The Compound Annual Growth Rate is a tool for calculating returns on an investment, or for any transaction involving compound interest. Use it for applying to loans, savings accounts, capital investments, and other financial speculations.
Compound Interest Formula
Compound Interest is unlike simple interest in that interest is earned not only on the original amount, but on the interest as well. This causes a sum to grow at a faster rate. The formula takes into account the number of times interest is compounded back into the base sum per year:
V = (1 + (r / n)) ^ (n * y)• V = the starting value
• r = the annual interest rate
• n = number of times per year the interest is compounded
• y = number of years in the investment's duration
The simple interest rate is also called the nominal interest rate. Compound interest is an example of a periodic function, where a formula is applied at specific times during a cycle. The CAGR, then, is the yearly rate of return on a sum of money vesting with compound interest.
In the simplest terms, a $1000 sum placed at a simple interest rate of 1%, paid only on the principal, would accumulate to $1100 after ten years. At a compound interest growth rate, ideally compounded daily, you add 1/365th of 1% to the principal, getting $1,105.17 after ten years.
So far that's not as impressive, but take that same sum of $1000, and invest it in such a way that you have the initial $1000 invested at the beginning, and every month you put another $100 away in the account. After just one year, this yields $16.57 in interest, and after ten years on the same schedule, you'd have $13,730.93, netting $730.93 in interest alone.
A Famous Experiment
None other than the American founding father Benjamin Franklin conducted a wily experiment with compound interest. Upon his death, his last will and testament bequeathed £1,000 (about $4.4K USD at the time) to each of the cities of Philadelphia and Boston, placing it in a trust fund that gathered compound interest for 100 years, starting from the year 1785.
The interest rate was to be 5% annual, and after 100 years he wished that three-fourths of the sum be spent on worthy causes while the remaining quarter be re-invested for another 100 years.
By 1985, the Philadelphia fund had accumulated more that $5 million, which was divided between small business loans in the city, mortgage loans, scholarships, and establishing the Franklin Institute of Boston.
Franklin, notorious for his sense of humor, doubtless got a kick out of seeding such a financial experiment to culminate two centuries after his passing. It was his way of motivating his beloved home city to preserve stability and order for two centuries - a going concern in that this was a young country at the time won from a revolution and with dubious prospects.
Florentine merchant Francesco Balducci Pegolotti is credited as being the first to publish a table of compound interest in 1340, giving the interest rate on 100 Italian lire, from 1% to 8% interest, for a maximum of 20 years.
In 1683, Swiss mathematician Jacob Bernoulli discovered the mathematical constant e when studying compound interest. The constant e (approximately 2.71828) is the base for the natural logarithm, and is the limit of (1 + (1 / n)). It is also an irrational, transitive number, in that it never repeats and never terminates.
This constant pops up in many scientific pursuits and is also used by the financial industry to estimate compound growth rates such as our focus here.