Simple Interest

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  Types of Interest

  Definition 01.2 (Simple Interest).

  Simple interest occurs when interest is only calculated using the original principal; that is, interest earned from prior periods is not required for interest calculations for consecutive periods.

  • It is characterized by $P$, the interest rate per measurement period $(i)$ and number of measurement periods $(t)$
  • The amount value is linear in $t$

  $$A(t)=P \cdot(1+it)$$

  Definition 01.3 (Compound Interest).

  Compound interest occurs when interest is accumulated on the principal in discrete subintervals of the measurement period.

  • It is characterized by $P$, the nominal interest rate $(r)$, the number of subdivisions for a single measurement period $(n)$, and the number of measurement periods $(t)$
  • The amount value is exponential in $t$ (assuming $n > 1$)

  The accumulated amount for one measurement period is:

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

  For multiple periods:

  $$ A(n, t) = P\left(1+\frac{r}{n}\right)^{nt}. $$

  The effective interest rate (ER) for one measurement period is

  $$ ER(n) = \left[ \left(1+\frac{r}{n}\right)^n - 1 \right] \cdot 100\%. $$

  For multiple periods:

  $$ ER(n,t) = \left[\left(1+\frac{r}{n}\right)^{nt} - 1\right] \cdot 100\%. $$

  Example: If $r=0.10$ is compounded semi-annually ($n=2$), then after one year ($t=1$):

  $$ A(2, 1) = P\left(1+\frac{0.10}{2}\right)^2 = 1.1025P, $$

  so:

  $$ ER(2,1) = 1.1025 - 1 = 0.1025 = 10.25\%. $$

• The effective interest rate $a(1) - 1$ is sometimes called the simple interest rate

  • The effective interest rate is decreasing in each consecutive period and converges to zero

• $n$th effective interest rate

Example 04.1.

(Example 4.1 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 32])

Absolute Interest Earned

• Absolute interest, $I_n=A(n)-A(n-1)$, is constant since the accumulation function is linear

$\mathbb{R}^+$ Extension of Simple Interest

• So far, we have treated simple interest as discrete unit steps, but we can make it integrable
• We can extend this definition to all positive real numbers
  • The interest earned for any time period $t+s$ is the sum of the interests earned in $t$ and $s$ (assuming a principal of $1)
  • Therefore, the simple interest equation holds for any nonnegative real number

• Simple interest is the only accumulation function in which the above property holds

Example 04.2.

(Example 4.2 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 34])

Example 04.3.

(Example 4.3 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 34])

Approximating Compound Interest

• For $t \in [0, 1]$, simple interest is approximately equal to compound interest

Example 04.4.

(Example 4.4 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 35])

• Need to use Accumulation Factor formula if there is a deposit/withdrawal midway and add that on (Example 4.5 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 36])

Practice