How do I figure average annual growth?

[Andy_B]

I know this seems like a basic question but it may not be (at least for me).

Say you have the following two successive years of %

year 1: -10%
year 2: +10%

The quick formula to use would simply be (year 1 + year 2)/2

But as anyone on here realizes, if you lose 10% in one year and then gain 10% back the next, you are not back to square 0.

Any help would be appreciated.

Andy

[peledre]

An easy example w/out any complicated math would be to simply attach a number to the percentage and do the computation by hand. I'm sure there's a formula to figure it out, but I don't have it at my reach right now.

Year 1:1000 -10%
Year 2:900 +10%
Year 3:990 +5%
Year 4:1039.50 +2%... and so on.

[Andy_B]

Hi Peledre,

I still don't understand. In your example, what is the average % annual growth?

I know its not (-10 + 10 + 5 + 2)/4


The reason I am wondering is that I know what percentage growth I have had for each of the last 7 years, but I am trying to figure out what my average annual growth is so that I can compare it to some growth models (7% annual growth, 8% annual growth etc).


Andy

[Pack87Man]

Hi Peledre,

I still don't understand. In your example, what is the average % annual growth?

I know its not (-10 + 10 + 5 + 2)/4


The reason I am wondering is that I know what percentage growth I have had for each of the last 7 years, but I am trying to figure out what my average annual growth is so that I can compare it to some growth models (7% annual growth, 8% annual growth etc).


Andy
Andy, the equation for geometric mean (which is what you're looking for. The one you know isn't right is called arithmetic mean) is very complicated. It involves cube roots and a few other things that would take forever to do by hand. I'm not sure where to find the actual equation, though.

[striker]

I have the same question, especially when I read mutual fund reports about their annual return over a number of years. Do they normally calculate this according to Andy_B's method or something more sophisticated?

[Andy_B]

Do they normally calculate this according to Andy_B's method or something more sophisticated?

No what I am saying is that I know what I showed was wrong. I was just trying to find the right formula to plug in.

Pack87man mentioned that I am looking for geometric mean so I will do a search on that. Thanks Pack!

Andy

[Pack87Man]

I have the same question, especially when I read mutual fund reports about their annual return over a number of years. Do they normally calculate this according to Andy_B's method or something more sophisticated?
They calculate it by Andy's method. Arithmetic mean is always going to be higher than geometric mean (for the simple reason that your portfolio can never drop more than 100%, and it can go up infinitely. If your portfolio drops 25% one year, then goes up 33% the next year, you're at the exact same place, but arithmetic mean will tell you that you have an average 4% gain a year). Since arithmetic mean is higher, and these people are in business, what do you think they're going to show you?

[Andy_B]

If your portfolio drops 25% one year, then goes up 33% the next year, you're at the exact same place, but arithmetic mean will tell you that you have an average 4% gain a year). Since arithmetic mean is higher, and these people are in business, what do you think they're going to show you?

Holy smokes. I never thought someone like Morningstar would do this. That sucks. Thank you for the heads up.

Andy

[striker]

They calculate it by Andy's method. Arithmetic mean is always going to be higher than geometric mean (for the simple reason that your portfolio can never drop more than 100%, and it can go up infinitely. If your portfolio drops 25% one year, then goes up 33% the next year, you're at the exact same place, but arithmetic mean will tell you that you have an average 4% gain a year). Since arithmetic mean is higher, and these people are in business, what do you think they're going to show you?

That was my impression. Thanks for confirming it.

[Pack87Man]

You're welcome. If'n I get the chance, I'll post up the difference in the S&P returns using the two different means tomorrow. It's eye opening.

[peledre]

Hi Peledre,

I still don't understand. In your example, what is the average % annual growth?

I know its not (-10 + 10 + 5 + 2)/4


The reason I am wondering is that I know what percentage growth I have had for each of the last 7 years, but I am trying to figure out what my average annual growth is so that I can compare it to some growth models (7% annual growth, 8% annual growth etc).


Andy
Take the total interest earned divided by the initial principal, divided by the # of years, and that would be your simple growth. So... $39.50 over 4 years equals about 1% per year growth.

[Andy_B]

Thanks Peledre, I will give that a shot.

Andy

[Pack87Man]

Thanks Peledre, I will give that a shot.

Andy
Remember, Andy, that that formula will get you simple growth. If you want compounded growth (i.e. how much you made that yearfrom where you began the year, rather than how much you made versus where you originally started) it gets more complicated.

[Andy_B]

Remember, Andy, that that formula will get you simple growth. If you want compounded growth (i.e. how much you made that yearfrom where you began the year, rather than how much you made versus where you originally started) it gets more complicated.

Shoot, now I am not sure what I want.

When I am running my models, I use X% yearly growth to look into the future.

Since I now have some good history to draw from, I want to compare the X% number with the actual number I am attaining right now.

Is that number simple or compounded? I think it may be compounded. From your explanation I am not sure how Peledre's equation would help me make the comparison I am looking for.

Andy

[MarioKempes]

Let's take a simple example

Principal = $1.00

1st Year Annual Growth Rate = -10% ==> Balance = $0.90

2nd Year Annual Growth Rate = 10% ==> Balance = $0.99

AVERAGE Annual Growth Rate = ((-10% + 10%)/2.0) * 100 = 0%

2 Year Return = ((0.99/1.00) - 1) * 100 = -1%

What you are really looking for is the compounded annual growth
rate, or CAGR.

n = # years
CAGR = (ending value/starting value)^(1/n) - 1


For our example,

n = 2
ending value = $0.99
starting value = $1

CAGR = (0.99/1)^(1/2) - 1 = -0.00501256..., or -0.501256%


$1.00 * (1 - 0.00501256) = $0.99499
$0.99499 * (1 - 0.00501256) = $0.99

[Andy_B]

ok thats some serious rep points there.

Thank you Mario!!

Time to plug that into excel!

Andy

[Pack87Man]

Ah, Mario beat me to it. I was about to post the same thing.

[MarioKempes]

Ah, Mario beat me to it. I was about to post the same thing.

Yeah sure you were. ;)

[Karl K]

You can quickly figure out Compouned Annual Growth Rate in Excel using the GEOMEAN function (for geometric mean)

Suppose I have a spreadsheet with the following numbers (row and column numbers)

...........A
1.......22.00%
2.......73.00%
3.......21.00%
4.......45.00%
5......-10.00%

To calculated the CAGR for this series of returns I use the following formula in Excel:

{=GEOMEAN(1+A1:A5)-1}

It's a bit tricky to get the brackets on the outside (you must have brackets).

Type:
=GEOMEAN(1+A1:A5)-1
in a cell...
but don't hit the Enter key ... yet!!
Hold down the Ctrl and Shift keys and NOW hit the Enter key.
It changes to {=GEOMEAN(1+A1:A25)-1} giving the CAGR of 27.22%

[Andy_B]

Thank you to Karl and to everyone that has helped me out! I very much appreciate it!

Andy