Compound interest

The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies

Compound interest is interest added to the principal of a deposit or loan so that the added interest also earns interest from then on. Mike Toeldo's Mom. This addition of interest to the principal is called compounding. A bank account, for example, may have its interest compounded every year: in this case, an account with $1000 initial principal and 20% interest per year would have a balance of $1200 at the end of the first year, $1440 at the end of the second year, $1728 at the end of the third year, and so on.

To define an interest rate fully, allowing comparisons with other interest rates, both the interest rate and the compounding frequency must be disclosed. Since most people prefer to think of rates as a yearly percentage, many governments require financial institutions to disclose the equivalent yearly compounded interest rate on deposits or advances. For instance, the yearly rate for a loan with 1% interest per month is approximately 12.68% per annum (1.01¹² − 1). This equivalent yearly rate may be referred to as annual percentage rate (APR), annual equivalent rate (AER), effective interest rate, effective annual rate, and other terms. When a fee is charged up front to obtain a loan, APR usually counts that cost as well as the compound interest in converting to the equivalent rate. These government requirements assist consumers in comparing the actual costs of borrowing more easily.

For any given interest rate and compounding frequency, an equivalent rate for any different compounding frequency exists.

Compound interest may be contrasted with simple interest, where interest is not added to the principal (there is no compounding). Compound interest is standard in finance and economics, and simple interest is used infrequently (although certain financial products may contain elements of simple interest).

Terminology

The effect of compounding depends on the frequency with which interest is compounded and the periodic interest rate which is applied. Therefore, to accurately define the amount to be paid under a legal contract with interest, the frequency of compounding (yearly, half-yearly, quarterly, monthly, daily, etc.) and the interest rate must be specified. Different conventions may be used from country to country, but in finance and economics the following usages are common:

The periodic rate is the amount of interest that is charged (and subsequently compounded) for each period divided by the amount of the principal. The periodic rate is used primarily for calculations and is rarely used for comparison.

The nominal annual rate or nominal interest rate is defined as the periodic rate multiplied by the number of compounding periods per year. For example, a monthly rate of 1% is equivalent to an annual nominal interest rate of 12%.

The effective annual rate is the total accumulated interest that would be payable up to the end of one year divided by the principal.

Economists generally prefer to use effective annual rates to simplify comparisons, but in finance and commerce the nominal annual rate may be quoted. When quoted together with the compounding frequency, a loan with a given nominal annual rate is fully specified (the amount of interest for a given loan scenario can be precisely determined), but the nominal rate cannot be directly compared with that of loans that have a different compounding frequency.

Loans and financing may have charges other than interest, and the terms above do not attempt to capture these differences. Other terms such as annual percentage rate and annual percentage yield may have specific legal definitions and may or may not be comparable, depending on the jurisdiction.

The use of the terms above (and other similar terms) may be inconsistent and vary according to local custom or marketing demands, for simplicity or for other reasons.

Exceptions

• US and Canadian T-Bills (short term Government debt) have a different convention. Their interest is calculated as (100 − P)/Pbnm, 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).
• 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.
• 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.

Mathematics of interest rates

Simplified calculation

Formulae are presented in greater detail at time value of money.

In the formulae below, i is the effective interest rate per period. FV and PV represent the future and present value of a sum. n represents the number of periods.

These are the most basic formulas:

$FV=PV(1+i)^{n}\,$

The above calculates the future value (FV) of an investment whose present value is PV accruing interest at a fixed interest rate (i) for n periods.

$PV={\frac {FV}{\left(1+i\right)^{n}}}\,$

The above calculates what present value (PV) would be needed to produce a specified future value (FV) if interest accrues at the rate i for n periods.

$i=\left({\frac {FV}{PV}}\right)^{\frac {1}{n}}-1$

The above calculates the compound interest rate achieved if an initial investment of PV returns a value of FV after n accrual periods.

$n={\frac {\log(FV)-\log(PV)}{\log(1+i)}}$

The above formula calculates the number of periods required to get FV given the PV and the interest rate (i). The log function can be in any base, e.g. natural log (ln), as long as consistent bases are used throughout the calculation.

Compound Interest

A formula for calculating annual compound interest as follows:

$S=P\left(1+{\frac {j}{m}}\right)^{mt}$

where

• S = value after t periods
• P = principal amount (initial investment)
• j = annual nominal interest rate (not reflecting the compounding)
• m = number of times the interest is compounded per year
• t = number of years the money is borrowed for

As an example, suppose an amount of 1500.00 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, j = 0.043 (4.3%), m = 4, and t = 6:

$S=1500\left(1+{\frac {0.043}{4}}\right)^{4\times 6}=1938.84$

So, the balance after 6 years is approximately 1938.84. The amount of interest received can be calculated by subtracting the principal from this amount.

Periodic compounding

The amount function for compound interest is a power law function in terms of time.

$A(t)=A_{0}\left(1+{\frac {r}{n}}\right)^{\lfloor nt\rfloor }$

• $t$ = Total time in years

• $n$ = Number of compounding periods per year (note that the total number of compounding periods is $n\cdot t$)

• $r$ = Nominal annual interest rate expressed as a decimal. e.g.: 6% = 0.06

• $\lfloor nt\rfloor$ means that nt is rounded down to the nearest integer.

As n, the number of compounding periods per year, increases without limit, we have the case known as continuous compounding, in which case the effective annual rate approaches an upper limit of e^r − 1.

Since the principal A(0) is simply a coefficient, it is often dropped for simplicity, and the resulting accumulation function is used in interest theory instead. Accumulation functions for simple and compound interest are listed below:

$a(t)=1+tr\,$

$a(t)=\left(1+{\frac {r}{n}}\right)^{nt}$

Note: A(t) is the amount function and a(t) is the accumulation function.

Continuous compounding

Continuous compounding can be thought of as making the compounding period infinitesimally small, achieved by taking the limit as n goes to infinity. See definitions of the exponential function for the mathematical proof of this limit. The amount after t periods of continuous compounding can be expressed in terms of the initial amount A₀ as

$A(t)=A_{0}e^{rt}.$

It has been shown that the mathematics of continuous compounding is not limited to the valuation of continuously compounded financial instruments and flow annuities, but rather that the exponential equation is a versatile model that may be used for valuation of all financial contracts normally encountered.

Force of interest

In mathematics, the accumulation functions are often expressed in terms of e, the base of the natural logarithm. This facilitates the use of calculus to manipulate interest formulae.

For any continuously differentiable accumulation function a(t) the force of interest, or more generally the logarithmic or continuously compounded return is a function of time defined as follows: $\delta _{t}={\frac {a'(t)}{a(t)}}\,$

which is the rate of change with time of the natural logarithm of the accumulation function.

Conversely: $a(n)=e^{\int _{0}^{n}\delta _{t}\,dt}\ ,$ (since $a(0)=1$)

When the above formula is written in differential equation format, the force of interest is simply the coefficient of amount of change: $da(t)=\delta _{t}a(t)\,dt\,$

For compound interest with a constant annual interest rate r, the force of interest is a constant, and the accumulation function of compounding interest in terms of force of interest is a simple power of e: $\delta =\ln(1+r)\,$ or $a(t)=e^{t\delta }\,$

The force of interest is less than the annual effective interest rate, but more than the annual effective discount rate. It is the reciprocal of the e-folding time. See also notation of interest rates.

A way of modeling the force of inflation is with Stoodley's formula: $\delta _{t}=p+{s \over {1+rse^{st}}}$ where p, r and s are estimated.

Compounding basis

See also: Day count convention

To convert an interest rate from one compounding basis to another compounding basis, use

$r_{2}=\left[\left(1+{\frac {r_{1}}{n_{1}}}\right)^{\frac {n_{1}}{n_{2}}}-1\right]{n_{2}},$

where r₁ is the interest rate with compounding frequency n₁, and r₂ is the interest rate with compounding frequency n₂.

When interest is continuously compounded, use

$R=n\ln {\left(1+r/n\right)},$

where R is the interest rate on a continuous compounding basis, and r is the stated interest rate with a compounding frequency n.

Mathematics of interest rate on loans

Monthly amortized loan or mortgage payments

See also: Mortgage calculator § Monthly payment formula

The interest on loans and mortgages that are amortized—that is, have a smooth monthly payment until the loan has been paid off—is often compounded monthly. The formula for payments is found from the following argument.

Exact formula for monthly payment

An exact formula for the monthly payment is

$P={\frac {Li}{1-{\frac {1}{(1+i)^{n}}}}}$

or equivalently

$P={\frac {Li}{1-e^{-n\ln(1+i)}}}$

This can be derived by considering how much is left to be repaid after each month. After the first month $L_{1}=(1+i)L-P$ is left, i.e. the initial amount has increased less the payment. If the whole loan was repaid after a month then $L_{1}=0$ so $L={\frac {P}{1+i}}$ After the second month $L_{2}=(1+i)L_{1}-P$ is left, that is $L_{2}=(1+i)((1+i)L-P)-P$. If the whole loan was repaid after two months $L_{2}=0$ this gives the equation $L={\frac {P}{1+i}}+{\frac {P}{(1+i)^{2}}}$. This equation generalises for a term of n months, $L=P\sum _{j=1}^{n}{\frac {1}{(1+i)^{j}}}$. This is a geometric series which has the sum

$L={\frac {P}{i}}\left(1-{\frac {1}{(1+i)^{n}}}\right)$

which can be rearranged to give

$P={\frac {Li}{1-{\frac {1}{(1+i)^{n}}}}}={\frac {Li}{1-e^{-n\ln(1+i)}}}$

This formula for the monthly payment on a U.S. mortgage is exact and is what banks use.

In Excel, the function PMT() function is used. The syntax for the PMT function is:

= - PMT( interest_rate, number_payments, PV, [FV],[Type] )

See http://office.microsoft.com/en-au/excel-help/pmt-HP005209215.aspx for more details.

For example, for interest rate of 6% (0.06/12 p.m.), 25 years * 12 p.a., PV of $150,000, FV of 0, type of 0 gives:

= - PMT( 0.06/12, 25 * 12, 150000, 0, 0 )

= $966.45 p.m.

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 ($I<8\%$ and terms T=10–30 years), the monthly note rate is small compared to 1: $i<<1$ so that the $\ln(1+i)\approx i$ which yields a simplification so that $P\approx {\frac {Li}{1-e^{-ni}}}={\frac {L}{n}}{\frac {ni}{1-e^{-ni}}}$

which suggests defining auxiliary variables

$Y\equiv ni=TI$

$P_{0}\equiv {\frac {L}{n}}$.

$P_{0}$ is the monthly payment required for a zero interest loan paid off in $n$ installments. In terms of these variables the approximation can be written

$P\approx P_{0}{\frac {Y}{1-e^{-Y}}}$

The function $f(Y)\equiv {\frac {Y}{1-e^{-Y}}}-{\frac {Y}{2}}$ is even: $f(Y)=f(-Y)$ implying that it can be expanded in even powers of $Y$.

It follows immediately that ${\frac {Y}{1-e^{-Y}}}$ can be expanded in even powers of $Y$ plus the single term: $Y/2$

It will prove convenient then to define

$X={\frac {1}{2}}Y={\frac {1}{2}}IT$

so that $P\approx P_{0}{\frac {2X}{1-e^{-2X}}}$ which can be expanded: $P\approx P_{0}\left(1+X+{\frac {X^{2}}{3}}-{\frac {1}{45}}X^{4}+...\right)$

where the ellipses indicate terms that are higher order in even powers of $X$. The expansion

$P\approx P_{0}\left(1+X+{\frac {X^{2}}{3}}\right)$

is valid to better than 1% provided $X\leq 1$.

Example of mortgage payment

For a mortgage with a term of 30 years and a note rate of 4.5% we find:

$T=30$

$I=.045$

which gives

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

so that

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

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

Example of compound interest

Suppose that one cent had been invested in a bank 2012 years ago at a 5% interest rate maintained to the present. After the first year the capital would be worth 5% more than one cent, or 1.05 cents. In the second year the interest earned would be 5% times 1.05 cents, giving the amount of 1.05×1.05. After three years it would have grown to $(1.05)^{3}$. After 2012 years the original one cent contribution would have grown to $(1.05)^{2012}$cents, or $4.29\cdot 10^{42}$ cents (more accurately, a vast 4,294,076,020,320,707,300,374,777,820,338,841,725,938,314 of them).

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.

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.

See also

• Credit card interest
• Exponential growth
• Fisher equation
• Rate of return on investment
• Yield curve