{"id":44,"date":"2023-11-02T06:17:58","date_gmt":"2023-11-02T06:17:58","guid":{"rendered":"https:\/\/openbook.ums.edu.my\/financialmathematicsineconomics\/chapter\/2-2-simple-interest\/"},"modified":"2024-09-25T08:32:04","modified_gmt":"2024-09-25T08:32:04","slug":"2-2-simple-interest","status":"publish","type":"chapter","link":"https:\/\/openbook.ums.edu.my\/financialmathematicsineconomics\/chapter\/2-2-simple-interest\/","title":{"raw":"2.2 Simple Interest","rendered":"2.2 Simple Interest"},"content":{"raw":"

Interest is the reward for lending capital to somebody for a period of time. There are various methods for computing the interest. For simple interest, is it the amount of interest in the product of three quantities which are: i) the rate of interest; ii) the principal; iii) the time period.<\/p>\r\n

From a lender\u2019s perspective, when money is borrowed for a loan, interest accumulates as a reward for the lenders. While, from the borrower\u2019s perspective, accumulated interest is a charge to the borrower for the financial transaction to take place. The amount of interest incurred by the borrower depends on the amount of money borrowed or invested, the principal, the interest rate, and time. The simple interest, $I$, accumulated on the principal, $P$, over an interval of t years at an annual interest rate of $r$, can be written as follows:<\/p>\r\n

[latexpage]<\/p>\r\n\r\n

\\begin{equation} \\label{eq:1}\r\nI = P \\times r \\times t\u00a0 \u00a0 \u00a0 \r\n\\end{equation}<\/pre>\r\n
Where the rate of interest must be expressed as a decimal for calculations, the interest rate is expressed in year or annually, and the interest rate is a flat rate where there are no changes happens for the interest rate during the tenure period. To note that, p.a. represents per annum or annual interest rate.<\/p>\r\n\r\n
2.2.1 Future Value and Present Value <\/strong><\/h1>\r\n
The total amount of money that must be repaid on a loan or the total value of an investment can be called the future value, $S$. The future value can be calculated using $S = P + I$, where $P$ is the principal or money borrowed for a loan or invested, and $I$ is the interest accumulated. The future value also can be calculated using the following formula, given information on the accumulated amount of the principal and interest after $t$ years:<\/p>\r\n
[latexpage]\r\n\\begin{equation} \\label{eq:2}\r\nS = P + I = P + (Prt) = P(1 + rt)\r\n\\end{equation}<\/p>\r\n
The principal is also called the present value of the discounted value of $S$. In Equation (\\ref{eq:2}), $(1+rt)$ is called the simple interest factor and $(1+rt)^{-1}$ \u00a0is called the present value discount factor at simple interest. The time, $t$, must be in years. When the time is given in months, then<\/p>\r\n
[latexpage]<\/p>\r\n\r\n
\\begin{equation} \\label{eq:3}\r\nt = \\tfrac{number of months}{12}\r\n\\end{equation}<\/pre>\r\n
When the time is given in weeks, then divide the weeks with 52, when the time is given in days, then divide the days with 365 days.<\/p>\r\n
The present value (or discounted value) of $S$ was calculated by using the present value factor at simple interest.\u00a0 The present value can be written as Equation (4) :<\/p>\r\n
[latexpage]<\/p>\r\n\r\n
\\begin{equation} \\label{eq:4}\r\nP = \\frac{S}{1+rt}\r\n\\end{equation}<\/pre>\r\n
\r\n
Example 2.1<\/p>\r\n\r\n<\/header>\r\n
\r\n
Find the simple interest on a RM1,000 investment made for 3 years at an interest rate of 5% per year. What is the accumulated amount?<\/p>\r\n[h5p id=\"3\"]\r\n\r\n<\/div>\r\n<\/div>\r\n
Example 2.2<\/header>\r\n
\r\n
A student borrows RM600 to buy a camera. The loan is over two years, and the simple interest rate is 6% per annum. How much will his\/her monthly repayments be?<\/p>\r\n[h5p id=\"4\"]\r\n\r\n<\/div>\r\n<\/div>\r\n
Example 2.3<\/header>\r\n
\r\n
Find the present value of RM800 at a simple interest rate of 10% p.a. for 8 months.<\/p>\r\n[h5p id=\"5\"]\r\n\r\n<\/div>\r\n<\/div>\r\n
2.2.2 The Time between Dates<\/strong><\/h1>\r\n[caption id=\"attachment_43\" align=\"aligncenter\" width=\"300\"]\"\" Figure 2
\"Calendar\"<\/a> by Nick Youngson<\/a> is licensed under CC BY-SA 3.0[\/caption]\r\n
In any financial transaction, loan terms are an important aspect to be considered before signing off. These include the loan\u2019s repayment period. The loan repayment period or time can be calculated using two ways which are i) the exact time, and ii) the approximate time.<\/p>\r\n
Exact time is found as the exact number of days including all days except the first. The exact time can be refereed using the table of the number of each year of the day (see Table 1). It is obtained as the difference between serial numbers of the given dates. For example, to find the exact time from April 18 to November 3 of the same year, see Figure 1. May 18 is the 108th day of the year and November 3 is the 307th day of the year. The exact time is 307 \u2013 108 = 199 days. Alternatively, use the Microsoft Excel.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Table 1<\/caption>\r\n
Date<\/td>\r\nMonth<\/td>\r\nDay<\/td>\r\n<\/tr>\r\n
November 3<\/td>\r\n10<\/td>\r\n33<\/td>\r\n<\/tr>\r\n
April 18<\/td>\r\n4<\/td>\r\n18<\/td>\r\n<\/tr>\r\n
Difference<\/td>\r\n6<\/td>\r\n15<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
Whereas, the approximate time is calculated by assuming that each month has 30 days.\u00a0 Using the same example as above, see Table 1, for the solution.<\/p>\r\n
Where we have borrowed 30 days from the 11th month. The approximate time is 6 months and 15 days, or (6 x 30 days) + 15 days = 195 days.<\/p>","rendered":"
Interest is the reward for lending capital to somebody for a period of time. There are various methods for computing the interest. For simple interest, is it the amount of interest in the product of three quantities which are: i) the rate of interest; ii) the principal; iii) the time period.<\/p>\n
From a lender\u2019s perspective, when money is borrowed for a loan, interest accumulates as a reward for the lenders. While, from the borrower\u2019s perspective, accumulated interest is a charge to the borrower for the financial transaction to take place. The amount of interest incurred by the borrower depends on the amount of money borrowed or invested, the principal, the interest rate, and time. The simple interest, \"I\", accumulated on the principal, \"P\", over an interval of t years at an annual interest rate of \"r\", can be written as follows:<\/p>\n
\n
<\/a> (1) <\/span>   <\/span>\"\begin{equation*} <\/pre>\n
Where the rate of interest must be expressed as a decimal for calculations, the interest rate is expressed in year or annually, and the interest rate is a flat rate where there are no changes happens for the interest rate during the tenure period. To note that, p.a. represents per annum or annual interest rate.<\/p>\n
2.2.1 Future Value and Present Value <\/strong><\/h1>\n
The total amount of money that must be repaid on a loan or the total value of an investment can be called the future value, \"S\". The future value can be calculated using \"S = P + I\", where \"P\" is the principal or money borrowed for a loan or invested, and \"I\" is the interest accumulated. The future value also can be calculated using the following formula, given information on the accumulated amount of the principal and interest after \"t\" years:<\/p>\n
\n<\/a><\/p>\n
 (2) <\/span>   <\/span>\"\begin{equation*} <\/p>\n
The principal is also called the present value of the discounted value of \"S\". In Equation (2<\/a>), \"(1+rt)\" is called the simple interest factor and \"(1+rt)^{-1}\" \u00a0is called the present value discount factor at simple interest. The time, \"t\", must be in years. When the time is given in months, then<\/p>\n
\n
<\/a> (3) <\/span>   <\/span>\"\begin{equation*} <\/pre>\n
When the time is given in weeks, then divide the weeks with 52, when the time is given in days, then divide the days with 365 days.<\/p>\n
The present value (or discounted value) of \"S\" was calculated by using the present value factor at simple interest.\u00a0 The present value can be written as Equation (4) :<\/p>\n
\n
<\/a> (4) <\/span>   <\/span>\"\begin{equation*} <\/pre>\n
\n
\n
Example 2.1<\/p>\n<\/header>\n
\n
Find the simple interest on a RM1,000 investment made for 3 years at an interest rate of 5% per year. What is the accumulated amount?<\/p>\n
\n