Exponential functions are incredibly powerful and are very important to understand. Compound interest, an example of an exponential function, plays an important role in investing. Albert Einstein once said, “Compound interest is the eighth wonder of the world.”
There are two types of interest, compound interest and simple interest. When one receives interest only on the principal amount invested, or the original investment, this is called simple interest. For example, if I buy bond ABC for $1,000 that has an interest rate of 10%, I would receive $100 or 10% of $1000 each year until the bond matures. If the same bond had annually compounded interest, I would receive interest on the principal amount plus previous interest payments. For example, I would receive an interest payment of $100 for the first year, but in the second year I would receive an interest payment of $110 or 10% of $1100. (Principal amount + First Year Interest Payment) In the third year, I would receive $121, or 10% of $1210. (Principal amount + First Year Interest Payment + Second Year Interest Payment) An additional $31 over three years may not seem like a big difference, but if this bond matured in 100 years, my original investment would be worth $137,806.34 compared to $11,000 if it was not compounded annually.
Compound Interest Formula: A=P(1+r/n)nt
A= Total amount of money accumulated after n years
P= Principal Investment (Initial amount of money invested or borrowed)
r= Interest Rate
n= Amount of times interest is compounded per year
t= Number of years the principal investment is invested for
Simple Interest Formula: A=P(1+rt)
A= Total amount of money accumulated after n years
P= Principal Investment (Initial amount of money invested or borrowed)
r= Interest rate
t= Number of years the principal investment is invested for
Compound interest is also important within the stock market. The S&P 500, which is an index of 500 of the largest publicly traded companies by market capitalization, had an annualized return of 10.26% for the last 25 years with reinvested dividends. If someone were to have invested $1,000 in the S&P 500 25 years ago, it would now be worth $11, 493. In this calculation, interest was compounded annually. Starting to invest at a young age, even with a small amount of money, can result in large profits due to compound interest.
To leave you with a final thought on compounded interest here is a story:
There is a story about an Emperor of China who was so excited about the game of chess that he offered the inventor of the game one wish. The inventor replied that he wanted one grain of rice on the first square of the chess board, two grains on the second square, four on the third and so on through the 64th square. The unwitting emperor agreed to the modest request. But two to the 64th power is 18 million trillion grains of rice - more than enough to cover the entire surface of the earth. The Emperor, realizing that he had been duped, had the inventor of the game beheaded.
Data for S&P 500 annualized returns: http://financeandinvestments.blogspot.com/2014/02/historical-annual-returns-for-s-500.html
