Periodic compounding The total accumulated value, including the principal sum P plus compounded interest I, is given by the formula: A=P\left(1+\frac{r}{n}\right)^{tn} where: •
A is the final amount •
P is the original principal sum •
r is the
nominal annual interest rate •
n is the compounding frequency (1: annually, 12: monthly, 52: weekly, 365: daily) •
t is the overall length of time the interest is applied (expressed using the same time units as
r, usually years). The total compound interest generated is the final amount minus the initial principal, since the final amount is equal to principal plus interest: I=P\left(1+\frac{r}{n}\right)^{tn}-P
Accumulation function Since the principal
P is simply a coefficient, it is often dropped for simplicity, and the resulting
accumulation function is used instead. The accumulation function shows what $1 grows to after any length of time. The accumulation function for compound interest is:a(t) = \left(1 + \frac {r} {n}\right) ^ {tn}
Continuous compounding When the number of compounding periods per year increases without limit, continuous compounding occurs, in which case the effective annual rate approaches an upper limit of . Continuous compounding can be regarded as letting the compounding period become infinitesimally small, achieved by taking the
limit as
n goes to
infinity. The amount after
t periods of continuous compounding can be expressed in terms of the initial amount
P0 as: P(t)=P_0 e ^ {rt}.
Force of interest As the number of compounding periods n tends to infinity in continuous compounding, the continuous compound interest rate is referred to as the force of interest \delta. 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 as follows: \delta_{t}=\frac{a'(t)}{a(t)}=\frac{d}{dt} \ln a(t) This is the
logarithmic derivative of the accumulation function. Conversely: a(t)=e^{\int_0^t \delta_s\, ds}\, , (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. 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 To convert an interest rate from one compounding basis to another compounding basis, so that \left(1+\frac{r_1}{n_1}\right)^{n_1} = \left(1+\frac{r_2}{n_2}\right)^{n_2} 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 \delta=n\ln{\left(1+\frac{r}{n}\right)}, where \delta is the interest rate on a continuous compounding basis, and
r is the stated interest rate with a compounding frequency
n.
Monthly amortized loan or mortgage payments 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 (c) is c = \frac{rP}{1-\frac{1}{(1+r)^n}} or equivalently c = \frac{rP}{1-e^{-n\ln(1+r)}} where: • c = monthly payment • P = principal • r = monthly interest rate • n = number of payment periods
Spreadsheet formula In spreadsheets, the
PMT() function is used. The syntax is: PMT(interest_rate, number_payments, present_value, future_value, [Type])
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 and terms T=10–30 years), the monthly note rate is small compared to 1. r so that the \ln(1+r)\approx r which yields the simplification: c\approx \frac{Pr}{1-e^{-nr}}= \frac{P}{n}\frac{nr}{1-e^{-nr}} which suggests defining auxiliary variables Y\equiv n r = ITc_0\equiv \frac{P}{n} . Here c_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 c\approx c_0 \frac{Y}{1-e^{-Y}}. Let X = \frac{1}{2}Y. The expansion c\approx c_0 \left(1 + X + \frac{X^2}{3}\right) is valid to better than 1% provided X\le 1 .
Example of mortgage payment For a $120,000 mortgage with a term of 30 years and a note rate of 4.5%, payable monthly, we find: T=30I=0.045c_0=\frac{$120,000}{360}=$333.33 which gives X=\frac{1}{2}IT=.675 so that c\approx c_0 \left(1 + X + \frac{1}{3}X^2 \right)=\$333.33 (1+.675+.675^2/3)=\$608.96 The exact payment amount is c=\$608.02 so the approximation is an overestimate of about a sixth of a percent.
Monthly deposits Given a principal deposit and a recurring deposit, the total return of an investment can be calculated via the compound interest gained per unit of time. If required, the interest on additional non-recurring and recurring deposits can also be defined within the same formula (see below). • P = principal deposit • r = rate of return (monthly) • M = monthly deposit, and • t = time, in months The compound interest for each deposit is: M'=M(1+r)^{t} Adding all recurring deposits over the total period t, (i starts at 0 if deposits begin with the investment of principal; i starts at 1 if deposits begin the next month): M'=\sum^{t-1}_{i=0}{M(1+r)^{t-i}} Recognizing the
geometric series: M'=M\sum^{t-1}_{i=0}(1+r)^{t}\frac{1}{(1+r)^{i}} and applying the
closed-form formula (common ratio: 1/(1+r)): P' = M\frac{(1+r)^{t}-1}{r}+P(1+r)^t If two or more types of deposits occur (either recurring or non-recurring), the compound value earned can be represented as \text{Value}=M\frac{(1+r)^{t}-1}{r}+P(1+r)^t+k\frac{(1+r)^{t-x}-1}{r}+C(1+r)^{t-y} where C is each lump sum and k are non-monthly recurring deposits, respectively, and x and y are the differences in time between a new deposit and the total period t is modeling. A practical estimate for reverse calculation of the
rate of return when the exact date and amount of each recurring deposit is not known, a formula that assumes a uniform recurring monthly deposit over the period, is: r=\left(\frac{P'-P-\sum{M}}{P+\sum{M}/2}\right)^{1/t} or r=\left(\frac{P'-\sum{M}/2}{P+\sum{M}/2}\right)^{1/t}-1 ==See also==