6.3 Simple Interest

Simple interest is a straightforward way to calculate how much you'll earn or owe on money over time. It's based on the initial amount, interest rate, and time period. This method is commonly used for short-term loans and basic savings accounts.

Understanding simple interest helps you make informed financial decisions. You can figure out loan payments, savings growth, and even how much to invest now to reach future goals. It's a fundamental concept in personal finance and basic investing.

Simple Interest Fundamentals

Calculation of simple interest

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• Simple interest formula calculates the interest earned on a principal amount over a given time period at a constant interest rate: $[I](https://www.fiveableKeyTerm:I) = Prt$
  • $I$ represents the total interest earned
  • $[P](https://www.fiveableKeyTerm:P)$ represents the principal or initial amount invested or borrowed
  • $[r](https://www.fiveableKeyTerm:r)$ represents the annual interest rate expressed as a decimal (6% = 0.06)
  • $[t](https://www.fiveableKeyTerm:t)$ represents the time in years the principal is invested or borrowed
• Future value formula determines the total amount after earning interest, found by adding the principal and interest: $[FV](https://www.fiveableKeyTerm:FV) = P + I$
  • $FV$ represents the future value or total amount after interest is added
• Converting time periods to years for use in simple interest calculations:
  • Months to years: divide the number of months by 12 (6 months = 0.5 years)
  • Days to years: divide the number of days by 365, or 360 for some financial institutions (90 days = 0.25 years)
• Example: $5,000 principal invested at 4% annual interest for 2 years
  • $I = 5000 * 0.04 * 2 = 400$
  • $FV = 5000 + 400 = 5400$

Loan balances and partial payments

• Remaining balance formula calculates the amount owed on a loan after making a partial payment, subtracting the payment from the future value: $[RB](https://www.fiveableKeyTerm:RB) = FV - [PP](https://www.fiveableKeyTerm:PP)$
  • $RB$ represents the remaining balance owed on the loan
  • $PP$ represents the partial payment amount applied to the loan
• Payoff amount is equivalent to the remaining balance, representing the total amount needed to fully repay the loan
• Example: $10,000 loan at 5$2,000 partial payment made after 2 years
  • $I_2 = 10000 * 0.05 * 2 = 1000$
  • $FV_2 = 10000 + 1000 = 11000$
  • $RB_2 = 11000 - 2000 = 9000$
  • $I_3 = 9000 * 0.05 * 1 = 450$
  • $FV_3 = 9000 + 450 = 9450$
  • Payoff amount after 3 years = $9,450

Monthly payments and present values

• Monthly payment formula determines the equal periodic payments needed to repay a loan over a given term, dividing the future value by the total number of payments: $MP = \frac{FV}{[n](https://www.fiveableKeyTerm:n)}$
  • $MP$ represents the monthly payment amount
  • $n$ represents the total number of monthly payments over the loan term
• Present value formula calculates the initial investment needed to reach a future goal amount, discounting the future value based on the interest rate and time: $[PV](https://www.fiveableKeyTerm:PV) = \frac{FV}{1 + rt}$
  • $PV$ represents the present value or initial investment required
• Example: $15,000 loan at 6% annual interest for 4 years
  • $I = 15000 * 0.06 * 4 = 3600$
  • $FV = 15000 + 3600 = 18600$
  • $n = 4 * 12 = 48$ monthly payments
  • $MP = \frac{18600}{48} = 387.50$ per month
• Example: $100,000 future retirement goal in 20 years, assuming 5% annual interest
  • $PV = \frac{100000}{1 + 0.05 * 20} = 37688.95$
  • Initial investment of $37,688.95 needed to reach$100,000 in 20 years at 5% interest

Additional Loan Terms

• Borrower: The individual or entity receiving the loan and responsible for repaying it with interest
• Lender: The financial institution or individual providing the loan funds
• Maturity date: The final date by which the loan must be fully repaid
• Finance charge: The total cost of borrowing, including interest and any additional fees
• Amortization: The process of gradually paying off a loan through regular payments that cover both principal and interest