### Author Topic: How to calc avg yearly return?  (Read 1004 times)

#### FIREin2018 ##### How to calc avg yearly return?
« on: May 02, 2020, 08:35:57 AM »
say i invested \$100k into a fund 9 years ago.
it's \$200k now.
100% gain.

is it simple division? 100%/9= 11.1% yearly gain?

but what about the 'Rule of 72' which states 72/#years it took to double= rate?
thus 72/9 = 8%

Which is right?
Where's the disconnect?

#### beltim

• Magnum Stache
•      • Posts: 2964 ##### Re: How to calc avg yearly return?
« Reply #1 on: May 02, 2020, 08:50:22 AM »
say i invested \$100k into a fund 9 years ago.
it's \$200k now.
100% gain.

is it simple division? 100%/9= 11.1% yearly gain?

but what about the 'Rule of 72' which states 72/#years it took to double= rate?
thus 72/9 = 8%

Which is right?
Where's the disconnect?

No, that's not right.  You need to take the nth root.  So in this case it would be (200/100)^(1/9) which equals 8.0%

#### MustacheAndaHalf

• Magnum Stache
•      • Posts: 2938 ##### Re: How to calc avg yearly return?
« Reply #2 on: May 02, 2020, 08:31:43 PM »
Each year's return builds on the prior year's return.  Starting at 100k you could earn 8% or 10%:

End of first year
100k x 1.08 = 108k
100k x 1.10 = 110k

The reason you can't use simple division is the second year math:
108k x 1.08 = 116.64k
110k x 1.10 = 1.21k

That's why you need to use the approach beltim suggested.
100k * (1.07 to the 10th power) = 197k
100k * (1.08 to the 10th power) = 216k
100k * (1.10 to the 10th power) = 259k

But the general rule gets close: 72/7 = 10 years

#### Financial.Velociraptor

• Handlebar Stache
•     • • Posts: 1620
• Age: 48
• Location: Houston TX ##### Re: How to calc avg yearly return?
« Reply #3 on: May 03, 2020, 04:03:18 PM »
The "simple average" formula for Compounded Annual Growth Rate (CAGR) is: (currentValue / originalValue)^(1 / ((currentDate - originalDate)/365)) - 1

MS Excel is your friend for computing that formula.

If you want the truly CAGR where the growth was different in each period, the math gets really pretty hairy.

#### GreatLaker ##### Re: How to calc avg yearly return?
« Reply #4 on: May 05, 2020, 11:15:00 AM »
This article from Million Dollar Journey shows how to use XIRR to calculate investment returns.
https://milliondollarjourney.com/how-to-calculate-annual-investment-returns-using-xirr.htm

This thread on CMF explains the same thing, with techniques for multiple accounts and multiple years.

XIRR can be used to calculate CAGR with irregular contributions. It is available in various spreadsheets including Excel, Google Sheets and Open Office. Note that if you use it for less than one year it will annualize the return, i.e. if it goes up 10% in 6 months it will give 20% as the rate of return. That's only a problem for periods <1 year.

#### FIREin2018 ##### Re: How to calc avg yearly return?
« Reply #5 on: May 26, 2020, 05:11:51 PM »
No, that's not right.  You need to take the nth root.  So in this case it would be (200/100)^(1/9) which equals 8.0%
so if i bought a house @\$135k and it's now worth \$400k 25yrs later, then
(400/135)^1/25 = 11.85%/yr appreciation?!

well, my mortgage was like 4% so 7.85%/yr appreciation?

#### hodedofome

• Handlebar Stache
•     • Posts: 1382
• Age: 40
• Location: Texas ##### Re: How to calc avg yearly return?
« Reply #6 on: May 26, 2020, 08:56:11 PM »
http://moneychimp.com/calculator/discount_rate_calculator.htm

All the calculators on that site are very useful.

#### beltim

• Magnum Stache
•      • Posts: 2964 ##### Re: How to calc avg yearly return?
« Reply #7 on: May 27, 2020, 12:09:39 AM »
No, that's not right.  You need to take the nth root.  So in this case it would be (200/100)^(1/9) which equals 8.0%
so if i bought a house @\$135k and it's now worth \$400k 25yrs later, then
(400/135)^1/25 = 11.85%/yr appreciation?!

well, my mortgage was like 4% so 7.85%/yr appreciation?

(400/135)^(1/25)= 4.4%/yr
It looks like you didnt do the parentheses correctly.

#### MustacheAndaHalf

• Magnum Stache
•      • Posts: 2938 ##### Re: How to calc avg yearly return?
« Reply #8 on: May 27, 2020, 01:23:32 AM »
I'd recommend using a website to make the calculation, but in case breaking it apart is helpful...

400/135 = 2.963  (the total growth)
Since it took 25 years, you want to know what number to the 25th power results in 2.963.

x ^ 25 = 2.963
to solve that exponential means taking the log of both sides.  I'm using "log base 10" for this:

log (x ^ 25) = 25 * log x     ;   log (2.963) = .4717    (aka, 10 ^ 0.4717 = 2.963)
25 * log x = 0.4717

divide both sides by the years:
log x = 0.4717 / 25 = 0.01887

To get rid of the (log x), we take 10 raised to the power of both sides of the equation:
10 ^ (log x) = x   ;   10 ^ 0.01887 = 1.044
x = 1.044

Given a growth of 2.983 times over 25 years, that is like an average growth of 4.4% per year.

#### vand

• Handlebar Stache
•     • Posts: 1014
• Location: UK ##### Re: How to calc avg yearly return?
« Reply #9 on: May 27, 2020, 01:52:06 AM »
Nope. Work out the total return and then caculate the nth rooth.

Total return = [(400/135) - 1]  = 1.93  ie 193%

Nth rooth = [(400/135) - 1]  ^ (1/25) = 1.0273, ie 2.73%pa

Check this by 1.0273 ^ 25 = 1.93 = 193% return

135 * 1.93 + 135 = 400

This is high-school maths

BUT

Looking at home prices is like looking at stock prices without accounting for dividends.

So if you are talking about your house then you need to factor in imputed rent that you have consumed from it. So while the capital appreciation is a rather unimpressive 2.73% if the fair market rental yield was say 5% of the price every year then your total return would have effectively been 7.73% annualised.

« Last Edit: May 27, 2020, 02:08:19 AM by vand »

#### mrmoonymartian ##### Re: How to calc avg yearly return?
« Reply #10 on: May 27, 2020, 02:40:04 AM »
Strange. I agree with the people who aren't currently in the UK:

((400/135)^1/25)-1 = 4.44% CAGR, according to the wikipedia formula.

Are we all using the same version of MS Paint to solve our fractional exponents?

#### beltim

• Magnum Stache
•      • Posts: 2964 ##### Re: How to calc avg yearly return?
« Reply #11 on: May 27, 2020, 02:58:48 AM »
Nope. Work out the total return and then caculate the nth rooth.

Total return = [(400/135) - 1]  = 1.93  ie 193%

Nth rooth = [(400/135) - 1]  ^ (1/25) = 1.0273, ie 2.73%pa

Check this by 1.0273 ^ 25 = 1.93 = 193% return

135 * 1.93 + 135 = 400

This is high-school maths

Where did you go to high school?  Your math is wrong, and the fact that your check comes up with the same number only means you can do reciprocal math - but it still gives the wrong answer.

To see an example, lets take something where we know the answer already. Something that has a 10% annual return will, after two years, be up by 21%. Lets plug that into your incorrect formulas and see what happens:
(1.21-1) = 0.21
0.21 ^ (1/2) = 0.46

Your formula makes a 10% return into a 46% return!

#### habanero

• Pencil Stache
•    • Posts: 636 ##### Re: How to calc avg yearly return?
« Reply #12 on: May 27, 2020, 03:03:15 AM »
Nope. Work out the total return and then caculate the nth rooth.

Nth rooth = [(400/135) - 1]  ^ (1/25) = 1.0273, ie 2.73%pa

This it where it goes wrong, you do not subtract 1 before thaking the n-th root. You do that afterwards.

Correct is

(400/135) ^ (1/25) - 1 = 0.0444 = 4.44% annualized return.

An to check it:

135 * 1.0444 ^ 25 = 400

#### vand

• Handlebar Stache
•     • Posts: 1014
• Location: UK ##### Re: How to calc avg yearly return?
« Reply #13 on: May 27, 2020, 06:31:03 AM »
Haha. OK, I fvcked up :)

#### hodedofome

• Handlebar Stache
•     • Posts: 1382
• Age: 40
• Location: Texas ##### Re: How to calc avg yearly return?
« Reply #14 on: June 03, 2020, 07:07:41 PM »
All you guys arguing over formulas, thats why I just go to moneychimp.com. No more jacking with stupid Excel formulas that can get jacked up with an extra parentheses.

#### MustacheAndaHalf

• Magnum Stache
•      • Posts: 2938 ##### Re: How to calc avg yearly return?
« Reply #15 on: June 03, 2020, 10:55:52 PM »
Nope. Work out the total return and then caculate the nth rooth.
...
This is high-school maths

Haha. OK, I fvcked up :)
Admitting a mistake is a good start, but I'd also suggest you avoid insulting me or others ("This is high-school maths").  Insults show a lack of something else to say, and when you additionally wind up being wrong, it makes me wonder if goal of your post was to insult someone. 