💰 Compound Interest

What compound interest is and how it differs from simple interest
Compounding frequency (yearly, monthly, daily, continuous)
Annual equivalent rates used to compare financial products
Real-world examples: bonds, Canadian and U.S. mortgages, derivatives
Calculation: periodic compounding formula and the accumulation function
Continuous compounding and the force of interest
Converting between compounding bases
Amortized loan and mortgage payments (exact and approximate formulas, worked example)
Monthly deposits and estimating the rate of return

📖 Definition

Compound interest (also called anatocism) is interest accumulated from a principal sum and previously accumulated interest.

It results from reinvesting or retaining interest that would otherwise be paid out.
It can also result from the accumulation of debts from a borrower.

Compound vs. simple interest

Feature	Simple interest	Compound interest
Previously accumulated interest	Not added to the principal of the current period	Added to the principal (capitalized)
Depends on	The interest rate	The simple interest rate applied and the frequency at which interest is compounded

🔁 Compounding Frequency

The compounding frequency is the number of times per given unit of time the accumulated interest is capitalized, on a regular basis.
Possible frequencies:
- Yearly
- Half-yearly
- Quarterly
- Monthly
- Weekly
- Daily
- Continuously
- Not at all until maturity
Example: monthly capitalization with interest expressed as an annual rate means the compounding frequency is 12, with time periods measured in months.

📊 Annual Equivalent Rate

Many countries require financial institutions to disclose the annual compound interest rate on deposits or advances on a comparable basis. This helps consumers compare retail financial products more fairly and easily.
The same idea goes by different names in different markets:
- Effective annual percentage rate (EAPR)
- Annual equivalent rate (AER)
- Effective interest rate
- Effective annual rate
- Annual percentage yield
Effective annual rate: the total accumulated interest that would be payable up to the end of one year, divided by the principal sum.
These rates are usually the annualised compound interest rate alongside charges other than interest, such as taxes and other fees.

🧾 Examples

Corporate and government bonds
- Interest is usually payable twice yearly.
- Each six-month payment is the disclosed interest rate divided by two, multiplied by the principal.
- The yearly compounded rate is therefore 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.
- An amortization schedule determines how payments are applied toward principal and interest.
- Interest on these loans is not added to the principal; it is paid off monthly as payments are applied.
Derivatives valuation: it is sometimes mathematically simpler to use continuous compounding.
- Continuous compounding is a natural consequence of Itô calculus.
- Financial derivatives are valued at ever-increasing frequency, until the limit is approached and the derivative is valued in continuous time.
Numerical example: suppose $1,000 earns 5% interest per year.
- After 1 year it is worth $1,050.
- After 2 years it is worth $1,102.50, because the second year's interest is calculated on a new value, not on the original principal.

🧮 Calculation

Periodic compounding

The total accumulated value, including the principal sum $P$ plus compounded interest $I$ , is:

$A = P\left(1+\frac{r}{n}\right)^{tn}$

where:

Symbol	Meaning
$A$	The final amount
$P$	The original principal sum
$r$	The nominal annual interest rate
$n$	The compounding frequency (1: annually, 12: monthly, 52: weekly, 365: daily)
$t$	The overall length of time the interest is applied (same time units as $r$ , usually years)

The total compound interest generated is the final amount minus the initial principal, since the final amount equals 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, and the resulting accumulation function is used instead.
The accumulation function shows what $1 grows to after any length of time. For compound interest:

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

Continuous compounding

When the number of compounding periods per year increases without limit, continuous compounding occurs.
In that case the effective annual rate approaches an upper limit of $e^r - 1$ .
It can be regarded as letting the compounding period become infinitesimally small, by taking the limit as $n$ goes to infinity.
The amount after $t$ periods of continuous compounding, in terms of the initial amount $P_0$ :

$P(t) = P_0 e^{rt}$

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 (more generally the logarithmic or continuously compounded return) is a function of time:

$\delta_t = \frac{a'(t)}{a(t)} = \frac{d}{dt}\ln a(t)$

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 written as a differential equation, the force of interest is simply the coefficient of the amount of change:

$da(t) = \delta_t\, a(t)\, dt$

Constant annual rate $r$ : the force of interest is a constant, and the accumulation function is a simple power of $e$ :

$\delta = \ln(1+r) \quad\text{or}\quad a(t) = e^{t\delta}$

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 Stoodley's formula, where $p$ , $r$ and $s$ are estimated:

$\delta_t = p + \frac{s}{1+rse^{st}}$

Compounding basis

To convert an interest rate from one compounding basis to another so that

$\left(1+\frac{r_1}{n_1}\right)^{n_1} = \left(1+\frac{r_2}{n_2}\right)^{n_2}$

use

$r_2 = \left[\left(1+\frac{r_1}{n_1}\right)^{n_1/n_2} - 1\right] n_2$

where $r_1$ is the interest rate with compounding frequency $n_1$ , and $r_2$ is the interest rate with compounding frequency $n_2$ .

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 compounding frequency $n$ .

🏠 Monthly Amortized Loan or Mortgage Payments

Loans and mortgages that are amortized (a smooth monthly payment until the loan is paid off) often have interest compounded monthly.
The payment formula is found from an argument about the compounding balance.

Exact formula for monthly payment

$c = \frac{rP}{1-\frac{1}{(1+r)^n}}$

or equivalently

$c = \frac{rP}{1-e^{-n\ln(1+r)}}$

Symbol	Meaning
$c$	Monthly payment
$P$	Principal
$r$	Monthly interest rate
$n$	Number of payment periods

Spreadsheet formula

In spreadsheets, the PMT() function is used. Syntax:

PMT(interest_rate, number_payments, present_value, future_value, [Type])

Approximate formula for monthly payment

A formula 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: $r \ll 1$ .
So $\ln(1+r)\approx r$ , which yields the simplification:

$c \approx \frac{Pr}{1-e^{-nr}} = \frac{P}{n}\,\frac{nr}{1-e^{-nr}}$

This suggests defining auxiliary variables:

$Y \equiv nr = IT$

$c_0 \equiv \frac{P}{n}$

Here $c_0$ is the monthly payment required for a zero-interest loan paid off in $n$ installments.
In these variables the approximation is $c \approx c_0 \dfrac{Y}{1-e^{-Y}}$ .

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:

$T = 30$
$I = 0.045$
$c_0$ = $120,000 / 360 = $333.33
$X = \dfrac{1}{2}IT = 0.675$
Approximate payment:

$c \approx c_0\left(1+X+\frac{1}{3}X^2\right) = \$333.33\,(1+0.675+0.675^2/3) = \$608.96$

The exact payment 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 be defined within the same formula.

Symbol	Meaning
$P$	Principal deposit
$r$	Rate of return (monthly)
$M$	Monthly deposit
$t$	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):

$M' = \sum_{i=0}^{t-1} M(1+r)^{t-i}$

Recognizing the geometric series:

$M' = M\sum_{i=0}^{t-1}(1+r)^t\,\frac{1}{(1+r)^i}$

and applying the closed-form formula (common ratio $1/(1+r)$ ):

$P' = M\,\frac{(1+r)^t-1}{r} + P(1+r)^t$

Several types of deposits

If two or more types of deposits occur (recurring or non-recurring), the compound value earned can be represented as:

$\text{Value} = M\,\frac{(1+r)^t-1}{r} + P(1+r)^t + k\,\frac{(1+r)^{t-x}-1}{r} + C(1+r)^{t-y}$

$C$ is each lump sum.
$k$ are non-monthly recurring deposits.
$x$ and $y$ are the differences in time between a new deposit and the total period $t$ being modeled.

Estimating the rate of return

A practical estimate for reverse calculation of the rate of return, when the exact date and amount of each recurring deposit is not known, assumes a uniform recurring monthly deposit over the period:

$r = \left(\frac{P'-\sum M/2}{P+\sum M/2}\right)^{1/t} - 1$

Compound Interest: Finance Study Notes