until they all averaged the same 9.9%. Then I calculated what would happen if I put $1500 in an account and added $1000 more each year.

The results? Well, a steady 9.9% came out $66,537.96. The wildest one was $63,784.95, $2753, or 96% of the steady investment. The milder one was $65,947.12 (less than $1000 and 1% off). The tiny one actually came out ahead…by about $8.

Point being? It’s actually not that bad if it all averages out. While over $2500 may seem like a big difference for the wild one, it was only 4%, which isn’t too bad. I guess the lesson is that we can’t take compounding at face value and assume that an average of 9.9% means we have a certain amount of money. But we can come close. Food for thought next time you’re looking over a long-term account, such as a Roth.

(Note: the rubric for this is take the previous year, add it to the interest it would be earning and then add $1000. So, for example, for Year 2, the formula is add Year 1 and Year 1*Year 2’s interest and $1000. The assumption is that the $1000 would be added at the end, for simplification and that the balance of any year is its year-end balance.)

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How are you doing the average? Because if they are the same CAGR, the end result should be the same. Sounds like you did arithmetic mean.

I’m sorry I didn’t actually look at the spreadsheet, but I wanted to take issue with “wild jumps from 4% APY to 17% APY”. 4% might be a very optimistic bottom end for the sort of ‘non-bank’ investments one should probably lean to if one were in their twenties, IMHO. This is just a fellow PF student’s half-ass opinion but knowing what I do now, I would say be prepared for more dramatic losses than “only” a 4% gain.

I lost a little money in the late 90’s tech boom, proof that advice like “invest in what you know” is not always foolproof. But it was a wonderful lesson in not taking such “exuberance” for granted and that it actually should be your second job to manage your money right, including learning all the boring details about investing. But now for better or worse I’m a little obsessed about it. 🙂

As risk averse as this experience made me, I’m sure you know already that there would be expectable down years in a properly aggressive retirement or long term investment portfolio for a person starting in their 20’s or 30’s. But it might be appropriate to mention in the context of this article.

That seems to be a big problem with beginning investors, being scared to lose money. That might be something to address, if you ever expect gains in an investment over 10%, be prepared for losses some years. [I just made that up, so don’t quote me. Unless its totally correct. 🙂 ]

@ Pinyo, you’re right, I did the arithmetic mean. Good point, that we can mean different things by “average.” If we looked at the end gain and said it came out to an average of x, that would be different.

@ Swamproot, I poked around a bit, but there wasn’t anything (and I can’t do intense internet research at work) which had a nice chart of the percentage returns by year for an index fund. 4-17 is indeed optimistic. It’s also a result of me saying “Hmm…(supposedly) random number generator…hmmm” 😉

When people talk about APY, they’re talking about CAGR (compound annual growth rate). I.e., if I invested $1000 and received a 9.9% APY over three years, that means I have $1327.37 in my investment after three years. How I got there isn’t relevant. I could have made 9.9% every year, or I could have lost 20% in year 1 and then made 28.8% each year over the next two years (which has an arithmetic mean of 12.5%, but the same APY of 9.9%), or any of an infinite number of other combinations.

Since you’re a spreadsheet geek, you might appreciate that APY is calculated as a mean, but as a geometric mean rather than an arithmetic mean.

The geometric mean is defined as the n-th root of the product of n numbers. To use my prior example, the cube root of .8 * 1.288 * 1.288 (losing 20%, and then two years of gaining 28.8%) is 1.099 or 9.9%. To calculate the APY then, we take the final value ($1327.37), divide it by the original investment ($1000) and then take the n-th root (n being number of years of investment, in this case the cube root). This gives us 1.099 or 9.9%.

For the truly geeky, geometric means can also be calculated as arithmetic means, but they’re the arithmetic mean of the natural logs (and then you exponentiate the final result to translate it back). I.e. exp((ln(.8)+ln(1.288)+ln(1.288))/3) = 1.099.

The reason this method is used to calculate APY is because what we really care about is the final yield, not the individual pieces that went into it. The good news is that APY does mean what you want it to mean. (The bad news is that past performance is no guarantee of future results.)

Very nice, Andrew. Though I must say it’s intimidating even for me. I suppose if I sat down with a book and a spreadsheet…but once logarithms come into play I start to look worried. Thanks, though, I think I see what you’re driving at, I just wouldn’t know how to do it.

Take heart. As John Von Neumann famously said, “In mathematics, you don’t understand things. You just get used to them.” Though if you understand exponents, you already understand logarithms and just don’t know it yet.