Business, Legal & Accounting Glossary
When interest is applied to capital and accrued up until that particular date. For example, a £1,000 loan with 20% interest will have a balance of £1,200 after the first year, then £1,440 at the end of the second year.
n. payment of interest upon principal and previously accumulated interest, which increases the amount paid for money use above simple interest. Thus, it can increase more rapidly if compounded daily, monthly or quarterly. The genius physicist Albert Einstein called compound interest man’s “greatest invention.” Most lenders agree.
Compound interest is the payment of interest on both principal as well past accrued interest. The opposite of compound interest is simple interest. Without compound interest, a $100 savings account at 10% per year earns a flat $10 in interest each year.
After 10 years that non-compound interest savings account is worth $200. With compound interest, that same $100 savings account earns $10 in interest the first year, but earns increasing amounts of interest in each subsequent year. The reason is that with compound interest, interest is paid on the previous years’ interest. After 10 years our $100 savings account with compound interest is worth more than $259. Albert Einstein called compound interest the eighth wonder of the world.
Compound interest is an important concept for any investor to understand. It truly can work wonders for your long-term gains if you understand its fundamental operation. Compounding interest involves adding the interest earned to your initial investment so that the interest itself earns interest. You can calculate compound interest using an equation with four essential factors.
To calculate what your investment will be worth in the future with a certain rate of compound interest, you begin with its present value. This is expressed as “PV” in the equation. It’s also often referred to as the initial investment.
The rate of interest you will earn on your investment is the next factor you need to know to perform your calculation. This is termed “i” in the equation. Of course, in a real-world situation, this often varies significantly over time, which is what makes investment formulas so complex and the outcome uncertain.
If you are adding interest back to your initial investment in order for it also to earn interest, you need to know how often this is going to happen. This is known as the compounding period. In a real-world situation, this can be annually, monthly, or, occasionally, daily. In the equation, you need to know the number of these compounding periods, expressed as “N”.
The last factor needed for the equation is the final value or “FV”, which is the value of the investment after compounding at the rate “i” for the number of periods “N”. Thus the equation to calculate compound interest is expressed: PV x (1+i) ^N = FV
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Compound interest is, in a nutshell, interest upon interest. That is, when an interest payment is added to the principal and then the whole thing (principal + interest) earns interest.
Compound interest has been called “the eighth wonder of the world” and is the engine that drives supersized returns for investors who start young or invest for a very long time. If an investment is paying “simple” interest, then it is paying the same amount every time, like an annuity and the total value increases at a steady, plodding pace. If it is paying compound interest, on the other hand, then it is playing an increasingly larger amount each time, and the total value increases exponentially.
Two factors determine the power of compound interest:
Time
Compounding period
When money is placed on deposit to earn interest, the interest can be paid out periodically as it is earned, or it can be left on deposit. If interest left on deposit does not itself earn interest, the deposit is said to earn simple interest.
Measure time in years. With simple interest, at any time t, the value of the deposit is given by the product
a(1 + rs t )
[1]
where a is the amount of the initial deposit, and rs is the simple rate of interest. For example, if USD 100 is left on deposit to earn an rs = .06 rate of simple interest, at the end three years, the deposit will be worth
100 (1 + (.06) 3) = 118
[2]
Formula [1] means that, when a deposit earns simple interest, its value grows linearly with time.
Exhibit 1: The value of a deposit earning simple interest grows linearly over time.
In practice, simple interest is rarely used for deposits held more than a year. An alternative is to credit interest, not based upon the initial value of the deposit, but based on its accumulated value. This approach is called compound interest. With it, interest is earned on both the initial deposit and on any interest that has already been earned but left on deposit—interest is earned on interest. At any time t, the value of the deposit is given by
[3]
where n is the compounding frequency—the number of times per year that interest is credited. The constant rn is the interest rate. Typical values for n include
For example, if USD 100 is left on deposit to earn an r2 = .06 rate of semiannually compounded interest, at the end of three years, the deposit will be worth
[4]
Formula [3] means that, when a deposit earns compound interest, its value grows exponentially with time.
Exhibit 2: The value of a deposit earning compound interest grows exponentially over time.
With compounding, larger values of n correspond to interest being credited more and more frequently. The limiting case of this is called continuous compounding where interest is credited on a continuous basis. The distinction is like the difference between getting water from a hand pump and getting water from a faucet. With the hand pump, the water flow is broken. With the faucet, it is continuous. The faucet does not necessarily deliver water any faster than the pump. It just delivers it continuously.
With continuous compounding, at any time t, the value of a deposit is given by
a exp(rc t )
[5]
where rc is the continuously compounded interest rate and e =2.71828182845…
For example, if USD 100 is left on deposit to earn an rc = .06 rate of continuously compounded interest, at the end of three years, the deposit will be worth
100exp(.06(3)) = 119.72
[6]
Interest is rarely compounded continuously in practice. Continuous compounding is more of a theoretical notion. It is used frequently in theoretical finance because it simplifies many formulas.
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This glossary post was last updated: 29th December, 2021 | 0 Views.