Final answer:
Continuous compounding is when interest is calculated and added to the principal continuously, leading to exponential growth. The effective annual rate for continuous compounding is expressed as EAR = e^r - 1, where e is the mathematical constant and r is the quoted interest rate. This concept highlights the powerful effect of compounding on savings and investments.
Step-by-step explanation:
Understanding Continuous Compound Interest
When talking about continuous compounding, we are referring to the scenario where interest is calculated and added to the principal an infinite number of times in a year. This is different from the more common periodic compounding where interest is added at set intervals such as annually, semi-annually, or monthly. In continuous compounding, the formula to calculate the effective annual rate (EAR) is expressed as:
EAR = er - 1
where e represents the base of the natural logarithm, approximately equal to 2.71828, and r is the quoted annual interest rate, expressed as a decimal. This is a direct result of the mathematical limit as the number of compounding periods goes to infinity. Through continuous compounding, even small rates can have significant effects on the growth of an investment over time due to the exponential nature of this growth.
Compound interest is crucial in finance because it calculates interest on both the principal and the accumulated interest, which can greatly increase savings or investments over time. This concept also extends to economic growth, illustrating how GDP can grow exponentially through the reinvestment of previous growth. The impact of continuous compounding can be immense, showcasing the power of exponential growth in financial contexts.