When it comes to borrowing and investing in finance, borrowing costs and returns are generally determined using percentages. Mathematically, it can be modelled using the exponential function. To fully understand the math behind this branch of mathematics, it is important to master the vocabulary and language associated with it.
Regardless if the situation refers to a monetary gain or loss, the situation described always indicates a precise period of time and a specific interest rate.
However, the method of calculating this interest may differ from one situation to another. Use the information given and key vocabulary in the problem statement to decide which method to use.
The interest period is the time elapsed, generally in years, between the start of an investment or loan and a future value.
Interest has a different meaning for the lender and the borrower.
For lenders, interest is an amount of money they receive in addition to the initial amount loaned.||\text{Final amount received} = \text{Initial loan} + \text{amount of interest}||
This means that when the loan comes to term, the lender recovers the initial amount loaned plus a premium, which is the interest.
A man lends his childhood friend |\$5000| to start a company. Five years later, his friend repays the debt by returning the |\$5000| and an extra |\$500| in interest as a thank-you gesture.
Here is a snapshot of the situation from the point of view of the lender after five years:||\begin{alignat}{1}\text{Final amount received} &=\text{Initial loan}&+\text{Interest amount}\\[3pt] \$5500&=\quad \$5000&+\quad \$500\end{alignat}||
The lender ends up with more money than was originally loaned.
On the other hand, a borrower must not only repay the entire initial amount, but must also pay interest.
For borrowers, interest is an amount of money they must pay to the lender in addition to the initial amount borrowed.||\text{Final amount paid} =\text{Initial loan} +\text{amount of interest}||
Therefore, when the loan comes to maturity (ends), the borrower repays the initial amount plus a premium, which is the interest. Let's consider the same example again from the borrower's point of view.
A man lends his childhood friend |\$5000| to start a company. Five years later, his friend repays the debt by returning the |\$5000| and an extra |\$500| in interest as a thank-you gesture.
Here is a snapshot of the situation from the point of view of the borrower after five years:||\begin{alignat}{1}\text{Final amount paid} &=\text{Initial loan}&+\text{Interest amount}\\[3pt] \$5500&=\quad \$5000&+\quad \$500\end{alignat}||
In the end, a borrower pays back more money than he or she originally borrowed.
More specifically, here is how lending and borrowing work:
Personal loans
Generally speaking, financial institutions and credit companies lend money to their members at an interest rate. In the end, the money generated by this interest rate represents a gain for the lenders, but a loss for the members.
Investment
When a person lends money to a financial institution, it is called an investment. In general, investments have a guaranteed capital, but a low return. In other words, you won't lose the initial amount invested, but the amount of money you earn through generated interest will be low. Financially speaking, it has low risk and a low potential return.
Stocks
When a person lends money to a company, it is a stock investment. In general, the company uses the money to grow and increase its revenue. If the company increases its income, part of the profit will be returned to the borrower. On the other hand, if the company makes no income and goes out of business, the investor risks losing the invested capital. In financial terms, the risks are high, but so are the potential returns.
The above examples help illustrate the concept of interest from the point of view of the lender and the borrower. However, in most financial situations, the interest calculated on the initial amount is rarely a fixed amount; it increases over time. This increase in interest over time is generally determined by interest rates.
A future value, generally denoted |C_n,| is the final amount obtained at the end of a loan.
Therefore, future value and capitalization are synonyms for the amount you receive at the end of the interest period.
A person wants to make an investment for his retirement. He invests |\$10\,000| for a period of 20 years at a |2.05\%| interest rate, compounded annually.
The future value of the |\$10\,000| investment is estimated to be |\$15\,005.84.|
As in the last example, the period and interest rate must be known to determine the future value.
Consult the concept sheets on simple interest and compound interest to understand the mathematics behind these numbers.
The current value, generally denoted |C_O,| is the initial amount lent or borrowed.
In financial terms, this amount is called capital.
Finding the current value consists of determining the present value of capital that leads to a given objective when the future value, the time period and interest rate are known.
After a lot of hard work, a father wants to increase his savings so he can take his family on a trip in exactly three years.
If he knows the trip will cost |\$7500| in total and the current interest rate is |2.55\%,| the initial amount invested must be |\$6954.31.|
In other words, the father must invest |\$6954.31| over a three-year interest period to go on the trip.