The Unpaid Balance At The Start Of A 28-Day Billing Cycle

The Unpaid Balance At The Start Of A 28 Day Billing Cycle Was 99

The assignment involves calculating interest charges and balances based on credit card and loan scenarios, considering variables such as initial unpaid balances, purchases, payments, interest rates, and billing cycle lengths. The specific tasks include: computing interest charges using the average daily balance method, determining remaining balances after payments, calculating annual interest rates from given charges, and solving for investment growth over different compounding periods. Additionally, the assignment covers amortized loan payments, mortgage balances, refinancing savings, and interest rate approximations using graphical or equation-based methods.

Paper For Above instruction

The problem set presented involves multiple facets of financial mathematics, primarily focusing on credit card interest calculations, loan amortization, investment growth, and mortgage analysis. Each question necessitates a clear understanding of the mathematical principles underpinning different financial products, as well as proficiency in applying formulas for compound interest, average daily balances, and loan amortization schedules. This paper will systematically analyze each scenario, demonstrate the use of relevant formulas, and interpret the results within a practical context.

Question 1: Calculating Interest on a Credit Card Balance with Transactions

In the first scenario, the initial unpaid balance is $997.81 at the start of a 28-day billing cycle. A purchase of $6,000 is made on day 1, increasing the balance, and a $100 payment occurs on day 21. The annual interest rate is 26.16%, and interest is calculated via the average daily balance method.

The principal method involves calculating the daily balance for each day in the cycle, then averaging these to determine the interest charged. Specifically, from day 1 to 1, the balance increases by $6,000; from day 2 onward, it continues to reflect the cumulative purchases minus payments, adjusted accordingly for each day's specific balance. The key steps include identifying the periods of different balances, calculating their duration, and applying the daily interest rate (annual rate divided by 365). The sum of the daily balances, divided by the number of days, yields the average daily balance. This average is then multiplied by the daily interest rate and the number of days for the cycle to find the interest charged at cycle's end.

Question 2: Balance Calculation after No Purchases and a Payment

Here, the initial unpaid balance is $615.37, with no further purchases, but a payment of the same amount is made on day 20. The annual interest rate is 20.36%. The calculation involves considering two periods: from the start to day 20, the balance remains at $615.37; after the payment, the balance drops to zero for the remaining days. The average daily balance is calculated by summing the products of each balance and its respective duration, then dividing by 30 days. Since no further transactions occur, the interest for the period is based only on the initial balance before the payment, and the subsequent zero balance.

Question 3: Impact of Purchases and Fees on Balance

With an initial balance of $791.99, a purchase of $59.81 occurs on day 15, and a late fee of $31 is charged on day 23. The annual interest rate is 20.39%. The calculation must incorporate these transactions at their respective periods by calculating the balance for each segment: from start to day 15, to day 23, and beyond, summing the products of balance and duration to find the average daily balance—then applying the interest formula accordingly.

Question 4: Calculating the Annual Interest Rate for a Payday Loan

Given a principal of $300 borrowed for 11 days with a maximum allowable finance charge of 17.7%, the task is to find the annual interest rate. The finance charge is computed as a percentage of the amount borrowed, scaled to a year by considering the ratio of days borrowed to days in a year. The formula involves dividing the finance charge by the principal, then annualizing this rate to find the equivalent annual interest rate.

Question 5: Investment Growth Under Different Compounding Frequencies

The scenario involves investing $700 at 8% interest, compounded annually, quarterly, and monthly, over four years. Calculations utilize the compound interest formula A = P(1 + r/n)^(nt), where P is principal, r is annual rate, n is the number of compounding periods per year, and t is time in years. The interest earned is the difference between the final amount and the initial investment. Results are computed separately for each compounding frequency, illustrating how more frequent compounding increases returns.

Question 6: Time to Tripling Investment at Different Compounding Rates

The question involves determining how long it takes for an investment to triple at 5% compounded monthly and continuously. The formulas involve solving for t in A = P(1 + r/n)^(nt) for monthly compounding, and using the continuous compounding formula A = Pe^(rt) for continuous compounding. Each solution involves logarithmic calculations to isolate t, providing the duration in years with appropriate rounding.

Question 7: Determining the Interest Rate of an Ordinary Annuity

The problem involves an annuity with annual payments and a future value, requiring the use of the future value of an ordinary annuity formula: FV = P * [((1 + r)^n - 1) / r], where FV is the accumulated amount, P is the periodic payment, r is the interest rate per period, and n is the number of periods.

To find the interest rate, an iterative or graphical method (such as an equation solver) can be used to approximate r that satisfies the given FV of $5720.98 after 5 years with $1000 annual payments.

Question 8: Calculating Nominal Rate from Monthly Deposits

A monthly deposit of $110 into an account reaching $1325.09 after one year involves solving for the monthly interest rate using the future value of an ordinary annuity formula. The annual nominal rate is then derived by multiplying the monthly rate by 12. To estimate this rate, numerical approximation methods or solver functions are applied, rounding the final rate to two decimal places.

Question 9: Loan Amortization and Interest Paid per Year

A woman borrows $4000 at 12% interest, amortized over three years with monthly payments. The analysis calculates the loan amortization schedule, focusing on the total interest paid in each year. Key steps include calculating the monthly payment, then tracking the unpaid balance after each year to determine interest paid, based on the remaining balance and the interest rate.

Question 10: Calculating Mortgage Payments and Remaining Balances

For a mortgage of $111,408 over 30 years at 6.3%, monthly payments are calculated using the standard mortgage formula. Remaining balances after specified periods (10, 20, 25 years) are derived by applying the amortization schedule calculations, reflecting total payments and interest accrued.

Question 11: Mortgage Payment Analysis with Extra Payments

Given a $92,529 loan at 6% over 25 years, the calculation involves computing the monthly payment and total interest. Adding an extra $100 monthly accelerates payoff, with the duration of repayment and interest savings determined via loan amortization formulas and iterative solutions.

Question 12: Annuity Payments, Withdrawals, and Accumulations

Scenario involves multiple phases: depositing $100 monthly for 30 years, then making withdrawals for 15 years to exhaust the balance. Using the future value of an ordinary annuity, the withdrawals are computed, as well as the total interest earned over the entire period. For the second part, calculating the initial deposit needed to sustain monthly withdrawals of $1500 over 15 years follows similar principles, working backward from the future value formula.

Question 13: Home Equity Loan Calculation

The homeowners' equity is used to determine a maximum loan amount of 70% of their home value. Key calculations involve the original mortgage principal, amortization over 15 years, current property value, and the remaining balance after 120 payments. The maximum loan is the lesser of 70% of the current property value and the remaining mortgage balance, requiring detailed mortgage amortization computations.

Question 14: Refinancing Savings Calculation

Comparison between interest paid on the original mortgage and the refinanced low-interest mortgage requires calculating the remaining unpaid balances at the time of refinancing, then computing interest savings over the respective periods, considering the different interest rates.

Question 15: Amortized Loan Interest Rate

This scenario involves an evenly paid stereo over 12 months with a total repayment amount, and calculations to determine the true annual nominal interest rate considering amortization effects. Using an equation solver or graphical approximation helps estimate the actual rate, which differs from the simple claim of 4.4% annual rate.

References

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  • Horan, T. (2020). Mortgage Banking: Principles and Practice. McGraw-Hill Education.
  • Investopedia. (2023). Understanding Average Daily Balance Method. Retrieved from https://www.investopedia.com
  • Ross, S. A., Westerfield, R. W., & Jordan, B. D. (2019). Fundamentals of Corporate Finance. McGraw-Hill Education.
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