Growth Rates

 

What Kind of Fool Am I

What kind of fool I am
What kind of mind is this
An empty shell
A lonely cell in which
An empty heart must dwell

What kind of growth is this?

It is common to express growth in terms of a Compound Annual Growth Rate, CAGR.  Savings accounts, loans, mortgages, inflation, and the Gross Domestic Product of the economy are often expressed this way.  This implicitly assumes that growth is geometric and follows the formula: 

Future= Base *(100%+CAGR)^(periods, years, between future and base).

This is considered a marked improvement over assuming a simple growth rate:

Future=base * (Period, years, between future and base)*(100% + simple growth rate).

The compound rate is also called a geometric growth rate.  It assumes a constant, but non-negative, growth rate (first derivative).  A problem with continuous growth is that it can surpass an absolute amount in the future.  An equally valid growth rate is exponential growth.  It is more commonly encountered as radioactive decay with a half life. For growth, the mirror of decay, the rate is often expressed as a doubling period.  Exponential growth will approach an absolute but will never exceed it.  It thus has a variable, not a constant, growth rate (first derivative).  This is consistent with a constant rate within an exponential function.

Future=base*(100% +exp(rate* period, years, between base and future))

where exp() is the exponential function.

Over the short term, less than 20% of the periods to the absolute, there is virtually no difference between exponential and compound growth.  Exponential growth is higher than compound growth, but less than 5% higher, over the medium term, 20% to 83% of the periods to the absolute.  It is only in the periods near the absolute ( 83% to 100% of the periods to the absolute), that the exponential growth becomes significantly larger.  However unlike compound growth, its future value will never exceed the absolute amount.