Every financial transaction involves two parties: a saver and a borrower. The first step in analyzing these transactions is to identify who the saver is and who the borrower is. This is crucial for making informed decisions.
For example, when you deposit money into a bank savings account, you are the saver, and the bank is the borrower. The bank will pay you interest for the use of your money. Conversely, if you take out a loan from the bank, you become the borrower and will need to pay interest.
The distinction between savers and borrowers is straightforward:
Savers earn interest – they aim to maximize the amount of interest earned.
Borrowers pay interest – they aim to minimize the amount of interest paid.
Introduction to Simple Interest
To grasp the basic concepts of financial decision-making, we begin with simple interest. Simple interest is rarely used by financial institutions for loans or savings accounts, as most modern accounts apply compound interest.
The Role of Simple Interest
While it’s uncommon to encounter simple interest loans today, understanding how simple interest works is essential. It lays the foundation for comprehending compound interest and is applied to the pricing of certain short-term financial securities, such as Treasury Notes.
Calculating Simple Interest
Simple interest is calculated on the original amount deposited or borrowed for a fixed period. This original amount is known as the principal or present value.
Formula: Simple Interest
I = P x r x T
Where:
I = Interest charged (or earned)
P = Principal (original amount invested or borrowed)
r = Simple interest rate per annum (expressed as a decimal)
T = Time of the investment or loan (expressed in years)
Converting Interest Rates
Interest rates are typically quoted per annum (p.a.). For instance, a 5% annual simple interest rate is written as 5% p.a., but in calculations, it is converted to a decimal (0.05).
**Be cautious with small and large interest rates. For example, 0.5% is not the same as 0.5 when expressed as a decimal. A decimal of 0.5 represents 50%, vastly different from 0.5%.**
Example: Interest Earned on an Investment
If you deposit $1000 in a savings account for 2 years with a simple interest rate of 1.25% p.a., the interest earned will be:
P = $1000
r = 0.0125
T = 2 years
I = 1000 x 0.0125 x 2 = $25
Example: Interest Charged on a Loan
Robert borrows $5000 at a simple interest rate of 5.5% p.a. from 1 September 2011 to 1 March 2013 (1.5 years). The interest payable is:
P = $5000
r = 0.055
T = 1.5 x years
I = 5000 x 0.055 x 1.5 = $412.50
Sometimes, we need to find the principal, rate, or time by rearranging the formula.
Restating the Formula
If the interest amount, rate, and time are known, the principal can be calculated as:
P = I / (r x T)
Example : Principal Investment Calculation
Adam’s savings account earns $79.22 in 180 days at a simple interest rate of 0.7% p.a. The original deposit is calculated as:
I = $79.22
r = 0.007
T = 180/365
P = 79.22 / (0.007 x 180/365) = $22,948.65
Time Calculation in Simple Interest
Example : Investment Duration
To find how long it takes to earn $90 interest on a $3000 investment at 1.2% p.a.:
I = $90
P = $3,000
r = 0.012
T = 90 / (3000 x 0.012) = 2.5 years
Simple Interest Rate Calculation
Example: Calculating Interest Rate
Kimmy borrows $7000 for 6 months and pays $80.50 in interest. The simple interest rate is:
I = $80.50
P = $7,000
T = 0.5 years
r = 80.50 / (7000 x 0.5) = 0.023 = 2.5% p.a.
Example: A 90-day term deposit of $55,000 earns simple interest at 3.6% p.a. How much interest was earned over the 90-day term?
Interest = P × r × t
where:
- is the principal amount ($55,000)
- is the annual interest rate (3.6% or 0.036)
- t is the time period in years
Example: A loan of $17,900 is due for repayment in 45 days. If simple interest is charged at a rate of 8.2% p.a., how much interest must be paid?
Interest = P × r × t
where:
P is the principle amount ($ 17,900)
r is the annual interest rate (8.2% or 0.082)
t is the time period in years
First, covert 45-day term to years
t = 45/365 = 0.01233
Let's calculate the interest now:
Interest = 17,900 x 0.082 x 0.01233 = $ 180.98
So, the interest that must be paid for the 45-day loan is approximately $180.98.
We need to solve for P (the principle amount), so we rearrange the formula to:
P = Interest / (r x t)
Given:
Interest = $115.85
r = 6.1% per annum = 0.061
t = 9 months = 9/12 years = 0.75 years
Now, substitute the values into the formula:
P = 115.85 / (0.061 x 0.75)
P = 15.85 / 0.04575
P = 2532.79
So, the size of Jessie's loan was approximately $2,532.79.
Calculating the Total Repayment on a Loan
Question:
How much will Sarah need to repay in total on 1 June 2024 if she borrowed $7,000 with an interest charge of $630?
Solution:
Having already calculated the interest on the loan, it's straightforward to figure out that Sarah will need to repay the original loan amount of $7,000 plus the interest charge of $630, resulting in a total repayment of $7,630.
This total repayment is known as the maturity value, or the accumulated value. It can be found by first determining the amount of interest and then adding this to the original principal. Alternatively, it can be directly calculated using a maturity value formula.
Formula : S = P (1 + r x T)
This is also known as the present value. By knowing the maturity value, interest rate, and time period, we can calculate the principal amount.
Where,
S = Maturity value (This is the principle and the interest amount)
P = Principle (original investment or borrowing)
r = Simple interest rate per annum
T = Time of the investment or loan
$2540 is deposited in a savings account for 6 months. If the account earns simple interest at a rate of 2.15% p.a., what is the accumulated value (maturity value) of the account?
Solution:
To find the accumulated value, or maturity value, of the account, we use the formula for simple interest:
S = P(1 + r × T)
Where:
S - is the maturity value
P - is the principal amount ($2540)
r - is the annual interest rate (2.15% or 0.0215 as a decimal)
T - is the time period in years (6 months or 0.5 years)
Plugging in the values:
S = 2540 × (1 + 0.0215 × 0.5)
S = 2540 × (1 + 0.01075)
S = 2540 × 1.01075
S = 2567.31
Therefore, the accumulated value (maturity value) of the account after 6 months is $2567.31
Calculating Principal with Simple Interest:
P = S / (1 + r × T)
Where:
P is the principal amount (the initial amount invested or borrowed)
S is the maturity value
r is the annual simple interest rate (expressed as a decimal)
T is the time period of the investment or loan (in years)
The above formula is a rearranged version of < S = P(1 + r × T) > that allows us to directly find the principal. Alternatively, it can be written as:
P = S / (1 + r × T)
Example:
Alan needs $10,500 in 3 years. How much does he need to deposit now to earn this amount with simple interest at a rate of 3.4% per year?
Solution:
S = $10,500
r = 0.034 (3.4 / 100)
T = 3 years
Formula: P = S / (1 + r × T)
P = 10500 / (1 + 0.034 × 3)
P = 10500 / (1 + 0.102)
P = 10500 / 1.102
P = 9528.13
So, if the simple interest rate is 3.4% per year, Alan would need to deposit $9528.13 today to accumulate $10,500 in 3 years.
Verify the reasonableness of answers. Since the maturity value is the principal plus interest, it logically follows that the maturity value must be larger than the principal, meaning S is always greater than P.
You can also perform reverse calculations to check the accuracy of your answers. For instance, finding the principal amount is essentially the reverse of finding the maturity value. Using a different example, suppose the principal amount is $8000. You can work backward to ensure the maturity value is correct. Let's say the interest rate is 4% per year and the time period is 5 years.
Using the formula for maturity value, we first calculate the expected maturity value:
S = P (1 + r × T) S = 8000 (1 + 0.04 × 5) S = 8000 (1 + 0.20) S = 8000 × 1.20 S = 9600
Therefore, the maturity value should be $9,600. By performing this reverse calculation, we can verify the accuracy and reasonableness of the principal and interest calculations.
Case Study:
Ella, a financial analyst at a well-established financial services company, has been given a series of tasks to assess the performance and viability of various investment products and loan arrangements. The company's goal is to make informed decisions about their offerings, so Ella needs to provide precise calculations and analyses for the following scenarios: Ella starts her day by tackling a client inquiry about a fixed deposit. The client has invested $8,000 in a term deposit for 180 days at an annual simple interest rate of 2.5%. Ella needs to determine both the total interest earned and the final maturity value of this deposit. Next, Ella receives a request from another client who wants to know how much they should deposit today to accumulate $50,000 in 4 years, with a simple interest rate of 3% per annum. Ella must not only calculate the required initial deposit but also verify the future value by working backward to ensure the accuracy of her result. Later in the afternoon, Ella is asked to analyze a loan agreement. A client is set to repay a total of $75,000 after 2 years and 3 months. The loan had a simple interest rate of 9.5% per annum. Ella’s task is to determine the original loan amount, making sure to convert the time period correctly into years for her calculations. Ella then shifts her focus to another investment scenario. She needs to calculate the simple interest earned on a $30,000 investment over a 60-day period at an annual interest rate of 4.2%. As the day progresses, Ella receives a query about a short-term loan. The loan amount is $22,500, and it’s due in 30 days at a simple interest rate of 6.8% per annum. Ella must compute the total interest that will be paid by the client. Another client seeks Ella’s expertise on a loan where $200 in interest was paid over 12 months at a simple interest rate of 5.4% per annum. Ella needs to find out the size of the original loan. Towards the end of her busy day, Ella is asked to determine the length of the term for a deposit of $5,000 that accrued $450 in interest, given a simple interest rate of 3.6% per annum. Finally, Ella is challenged with calculating the annual simple interest rate for an investment of $15,000 that earned $180 in interest over 180 days. Ella must provide all necessary calculations for each scenario, ensure accurate conversion of time periods into years where applicable, verify results, especially in cases requiring reverse calculations, and present findings clearly and with detailed explanations.
For the fixed deposit of $8,000 with a term of 180 days at an annual simple interest rate of 2.5%, the interest earned is calculated as follows:
Interest = Principal × Rate × Time.
Time in years is 180/365 = 0.493.
So, Interest = 8000 × 0.025 × 0.493 = 245.80.
The maturity value is Principal + Interest = 8000 + 245.80 = 8245.80. To find out how much should be deposited today to accumulate $50,000 in 4 years with a simple interest rate of 3% per annum, use the formula:
Principal = Maturity Value / (1 + Rate × Time).
Time is 4 years.
So, Principal = 50000 / (1 + 0.03 × 4) = 50000 / 1.12 = 44642.86.
To verify, calculate:
Interest = 44642.86 × 0.03 × 4 = 5357.14,
and check Maturity Value = 44642.86 + 5357.14 = 50000. For the loan that will be repaid with $75,000 after 2 years and 3 months (which is 2.25 years) at a simple interest rate of 9.5% per annum, calculate the original loan amount:
Principal = Maturity Value / (1 + Rate × Time).
So, Principal = 75000 / (1 + 0.095 × 2.25) = 75000 / 1.21375 = 61753.82. For the $30,000 investment over 60 days at a simple interest rate of 4.2% per annum, the interest is:
Interest = Principal × Rate × Time.
Time in years is 60/365 = 0.164.
So, Interest = 30000 × 0.042 × 0.164 = 205.56. For the loan of $22,500 due in 30 days at 6.8% per annum, the interest is:
Interest = Principal × Rate × Time.
Time in years is 30/365 = 0.082.
So, Interest = 22500 × 0.068 × 0.082 = 382.05. To find out the size of Jessie's loan where $200 interest was paid over 12 months (1 year) at 5.4% per annum, use:
Principal = Interest / (Rate × Time).
So, Principal = 200 / (0.054 × 1) = 3703.70. For the $5,000 deposit that earned $450 in interest at a simple interest rate of 3.6% per annum, the term length in years is:
Time = Interest / (Principal × Rate).
So, Time = 450 / (5000 × 0.036) = 2.5 years. Finally, for the investment of $15,000 that earned $180 in interest over 180 days, the annual simple interest rate is:
Rate = Interest / (Principal × Time).
Time in years is 180/365 = 0.493.
So, Rate = 180 / (15000 × 0.493) = 0.024 or 2.4%.
Updated: 31/07/2024
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