How to convert annual rate to monthly rate for personal investment?

Question

If we have an annual rate of return of 7% for our stock investment, how much is the monthly rate?
First of all, why might we want to use monthly rate: that's because we may do monthly contribution of $300 to the account, so we need to calculate it month by month.
I have seen articles that talk about just to divide the annual rate by 12.
However, let's say, we look at $1000.
Assume we don't have money to contribute for the month, for simplicity of calculation.
Divide by 12 method:
$1000 * 7 / 100 / 12 = 0.583333 %

using this method, we have $1072.2901 at the end of 12 months. (please see spreadsheet at the end of question).
30 years later by the same calculation method in the spreadsheet, it is $8116.4975
Exponential method:
rate is really:
(1 + 0.07) ** (1 / 12) - 1 = 0.5654145387405274 %

(the ** above means "to the power of")
using this method, we have $1070.0000 at the end of 12 months. (please see spreadsheet at the end of question).
30 years later by the same calculation method in the spreadsheet, it is $7612.2550
What it really is, $1000 at 7% for 30 years:
1000 * (1.07) ** 30 = 7612.255042662042

so it looks like we really should use method 2?  The bank giving us the rate of 7% / 12 = 0.583333 % would be a "nominal value" for us to divide by 12, if the compounding is by monthly, and everybody does it by dividing by 12.  But if it is our investment calculations, then it seems it really should be by method 2?
By Method 1:

By Method 2:

D Stanley · Answer

Rates for loans and fixed income deposits (savings accounts, bonds) are typically quoted as an annualized rate without compounding, so when you see a loan at 12% interest, it's really a loan at 1% monthly. Quoting that way makes different compounding periods easily comparable and easier for consumers to understand the actual rate used without having to go into the complex math of compounding.
But that's not what you're looking at. You're looking at how much you are earning monthly taking compounding into consideration, so taking your annual gain to the 1/12 exponent is appropriate, meaning that you'll get the same return compounding the monthly return over 30 years as the annual return (1.56541453874 ^ (30*12) == (1.07)^30).
Or, put another way, what's the monthly growth you'll need to end up with a 7% return at the end of one year? You've shown that method 2 is the correct way to calculate that.

mhoran_psprep · Answer

I would go for the simpler method when doing the calculations. I base that a key assumption you have made in your question:

annual rate of return of 7% for our stock investment.

That is not guaranteed. It is in fact the long term average. You will have years where it is in the double digits, and years where it is negative. That is even more true if you try to break it down by month, during the 12 months there will be weeks or months where the value goes down. Think of the crash in the spring of 2020, but the S&P 500 still was up more than 15% for the year.
When looking at investments that will stretch out for decades you will not need to break it down by month, annual numbers work just as well as a guess.
If you are looking at an investment that will only last a few months then annual numbers have no meaning.

How to convert annual rate to monthly rate for personal investment?

Divide by 12 method:

Exponential method:

What it really is, $1000 at 7% for 30 years:

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