### Note on Computation of Annualized Growth Rates

There are several different ways in which the annualized rate of growth can be calculated. None of these ways is "the" correct way; but we believe that the method we adopted is the most appropriate for the users of this calculator. To outline the alternative methods, consider the following notation:

B = value of variable in beginning year.

E = value of variable in ending year.

N = number of years between beginning and ending year.

For example, if the beginning year is 1950 and the ending year is 1965, then N = 15.

#### Method 1 *(this is the computation that we adopt)*

The annualized rate of growth is 100·X, where X is the solution to the equation:
B · (1 + X) ^{N} = E. One plugs in the values of B, E, and N, and solves for X. As an easy example,
suppose that B = 100, E = 110.25, and N = 2. Then X is obtained via the equation (1 + X) ^{N} = 110.25/100,
the solution of which is X = 0.05. Therefore the annualized growth rate is 100 times 0.05, which is 5 percent.

This method can be described as "annualized compounding in the direction of time".
The advantage of this method is its intuitive nature, that accords with common sense. In the example,
five percent of 100 **(i.e., 5**) added to 100 is 105, and five percent of 105 **(i.e., 5.5)** added to 105 is 110.25. This method is
superior to all others in situations in which the direction of movement is always in the direction of
the movement of time (toward the future), never in the opposite direction (toward the past).

#### Method 2

Suppose that the beginning and ending years are reversed. For most purposes, one would
not want to compute the growth rate in the opposite direction to time; but let us do so. Then, retaining
the above simple example, B = 110.25, E = 100, N = 2. Solving the equation (1 + X) ^{N} = 100/110.25 for X
results in X = - .047619 (approximately). Multiplying by 100, the annualized growth rate is 4.7619
percentnegative if one goes from 110.25 to 100. This method is "annualized compounding in the opposite direction of time".

#### Method 3

One could apply straight-line growth rather than compounding.
The formula becomes X = (1/N) · (E - B)/B. In our simple example, X = (1/2) · (110.25 - 100)/100 = .05125;
and the annualized growth rate is 5.125 percent. In most economic applications, it would be inappropriate to
ignore the compounding aspect of growth (or decline). So method 3 is usually inappropriate.

#### Method 4

One could apply straight-line growth in the opposite direction to time.
The formula is then X = (1/N) · (B - E)/E. In the example X = (1/2) · (100 - 110.25)/110.25 = - 0465
(approximately). The rate of growth is 4.65 percent. Ignoring the compounding of growth is usually
inappropriate in this direction as well.

#### Method 5

For some purposes, one might want to have the same growth rate (except for sign)
whether one goes in the direction of time or in the opposite direction. The formula X = (1/N) · log(E/B)
has this property; for (1/N) · log(E/B) = - (1/N) · log(B/E), where log denotes the natural logarithm. Taking
the example, X = (1/2) · log(110.25/100) = .0488 percent (approximately). Going opposite to time,
X = (1/2) · log(100/110.25) = - .0488. Of course, one multiplies by 100 to obtain the annualized growth rate,
which is 4.88 percent - positive or negative, depending on the direction of the computation.

The same magnitude growth rate irrespective of direction could be a pleasing property
of a growth formula. Another desirable property is that the resulting growth rate is always between the
results of methods 3 and 4. In the example, 4.65 < 4.88 < 5.125. The disadvantages of method 5 are that,
for most applications, it is counterintuitive in two respects: it applies straight-line growth rather than
annualized compounding, and the growth rate is below that obtained by pure movement in the direction of time.

Please let us know if and how this discussion has assisted you in using our calculators.