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The Gory Details: When is AADGR the same as DGR?

When is the Average Annual Dividend Growth Rate (AADGR) the same as the Dividend Growth Rate (DGR) for the entire time period?
In other words, when is:


        AVERAGE(divtn/divtn-1-1;
             divtn-1/divtn-2-1;
             …;
             divt0+1/divt0-1) =
        (divtn/divt0)^(1/(tn-t0))-1

 


Where divt
m is the dividend payment for time period m.

 

Answer:

        AVERAGE(a; b; c; d; …; z)

is a shorthand notation for:

        (a + b + c + d + … + z)
        / (number_of_terms)

 

But if the individual terms, a, b, c, …, z  are all equal, then:

        (a + b + c + d + … + z)
        / (number_of_terms) =
        (a + a + a + a + … + a)
        / (number_of_terms)

 

And

        (a + a + a + a + … + a)
        / (number_of_terms) = a

 

Thus,

        divtn/divtn-1-1 =
        divtn-1/divtn-2-1 =
        … =
        divt0+1/divt0-1 =
        (divtn/divt0)^(1/(tn-t0))-1

 

Or for any sequential dividend payments, divtm and divtm-1

        divtm/divtm-1-1 =
        (divtn/divt0) ^(1/(tn-t0)) -1

 

Summary: If the AADGR is composed of a series of exactly equal annual DGRs, then the AADGR for the time period is the same as any individual annual DGR which is also the same as the DGR for the entire time period.