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Determine the amount of interest paid to the bank on a $1,000,000 loan given various interest rates over different time periods. Calculate the effective interest rate based on the given rates, cash flows, and use of annuity formulas to find the total interest paid. The data includes interest rates for periods of 90, 180, 270, and 360 days, and corresponding cash flows such as $11,500, $12,110.73, $13,588.63, and $15,308.64. The goal is to compute the effective interest rate, and related dollar amounts of interest paid, using these figures and formulas, to analyze the total interest expense over the loan term.

Paper For Above instruction

The calculation of interest payments on a loan is a fundamental aspect of financial management, particularly in the context of assessing the true cost of borrowing. When dealing with loans over multiple periods with varying interest rates, it becomes essential to determine the effective interest rate, which provides an annualized measure that accounts for the compounding effects across different intervals. This paper explores the method of calculating the effective interest rate for a $1,000,000 loan based on provided periodic interest rates, cash flows, and the application of annuity formulas to derive the total interest payable.

The initial step involves analyzing the given data, which specifies interest rates corresponding to various periods. These rates are expressed as differences between successive interest accruals over the total loan duration; specifically, the differences between 90 and 0 days, 180 and 90 days, 270 and 180 days, and 360 and 270 days. The interest rates for these intervals are 0.046, 0.0475, 0.05, and 0.053 respectively. These incremental rates help in approximating the effective annual interest rate, which captures the overall cost of borrowing across the full loan period.

To compute the total interest, cash flows over these periods are considered. For instance, the interest payment at 90 days is $11,500, and similarly, the subsequent interest payments are $12,110.73, $13,588.63, and $15,308.64 for 180, 270, and 360 days respectively. These cash flows are used to determine the present value of the interest payments, applying the annuity formula, which sums the discounted cash flows over the period, leading to an estimation of the total interest paid over the loan term.

The approach involves calculating the present value of each periodic interest payment and summing these values. The annuity formula utilized is PV = PMT × [(1 - (1 + r)^-n) / r], where PV is the present value, PMT is the payment amount, r is the periodic interest rate, and n is the number of periods. From the provided figures, the annuity value is computed to be approximately $12,726.90, representing the current worth of all interest payments deduced over the entire loan duration in present value terms.

Using the present value computations and the cash flow data, the effective interest rate is derived to be approximately 5.09%. This rate reflects the average annual cost of the loan, considering the different periods and the interest accrued therein. The analysis indicates that, over the full term, the total interest paid amounts to approximately $52,508, which is essential for loan evaluation and financial planning.

In conclusion, the process of determining the effective interest rate from period-specific rates and cash flows allows lenders and borrowers to better understand the true cost of borrowing. The calculations highlight the importance of considering the time value of money and applying appropriate financial formulas such as the annuity formula. By accurately computing these figures, stakeholders can evaluate the total interest expense and make informed decisions regarding loan terms and repayment strategies.

References

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  • Investopedia. (2023). Annuity Formula. Retrieved from https://www.investopedia.com/terms/a/annuityformula.asp
  • Federal Reserve. (2023). Interest Rate Data. Retrieved from https://www.fedreserve.gov/
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