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The S&P500 has a yearly rate of return of 9.5%, so I assume this means if I invest \$100 on January 1 I'll have \$109.50 on January 1 of the following year, correct? So how much does that translate to per month? I'm pretty sure it's not just 0.095/12, so how do you solve this?
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I sort of have two answers for you. The first is the mathematically precise answer you're looking for and the second is a real world answer that will probably not excite you all that much.

1. First answer. The way to find the compounded average return for any time period is to use this formula:

1 - ((ending value/beginning value)^1/compounding periods)

So if we plug things in we get:

1- ((\$109.50/\$100)^1/12) = 0.76% monthly return

That's a little bit less than the 0.79% return if you simply divided 9.5% by 12. That makes sense because the way I figured it takes into effect compounding. You can double check things by doing this:

((1 + 0.76%)^12) - 1 = 9.5%

2. Second answer. My first answer is total false precision. If you look at the last 60 years, I think the number of times the market has returned the average compounded return of those 60 years in any one year was something like 2 years out of 60. In other words, though the average compounded annual return of the S&P 500 index might be something like 8% (not 9.5% as far as I know), the chance that it actually returned pretty much exactly 8% in any one of those 60 years was exceedingly low. More often it was up 20%, down 15%, flat, down 40%, up 50%, and that all averaged out to something like 8%.

So that said, trying to get even more granularity to see what monthly compounding would be on the S&P 500 index seems not so useful. When you're looking at spans of time of around 2 years or less the returns are largely unpredictable noise.

Mike
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What is the name of the formula in answer 1? Is that on Wikipedia?
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Fantastic! Thank you so much!
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```I see that given an initial investment of \$100 on 1/1/2015 with a
CAGR of 9.5% or a CMGR of 0.7591534291% results in a 1/1/2016 balance
of \$109.50 in both cases:

However, if we introduce additional \$100 contributions at the first
of every month, then the 1/1/2016 results change. The 9.5% CAGR gives
a 1/1/2016 balance of \$1,314.00 and the 0.7591534291% CMGR gives a
1/1/2016 balance of \$1,260.89. Why are these values different, and
which is correct?

Date        Contribution     Using 9.5% CAGR    Using 0.759... CMGR
1/1/2015            \$100                  \$0                  \$0.76
2/1/2015            \$100                  \$0                  \$1.52
3/1/2015            \$100                  \$0                  \$2.29
4/1/2015            \$100                  \$0                  \$3.07
5/1/2015            \$100                  \$0                  \$3.85
6/1/2015            \$100                  \$0                  \$4.64
7/1/2015            \$100                  \$0                  \$5.44
8/1/2015            \$100                  \$0                  \$6.24
9/1/2015            \$100                  \$0                  \$7.04
10/1/2015           \$100                  \$0                  \$7.86
11/1/2015           \$100                  \$0                  \$8.67
12/1/2015           \$100                \$114                  \$9.50
1/1/2016 Balance:                     \$1,314              \$1,260.89```