Annualized Rate of Return

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TJNeedsHelp

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May 29, 2022
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The formula for Annualized Rate of Return:
P = principal, or initial investment
G = gains or losses
n = number of days
AP = annualized performance rate
AP = ((P + G) / P) ^ (1 / n) - 1

So using :
P = 74.24
G = 1.95
n = 9
AP = ?
AP = ((74.24 + 1.95) / 74.24) ^ (1 / 9) - 1
AP = 1.025917 ^ 40.555 - 1
AP = 1.822 or 182.2%
No problem there

I need help with the equation to solve for G
1.822 = ((74.24 + G) / 74.24) ^ (1 / 9) - 1

And for knowledge I would like to be able to solve for n.
1.822 = ((74.24 + 1.95) / 74.24) ^ (1 / n) - 1


Everything I have tried has failed and I know the answer is easy and I will feel foolish. Can someone please show me the equation to solve for G and n. I am trying to do some projections for what Gain is needed for known number of days at a known AP with a known principle.

I am grateful for any help.

1.0259170654​
 
TJNeedsHelp said:
I need help with the equation to solve for G
1.822 = ((74.24 + G) / 74.24) ^ (1 / 9) - 1
Click to expand...
First step of solving a variable is to isolate the numbers from variables. Here first step of first step would be:

1.822 + 1 = ((74.24 + G) / 74.24) ^ (1 / 9) - 1+1 → 2.822 = ((74.24 + G) / 74.24) ^ (1 / 9)

To further isolate 'G' - I would need to do eliminate the ^(1/9)

How can I/you do that?
 
Major error in the above equation:
1.822 = ((74.24 + G) / 74.24) ^ (1 / 9) - 1
should be
1.822 = ((74.24 + G) / 74.24) ^ (365 / 9) - 1


I tried and failed the following to solve for G:
1.822 = ((74.24 + G) / 74.24) ^ (365 / 9) - 1


(1.822 + 1)^(1 / (365 / 9) = G = 1.0295 (Not Correct, Should be 1.822)
and
1.822 ^ (1 / (365 / 9) + 1 = G = 2.0149 (Not Correct, Should be 1.822)
 
Geeze, another error. There should be an edit button.

I tried and failed the following to solve for G:
1.822 = ((74.24 + G) / 74.24) ^ (365 / 9) - 1


(1.822 + 1)^(1 / (365 / 9) = G = 1.0295 (Not Correct, Should be 1.95)
and
1.822 ^ (1 / (365 / 9) + 1 = G = 2.0149 (Not Correct, Should be 1.95)
 
TJNeedsHelp said:
Major error in the above equation:
1.822 = ((74.24 + G) / 74.24) ^ (1 / 9) - 1
should be
1.822 = ((74.24 + G) / 74.24) ^ (365 / 9) - 1


I tried and failed the following to solve for G:
1.822 = ((74.24 + G) / 74.24) ^ (365 / 9) - 1
Click to expand...
1.822 = ((74.24 + G) / 74.24) ^ (365 / 9) - 1

1.822 + 1 = ((74.24 + G) / 74.24) ^40.55555556

2.822 = ((74.24 + G) / 74.24) ^40.5556

Now take "log" of both sides -

what do you get?​
 
By the way, there is both a preview button and an edit button.

You do realize that the formula you are using is derived by applying correct mathematics to an economically implausible assumption.
 
And the correct equation is:

G = (((AP+1)^(1/(365/n)) * P) – P

For JeffM
"You do realize that the formula you are using is derived by applying correct mathematics to an economically implausible assumption."

I totally agree with your statement when trying to project a stock price into the future. However it is useful to look into the future and evaluate whether or not a stock is meeting my investment criteria on an Annualized Basis. Now that I have the correct equation I can write the code to draw a trend line on my stock charting software.
 
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