- Compound interest is the addition of interest to the principal sum of a loan or deposit, where interest on principal plus interest is earned in the next period.
- It is standard in finance and economics and is contrasted with simple interest, where previously accumulated interest is not added to the principal amount.
- The simple annual interest rate is also known as the nominal interest rate, but it is not adjusted for inflation.
- The compounding frequency is the number of times per year the accumulated interest is paid out or capitalized. It can be yearly, half-yearly, quarterly, monthly, weekly, daily, or continuously.
- To compare interest-bearing financial instruments, both the nominal interest rate and the compounding frequency are required.
- Many countries require financial institutions to disclose the annual compound interest rate on deposits or advances on a comparable basis, known as the annual equivalent rate (AER), effective annual percentage rate (EAPR), or other terms.
- The effective annual rate is the total accumulated interest that would be payable up to the end of one year, divided by the principal sum.
- The effect of compounding depends on the nominal interest rate and the frequency interest is compounded.
- Examples include compound interest of 15% on initial $10,000 investment over 40 years, an annual dividend of 1.5% on initial $10,000 investment, and inflation compounded over 40 years at different rates.
- The effect of fees or taxes which the customer is charged may be included in the AER, but it varies by country and may not be comparable between different jurisdictions.
The mathematics behind compound interest can be represented by the following formula:
A = P(1 + r/n)^(nt)
Where: A = the final amount P = the principal amount (initial investment) r = the interest rate (expressed as a decimal) n = the number of times the interest is compounded per year t = the number of years the money is invested
For example, if an individual invests $1000 at a 5% interest rate compounded annually for 10 years, the final amount would be:
A = $1000(1 + 0.05)^(1*10) = $1628.89
In the case of continuous compounding, the formula is slightly different:
A = Pe^(rt)
Where: e = the base of the natural logarithm (approximately 2.718)
Using the same example as above, but with continuous compounding:
A = $1000e^(0.05*10) = $1648.15
As you can see, the final amount is greater with continuous compounding than with annual compounding.
It’s also worth noting that the formula for the effective annual interest rate, which accounts for compounding, is:
(1 + r/n)^n – 1
This formula can be used to compare the effective annual interest rate between different compounding frequencies.
It’s also worth mentioning that for compound interest calculations over a long period of time, the rule of 70 can be used as a rough approximation of time required for an investment to double in value, which is a way of finding out how long it will take for an investment to double in value, given the interest rate. For example, if the interest rate is 8% per year, it will take about 9 years for an investment to double in value. The formula for this is 70/r.
Euler’s Number, exponential function and natural logarithm
- The number e, also known as Euler’s number, is a mathematical constant approximately equal to 2.71828.
- It can be characterized in many ways, including as the base of the natural logarithms and the limit of (1 + 1/n)^n as n approaches infinity, an expression that arises in the study of compound interest.
- It can also be calculated as the sum of the infinite series e = ∑(n=0 to ∞) (1/n!) = 1 + (1/1) + (1/1*2) + (1/1*2*3) + …
- It is also the unique positive number a such that the graph of the function y = a^x has a slope of 1 at x = 0.
- The exponential function f(x) = e^x is the unique function f that equals its own derivative and satisfies the equation f(0) = 1; hence one can also define e as f(1).
- The natural logarithm, or logarithm to base e, is the inverse function to the natural exponential function.
- The natural logarithm of a number k > 1 can be defined directly as the area under the curve y = 1/x between x = 1 and x = k, in which case e is the value of k for which this area equals one.
- The number e is of great importance in mathematics, alongside 0, 1, π, and i. All five appear in one formulation of Euler’s identity e^(i*π) + 1 = 0 and play important and recurring roles across mathematics.
- Like the constant π, e is irrational (it cannot be represented as a ratio of integers) and transcendental (it is not a root of any non-zero polynomial with rational coefficients).
- The constant was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest. The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier.
- The letter e was first used to represent the constant by Leonhard Euler in a letter to Christian Goldbach on 25 November 1731.