Chapter 13, Problem 1ES

To determine

To calculate: The compound interest paid on the loan of value $\$1000$ at $8\%$ compounded annually for two years.

Answer to Problem 1ES

Solution:

The compound interest on the loan is $\underset{\_}{\text{\$166}\text{.4}}$.

Explanation of Solution

Given Information:

Loan of value $\$1000$ at $8\%$ compounded annually for two years.

Formula used:

Steps to find the future value using the simple interest formula:

1. Find the periodic interest rate using the formula

$\text{Period Interest rate}=\frac{\text{annual interest rate}}{\text{number of interest periods per year}}$

2. Then multiply the original principal by the sum of $1$ and the periodic interest rate as

$\text{First end-of-period principal}=\text{origianl principal}\times \left(1\text{}+\text{periodic interest rate}\right)$ or $A=P\left(1+R\right)$

3. Then for each remaining period find the next end of period principal for that multiply the previous period principal by the sum of $1$ and the periodic interest rate as

$\text{End-of-period principal}=\text{previous end-of-period principal}\times \left(1+\text{}\text{periodic interest rate}\right)$

4. Then the last end of period principal will be the future value as

$\text{Futur value or Compound Amount}=\text{last end-of-period principal}$

Determine the compound interest by subtracting original value from future value as

$\text{Compund Interest}=\text{Future Value}-\text{Original Principal}$

Calculation:

Consider the loan of value $\$1000$ at $8\%$ compounded annually for two years.

Because the loan is compounded annually, there is one interest period per year. So, the period interest rate is $8\%$ or $0.08$. There are $2$ interest periods one for each of the two years.

Original Principal Amount is $\$1000$

Therefore,

$\begin{array}{c}\text{First end-of-period principal}=\text{origianl principal}\times \left(1\text{}+\text{periodic interest rate}\right)\\ =\$1000\left(1+.08\right)\\ =\$1000\left(1.08\right)\\ =\$1080\end{array}$

Now

$\begin{array}{c}\text{End-of-period principal}=\text{previous end-of-period principal}\times \left(1+\text{}\text{periodic interest rate}\right)\\ =\$1080\left(1+0.08\right)\\ =\$1080\left(1.08\right)\\ =\$1166.4\end{array}$

And

$\begin{array}{c}\text{Futur value or Compound Amount}=\text{last end-of-period principal}\\ =\$1166.4\end{array}$

And the compound interest amount is,

$\begin{array}{c}\text{Compund Interest}=\text{Future Value}-\text{Original Principal}\\ =\$1166.4-\$1000\\ =\$166.4\end{array}$

Hence the compound interest on the loan is $\text{\$166}\text{.4}$.