The answer is described in detail in my paper "a new discounting model ....". Please take a look and let me know if it answered your question to your satisfaction.

You may refer Bartlett, A.A. (1993).The arithmetic of growth: methods of calculation, population and environment, A Journal of Interdisciplinary Studies, Volume 14, Number 4, March 1993.

If the term exponential is used loosely, they are the same thing. Annualization calculations are at the heart of any applied macroeconomics. For example, US GDP growth is stated in annualized terms ((Q2/Q1)^4-1)*100. Interest rates get compounded at semi-ammual, quarterly, monthly, or even sometimes daily rate. ((1+r/(365*100))^365-1)*100, so, for example, 5% compounded daily translates to a true APR of 5.13% (so when Reg Q put a 5% ceiling on deposit rates, daily compounding could add an extra 13bps). Now true exponential growth of the form e^(rt) is simply compounding in continuous time--in other words it is the limit of CAGR or APR when the compounding interval tends to zero. So $1000 compounded daily at 5% for two years comes to $1105.16 = 1000*(1+5/(36500))^(365*2) whereas compounded in continuous time for two years it comes out at $1105.17 = 1000*(2.718281828)^(.05*2) So the truth is the choice depends on what is convenient for the most part. I deal with data more than modeling and the continuous time version doesn't help me much. For formal modeling, I am sure the continuous time version makes the math easier. However, given all the uncertainties in economics, this is not a choice worth losing a minute's sleep over.

Hi John,

So, would it be better to go for CAGR when the time interval is "annual" rather than quarter or something ? I have annual data series. My other doubt is that both of these functions produce same curves or different (it would be different i believe)? Given the scatter plot of fairly exponential shape, which among the two would serve better?

Thanks in advance.

Annual data would, to me, suggest CAGR as would quarterly or monthly data. If I have a monthly simple percent change of x then my CAGR is ((1+x/100)^12-1)*100. I don't think as a matter of applied economics I ever use the e^(rt) approach.

Ok John. And a last question. Like we get some curves for each functions like a straight line for a linear, the curves we get for CAGR & exponential functions will be different. The growth rate if we find for same data set will be different through these methods. Is that right ?

SJ It is not clear to me what you are doing, so I don't think I can answer your question as posed. The exponential to me is merely the limit of the compound growth method as the compounding interval shrinks. For all practical purposes, I can't imagine it makes a difference. I need a bit more color to be more helpful.

Hi John,

I have annual data for food grain production in India from 1960-2010. Scatter plot against year shows the shape is like exponential. Much common procedure is to find out CAGR with form Yt = Yo. (1+b)^t and find out "b" which is growth rate. I wish to know what is the difference between the above functional form and Yt = Yo.e^bt. Both are different forms of exponential curves. And what makes them different in terms of "shape" of the curve? I'm sure estimated values of "b" in above two equations deliver different growth estimates. If i wish to find out the growth rate, which of the above function should i prefer?

Second equation may be useful when time is an independent variable. Refer some research papers on the same or you can look into IASRI Website also.

Hi Naveen, even in first equation, time variable is the independent variable...... would that make any difference ? I'm not sure, and let me check for the statement......

Dear Balaji, It would make the sense, but usually the first equation may be used when the positive growth rate is expected.

OK I get it now. So you have a time series of grain output. Y1, Y2, ..... for periods t1, t2, etc. So take the form Yt = Y0*e^gt. Take logs and you have logY = A + gt, which is your regression. Run it in OLS and the coefficient on t is your growth rate.

Suggestion: Try going into Excel and writing two formulas in two different cells,

in one cell, [1+.(01/n) raised to the power n, where n equals (say) 2000 (put the n in a separate cell)

and

exp(1) raised to the power .01.

If you keep raising the value of n, from 2000 to 200000 to 2000000000 and so on, you should see the two values converge to be the same.