## Tuesday, February 05, 2008

### CAGR

Today I learnt much about CAGR.

No, it's not a new brand of cigar. CAGR stands for compound annual growth rate. It's a nifty mathematical formula to calculate the rate at which an investment grows linearly in a given period.
It all springs from the basic future value/present value equation:

Future value = Present value x (1 + i)^(t)

where i - interest rate compounded per per year
where t - time in years

Appropriate adjustment can be made if the compounding period is less than a year. Just change the appropriate figure for the time.

I saw CAGR all the time in brokerage report, yet I do not understand what it is until I was forced to come to terms with it while analysing some presentation materials for china milk. It helps that prior to that, I was reading intensively on a lot of books that give calculates the growth rate for their investment using this method (though they might not call it by the same name).

A good example should be able to illustrate the finer points of the formula:

Let's say that in 1st Jan 2005, I started my investment portfolio with \$10,000
In 1st Jan 2006, I made a loss, so my portfolio drops to \$8000
In 1st Jan 2007, I recouped my loss somewhat and my portfolio stands at \$9500
In 1st Jan 2008, my portfolio ends at \$12,000

To calculate my CAGR, I just plug it into the formula:

Present value = 10,000
Future value = 12,000
time period = 3 years

i or CAGR = 6.27% compounded annually

When CAGR is used, one must be careful of the assumptions behind the formula:

1. The maths of the formula assumes that the value of the investment is increasing steadily from an initial value of \$10,000, compounded annually for 3 years to reach \$12,000

Based on CAGR of 6.27%, my linear returns are as follows:

1st Jan 2005 - \$10,000
1st Jan 2006 - \$10,627
1st Jan 2007 - \$11,293
1st Jan 2008 - \$12,001

As you can see, this is vastly different from the actual portfolio returns I have at the start of each period. Therein lies the problem...CAGR does not take into account the stability of the portfolio value. CAGR cares more about the destination rather than the journey to reach it. In other words, a CAGR of 10% can be more volatile than a CAGR of 10% in another investment. One still have to look at the year to year figures to see it one can stomach the ups and downs of the value, given the same CAGR.

2. For calculations of CAGR, the time period have to be the same. Otherwise, there is really no basis for computation. For example, if I want to find the value at 5th May, 2008, it's hard. We can make a linear extrapolation to generate the results, but seriously it's pointless because CAGR assumes a linear trend, which doesn't really happen in real life. Manipulations can be done to compare quarter to quarter, half year to half year or annually period to get different CAGR to amplify one's point.

3. The formula also does not take into account of what happens when I inject another sum of capital on top of the initial amount invested.

Let's say that in 1st Jan 2005, I started my investment portfolio with \$10,000
In 1st Jan 2006, I made a loss, so my portfolio drops to \$8000
In 1st May 2006, I injected another \$5000, into my portfolio and it ends at \$14,000
In 1st Jan 2007, I recouped my loss somewhat and my portfolio stands at \$13,800
In 1st Jan 2008, my portfolio ends at \$15,500

How to account for that? I do not know yet. Perhaps CAGR no longer applies, and I have to calculate my rate of returns instead.

One can also extend this argument to a company. If I want to calculate the CAGR of revenues coming in for a period of 5 years, and in the period the company bought more assets which can be used to generate more revenues...CAGR no longer makes sense anymore. What to make of this? I do not know.

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** Here, I would like to appeal for help to enlighten me on this issue. Any comments will be answered and thanked for!

Steadybull mentioned a very interesting insight...that of using EPS to calculate and compare one's return, where earnings refer to capital gains (loss) plus dividend received. By doing that, the EPS will be automatically reflecting any changes in my working capital (i.e. if there is an injection or withdrawal of funds). Perhaps one can do an EPS annually, then do a CAGR on each of the EPS for the period to be calculated.

But I foresee 2 problems - one is that the problem of different period won't be solved if I use EPS. Second problem...what if there are 'negative' earnings? Meaning that there are capital losses?

There should be a more elegant way to dealing with this.

Grey said...

Just to share what I know;

I think it's better to generalise when using the FV-PV formula to:

FV = PV*(1 + r)^t, where r is the rate of returns;

By thinking in more general terms, you might be able to think of more opportunities in the real world;

Suppose you only remember the formula's r as interest, you might gravitate towards bonds, fixed deposits, etc, which pays interest, when better returns could be achieved elsewhere, if you only think of it;

I think, a subtle difference in thinking can result in drastic change in real world results; It's like when you merely slightly misalign the rifle sights when shooting a target, you miss by a large margin;

Ok, seems like I just made a mountain out of a molehill;

Createwealth8888 said...

Hi,

I track the return on trading capital simply on capital growth rate relatively to ROC, ROC growth rate, FD interest rate, STI growth rate. Hope that you may find a better method and share with us.

la papillion said...

Hi grey,

Thks for pointing that out. I am actually thinking of rate of returns, but couldn't find a better name for it, hence i called it interest. I do agree that interest isn't the best word to use here :)

Actually for the rifle sight, the error will be greater if the target is further. There is no error if it is point blank. I think this analogy relates to real life as well :P

la papillion said...

Hi creatwealth8888,

Can share more on what you meant by capital growth rate? Do you mean that you check your final portfolio value (initial amt + loss/gains on capital + dividends) against the initial and measure the rate? Isn't that CAGR?

But I like the idea of tracking with reference to the ones that you mentioned. It gives a comparison of your investment versus other possibilites.

Createwealth8888 said...

Hi la papillion,

You have already mentioned it in your point 3, when we add more capital, then CAGR no longer applies.

Capital growth rate means % of capital added annually to the total trading capital. You may wish to visit my blog to view my tabulated portfolio performance.

la papillion said...

Hi creatwealth8888,

Okay, I got what you mean when I read your blog :) Thks! I think you have a very systematic way of tracking your portfolio. Makes it very easy to track it.

I think I should have a proper system too :)

Thks!