The amount of money earned for the use of borrowed capital is called **interest**. From the borrower’s point of view, interest is the amount of money *paid* for the capital. For the lender, interest is the *income* generated by the capital which he has lent.

There are two types of interest, *simple* and *compound*.

## Simple Interest

In simple interest, only the original principal bears interest and the interest to be paid varies directly with time.

The formula for simple interest is given by

The future amount is

$F = P + I$

$F = P + Prt$

Where

$I$ = interest

$P$ = principal, present amount, capital

$F$ = future amount, maturity value

$r$ = rate of simple interest expressed in decimal form

$t$ = time in years, term in years

**Ordinary and Exact Simple Interest**

In an instance when the time *t* is given in number of days, the fractional part of the year will be computed with a denominator of 360 or 365 or 366. With *ordinary simple interest*, the denominator is 360 and in *exact simple interest*, the denominator is either 365 or 366. We can therefore conclude that ordinary interest is greater than exact interest.

Ordinary simple interest is computed on the basis of banker’s year.

Exact simple interest is based on the actual number of days in a year. One year is equivalent to 365 days for ordinary year and 366 days for leap year. A leap year is when the month of February is 29 days, and ordinary year when February is only 28 days. Leap year occurs every four years.

*Note:*

Leap years are those which are exactly divisible by 4 except century years, but those century years that are exactly divisible by 400 are also leap years.

*d*is the number of days, then ...

- $t = \dfrac{d}{360}$

- $t = \dfrac{d}{365}$ (for ordinary year)

- $t = \dfrac{d}{366}$ (for leap year)

## Compound Interest

In compound interest, the interest earned by the principal at the end of each interest period (compounding period) is added to the principal. The sum (principal + interest) will earn another interest in the next compounding period.

Consider \$1000 invested in an account of 10% per year for 3 years. The figures below shows the contrast between simple interest and compound interest.

At 10% simple interest, the \$1000 investment amounted to \$1300 after 3 years. Only the principal earns interest which is \$100 per year.

At 10% compounded yearly, the \$1000 initial investment amounted to \$1331 after 3 years. The interest also earns an interest.

**Elements of Compound Interest**

$P$ = principal, present amount

$F$ = future amount, compound amount

$i$ = interest rate per compounding period

$r$ = nominal annual interest rate

$n$ = total number of compounding in t years

$t$ = number of years

$m$ = number of compounding per year

Future amount,

The factor $(1 + i)^n$ is called single-payment compound-amount factor and is denoted by $(F/P, \, i, \, n)$.

Present amount,

The factor $\dfrac{1}{(1 + i)^n}$ is called single-payment present-worth factor and is denoted by $(P/F, \, i, \, n)$.

Number of compounding periods,

Interest rate per compounding period,

**Values of $i$ and $n$**

In most problems, the number of years $t$ and the number of compounding periods per year $m$ are given. The example below shows the value of $i$ and $n$.

*Example*

Number of years, $t = 5 \text{ years}$

Nominal rate, $r = 18\%$

- Compounded annually ($m = 1$)
$n = 1(5) = 5$

$i = 0.18 / 1 = 0.18$

- Compounded semi-annually ($m = 2$)
$n = 2(5) = 10$

$i = 0.18 / 2 = 0.09$

- Compounded quarterly ($m = 4$)
$n = 4(5) = 20$

$i = 0.18 / 4 = 0.045$

- Compounded semi-quarterly ($m = 8$)
$n = 8(5) = 40$

$i = 0.18 / 4 = 0.0225$

- Compounded monthly ($m = 12$)
$n = 12(5) = 60$

$i = 0.18 / 12 = 0.015$

- Compounded bi-monthly ($m = 6$)
$n = 6(5) = 30$

$i = 0.18 / 6 = 0.03$

- Compounded daily ($m = 360$)
$n = 360(5) = 1800$

$i = 0.18 / 360 = 0.0005$

**Continuous Compounding (m → ∞)**

In continuous compounding, the number of interest periods per year approaches infinity. From the equation

$F = \left( 1 + \dfrac{r}{m} \right)^{mt}$

when $m \to \infty$, $mt = \infty$, and $\dfrac{r}{m} \to 0$. Hence,

$\displaystyle F = P \lim_{m \to \infty}\left( 1 + \dfrac{r}{m} \right)^{mt}$

Let $x = \dfrac{r}{m}$. When $m \to \infty$, $x \to 0$, and $m = \dfrac{r}{x}$.

$\displaystyle F = P \lim_{x \to 0}(1 + x)^{\frac{r}{x}t}$

$\displaystyle F = P \lim_{x \to 0}(1 + x)^{\frac{1}{x}rt}$

From Calculus, $\displaystyle \lim_{x \to \infty}(1 + x)^{1/x} = e$, thus,