Derivation of Formula for calculating compound annual growth rate for period when two consecutive growth rates are provided with.

Formula for calculating compound annual growth rate is

r1=[{V 1V 0 }1 /n1

−1]∗100........................................(1)

Here r1 is the compound annual growth rate for n1 duration for first period and V1 and V0 are the values for first period and initial time respectively.

Similarly we can write

r2=[{V 2V 0 }1 /n2

−1]∗100......................................(2)

Note that both the growth rates r1 and r2 are calculated from base year value of V0.

By rearrangement of (1) we can write

r1100

+1=[ V 1V 0 ]1/n1

Let

r1100

+1=R1 and therefore

R1=[V 1V 0 ]1/n1

R1n1=[V 1

V 0]

...................(3)

Similarly we can write for (2)

R2n2=[ V 2V 0

]........................(4)

Upon dividing (3) by (4) result is

R1n1

R2n2

=V 1/V 0

V 2/V 0

R1n1

R2n2

=V 1V 2

V 1V 2

=R1n1

R2n2

[V 1V 2 ]

1n2−n1=[ R1n1R2n2 ]

1n2−n1

.....................(5)

Let n2−n1=e and always that n2 > n1 and e>0

Now (5) can be rewritten as following

{[ V 2V 1 ]1e−1}∗100={[ R2n2R1n1 ]

1e

−1}∗100...............(6)

Now left hand side (LHS) of (6) can be written as gw because LHS is the growth rate in values from V1

to V2 during duration ‘e’ and therefore (6) can be rewritten as following

gw={[ R2n2R1n1 ]1e

−1}∗100

gw={[ R2e+n1R1n1 ]

1e

−1}∗100

gw={[ R2e R2n1R1n1 ]

1e

−1}∗100

gw={[R2e( R2R1 )n1]1e−1}∗100

and

Then

gw={R2 (R2R1 )n1e−1}∗100

Now how to use this formula?Suppose you have been provided with data on growth rates as following

For period 2000 to 2004 compound annual growth rate of a firm was 6% per annum and for period 2000 to 2011 the growth rate was 4%. Then how do we know that what was the growth rate during period 2004-2011?

In this problem, r2 =4% and r1=6% therefore R2 = 1.04and R1=1.06 and n1=5, n2 =12 and e=7.

Now we apply the formula

gw={1 .04 ( 1.041 .06 )57−1}∗100

gw= {1.04 (0 .981 )0 .714−1}∗100

gw= {1 .04∗0 .986−1 }∗100 gw= {1 .02594−1 }∗100=2.594%

What is the alternative way?Another way of solving this problem without formula is first calculate the value of 100 after 5 years with growth rate of 6% per annum.

=100*(1.06)^5=133.8226 then calculate the value of 100 after 12 years with growth rate of 4% per annum.

=100*(1.04) ^12=160.103222

Here n2= e+n1

Now again apply compounding formula and this time future value is 160.103222 and present value is 133.8226.Now rate of growth is unknown and duration is 7 years.

160.103222=133.8226(1+r) ^7

(160.103222/133.8226)= (1+r) ^7

(1.19638441) = (1+r) ^7

(1.19638441)^ (1/7) = (1+r)

1.02594574=1+r

r=0.2594Hence growth rate was 2.594%