Compound interest

Effective interest rates

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

The addition of interest to the principal sum of a loan or deposit is called compounding. Compound interest is 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 nominal as opposed to real interest rates).

Example

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.

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 years.

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

• 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.
• 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, 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).

Calculation of compound interest

The total accumulated value, including the principal sum plus compounded interest, is given by the formula:

$P\left(1+{\frac {i}{n}}\right)^{nt}$

where:

P is the principal sum

i is the nominal 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:

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

Example 1

Suppose an 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, i = 4.3% = 0.043, n = 4, and t = 6:

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

So the balance after 6 years is approximately 1,938.84.

Subtracting the principal from this amount gives the amount of interest received:

$1,938.84-1,500=438.84$

Example 2

Suppose the same amount 1,500 is compounded biennially.
Then the balance after 6 years is found by using the formula above, with P = 1500, i = 0.043 (4.3%), n = 1/2 = 0.5 (the interest is compounded every two years), and t = 6:

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

So, the balance after 6 years is approximately 1,921.24.

The amount of interest received can be calculated by subtracting the principal from this amount.

$1,921.24-1,500=421.24$

The interest is less compared with the previous case, as a result of the lower compounding frequency.

Periodic compounding

The amount function for compound interest is an exponential 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 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. In particular, any given interest rate (r) and compounding frequency (n) can be expressed in terms of a continuously compounded rate $r_{0}$:

$r_{0}=n\,\ln \left(1+{\frac {r}{n}}\right)$

which will also hold true for any other interest rate and compounding frequency. All formulas involving specific interest rates and compounding frequencies may be expressed in terms of the continuous interest rate and the compounding frequencies.

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 $a(0)=1$; this can be viewed as a particular case of a product integral)

When the above formula is written in differential equation format, then 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

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

• $P$ = monthly payment
• $L$ = principal
• $i$ = monthly interest rate
• $n$ = number of payment periods

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

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

See https://support.office.com/en-us/article/PMT-function-0214da64-9a63-4996-bc20-214433fa6441 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 $10,000 mortgage with a term of 30 years and a note rate of 4.5%, payable yearly, we find:

$T=30$

$I=0.045$

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 at year 0 at a constant annual interest rate of 2%. After the first year, this interest rate was applied to the initial principal of one cent and the capital grew to 1.02 cent. In the second year, the interest earned was again 2%. However, from that time onwards, it was not applied to the principal only but to the compound capital value (i.e., 1.02 cent). Thus, after the second year, the capital increased to 1.02×1.02 cent. After the third year, the capital grew to 1.02³ cent. After 2015 years, the capital has eventually grown to 1.02²⁰¹⁵ cent, which is roughly equal to 2.13x10¹⁷ cent or, more precisely, 213,474,546,813,926,768.7 cent.

Compare this figure to a similar investment using simple interest rather than compound interest. Suppose again that 1 cent is invested for a period of 2015 years at a constant annual interest rate of 2%. In this case, after 2015 years, the final capital is only 41.3 cent. This comparison highlights the effect of compounding, especially for long-term investments.

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
• Interest
• Interest rate
• Rate of return
• Rate of return on investment
• Real versus nominal value (economics)
• Yield curve
• e (mathematical constant)