Business Math And Statistical Measures Unit 4 Instruc 768977

Mm255 Business Math And Statistical Measuresunit 4 Instructor Graded

Explain the concept of simple interest, including how it is calculated, and provide examples demonstrating the calculation of interest paid, principal adjustments after partial payments, and the derivation of interest rates from given loan terms. Include the steps involved in solving each example, referencing reliable sources for the formulas used. Show all work clearly, including calculations of days, application of the simple interest formula, and conversions between decimal and percentage forms.

Paper For Above instruction

Understanding simple interest is fundamental in financial mathematics, especially when dealing with loans and investments. Simple interest is calculated on the original principal amount throughout the duration of the loan or investment, regardless of any payments made or interest accrued over previous periods. The primary formula for calculating simple interest and total repayment is:

A = P(1 + rt)

where:

• A = Total amount to be repaid at the end of the period
• P = Principal amount borrowed or invested
• r = Annual interest rate (in decimal form)
• t = Time in years

This formula allows calculation of the total amount owed or accrued interest over a specific period. To determine interest explicitly, subtract the principal from the total amount:

Interest = A - P

In applying the simple interest formula, it is important to accurately determine the time period in years, which can be derived from the number of days the loan is outstanding. Often, specific days are counted based on the calendar, and the resulting fraction of a year is used in the formula to account for time variations.

Example 1: Calculating Interest on a Loan

Suppose an individual borrows $2000 at an annual interest rate of 10%. The loan starts on April 30, 2013, and the borrower repays the loan on December 15, 2013. To calculate the interest paid, first determine the number of days the loan was outstanding. Based on standard day counts (as per the textbook table), November 4 is the 308th day of the year, and December 15 is the 349th day, thus the loan duration is 41 days.

Next, convert this period into years: 41 days divided by 365 days in a year equals approximately 0.1123 years. Applying the formula:

A = 2000 (1 + 0.10 0.1123) = 2000 (1 + 0.01123) = 2000 1.01123 = $2022.47

The total repayment is approximately $2022.47. The interest paid is the difference between the total amount and the principal:

$2022.47 - $2000 = $22.47.

This calculation demonstrates how simple interest on a short-term loan can be derived using precise day counts and the formula.

Example 2: Adjusting Principal after Partial Payments

Consider a $15,000 loan at 5% annual interest. The borrower makes a partial payment of $5,000 after 30 days. To find the new principal after this payment, first calculate the amount owed after 30 days:

A = 15000 (1 + 0.05 (30/365)) = 15000 (1 + 0.05 0.0822) = 15000 (1 + 0.00411) = 15000 1.00411 = $15,061.64

After making the $5,000 payment: 15,061.64 - 5,000 = $10,061.64, which becomes the new principal. The borrower then waits another 75 days before making a second payment.

The amount owed on the new principal after 75 days is:

A = 10061.64 (1 + 0.05 (75/365)) = 10061.64 (1 + 0.05 0.2055) = 10061.64 (1 + 0.01028) = 10061.64 1.01028 = $10,165.01

After paying another $5,000, the remaining principal is:

$10,165.01 - $5,000 = $5,165.01.

Example 3: Deriving Interest Rate from Loan Terms

Suppose someone needs $250 and writes a check for $350, which will be cashed in 3 weeks. To find the interest rate paid in this transaction, set up the simple interest formula assuming the borrowed amount is the principal:

350 = 250 (1 + r (3/52))

Here, 3 weeks out of 52 weeks in a year give the time period. Solving for r:

Divide both sides by 250:

1.4 = 1 + r * (3/52)

Subtract 1: 0.4 = r * (3/52)

Multiply both sides by 52/3 to isolate r:

r = 0.4 (52/3) ≈ 0.4 17.33 = 6.93

Convert to percentage: 6.93 or approximately 693.33% annual interest rate. This extremely high rate reflects the short-term nature and the high-cost lending at a check cashing place, illustrating the concept of effective annual interest rate.

Conclusion

Simple interest calculations are vital in understanding basic financial transactions involving loans, investments, and short-term borrowing. Accurate day counts, precise application of formulas, and proper conversions between time units and interest rates are critical for correct financial analysis. These examples underscore the importance of systematic calculation methods and thorough understanding of the simple interest model.

References

• Brigham, E. F., & Ehrhardt, M. C. (2016). Financial Management: Theory & Practice (15th ed.). Cengage Learning.
• Gordon, R. (2014). Financial Mathematics: A Core Course. Springer.
• Ross, S. A., Westerfield, R., & Jordan, B. D. (2018). Essentials of Corporate Finance (10th ed.). McGraw-Hill Education.
• Damodaran, A. (2012). Investment Valuation: Tools and Techniques for Determining the Value of Any Asset (3rd ed.). Wiley.
• Hull, J. C. (2018). Options, Futures, and Other Derivatives (10th ed.). Pearson.
• Excel, Microsoft. (2023). Financial functions in Excel: FV, PV, RATE, NPER. Microsoft Support.
• Kothari, C. R. (2004). Research Methodology: Methods And Techniques. New Age International.
• Levine, R., & Zervos, S. (1998). Stock Market Liberalization and Economic Development. World Bank Economic Review, 12(2), 223–249.
• Investopedia. (2023). Simple interest Definition. https://www.investopedia.com/terms/s/simpleinterest.asp
• Federal Reserve Bank. (2023). Understanding Interest Rates. https://www.federalreserve.gov/education.htm