BUSN 5200 Discussion Questions: Ear Vs APR Back When Interes

Busn 5200discussion Questions1 Ear Vs Aprback When Interest Rates And

BUSN 5200 Discussion questions 1 EAR vs APR Back when interest rates and inflation were about 18%, Congress passed the Truth in Lending Law, requiring lenders to show both figures to prospective borrowers. A. If the APR on a car loan is 6%, with interest compounded annually, what is the EAR? 1. What’s EAR if interest is compounded monthly? 2. If interest is compounded daily? 3. Why are these 3 answers different? 4. You are considering a 3-year car loan for $12,000; the car dealer states the APR is 8%. 5. If you make 3 equal end-of-year payments, how much is each payment? 6. If you make 36 equal end-of-month payments, how much is each payment? 7. Your parents plan to buy an annuity for their grandchild by paying $10,000 into a bank to buy a 10-year, $100/month annuity. What is their rate of return or yield on this annuity purchase?

Paper For Above instruction

The concepts of Effective Annual Rate (EAR) and Annual Percentage Rate (APR) are fundamental in understanding the true cost of borrowing and the return on investments over different compounding periods. They are especially relevant in the context of consumer loans, such as car and personal loans, as well as investments like annuities. This paper aims to clarify these concepts through detailed calculations and analysis, contextualized within the framework of historical interest rates and consumer protection laws.

Initially, it is imperative to understand that APR is a nominal interest rate that does not account for compounding frequency, whereas EAR reflects the actual annual rate earned or paid, considering the effects of compounding. During periods of high inflation—such as 18% as noted in the scenario—these distinctions become significant because they influence borrowing costs and investment yields. Legislative measures like the Truth in Lending Act mandated transparency by requiring lenders to disclose both rates to enable consumers to make informed decisions.

Calculations of EAR based on APR with different compounding frequencies

Given an APR of 6%, the EAR can be calculated for different compounding intervals. The general formula for EAR is:

EAR = (1 + APR / n)^{n} − 1

where n is the number of compounding periods per year.

Annually Compounded Interest:

n = 1

EAR = (1 + 0.06/1)^{1} − 1 = 6% — This is straightforward since compounding once per year yields the same rate as the APR.

Monthly Compounding:

n = 12

EAR = (1 + 0.06/12)^{12} − 1 ≈ (1 + 0.005)^{12} − 1 ≈ 1.005^{12} − 1 ≈ 1.0617 − 1 ≈ 0.0617 or 6.17%.

Daily Compounding:

n = 365

EAR = (1 + 0.06/365)^{365} − 1 ≈ (1 + 0.0001644)^{365} − 1 ≈ 1.0001644^{365} − 1 ≈ 1.06481 − 1 ≈ 0.0648 or 6.48%.

Analysis of Differences in EAR with various compounding frequencies

The differences arise because more frequent compounding periods increase the effect of interest accumulation within a given year. The annual compounding results in no additional interest beyond the nominal rate, while monthly and daily compounding incorporate interest on previously accumulated interest more frequently, leading to a higher effective rate.

Loan payment calculations

Considering the $12,000 car loan with an 8% APR over three years, the monthly and yearly payment calculations utilize the amortization formula:

Payment = P * [r(1 + r)^{n}]/[(1 + r)^{n} − 1]

Where P = principal, r = periodic interest rate, n = total number of payments.

Annual Payments:

r = 8% / 1 = 0.08

n = 3

Payment = 12000 [0.08(1 + 0.08)^{3}]/[(1 + 0.08)^{3} − 1] ≈ 12000 [0.08 1.2597]/(1.2597 − 1) ≈ 12000 0.1008 / 0.2597 ≈ 12000 * 0.389 ≈ $4,668.91 per year.

Monthly Payments:

r = 8% / 12 ≈ 0.0066667

n = 36

Payment = 12000 [0.0066667(1 + 0.0066667)^{36}]/[(1 + 0.0066667)^{36} − 1] ≈ 12000 [0.0066667 1.2833]/(1.2833 − 1) ≈ 12000 0.008555/0.2833 ≈ 12000 * 0.0302 ≈ $362.40 per month.

Evaluating the Rate of Return on the Annuity Purchase

The parents' plan involves depositing $10,000 to purchase a 10-year annuity paying $100/month. To determine the yield, we treat this as finding the internal rate of return (IRR) for a series of cash flows.

Cash outflow at the start: −$10,000

Monthly inflows: $100 for 120 months

Applying the present value of an annuity formula:

PV = PMT * [1 − (1 + r)^{−n}]/r

Where PV is the initial investment ($10,000), PMT is $100, n = 120 months, and r is the monthly rate of return.

Solving for r involves iterative techniques or financial calculator functions. Approximate calculations suggest a monthly rate r ≈ 0.005 or 0.5%, leading to an annualized rate of about 6%.

Thus, their approximate yield (rate of return) is around 6% annually, aligning with typical market rates for such investments.

Conclusion

Understanding the distinction between APR and EAR is crucial for consumers and investors to accurately compare different financial products. The frequency of compounding significantly influences the effective rate, which impacts loan payments and investment yields. Accurate calculations, including amortization formulas and IRR estimation, enable consumers to make informed financial decisions, especially during periods of high inflation or volatile interest rates. Legislation like the Truth in Lending Act has enhanced transparency, helping consumers grasp the true costs and returns associated with borrowing and investing.

References

• Brigham, E. F., & Ehrhardt, M. C. (2016). Financial Management: Theory & Practice. Cengage Learning.
• Fabozzi, F. J. (2018). Bond Markets, Analysis, and Strategies. Pearson Education.
• Investopedia. (2023). Effective Annual Rate (EAR). https://www.investopedia.com/terms/e/ear.asp
• Investopedia. (2023). Annual Percentage Rate (APR). https://www.investopedia.com/terms/a/apr.asp
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• Congressional Research Service. (2014). The Truth in Lending Act and Regulation Z. CRS Report.
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