Compounding
Compounding is the process in which an asset's earnings, from either capital gains or interest, are reinvested to generate additional earnings over time. The resulting future value, based on a varying number of compounding periods, is: Annual compounding (n = 1): FV = $1,000,000 x \[1 + (20%/1)\] (1 x 1) = $1,200,000 Semi-annual compounding (n = 2): FV = $1,000,000 x \[1 + (20%/2)\] (2 x 1) = $1,210,000 Quarterly compounding (n = 4): FV = $1,000,000 x \[1 + (20%/4)\] (4 x 1) = $1,215,506 Monthly compounding (n = 12): FV = $1,000,000 x \[1 + (20%/12)\] (12 x 1) = $1,219,391 Weekly compounding (n = 52): FV = $1,000,000 x \[1 + (20%/52)\] (52 x 1) = $1,220,934 Daily compounding (n = 365): FV = $1,000,000 x \[1 + (20%/365)\] (365 x 1) = $1,221,336 As evident, the future value increases by a smaller margin even as the number of compounding periods per year increases significantly. The generalized formula for compound interest is: F V \= P V × ( 1 \+ i ) n where: F V \= Future value P V \= Present value i \= Annual interest rate \\begin{aligned}&FV=PV\\times(1+i)^n\\\\&\\textbf{where:}\\\\&FV=\\text{Future value}\\\\&PV=\\text{Present value}\\\\&i=\\text{Annual interest rate}\\\\&n=\\text{Number of compounding periods per year}\\end{aligned} FV\=PV×(1+i)nwhere:FV\=Future valuePV\=Present valuei\=Annual interest rate The effects of compounding strengthen as the frequency of compounding increases. The more compounding periods throughout this one year, the higher the future value of the investment, so naturally, two compounding periods per year are better than one, and four compounding periods per year are better than two. To illustrate this effect, consider the following example given the above formula. It takes into account the present value of an asset, the annual interest rate, and the frequency of compounding (or the number of compounding periods) per year and the total number of years.
What Is Compounding?
Compounding is the process in which an asset's earnings, from either capital gains or interest, are reinvested to generate additional earnings over time. This growth, calculated using exponential functions, occurs because the investment will generate earnings from both its initial principal and the accumulated earnings from preceding periods. Compounding, therefore, differs from linear growth, where only the principal earns interest each period.
Understanding Compounding
Compounding typically refers to the increasing value of an asset due to the interest earned on both a principal and accumulated interest. This phenomenon, which is a direct realization of the time value of money (TMV) concept, is also known as compound interest.
Compound interest works on both assets and liabilities. While compounding boosts the value of an asset more rapidly, it can also increase the amount of money owed on a loan, as interest accumulates on the unpaid principal and previous interest charges.
To illustrate how compounding works, suppose $10,000 is held in an account that pays 5% interest annually. After the first year or compounding period, the total in the account has risen to $10,500, a simple reflection of $500 in interest being added to the $10,000 principal. In year two, the account realizes 5% growth on both the original principal and the $500 of first-year interest, resulting in a second-year gain of $525 and a balance of $11,025. After 10 years, assuming no withdrawals and a steady 5% interest rate, the account would grow to $16,288.95.
Special Considerations
The formula for the future value (FV) of a current asset relies on the concept of compound interest. It takes into account the present value of an asset, the annual interest rate, and the frequency of compounding (or the number of compounding periods) per year and the total number of years. The generalized formula for compound interest is:
F V = P V × ( 1 + i ) n where: F V = Future value P V = Present value i = Annual interest rate \begin{aligned}&FV=PV\times(1+i)^n\\&\textbf{where:}\\&FV=\text{Future value}\\&PV=\text{Present value}\\&i=\text{Annual interest rate}\\&n=\text{Number of compounding periods per year}\end{aligned} FV=PV×(1+i)nwhere:FV=Future valuePV=Present valuei=Annual interest rate
Increased Compounding Periods
The effects of compounding strengthen as the frequency of compounding increases. Assume a one-year time period. The more compounding periods throughout this one year, the higher the future value of the investment, so naturally, two compounding periods per year are better than one, and four compounding periods per year are better than two.
To illustrate this effect, consider the following example given the above formula. Assume that an investment of $1 million earns 20% per year. The resulting future value, based on a varying number of compounding periods, is:
As evident, the future value increases by a smaller margin even as the number of compounding periods per year increases significantly. The frequency of compounding over a set length of time has a limited effect on an investment's growth. This limit, based on calculus, is known as continuous compounding and can be calculated using the formula:
F V = P × e r t where: e = Irrational number 2.7183 r = Interest rate \begin{aligned}&FV=P\times e^{rt}\\&\textbf{where:}\\&e=\text{Irrational number 2.7183}\\&r=\text{Interest rate}\\&t=\text{Time}\end{aligned} FV=P×ertwhere:e=Irrational number 2.7183r=Interest rate
In the above example, the future value with continuous compounding equals: FV = $1,000,000 x 2.7183 (0.2 x 1) = $1,221,403.
Example of Compounding
Compounding is crucial in finance, and the gains attributable to its effects are the motivation behind many investing strategies. For example, many corporations offer dividend reinvestment plans that allow investors to reinvest their cash dividends to purchase additional shares of stock. Reinvesting in more of these dividend-paying shares compounds investor returns because the increased number of shares will consistently increase future income from dividend payouts, assuming steady dividends.
Investing in dividend growth stocks on top of reinvesting dividends adds another layer of compounding to this strategy that some investors refer to as "double compounding." In this case, not only are dividends being reinvested to buy more shares, but these dividend growth stocks are also increasing their per-share payouts.
Related terms:
Asset
An asset is a resource with economic value that an individual or corporation owns or controls with the expectation that it will provide a future benefit. read more
Automatic Reinvestment Plan
An automatic reinvestment plan is a mutual fund plan that automatically reinvests capital gains back into the fund. read more
Capital Gain
Capital gain refers to an increase in a capital asset's value and is considered to be realized when the asset is sold. read more
Continuous Compounding & Formula
Continuous compounding is the process of calculating interest and reinvesting it into an account's balance over an infinite number of periods. read more
Discrete Compounding
Discrete compounding refers to the method by which interest is calculated and added to the principal at certain set points in time. read more
Dividend
A dividend is the distribution of some of a company's earnings to a class of its shareholders, as determined by the company's board of directors. read more
Dividend Reinvestment Plan—DRIP
A dividend reinvestment plan (DRIP) is an arrangement that allows shareholders to automatically reinvest a stock's cash dividends into additional or fractional shares of the underlying company. It is offered by a public company free or for a nominal fee, though minimum investment amounts may apply. read more
Effective Annual Interest Rate
The effective annual interest rate is the real return on an investment, accounting for the effect of compounding over a given period of time. read more
Euler's Constant
Euler's constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter gamma (γ). read more