CAGR Formula

CFA Level II Quant
#1

Is the formula for CAGR:

Ending Value/Beginning Value ^(1/# of years) - 1

OR

Ending Value/Beginning Value ^(1/(# of years - 1)) - 1

Essentially, what is the correct denominator in the exponent to use? I have seen it presented different ways and am just wondering what the correct way to think about this is.

For example:

Stock A Performance:

2000: +10%

2001: +15%

2002: +3 %

Is the CAGR formula (130/100)^(1/3)-1

OR:

(130/100)^(1/2) - 1

Does it depend on the start date and go through the end date? For example, if we are measuring a stocks returns starting 1/1/2000 and ending on 12/31/2002, as exemplified above, wouldn’t that be three periods so (1/3) as the exponent or would it still be the intermittent periods so (1/2)?

#2

n=number of years after t0

2001, 2002, 2003

100, 110, 140

would be (140/100)^1/2 -1

#3

Where are you getting 140?

#4

It’s the made up value in his example that is below the year 2003

#5

Using this example:

Does it depend on the start date and go through the end date? For example, if we are measuring a stocks returns starting 1/1/2000 and ending on 12/31/2002, as exemplified above, wouldn’t that be three periods so (1/3) as the exponent or would it still be the intermittent periods so (1/2)?

What would the formula for CAGR be? I’m trying to understand how the exact start dates and end dates as far as when we are actually measuring this apply to the exponent in the formula.

#6

beginning value on 1/1/2000 is the same as the ending value on 12/31/1999…

2002 - 1999 = 3

So, the CAGR would be (EndValue2002 / EndValue1999) ^ (1/3) - 1

#7

Corrected

#8

Two different answers above…

If the number is compounding for all of 2000 (start day 1/1/2000), all of 2001, and all of 2002 (end day 12/31/2002), wouldn’t the CAGR from 100 to 130 be roughly 9% so n=3, not 2?

100 x 1.09 x 1.09 x 1.09 = 130

I guess it just depends on whether the beginning day is the first day of the year or last day of the same year for the n or n-1 # of years in the denominator of the exponent?

#9

You’re right, it’s 3 compounding periods. Assume it’s the start of 2000 to the start of 2003, so 3 years.

#10

Wait a second.

I will give an example:

2011 total sales = 1,200 million

2012 total sales = 1,290 million

2013 total sales = 1,375 million

What is the CAGR 2011 - 2013 ?

( 1,375 / 1,200 ) ^ (1 / 2) - 1 … not (1/3)

Why 2 periods? Because 1200 is end of year 2011 and 1375 is end of year 2013, so it is only 2 years. 2011 is not counted as a year because the 1200 value is only actual at the END of year 2011, so the previous 12 month (1year) must not be counted.

If they give you Total sales 2010 you would use 1 / 3 for cagr 2010 - 2013

Any question please ask. Regards.

#11

Well… in your example, you’re giving a base level in the first year (i.e. $1,200 in sales in 2011) versus us giving him a level of growth in the first year (i.e. 20% in 2011). Thus, when given a rate of growth in 2011, we are 20% higher than 2010. Therefore, you need one less year of sales data than you do growth rate data since your first year of growth is relative to the previous year. Let us combine examples, shall we…

2010 Sales = 1000

2011 Sales = 1200 (20% Growth)

2012 Sales = 1290 (7.50% growth)

2013 Sales = 1375 (6.589% growth)

Using the level of sales, we get: (1375/ 1000 )^(1/3) -1 = 11.199% … notice that the base level of sales was from 2010

Using the growth rates we get [( 1.20 )(1.075)(1.06589)]^(1/3) - 1 = 11.199% … notice the first growth rate used was the growth in 2011, however, it’s growth relative to 2010.

#12

yr0 - 100

yr1 - 110 (10%)

yr2 - 121 (10%)

yr3 - 133.1 (10%)

yr4 - 146.4 (10%)

146.4/100^4 - 1

CAGR = 10%

even though there are “five” numbers, there are only 4 compounding periods

#13

I’m totally agree with you, when using 3 growth rates you are assuming 3 years of growth, so cagr 2010-2013 will indeed be based on 3 years too. The issue is when you get 3 numbers (like my example), there you have 2 years of growth only. Notice that getting 3 growth rates you have 4 year numbers.