# Compound interest

(Redirected from Continuously compounded interest)
Effective interest rates
The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies Compound interest is the addition of interest to the principal sum of a loan or deposit, or in other words, interest on interest. It is the result of reinvesting interest, rather than paying it out, so that interest in the next period is then earned on the principal sum plus previously-accumulated interest. Compound interest is standard in finance and economics. Compound interest may be contrasted with simple interest, where interest is not added to the principal, so there is no compounding. The simple annual interest rate is the interest amount per period, multiplied by the number of periods per year. The simple annual interest rate is also known as the nominal interest rate (not to be confused with the real, or inflation-adjusted, rate). ## Compounding frequency The compounding frequency is the number of times per year (or other unit of time) the accumulated interest is paid out, or capitalized (credited to the account), on a regular basis. The frequency could be yearly, half-yearly, quarterly, monthly, weekly, daily (or not at all, until maturity). For example, monthly capitalization with annual rate of interest means that the compounding frequency is 12, with time periods measured in months. The effect of compounding depends on: 1. The nominal interest rate which is applied and 2. The frequency interest is compounded. ## Annual equivalent rate The nominal rate cannot be directly compared between loans with different compounding frequencies. Both the nominal interest rate and the compounding frequency are required in order to compare interest-bearing financial instruments. To assist consumers compare retail financial products more fairly and easily, many countries require financial institutions to disclose the annual compound interest rate on deposits or advances on a comparable basis. The interest rate on an annual equivalent basis may be referred to variously in different markets as annual percentage rate (APR), annual equivalent rate (AER), effective interest rate, effective annual rate, annual percentage yield and 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. There are usually two aspects to the rules defining these rates: 1. The rate is the annualised compound interest rate, and 2. There may be charges other than interest. The effect of fees or taxes which the customer is charged, and which are directly related to the product, may be included. Exactly which fees and taxes are included or excluded varies by country. may or may not be comparable between different jurisdictions, because the use of such terms may be inconsistent, and vary according to local practice. ## Examples • 1,000 Brazilian real (BRL) is deposited into a Brazilian savings account paying 20% per annum, compounded annually. At the end of one year, 1,000 x 20% = 200 BRL interest is credited to the account. The account then earns 1,200 x 20% = 240 BRL in the second year. • A rate of 1% per month is equivalent to a simple annual interest rate (nominal rate) of 12%, but allowing for the effect of compounding, the annual equivalent compound rate is 12.68% per annum (1.0112 − 1). • The interest on corporate bonds and government bonds is usually payable twice yearly. The amount of interest paid (each six months) is the disclosed interest rate divided by two and multiplied by the principal. The yearly compounded rate is higher than the disclosed rate. • Canadian mortgage loans are generally compounded semi-annually with monthly (or more frequent) payments.[1] • U.S. mortgages use an amortizing loan, not compound interest. With these loans, an amortization schedule is used to determine how to apply payments toward principal and interest. Interest generated on these loans is not added to the principal, but rather is paid off monthly as the payments are applied. • It is sometimes mathematically simpler, e.g. in the valuation of derivatives, to use continuous compounding, which is the limit as the compounding period approaches zero. Continuous compounding in pricing these instruments is a natural consequence of Itō calculus, where financial derivatives are valued at ever increasing frequency, until the limit is approached and the derivative is valued in continuous time. ## Discount instruments • US and Canadian T-Bills (short term Government debt) have a different convention. Their interest is calculated on a discount basis as (100 − P)/Pbnm,[clarification needed] where P is the price paid. Instead of normalizing it to a year, the interest is prorated by the number of days t: (365/t)×100. (See day count convention). ## Mathematics of interest rate on loans ### Calculation of compound interest The total accumulated value, including the principal sum ${\displaystyle P}$ plus compounded interest ${\displaystyle I}$, is given by the formula: Fv=Pv(r/n)^nt where: P is the original principal sum P' is the new principal sum r is the nominal annual interest rate n is the compounding frequency t is the overall length of time the interest is applied (usually expressed in years). The total compound interest generated is: ${\displaystyle P'=P+I}$ ${\displaystyle I=P\left(1+{\frac {r}{n}}\right)^{nt}-P}$ #### Example 1 Suppose a principal amount of$1,500 is deposited in a bank paying an annual interest rate of 4.3%, compounded quarterly.
Then the balance after 6 years is found by using the formula above, with P = 1500, r = 0.043 (4.3%), n = 4, and t = 6:

${\displaystyle P'=1,500\times \left(1+{\frac {0.043}{4}}\right)^{4\times 6}\approx 1,938.84}$

So the new principal ${\displaystyle P'}$ after 6 years is approximately $1,938.84. Subtracting the original principal from this amount gives the amount of interest received: ${\displaystyle 1,938.84-1,500=438.84}$ #### Example 2 Suppose the same amount$1,500 is compounded biennially (every 2 years).
Then the balance after 6 years is found by using the formula above, with P = 1500, r = 0.043 (4.3%), n = 1/2 = 0.5 (the interest is compounded every two years), and t = 6:

${\displaystyle P'=1,500\times \left(1+{\frac {0.043}{0.5}}\right)^{0.5\times 6}\approx 1,921.24}$

= $966.45 ### Approximate formula for monthly payment A formula that is accurate to within a few percent can be found by noting that for typical U.S. note rates (${\displaystyle I<8\%}$ and terms ${\displaystyle T}$=10–30 years), the monthly note rate is small compared to 1: ${\displaystyle r<<1}$ so that the ${\displaystyle \ln(1+r)\approx r}$ which yields a simplification so that ${\displaystyle c\approx {\frac {Pr}{1-e^{-nr}}}={\frac {P}{n}}{\frac {nr}{1-e^{-nr}}}}$ which suggests defining auxiliary variables ${\displaystyle Y\equiv nr=IT}$ ${\displaystyle c_{0}\equiv {\frac {P}{n}}}$. ${\displaystyle c_{0}}$ is the monthly payment required for a zero interest loan paid off in ${\displaystyle n}$ installments. In terms of these variables the approximation can be written ${\displaystyle c\approx c_{0}{\frac {Y}{1-e^{-Y}}}}$ The function ${\displaystyle f(Y)\equiv {\frac {Y}{1-e^{-Y}}}-{\frac {Y}{2}}}$ is even: ${\displaystyle f(Y)=f(-Y)}$ implying that it can be expanded in even powers of ${\displaystyle Y}$. It follows immediately that ${\displaystyle {\frac {Y}{1-e^{-Y}}}}$ can be expanded in even powers of ${\displaystyle Y}$ plus the single term: ${\displaystyle Y/2}$ It will prove convenient then to define ${\displaystyle X={\frac {1}{2}}Y={\frac {1}{2}}IT}$ so that ${\displaystyle c\approx c_{0}{\frac {2X}{1-e^{-2X}}}}$ which can be expanded: ${\displaystyle c\approx c_{0}\left(1+X+{\frac {X^{2}}{3}}-{\frac {1}{45}}X^{4}+...\right)}$ where the ellipses indicate terms that are higher order in even powers of ${\displaystyle X}$. The expansion ${\displaystyle P\approx P_{0}\left(1+X+{\frac {X^{2}}{3}}\right)}$ is valid to better than 1% provided ${\displaystyle X\leq 1}$. #### Example of mortgage payment For a$10,000 mortgage with a term of 30 years and a note rate of 4.5%, payable yearly, we find:

${\displaystyle T=30}$

${\displaystyle I=0.045}$

which gives

${\displaystyle X={\frac {1}{2}}IT=.675}$

so that

${\displaystyle P\approx P_{0}\left(1+X+{\frac {1}{3}}X^{2}\right)=\333.33(1+.675+.675^{2}/3)=\608.96}$

The exact payment amount is ${\displaystyle P=\608.02}$ so the approximation is an overestimate of about a sixth of a percent.

## History

Compound interest was once regarded as the worst kind of usury and was severely condemned by Roman law and the common laws of many other countries.[2]

The Florentine merchant Francesco Balducci Pegolotti provided a table of compound interest in his book Pratica della mercatura of about 1340. It gives the interest on 100 lire, for rates from 1% to 8%, for up to 20 years.[3] The Summa de arithmetica of Luca Pacioli (1494) gives the Rule of 72, stating that to find the number of years for an investment at compound interest to double, one should divide the interest rate into 72.

Richard Witt's book Arithmeticall Questions, published in 1613, was a landmark in the history of compound interest. It was wholly devoted to the subject (previously called anatocism), whereas previous writers had usually treated compound interest briefly in just one chapter in a mathematical textbook. Witt's book gave tables based on 10% (the then maximum rate of interest allowable on loans) and on other rates for different purposes, such as the valuation of property leases. Witt was a London mathematical practitioner and his book is notable for its clarity of expression, depth of insight and accuracy of calculation, with 124 worked examples.[4][5]

Jacob Bernoulli discovered the constant ${\displaystyle e}$ in 1683 by studying a question about compound interest.

### Trivia

Albert Einstein is apocryphally quoted as saying "Compound interest is the eighth wonder of the world. He who understands it, earns it ... he who doesn't ... pays it.[6]